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"content": "\u003cp>When Nicole Thomson first heard about the importance of teaching spatial reasoning and geometry in her kindergarten math curriculum she had already been teaching for several years. Her teacher training program hadn’t mentioned these skills, and yet at a professional development session for math teachers a group of researchers from the University of Toronto explained the \u003ca href=\"http://tmerc.ca/publications/#m4yc\">large body of research\u003c/a> that ties spatial reasoning skills to future success in math and reading. The National Council of Teachers of Mathematics (NCTM) \u003ca href=\"http://www.nctm.org/Publications/teaching-children-mathematics/2006/Vol13/Issue3/Curriculum-Focal-Points-for-Pre-K%E2%80%93Grade-8-Mathematics_-A-Quest-for-Coherence/\">recommends \u003c/a>that spatial reasoning should be a large focus of preK - 8th grade math education.\u003c/p>\n\u003cp>“At first we had no idea what it meant,” Thomson said, but as the \u003ca href=\"https://ww2.kqed.org/mindshift/2017/01/20/five-compelling-reasons-to-teach-spatial-reasoning-to-young-children/\">researchers explained cognitive science studies\u003c/a> showing the power of spatial reasoning in the early grades they were gradually convinced that it was worth trying. Early elementary teachers like Thomson in select Rainy River District schools began using \u003ca href=\"http://www.mathforyoungchildren.ca/\">Math For Young Children\u003c/a> lessons designed by the researchers.\u003c/p>\n\u003cfigure id=\"attachment_47320\" class=\"wp-caption alignleft\" style=\"max-width: 640px\">\u003cimg class=\"wp-image-47320 size-large\" src=\"http://ww2.kqed.org/mindshift/wp-content/uploads/sites/23/2017/01/symmetry-e1484178973775-1020x1360.jpg\" alt=\"Students in Cristol Bailey's early years class show off their symmetry.\" width=\"640\" height=\"853\" srcset=\"https://ww2.kqed.org/app/uploads/sites/23/2017/01/symmetry-e1484178973775-1020x1360.jpg 1020w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/symmetry-e1484178973775-160x213.jpg 160w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/symmetry-e1484178973775-800x1067.jpg 800w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/symmetry-e1484178973775-768x1024.jpg 768w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/symmetry-e1484178973775-1180x1573.jpg 1180w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/symmetry-e1484178973775-960x1280.jpg 960w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/symmetry-e1484178973775-240x320.jpg 240w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/symmetry-e1484178973775-375x500.jpg 375w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/symmetry-e1484178973775-520x693.jpg 520w\" sizes=\"(max-width: 640px) 100vw, 640px\">\u003cfigcaption class=\"wp-caption-text\">Students in Cristol Bailey's early years class show off their symmetry. \u003ccite>(Cristol Bailey)\u003c/cite>\u003c/figcaption>\u003c/figure>\n\u003cp>The lessons focus on specific spatial reasoning skills like mental rotation, visual spatial reasoning, and spatial vocabulary all done in a playful, exploratory style that is developmentally appropriate for students ages four to eight.\u003c/p>\n\u003cp>“On day one of our professional development, we would work with kids and directly show how these ideas play out in classrooms or with kids,” said Zachary Hawes, a doctoral candidate in the \u003ca href=\"http://www.numericalcognition.org/\">Numerical Cognition Laboratory\u003c/a> at the University of Western Ontario. He is one of the Math For Young Children researchers along with Joan Moss, \u003ca href=\"https://www.trentu.ca/education/faculty-and-research/dean\" target=\"_blank\">Cathy Bruce\u003c/a>, Bev Caswell, and Tara Flynn. Since 2011, these researchers led by Moss and Bruce have been conducting research at several sites around Ontario. They felt including students in the professional development trainings would help give teachers a chance to see the lessons in action and help them imagine how they could bring them back to their classrooms.*\u003c/p>\n\u003cp>“We would take those lessons and games and think about what else we could do with these. How could we extend it, what could we try?” Thomson said. She and her colleagues would take the lessons researchers developed in a lab and try them out in their classrooms, returning to the next professional learning session with feedback and examples of how they’d modified or extended activities.\u003c/p>\n\u003cp>[ad fullwidth]\u003c/p>\n\u003cp>“Everything is supposed to be exploratory and it comes from the kids,” Thomson said. She noted they particularly love pattern blocks, which are like puzzles to them and tend to calm them down. She doesn’t ever lecture her students on how to use the spatial reasoning tools, but rather sets kids a challenge and lets them figure out how to put the blocks together. Often she’ll lead them in one group activity and then leave the materials out around the room so kids can play with them during free time as well.\u003c/p>\n\u003cp>https://www.youtube.com/watch?v=vHbpq8rnrk8\u003c/p>\n\u003cp>A favorite lesson is the “magic key” activity where she puts on a witch hat and explains to her kindergarteners that a witch has hidden a treasure behind a door, casting a spell to lock it. She then gives them a set of \u003ca href=\"https://en.wikipedia.org/wiki/Pentomino\">pentominoes\u003c/a> which contains five squares, and tells students their job is to find as many ways to combine the squares with one full side touching as they can. The more combinations they find the better their chances are of locating the key.\u003c/p>\n\u003cp>“They discover within the half hour that there are 12 of these keys and we can’t make more,” Thomson said. As she and her colleagues experimented with spatial reasoning activities like this one, they were consistently amazed at how much more young students could do than they expected. And because the activities largely deal with manipulating shapes, practicing mental rotations and talking about positional language, kids who struggle with more traditional numeracy exercises were shining.\u003c/p>\n\u003cp>“We started making sure we labeled this as math,” Thomson said. Before, kids thought math was just numbers, but when she worked to broaden the definition to include spatial reasoning tasks and toys they suddenly started to really enjoy math time, often choosing to play with materials during choice time.\u003c/p>\n\u003cp>Thomson said she was so impressed with the results she was getting that she focused almost exclusively on spatial reasoning, neglecting other kindergarten concepts like patterning and numeracy. That made her a little nervous, so she was surprised and delighted when her students still performed well on those more traditional math concepts by the end of the year. That direct experience of success validated the research the Math For Young Children team presented.\u003c/p>\n\u003cp>https://www.youtube.com/watch?v=JnIR_GwoHtY\u003c/p>\n\u003cp>“It was good for me to see how important it was,” Thomson said. She’s now pulling the spatial reasoning tasks in more, connecting numeracy concepts like the number line to spatial and geometry concepts. She’s has students use blocks on number lines to help them understand the concept of magnitude, for example.\u003c/p>\n\u003cp>Cristol Bailey also began using spatial reasoning in her classes several years ago. At that time she taught at a rural school with a high First Nations population. Bailey taught special education, but many of the students were underachieving even without that categorization. She says she was skeptical of spatial reasoning, but it was a “seeing is believing situation” for her.\u003c/p>\n\u003cp>“The lower achieving kids had such a high degree of success with these activities and showed strengths that more standardized number sense lesson plans would never have brought out,” Bailey said. “For them to be successful in math -- and successful to the degree they were -- was mind boggling.”\u003c/p>\n\u003cp>She began to see her entire math program through a spatial and geometry lens. Even when students were doing number sense activities she would encourage them to gesture with their hands or visualize the number line. She found often kids didn’t have the language to describe spatial positioning, but as they used their hands to gesture they began to find the words.\u003c/p>\n\u003cfigure id=\"attachment_47323\" class=\"wp-caption aligncenter\" style=\"max-width: 640px\">\u003cimg class=\"size-large wp-image-47323\" src=\"http://ww2.kqed.org/mindshift/wp-content/uploads/sites/23/2017/01/3D-number-line-1020x680.jpg\" alt=\"Student plays with a three-dimensional number line.\" width=\"640\" height=\"427\" srcset=\"https://ww2.kqed.org/app/uploads/sites/23/2017/01/3D-number-line-1020x680.jpg 1020w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/3D-number-line-160x107.jpg 160w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/3D-number-line-800x533.jpg 800w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/3D-number-line-768x512.jpg 768w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/3D-number-line-1180x786.jpg 1180w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/3D-number-line-960x640.jpg 960w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/3D-number-line-240x160.jpg 240w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/3D-number-line-375x250.jpg 375w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/3D-number-line-520x347.jpg 520w\" sizes=\"(max-width: 640px) 100vw, 640px\">\u003cfigcaption class=\"wp-caption-text\">Student plays with a three-dimensional number line. \u003ccite>(Zach Pedersen/Robertson Program)\u003c/cite>\u003c/figcaption>\u003c/figure>\n\u003cp>“We went into it with a sort of learning trajectory in mind and most of the time they far surpassed what we thought they’d be capable of,” Bailey said. She now teaches Grade 2 students, most of whom have had spatial reasoning lessons since kindergarten. They’ve mastered many of the tasks, but she still finds more difficult ones to grow their skills. One favorite is the “hole punch symmetry challenge,” in which students imagine punching a hole in a folded up piece of paper. As the paper unfolds, where will the holes be?\u003c/p>\n\u003cp>“It is my struggling paper and pencil kids who nailed it right off the bat, which was really surprising and great because I was not expecting that,” she said.\u003c/p>\n\u003cp>In Ontario, students take an important standardized test in Grade 3 called the \u003ca href=\"http://www.eqao.com/en\">EQAO\u003c/a> that determines whether they are on grade level. That means that even in Grade 2 there’s pressure to cover a broad array of topics and anxiety that kids won’t be ready. Teachers go over diagnostic data at divisional meetings, creating lessons to target concepts and skills that students haven’t mastered. Bailey has noticed that students often struggle with tasks that involve spatial sense, a further indicator to her that spatial reasoning should be the norm in every early elementary classroom.\u003c/p>\n\u003cp>https://www.youtube.com/watch?v=jNCCW7pyMww\u003c/p>\n\u003cp>While she still uses spatial reasoning in her Grade 2 classroom, Bailey admits that without the support of colleagues working to adapt the materials to this grade level it’s more of a challenge. She thinks her experience with the Math For Young Children team and curriculum has changed her teaching forever, but wishes it was more of a priority even as kids get older. Perhaps just as important, the experience of working with math researchers and colleagues to refine lessons has her thinking about going back to school for another degree on how to better teach math.\u003c/p>\n\u003cp>\u003cstrong>MATH FOR YOUNG CHILDREN\u003c/strong>\u003c/p>\n\u003cp>There’s a well-known rift between those who believe the only type of developmentally appropriate early childhood education is a play-based one, and those concerned that relying solely on any learning that comes out of play could put students coming from impoverished backgrounds at a disadvantage. Research has shown that students from lower socioeconomic groups enter school with significantly less mathematical knowledge, and it is difficult to overcome that gap without intentional mathematics programming. But, at the same time, traditional teacher-led instruction often isn’t developmentally appropriate for five-year-olds.\u003c/p>\n\u003cp>“This project started as a way to show young children engaged in rigorous mathematics in ways that were play,” said Joan Moss, Associate Professor Emerita at the University of Toronto’s \u003ca href=\"http://www.oise.utoronto.ca/oise/Home/index.html\">Ontario Institute for Studies in Education\u003c/a>. She stresses that while math learning doesn’t only emerge from play, as some insist, the activities are still developmentally appropriate because they are presented playfully; students have lots of choice, there are many entry points, and while there are right answers, teachers build a culture in which getting a wrong answer isn’t bad.\u003c/p>\n\u003cfigure id=\"attachment_47324\" class=\"wp-caption alignright\" style=\"max-width: 640px\">\u003cimg class=\"wp-image-47324 size-large\" src=\"http://ww2.kqed.org/mindshift/wp-content/uploads/sites/23/2017/01/drawing-3d-shape-1020x1360.jpg\" alt=\"Student draws a 3D shape.\" width=\"640\" height=\"853\" srcset=\"https://ww2.kqed.org/app/uploads/sites/23/2017/01/drawing-3d-shape-1020x1360.jpg 1020w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/drawing-3d-shape-160x213.jpg 160w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/drawing-3d-shape-800x1067.jpg 800w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/drawing-3d-shape-768x1024.jpg 768w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/drawing-3d-shape-1180x1573.jpg 1180w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/drawing-3d-shape-960x1280.jpg 960w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/drawing-3d-shape-240x320.jpg 240w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/drawing-3d-shape-375x500.jpg 375w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/drawing-3d-shape-520x693.jpg 520w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/drawing-3d-shape.jpg 1500w\" sizes=\"(max-width: 640px) 100vw, 640px\">\u003cfigcaption class=\"wp-caption-text\">Student draws a 3D shape. \u003ccite>(Bev Caswell/Robertson Program)\u003c/cite>\u003c/figcaption>\u003c/figure>\n\u003cp>For example, in the “quick image activity” the teacher flashes a complex pattern made out of pattern blocks. Students see it for a very brief time and then try to recreate it themselves. After working for a bit they get to see the original image again and make fixes to their original attempts. Hawes and Moss say a lot of learning happens in the fixing.\u003c/p>\n\u003cp>In addition to demonstrating that well-trained teachers can teach math concepts in developmentally appropriate and playful ways, the Math For Young Children project has been an experiment in a more collaborative type of professional development. The university researchers are working alongside classroom teachers to fine tune lessons and evaluate how well they work. The Rainy River School Board teachers who were the first participants kept logs of when they used spatial reasoning activities, how long they took, and the tweaks they made. They brought feedback from the classroom back to researchers, and used a lesson study approach to improving the lessons together.\u003c/p>\n\u003cp>Joan Moss says this collaborative model of professional development, featuring teachers working alongside researchers to build quality activities grounded in research and classroom practice has been thrilling and a huge part of the program’s success. Teachers agree: “To be able to get together with people with that much math knowledge, it was an amazing experience,” Cristol Bailey said.\u003c/p>\n\u003cp>“It changed my teaching in the fact that I think of myself as a teacher-researcher, as they call us,” Thomson added. She now approaches every classroom activity as a mini experiment, tweaking and adjusting along the way. “I’m a lot more reflective in what I’m doing and what I put out there. It’s a neat lens to look through.”\u003c/p>\n\u003cp>The University of Toronto team \u003ca href=\"http://link.springer.com/article/10.1007/s11858-014-0637-4\">evaluated\u003c/a> the Math For Young Children program as it was being implemented in the Rainy River schools. Since teachers of the experimental group were engaged in inquiry-based professional development with researchers around spatial reasoning, the control group’s teachers also had interaction with researchers on a different topic. This was meant to make the groups more similar in exposure, but with different focuses.\u003c/p>\n\u003cp>After the first year, students in the experimental group made \u003ca href=\"http://link.springer.com/article/10.1007/s11858-014-0637-4\">significant gains on assessments \u003c/a>of geometry, spatial reasoning and numerical skills compared to the control group. In the second year, researchers decided to test students on the KeyMath measures, which are used to assess school-based mathematical concepts and skills. Students in the experimental group showed significant gains on those more traditional measures as well (a paper with these findings will soon be published in \u003ca href=\"http://www.tandfonline.com/toc/hcgi20/current\">Cognition and Instruction\u003c/a>). The first class of students will take the EQAO this year, and researchers hope they will show increased learning over peers in the rest of the province.\u003c/p>\n\u003cp>[ad floatright]\u003c/p>\n\u003cp>The Ontario Ministry of Education is interested in spreading the spatial reasoning work that the researchers started. Hawes and Flynn wrote a document titled \u003ca href=\"http://www.edu.gov.on.ca/eng/literacynumeracy/LNSPayingAttention.pdf\">\"Paying Attention to Spatial Reasoning\"\u003c/a> that the ministry distributed to educators across the district.** Individual school boards are also showing interest in training and implementation.\u003cbr>\n\u003cem>\u003cbr>\n*The article has been updated to note that Cathy Bruce helped lead the Math For Young Children research, which is taking place in several locations around Ontario.\u003cbr>\n**The article has been updated to include Tara Flynn’s contribution to the document. We regret these errors.\u003c/em>\u003c/p>\n\n",
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"content": "\u003cdiv class=\"post-body\">\u003cp>\u003cp>When Nicole Thomson first heard about the importance of teaching spatial reasoning and geometry in her kindergarten math curriculum she had already been teaching for several years. Her teacher training program hadn’t mentioned these skills, and yet at a professional development session for math teachers a group of researchers from the University of Toronto explained the \u003ca href=\"http://tmerc.ca/publications/#m4yc\">large body of research\u003c/a> that ties spatial reasoning skills to future success in math and reading. The National Council of Teachers of Mathematics (NCTM) \u003ca href=\"http://www.nctm.org/Publications/teaching-children-mathematics/2006/Vol13/Issue3/Curriculum-Focal-Points-for-Pre-K%E2%80%93Grade-8-Mathematics_-A-Quest-for-Coherence/\">recommends \u003c/a>that spatial reasoning should be a large focus of preK - 8th grade math education.\u003c/p>\n\u003cp>“At first we had no idea what it meant,” Thomson said, but as the \u003ca href=\"https://ww2.kqed.org/mindshift/2017/01/20/five-compelling-reasons-to-teach-spatial-reasoning-to-young-children/\">researchers explained cognitive science studies\u003c/a> showing the power of spatial reasoning in the early grades they were gradually convinced that it was worth trying. Early elementary teachers like Thomson in select Rainy River District schools began using \u003ca href=\"http://www.mathforyoungchildren.ca/\">Math For Young Children\u003c/a> lessons designed by the researchers.\u003c/p>\n\u003cfigure id=\"attachment_47320\" class=\"wp-caption alignleft\" style=\"max-width: 640px\">\u003cimg class=\"wp-image-47320 size-large\" src=\"http://ww2.kqed.org/mindshift/wp-content/uploads/sites/23/2017/01/symmetry-e1484178973775-1020x1360.jpg\" alt=\"Students in Cristol Bailey's early years class show off their symmetry.\" width=\"640\" height=\"853\" srcset=\"https://ww2.kqed.org/app/uploads/sites/23/2017/01/symmetry-e1484178973775-1020x1360.jpg 1020w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/symmetry-e1484178973775-160x213.jpg 160w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/symmetry-e1484178973775-800x1067.jpg 800w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/symmetry-e1484178973775-768x1024.jpg 768w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/symmetry-e1484178973775-1180x1573.jpg 1180w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/symmetry-e1484178973775-960x1280.jpg 960w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/symmetry-e1484178973775-240x320.jpg 240w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/symmetry-e1484178973775-375x500.jpg 375w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/symmetry-e1484178973775-520x693.jpg 520w\" sizes=\"(max-width: 640px) 100vw, 640px\">\u003cfigcaption class=\"wp-caption-text\">Students in Cristol Bailey's early years class show off their symmetry. \u003ccite>(Cristol Bailey)\u003c/cite>\u003c/figcaption>\u003c/figure>\n\u003cp>The lessons focus on specific spatial reasoning skills like mental rotation, visual spatial reasoning, and spatial vocabulary all done in a playful, exploratory style that is developmentally appropriate for students ages four to eight.\u003c/p>\n\u003cp>“On day one of our professional development, we would work with kids and directly show how these ideas play out in classrooms or with kids,” said Zachary Hawes, a doctoral candidate in the \u003ca href=\"http://www.numericalcognition.org/\">Numerical Cognition Laboratory\u003c/a> at the University of Western Ontario. He is one of the Math For Young Children researchers along with Joan Moss, \u003ca href=\"https://www.trentu.ca/education/faculty-and-research/dean\" target=\"_blank\">Cathy Bruce\u003c/a>, Bev Caswell, and Tara Flynn. Since 2011, these researchers led by Moss and Bruce have been conducting research at several sites around Ontario. They felt including students in the professional development trainings would help give teachers a chance to see the lessons in action and help them imagine how they could bring them back to their classrooms.*\u003c/p>\n\u003cp>“We would take those lessons and games and think about what else we could do with these. How could we extend it, what could we try?” Thomson said. She and her colleagues would take the lessons researchers developed in a lab and try them out in their classrooms, returning to the next professional learning session with feedback and examples of how they’d modified or extended activities.\u003c/p>\n\u003cp>\u003c/p>\u003c/div>",
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"content": "\u003cdiv class=\"post-body\">\u003cp>\u003c/p>\n\u003cp>“Everything is supposed to be exploratory and it comes from the kids,” Thomson said. She noted they particularly love pattern blocks, which are like puzzles to them and tend to calm them down. She doesn’t ever lecture her students on how to use the spatial reasoning tools, but rather sets kids a challenge and lets them figure out how to put the blocks together. Often she’ll lead them in one group activity and then leave the materials out around the room so kids can play with them during free time as well.\u003c/p>\u003c/p>\u003cp>\u003cspan class='utils-parseShortcode-shortcodes-__youtubeShortcode__embedYoutube'>\n \u003cspan class='utils-parseShortcode-shortcodes-__youtubeShortcode__embedYoutubeInside'>\n \u003ciframe\n loading='lazy'\n class='utils-parseShortcode-shortcodes-__youtubeShortcode__youtubePlayer'\n type='text/html'\n src='//www.youtube.com/embed/vHbpq8rnrk8'\n title='//www.youtube.com/embed/vHbpq8rnrk8'\n allowfullscreen='true'\n style='border:0;'>\u003c/iframe>\n \u003c/span>\n \u003c/span>\u003c/p>\u003cp>\u003cp>A favorite lesson is the “magic key” activity where she puts on a witch hat and explains to her kindergarteners that a witch has hidden a treasure behind a door, casting a spell to lock it. She then gives them a set of \u003ca href=\"https://en.wikipedia.org/wiki/Pentomino\">pentominoes\u003c/a> which contains five squares, and tells students their job is to find as many ways to combine the squares with one full side touching as they can. The more combinations they find the better their chances are of locating the key.\u003c/p>\n\u003cp>“They discover within the half hour that there are 12 of these keys and we can’t make more,” Thomson said. As she and her colleagues experimented with spatial reasoning activities like this one, they were consistently amazed at how much more young students could do than they expected. And because the activities largely deal with manipulating shapes, practicing mental rotations and talking about positional language, kids who struggle with more traditional numeracy exercises were shining.\u003c/p>\n\u003cp>“We started making sure we labeled this as math,” Thomson said. Before, kids thought math was just numbers, but when she worked to broaden the definition to include spatial reasoning tasks and toys they suddenly started to really enjoy math time, often choosing to play with materials during choice time.\u003c/p>\n\u003cp>Thomson said she was so impressed with the results she was getting that she focused almost exclusively on spatial reasoning, neglecting other kindergarten concepts like patterning and numeracy. That made her a little nervous, so she was surprised and delighted when her students still performed well on those more traditional math concepts by the end of the year. That direct experience of success validated the research the Math For Young Children team presented.\u003c/p>\u003c/p>\u003cp>\u003cspan class='utils-parseShortcode-shortcodes-__youtubeShortcode__embedYoutube'>\n \u003cspan class='utils-parseShortcode-shortcodes-__youtubeShortcode__embedYoutubeInside'>\n \u003ciframe\n loading='lazy'\n class='utils-parseShortcode-shortcodes-__youtubeShortcode__youtubePlayer'\n type='text/html'\n src='//www.youtube.com/embed/JnIR_GwoHtY'\n title='//www.youtube.com/embed/JnIR_GwoHtY'\n allowfullscreen='true'\n style='border:0;'>\u003c/iframe>\n \u003c/span>\n \u003c/span>\u003c/p>\u003cp>\u003cp>“It was good for me to see how important it was,” Thomson said. She’s now pulling the spatial reasoning tasks in more, connecting numeracy concepts like the number line to spatial and geometry concepts. She’s has students use blocks on number lines to help them understand the concept of magnitude, for example.\u003c/p>\n\u003cp>Cristol Bailey also began using spatial reasoning in her classes several years ago. At that time she taught at a rural school with a high First Nations population. Bailey taught special education, but many of the students were underachieving even without that categorization. She says she was skeptical of spatial reasoning, but it was a “seeing is believing situation” for her.\u003c/p>\n\u003cp>“The lower achieving kids had such a high degree of success with these activities and showed strengths that more standardized number sense lesson plans would never have brought out,” Bailey said. “For them to be successful in math -- and successful to the degree they were -- was mind boggling.”\u003c/p>\n\u003cp>She began to see her entire math program through a spatial and geometry lens. Even when students were doing number sense activities she would encourage them to gesture with their hands or visualize the number line. She found often kids didn’t have the language to describe spatial positioning, but as they used their hands to gesture they began to find the words.\u003c/p>\n\u003cfigure id=\"attachment_47323\" class=\"wp-caption aligncenter\" style=\"max-width: 640px\">\u003cimg class=\"size-large wp-image-47323\" src=\"http://ww2.kqed.org/mindshift/wp-content/uploads/sites/23/2017/01/3D-number-line-1020x680.jpg\" alt=\"Student plays with a three-dimensional number line.\" width=\"640\" height=\"427\" srcset=\"https://ww2.kqed.org/app/uploads/sites/23/2017/01/3D-number-line-1020x680.jpg 1020w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/3D-number-line-160x107.jpg 160w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/3D-number-line-800x533.jpg 800w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/3D-number-line-768x512.jpg 768w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/3D-number-line-1180x786.jpg 1180w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/3D-number-line-960x640.jpg 960w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/3D-number-line-240x160.jpg 240w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/3D-number-line-375x250.jpg 375w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/3D-number-line-520x347.jpg 520w\" sizes=\"(max-width: 640px) 100vw, 640px\">\u003cfigcaption class=\"wp-caption-text\">Student plays with a three-dimensional number line. \u003ccite>(Zach Pedersen/Robertson Program)\u003c/cite>\u003c/figcaption>\u003c/figure>\n\u003cp>“We went into it with a sort of learning trajectory in mind and most of the time they far surpassed what we thought they’d be capable of,” Bailey said. She now teaches Grade 2 students, most of whom have had spatial reasoning lessons since kindergarten. They’ve mastered many of the tasks, but she still finds more difficult ones to grow their skills. One favorite is the “hole punch symmetry challenge,” in which students imagine punching a hole in a folded up piece of paper. As the paper unfolds, where will the holes be?\u003c/p>\n\u003cp>“It is my struggling paper and pencil kids who nailed it right off the bat, which was really surprising and great because I was not expecting that,” she said.\u003c/p>\n\u003cp>In Ontario, students take an important standardized test in Grade 3 called the \u003ca href=\"http://www.eqao.com/en\">EQAO\u003c/a> that determines whether they are on grade level. That means that even in Grade 2 there’s pressure to cover a broad array of topics and anxiety that kids won’t be ready. Teachers go over diagnostic data at divisional meetings, creating lessons to target concepts and skills that students haven’t mastered. Bailey has noticed that students often struggle with tasks that involve spatial sense, a further indicator to her that spatial reasoning should be the norm in every early elementary classroom.\u003c/p>\u003c/p>\u003cp>\u003cspan class='utils-parseShortcode-shortcodes-__youtubeShortcode__embedYoutube'>\n \u003cspan class='utils-parseShortcode-shortcodes-__youtubeShortcode__embedYoutubeInside'>\n \u003ciframe\n loading='lazy'\n class='utils-parseShortcode-shortcodes-__youtubeShortcode__youtubePlayer'\n type='text/html'\n src='//www.youtube.com/embed/jNCCW7pyMww'\n title='//www.youtube.com/embed/jNCCW7pyMww'\n allowfullscreen='true'\n style='border:0;'>\u003c/iframe>\n \u003c/span>\n \u003c/span>\u003c/p>\u003cp>\u003cp>While she still uses spatial reasoning in her Grade 2 classroom, Bailey admits that without the support of colleagues working to adapt the materials to this grade level it’s more of a challenge. She thinks her experience with the Math For Young Children team and curriculum has changed her teaching forever, but wishes it was more of a priority even as kids get older. Perhaps just as important, the experience of working with math researchers and colleagues to refine lessons has her thinking about going back to school for another degree on how to better teach math.\u003c/p>\n\u003cp>\u003cstrong>MATH FOR YOUNG CHILDREN\u003c/strong>\u003c/p>\n\u003cp>There’s a well-known rift between those who believe the only type of developmentally appropriate early childhood education is a play-based one, and those concerned that relying solely on any learning that comes out of play could put students coming from impoverished backgrounds at a disadvantage. Research has shown that students from lower socioeconomic groups enter school with significantly less mathematical knowledge, and it is difficult to overcome that gap without intentional mathematics programming. But, at the same time, traditional teacher-led instruction often isn’t developmentally appropriate for five-year-olds.\u003c/p>\n\u003cp>“This project started as a way to show young children engaged in rigorous mathematics in ways that were play,” said Joan Moss, Associate Professor Emerita at the University of Toronto’s \u003ca href=\"http://www.oise.utoronto.ca/oise/Home/index.html\">Ontario Institute for Studies in Education\u003c/a>. She stresses that while math learning doesn’t only emerge from play, as some insist, the activities are still developmentally appropriate because they are presented playfully; students have lots of choice, there are many entry points, and while there are right answers, teachers build a culture in which getting a wrong answer isn’t bad.\u003c/p>\n\u003cfigure id=\"attachment_47324\" class=\"wp-caption alignright\" style=\"max-width: 640px\">\u003cimg class=\"wp-image-47324 size-large\" src=\"http://ww2.kqed.org/mindshift/wp-content/uploads/sites/23/2017/01/drawing-3d-shape-1020x1360.jpg\" alt=\"Student draws a 3D shape.\" width=\"640\" height=\"853\" srcset=\"https://ww2.kqed.org/app/uploads/sites/23/2017/01/drawing-3d-shape-1020x1360.jpg 1020w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/drawing-3d-shape-160x213.jpg 160w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/drawing-3d-shape-800x1067.jpg 800w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/drawing-3d-shape-768x1024.jpg 768w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/drawing-3d-shape-1180x1573.jpg 1180w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/drawing-3d-shape-960x1280.jpg 960w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/drawing-3d-shape-240x320.jpg 240w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/drawing-3d-shape-375x500.jpg 375w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/drawing-3d-shape-520x693.jpg 520w, https://ww2.kqed.org/app/uploads/sites/23/2017/01/drawing-3d-shape.jpg 1500w\" sizes=\"(max-width: 640px) 100vw, 640px\">\u003cfigcaption class=\"wp-caption-text\">Student draws a 3D shape. \u003ccite>(Bev Caswell/Robertson Program)\u003c/cite>\u003c/figcaption>\u003c/figure>\n\u003cp>For example, in the “quick image activity” the teacher flashes a complex pattern made out of pattern blocks. Students see it for a very brief time and then try to recreate it themselves. After working for a bit they get to see the original image again and make fixes to their original attempts. Hawes and Moss say a lot of learning happens in the fixing.\u003c/p>\n\u003cp>In addition to demonstrating that well-trained teachers can teach math concepts in developmentally appropriate and playful ways, the Math For Young Children project has been an experiment in a more collaborative type of professional development. The university researchers are working alongside classroom teachers to fine tune lessons and evaluate how well they work. The Rainy River School Board teachers who were the first participants kept logs of when they used spatial reasoning activities, how long they took, and the tweaks they made. They brought feedback from the classroom back to researchers, and used a lesson study approach to improving the lessons together.\u003c/p>\n\u003cp>Joan Moss says this collaborative model of professional development, featuring teachers working alongside researchers to build quality activities grounded in research and classroom practice has been thrilling and a huge part of the program’s success. Teachers agree: “To be able to get together with people with that much math knowledge, it was an amazing experience,” Cristol Bailey said.\u003c/p>\n\u003cp>“It changed my teaching in the fact that I think of myself as a teacher-researcher, as they call us,” Thomson added. She now approaches every classroom activity as a mini experiment, tweaking and adjusting along the way. “I’m a lot more reflective in what I’m doing and what I put out there. It’s a neat lens to look through.”\u003c/p>\n\u003cp>The University of Toronto team \u003ca href=\"http://link.springer.com/article/10.1007/s11858-014-0637-4\">evaluated\u003c/a> the Math For Young Children program as it was being implemented in the Rainy River schools. Since teachers of the experimental group were engaged in inquiry-based professional development with researchers around spatial reasoning, the control group’s teachers also had interaction with researchers on a different topic. This was meant to make the groups more similar in exposure, but with different focuses.\u003c/p>\n\u003cp>After the first year, students in the experimental group made \u003ca href=\"http://link.springer.com/article/10.1007/s11858-014-0637-4\">significant gains on assessments \u003c/a>of geometry, spatial reasoning and numerical skills compared to the control group. In the second year, researchers decided to test students on the KeyMath measures, which are used to assess school-based mathematical concepts and skills. Students in the experimental group showed significant gains on those more traditional measures as well (a paper with these findings will soon be published in \u003ca href=\"http://www.tandfonline.com/toc/hcgi20/current\">Cognition and Instruction\u003c/a>). The first class of students will take the EQAO this year, and researchers hope they will show increased learning over peers in the rest of the province.\u003c/p>\n\u003cp>\u003c/p>\u003c/div>",
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"content": "\u003cdiv class=\"post-body\">\u003cp>\u003c/p>\n\u003cp>The Ontario Ministry of Education is interested in spreading the spatial reasoning work that the researchers started. Hawes and Flynn wrote a document titled \u003ca href=\"http://www.edu.gov.on.ca/eng/literacynumeracy/LNSPayingAttention.pdf\">\"Paying Attention to Spatial Reasoning\"\u003c/a> that the ministry distributed to educators across the district.** Individual school boards are also showing interest in training and implementation.\u003cbr>\n\u003cem>\u003cbr>\n*The article has been updated to note that Cathy Bruce helped lead the Math For Young Children research, which is taking place in several locations around Ontario.\u003cbr>\n**The article has been updated to include Tara Flynn’s contribution to the document. We regret these errors.\u003c/em>\u003c/p>\n\n\u003c/div>\u003c/p>",
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"disqusTitle": "Five Compelling Reasons For Teaching Spatial Reasoning To Young Children",
"title": "Five Compelling Reasons For Teaching Spatial Reasoning To Young Children",
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"content": "\u003cp>\u003cem>Excerpted from \"\u003ca href=\"http://www.pearsoncanadaschool.com/index.cfm?locator=PS2uLd\">Taking Shape: Activities to Develop Geometric and Spatial Thinking\u003c/a>\" by Joan Moss, Catherine D. Bruce, Bev Caswell, Tara Flynn, and Zachary Hawes. Published by Pearson Canada Inc., 2016, pp. 13–16. Reprinted with permission by Pearson Canada Inc.\u003c/em>\u003cbr>\n\u003cstrong>\u003cbr>\nBy Joan Moss, Catherine D. Bruce, Bev Caswell, Tara Flynn, and Zachary Hawes\u003c/strong>\u003c/p>\n\u003cp>Our journey began when we conducted an extensive literature review at the outset of the project (Bruce, Flynn, & Moss, 2012) and learned about the crucial importance of spatial reasoning. This theme was consistent across many research disciplines, including biology, cognitive sciences, psychology, developmental sciences, education, as well as educational neuroscience—an emerging transdisciplinary field which sits at the intersection of these other disciplines and aims for a collaborative approach in which educational theory and practice are informed by new findings in the cognitive sciences, and vice versa (Fisher, 2009). We also learned—and have experienced in our careers as mathematics educators and researchers—that spatial reasoning is a curiously unacknowledged and neglected area of the curriculum. During our involvement with the M4YC project, we have become more and more convinced of reasons why we should pay attention to spatial reasoning in early years mathematics. Below we offer our Top Five reasons why, as educators, we should care about spatial thinking when we plan, observe, and assess mathematics in our classrooms.\u003c/p>\n\u003cp>\u003cstrong>1. Spatial reasoning and mathematical thinking are intimately linked.\u003c/strong>\u003c/p>\n\u003cp>There are numerous research studies that demonstrate the relationship between spatial reasoning and what we typically think of as mathematical ability. For example, one research study found that the quality of block play at four years of age was a predictor of high school mathematics achievement (Wolfgang et al., 2001). Another study found a relationship between young children’s construction skills (such as playing with jigsaw puzzles and blocks) and strong number sense and success in solving mathematical word problems (Nath & Szücs, 2014). In fact, as Mix and Cheng (2012) report, “The relation between spatial ability and mathematics is so well established that it no longer makes sense to ask whether they are related” (p. 206). Researchers have underlined that the link between spatial reasoning and math is so strong that it is “almost as if they are one and the same thing” (Dehaene, 1997, p. 125). Reflecting on the strength of this relationship, others have noted that “spatial instruction will have a two-for-one effect” that yields benefits in mathematics as well as the spatial domain (Verdine, Golinkoff, Hirsh-Pasek, & Newcombe, 2013, p. 13). Of course, the practices of mathematicians also benefit from spatial reasoning; many mathematicians stress that their work relies strongly on visual and spatial representations and forms of understanding (Whiteley, Sinclair, & Davis, 2015).\u003c/p>\n\u003cp>\u003ca href=\"https://www.amazon.ca/Taking-Shape-Activities-Geometric-Thinking/dp/0134153499\">\u003cimg class=\"alignright wp-image-47302 size-full\" src=\"http://ww2.kqed.org/mindshift/wp-content/uploads/sites/23/2017/01/taking-shape-e1483996684549.jpg\" alt=\"taking-shape\" width=\"250\" height=\"328\">\u003c/a>We can see how the various strands of mathematics are inherently spatial. Think about what happens when we compare the area of two polygons, such as a rhombus and a rectangle. To be successful, we can draw on spatial strategies such as composition and decomposition of 2D shapes, mental rotation, and visualization. In fact, research shows that spatial reasoning is linked to performance within many strands of mathematics including: basic magnitude and counting skills (Thompson, Nuerk, Moeller, & Cohen Kadosh, 2013), mental arithmetic (Kyttälä & Lehto, 2008), word problems (Hegarty & Kozhevnikov, 1999), algebra (Tolar, Lederberg, & Fletcher, 2009), calculus (Sorby, Casey, Veurink, & Dulaney, 2013), and advanced mathematics (Wei, Yuan, Chen, & Zhou, 2012).\u003cbr>\nIn one of the first studies of its kind to show specific links between spatial and mathematical skills, Cheng and Mix (2013) assessed children in both spatial and math skills. Children were randomly assigned to one of two groups: one group engaged in spatial training involving mental rotations, and the other group spent the equivalent amount of time working on crossword puzzles. Both groups of children completed pre- and post-tests involving a range of math and spatial skills. Children in the spatial training group outperformed those in the crossword puzzle group, demonstrating significant improvements in their calculation skills.\u003c/p>\n\u003cp>[ad fullwidth]\u003c/p>\n\u003cp>In another study, Verdine, Irwin, Golinkoff, and Hirsh-Pasek (2014) found that a child’s spatial skill at age three was a reliable predictor of the child’s grasp of number concepts such as more, less, equal, and second, as well as overall number knowledge skills. Taken together, research suggests that spatial instruction offers a potentially powerful means of supporting children’s mathematical thinking and learning.\u003cbr>\n\u003cstrong>\u003cbr>\n2. Spatial reasoning can be improved. Education matters!\u003c/strong>\u003c/p>\n\u003cp>Spatial reasoning is malleable; that is, it can be improved. Spatial reasoning is not a biologically determined cognitive trait as was once thought to be the case. A recent meta-analysis of 217 studies, representing more than two decades of research on spatial training, found that a variety of activities improve spatial reasoning across all age groups (Uttal et al., 2013). Not only did the authors find that spatial training led to improvements on spatial tasks closely related to the training task, but improvements were also seen on other types of tasks that were not part of the training. More research is needed to discover how and why this is the case. In the meantime, the finding that spatial ability can be improved at any age has massive implications for educators, particularly given that spatial reasoning is proving to be an important domain with strong connections to mathematical achievement.\u003c/p>\n\u003cp>\u003cstrong>3. Spatial thinking is an important predictor of achievement in STEM careers.\u003c/strong>\u003c/p>\n\u003cp>Research shows that spatial thinking is an important predictor of achievement in the STEM disciplines—science, technology, engineering, and mathematics (Wai, Lubinski, & Benbow, 2009). Sometimes these are called “STEAM” to reflect the inclusion of the arts. In addition, recent research indicates that early attention to developing children’s spatial thinking increases achievement in math and science and can promote skill and interest in future careers in STEM disciplines (Newcombe, 2010). Currently, many countries are concerned by the low numbers of post-secondary students, particularly female students, entering these disciplines. For example, a 2013 report found that fewer than 50 percent of Canadian secondary school students were graduating with senior-level STEM credits, while 70 percent of the highest-paying jobs require expertise in these disciplines (Let’s Talk Science [with Amgen Canada Inc.], 2013).\u003c/p>\n\u003cp>Geometry spans mathematics and science and plays a central role in disciplines such as surveying, astronomy, chemistry and physics, biology, geography and geology, art and architecture (Wai, Lubinski, & Benbow, 2009).\u003cbr>\n\u003cstrong>\u003cbr>\n4. Spatial reasoning is currently an underserved area of mathematics instruction.\u003c/strong>\u003c/p>\n\u003cp>The National Council of Teachers of Mathematics recommends that at least 50 percent of mathematics instruction focus on spatial reasoning (National Council of Teachers of Mathematics [NCTM], 2006, 2010). Despite calls to bring geometry and spatial thinking to the forefront of early math curricula, local and international studies reveal that geometry and spatial sense receive the least amount of attention in early years math (Bruce, Flynn, & Moss, 2012; Sarama & Clements, 2009a), making it an underserved area of mathematics instruction. Spatial thinking is important in many areas of mathematics and beyond; most subjects in school—art, geography, science, language, and physical education to name a few—rely on at least some aspects of spatial thinking. Yet spatial reasoning itself is rarely, if ever, paid explicit attention. The National Research Council (2006) has highlighted this as a “major blind spot” in education and calls on educators and researchers to pay attention to spatial reasoning. Otherwise, the Council warns, spatial reasoning “will remain locked in a curious educational twilight zone: extensively relied on across the K–12 curriculum but not explicitly and systematically instructed in any part of the curriculum” (p. 7). Geometry and spatial reasoning in the early years typically focus on having children label and sort shapes (Clements, 2004), yet cognitive science and educational research, including the M4YC research, shows us that young children are capable of—and interested in—more dynamic and complex spatial thinking.\u003c/p>\n\u003cp>\u003cstrong>5. Spatial reasoning provides multiple entry points and equitable access to mathematics.\u003c/strong>\u003c/p>\n\u003cp>Many educators in our research classrooms have found that a focus on spatial reasoning provides multiple entry points for children to explore mathematics in an accessible and inclusive way. In fact, many educators have reported to us that, through using the activities that now appear in this book, they have been able to see their students in a new light. This, in turn, gives children the opportunity to participate in the mathematics and to contribute to mathematical discussions in the classroom, building their identities as mathematicians. For example, educators have found that some children who may be struggling in the area of number sense may excel in the area of spatial reasoning. For most children, a spatial approach enhances their developing sense of number. According to Baroody, Lai, and Mix (2006), “Most individual differences [in math ability] are probably due to the lack of opportunity” (p. 200). When we focus on spatial reasoning, we highlight and invite the diverse strengths that children bring to school (Flynn and Hawes, 2014).\u003c/p>\n\u003cp>\u003cem>Joan Moss is an Associate Professor in the Department of Applied Psychology and Human Development at the Dr. Eric Jackman Institute of Child Study at the Ontario Institute for Studies in Education of the University of Toronto.\u003c/em>\u003c/p>\n\u003cp>\u003cem>Catherine D. Bruce is a Professor and Dean of the School of Education and Professional Learning and Director of the Centre for Teaching and Learning at Trent University.\u003c/em>\u003c/p>\n\u003cp>\u003cem>Bev Caswell is the Director of the Robertson Program for Inquiry-Based Teaching in Mathematics and Science at the Dr. Eric Jackman Institute of Child Study and Assistant Professor, Teaching Stream at the Ontario Institute for Studies in Education of the University of Toronto.\u003c/em>\u003c/p>\n\u003cp>\u003cem>Tara Flynn is an educator, author, and editor, and Project Manager and Research Officer for Dr. Cathy Bruce at the Trent University School of Education and Professional Learning.\u003c/em>\u003c/p>\n\u003cp>[ad floatright]\u003c/p>\n\u003cp>\u003cem>Zachary Hawes is a Ph.D. candidate in the Numerical Cognition Laboratory at the University of Western Ontario. Prior to this, he completed his M.A. and teacher training at the University of Toronto’s Dr. Eric Jackman Institute of Child Study.\u003c/em>\u003c/p>\n\n",
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"content": "\u003cdiv class=\"post-body\">\u003cp>\u003cp>\u003cem>Excerpted from \"\u003ca href=\"http://www.pearsoncanadaschool.com/index.cfm?locator=PS2uLd\">Taking Shape: Activities to Develop Geometric and Spatial Thinking\u003c/a>\" by Joan Moss, Catherine D. Bruce, Bev Caswell, Tara Flynn, and Zachary Hawes. Published by Pearson Canada Inc., 2016, pp. 13–16. Reprinted with permission by Pearson Canada Inc.\u003c/em>\u003cbr>\n\u003cstrong>\u003cbr>\nBy Joan Moss, Catherine D. Bruce, Bev Caswell, Tara Flynn, and Zachary Hawes\u003c/strong>\u003c/p>\n\u003cp>Our journey began when we conducted an extensive literature review at the outset of the project (Bruce, Flynn, & Moss, 2012) and learned about the crucial importance of spatial reasoning. This theme was consistent across many research disciplines, including biology, cognitive sciences, psychology, developmental sciences, education, as well as educational neuroscience—an emerging transdisciplinary field which sits at the intersection of these other disciplines and aims for a collaborative approach in which educational theory and practice are informed by new findings in the cognitive sciences, and vice versa (Fisher, 2009). We also learned—and have experienced in our careers as mathematics educators and researchers—that spatial reasoning is a curiously unacknowledged and neglected area of the curriculum. During our involvement with the M4YC project, we have become more and more convinced of reasons why we should pay attention to spatial reasoning in early years mathematics. Below we offer our Top Five reasons why, as educators, we should care about spatial thinking when we plan, observe, and assess mathematics in our classrooms.\u003c/p>\n\u003cp>\u003cstrong>1. Spatial reasoning and mathematical thinking are intimately linked.\u003c/strong>\u003c/p>\n\u003cp>There are numerous research studies that demonstrate the relationship between spatial reasoning and what we typically think of as mathematical ability. For example, one research study found that the quality of block play at four years of age was a predictor of high school mathematics achievement (Wolfgang et al., 2001). Another study found a relationship between young children’s construction skills (such as playing with jigsaw puzzles and blocks) and strong number sense and success in solving mathematical word problems (Nath & Szücs, 2014). In fact, as Mix and Cheng (2012) report, “The relation between spatial ability and mathematics is so well established that it no longer makes sense to ask whether they are related” (p. 206). Researchers have underlined that the link between spatial reasoning and math is so strong that it is “almost as if they are one and the same thing” (Dehaene, 1997, p. 125). Reflecting on the strength of this relationship, others have noted that “spatial instruction will have a two-for-one effect” that yields benefits in mathematics as well as the spatial domain (Verdine, Golinkoff, Hirsh-Pasek, & Newcombe, 2013, p. 13). Of course, the practices of mathematicians also benefit from spatial reasoning; many mathematicians stress that their work relies strongly on visual and spatial representations and forms of understanding (Whiteley, Sinclair, & Davis, 2015).\u003c/p>\n\u003cp>\u003ca href=\"https://www.amazon.ca/Taking-Shape-Activities-Geometric-Thinking/dp/0134153499\">\u003cimg class=\"alignright wp-image-47302 size-full\" src=\"http://ww2.kqed.org/mindshift/wp-content/uploads/sites/23/2017/01/taking-shape-e1483996684549.jpg\" alt=\"taking-shape\" width=\"250\" height=\"328\">\u003c/a>We can see how the various strands of mathematics are inherently spatial. Think about what happens when we compare the area of two polygons, such as a rhombus and a rectangle. To be successful, we can draw on spatial strategies such as composition and decomposition of 2D shapes, mental rotation, and visualization. In fact, research shows that spatial reasoning is linked to performance within many strands of mathematics including: basic magnitude and counting skills (Thompson, Nuerk, Moeller, & Cohen Kadosh, 2013), mental arithmetic (Kyttälä & Lehto, 2008), word problems (Hegarty & Kozhevnikov, 1999), algebra (Tolar, Lederberg, & Fletcher, 2009), calculus (Sorby, Casey, Veurink, & Dulaney, 2013), and advanced mathematics (Wei, Yuan, Chen, & Zhou, 2012).\u003cbr>\nIn one of the first studies of its kind to show specific links between spatial and mathematical skills, Cheng and Mix (2013) assessed children in both spatial and math skills. Children were randomly assigned to one of two groups: one group engaged in spatial training involving mental rotations, and the other group spent the equivalent amount of time working on crossword puzzles. Both groups of children completed pre- and post-tests involving a range of math and spatial skills. Children in the spatial training group outperformed those in the crossword puzzle group, demonstrating significant improvements in their calculation skills.\u003c/p>\n\u003cp>\u003c/p>\u003c/div>",
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"content": "\u003cdiv class=\"post-body\">\u003cp>\u003c/p>\n\u003cp>In another study, Verdine, Irwin, Golinkoff, and Hirsh-Pasek (2014) found that a child’s spatial skill at age three was a reliable predictor of the child’s grasp of number concepts such as more, less, equal, and second, as well as overall number knowledge skills. Taken together, research suggests that spatial instruction offers a potentially powerful means of supporting children’s mathematical thinking and learning.\u003cbr>\n\u003cstrong>\u003cbr>\n2. Spatial reasoning can be improved. Education matters!\u003c/strong>\u003c/p>\n\u003cp>Spatial reasoning is malleable; that is, it can be improved. Spatial reasoning is not a biologically determined cognitive trait as was once thought to be the case. A recent meta-analysis of 217 studies, representing more than two decades of research on spatial training, found that a variety of activities improve spatial reasoning across all age groups (Uttal et al., 2013). Not only did the authors find that spatial training led to improvements on spatial tasks closely related to the training task, but improvements were also seen on other types of tasks that were not part of the training. More research is needed to discover how and why this is the case. In the meantime, the finding that spatial ability can be improved at any age has massive implications for educators, particularly given that spatial reasoning is proving to be an important domain with strong connections to mathematical achievement.\u003c/p>\n\u003cp>\u003cstrong>3. Spatial thinking is an important predictor of achievement in STEM careers.\u003c/strong>\u003c/p>\n\u003cp>Research shows that spatial thinking is an important predictor of achievement in the STEM disciplines—science, technology, engineering, and mathematics (Wai, Lubinski, & Benbow, 2009). Sometimes these are called “STEAM” to reflect the inclusion of the arts. In addition, recent research indicates that early attention to developing children’s spatial thinking increases achievement in math and science and can promote skill and interest in future careers in STEM disciplines (Newcombe, 2010). Currently, many countries are concerned by the low numbers of post-secondary students, particularly female students, entering these disciplines. For example, a 2013 report found that fewer than 50 percent of Canadian secondary school students were graduating with senior-level STEM credits, while 70 percent of the highest-paying jobs require expertise in these disciplines (Let’s Talk Science [with Amgen Canada Inc.], 2013).\u003c/p>\n\u003cp>Geometry spans mathematics and science and plays a central role in disciplines such as surveying, astronomy, chemistry and physics, biology, geography and geology, art and architecture (Wai, Lubinski, & Benbow, 2009).\u003cbr>\n\u003cstrong>\u003cbr>\n4. Spatial reasoning is currently an underserved area of mathematics instruction.\u003c/strong>\u003c/p>\n\u003cp>The National Council of Teachers of Mathematics recommends that at least 50 percent of mathematics instruction focus on spatial reasoning (National Council of Teachers of Mathematics [NCTM], 2006, 2010). Despite calls to bring geometry and spatial thinking to the forefront of early math curricula, local and international studies reveal that geometry and spatial sense receive the least amount of attention in early years math (Bruce, Flynn, & Moss, 2012; Sarama & Clements, 2009a), making it an underserved area of mathematics instruction. Spatial thinking is important in many areas of mathematics and beyond; most subjects in school—art, geography, science, language, and physical education to name a few—rely on at least some aspects of spatial thinking. Yet spatial reasoning itself is rarely, if ever, paid explicit attention. The National Research Council (2006) has highlighted this as a “major blind spot” in education and calls on educators and researchers to pay attention to spatial reasoning. Otherwise, the Council warns, spatial reasoning “will remain locked in a curious educational twilight zone: extensively relied on across the K–12 curriculum but not explicitly and systematically instructed in any part of the curriculum” (p. 7). Geometry and spatial reasoning in the early years typically focus on having children label and sort shapes (Clements, 2004), yet cognitive science and educational research, including the M4YC research, shows us that young children are capable of—and interested in—more dynamic and complex spatial thinking.\u003c/p>\n\u003cp>\u003cstrong>5. Spatial reasoning provides multiple entry points and equitable access to mathematics.\u003c/strong>\u003c/p>\n\u003cp>Many educators in our research classrooms have found that a focus on spatial reasoning provides multiple entry points for children to explore mathematics in an accessible and inclusive way. In fact, many educators have reported to us that, through using the activities that now appear in this book, they have been able to see their students in a new light. This, in turn, gives children the opportunity to participate in the mathematics and to contribute to mathematical discussions in the classroom, building their identities as mathematicians. For example, educators have found that some children who may be struggling in the area of number sense may excel in the area of spatial reasoning. For most children, a spatial approach enhances their developing sense of number. According to Baroody, Lai, and Mix (2006), “Most individual differences [in math ability] are probably due to the lack of opportunity” (p. 200). When we focus on spatial reasoning, we highlight and invite the diverse strengths that children bring to school (Flynn and Hawes, 2014).\u003c/p>\n\u003cp>\u003cem>Joan Moss is an Associate Professor in the Department of Applied Psychology and Human Development at the Dr. Eric Jackman Institute of Child Study at the Ontario Institute for Studies in Education of the University of Toronto.\u003c/em>\u003c/p>\n\u003cp>\u003cem>Catherine D. Bruce is a Professor and Dean of the School of Education and Professional Learning and Director of the Centre for Teaching and Learning at Trent University.\u003c/em>\u003c/p>\n\u003cp>\u003cem>Bev Caswell is the Director of the Robertson Program for Inquiry-Based Teaching in Mathematics and Science at the Dr. Eric Jackman Institute of Child Study and Assistant Professor, Teaching Stream at the Ontario Institute for Studies in Education of the University of Toronto.\u003c/em>\u003c/p>\n\u003cp>\u003cem>Tara Flynn is an educator, author, and editor, and Project Manager and Research Officer for Dr. Cathy Bruce at the Trent University School of Education and Professional Learning.\u003c/em>\u003c/p>\n\u003cp>\u003c/p>\u003c/div>",
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"content": "\u003cp>Every year there are some topics and conversations that grab readers’ attention more than others. In 2016, MindShift readers engaged most often and deeply with stories about the tricky job of motivating learners, especially when circumstances like poverty, learning differences and trauma complicate classroom dynamics. Educators are looking for ways to reach all facets of the complicated learners that sit in their classrooms, diving deeply into research about self-control, mindfulness programs and teaching strategies to give students structures for their thinking. And, since educating a child is a partnership between schools, families and communities, many classroom teachers and parents alike are increasingly concerned about the role parents play in nurturing and supporting students.\u003c/p>\n\u003cp>\u003cstrong>MOTIVATION AND ENGAGEMENT\u003c/strong>\u003c/p>\n\u003cp>Motivating students is a perennially difficult aspect of teaching, so it’s no wonder that there is robust interest in the \u003ca href=\"https://ww2.kqed.org/mindshift/2016/03/04/how-to-turn-on-the-part-of-your-brain-that-controls-motivation/\" target=\"_blank\">neuroscience behind motivation\u003c/a>. Researchers found that when test subjects could see how their brains were reacting to different motivational strategies on MRI images, they got better using successful approaches. But they also found it exhausting. While not yet applicable to the classroom setting, this neuroscience does offer educators insights into strategies that did and didn’t work, as well as how tiring the process can be.\u003c/p>\n\u003cp>On a more practical note, an article featuring \u003ca href=\"https://ww2.kqed.org/mindshift/2016/03/03/20-strategies-for-motivating-reluctant-learners/\" target=\"_blank\">20 tips to engage even the most seemingly reluctant\u003c/a> students also grabbed readers’ attention. No teaching approach is going to reach every student, so teachers need lots of strategies. When teachers have many ways to present information, to offer varying points of entry, and know how to demonstrate concepts from multiple viewpoints, they can better serve the different needs of their students.\u003c/p>\n\u003cp>\u003cstrong>SELF-REGULATION AND TRAUMA\u003c/strong>\u003c/p>\n\u003cp>[ad fullwidth]\u003c/p>\n\u003cp>Increasingly, teachers are being asked to do far more than deliver content, and that shift requires a new set of strategies and a compassionate approach to the job. Often educators are looking for guidance on how they can help kids improve self-control and behavior, as well as address their social and emotional needs.\u003c/p>\n\u003cp>Managing the behavior of 30 kids in an enclosed space is one of the most difficult aspects of teaching, so it’s no surprise that no teacher knows exactly how to respond to every situation. Yet acting out is a form of communication that can easily be misinterpreted as intentional disobedience or malice. That’s why \u003ca href=\"https://ww2.kqed.org/mindshift/2016/04/21/20-tips-to-help-de-escalate-interactions-with-anxious-or-defiant-students/\" target=\"_blank\">tips to de-escalate situations\u003c/a> with anxious or defiant students, presented by an experienced behavior analyst, was so helpful to educators.\u003c/p>\n\u003cp>Similarly, more and more educators are beginning to realize how much trauma their students have endured and how their behavior is often a symptom of those experiences. Educators are gravitating to workshops on \u003ca href=\"https://ww2.kqed.org/mindshift/2016/06/06/how-trauma-informed-teaching-builds-a-sense-of-safety-and-care/\" target=\"_blank\">how to teach with a trauma-informed lens\u003c/a>, and are seeking support as they deal with the taxing work of educating children who are suffering intensely.\u003c/p>\n\u003cp>One school turned to a \u003ca href=\"https://ww2.kqed.org/mindshift/2016/03/30/what-changes-when-a-school-embraces-mindfulness/\" target=\"_blank\">program that combines mindfulness and education about the brain\u003c/a> to deal with residual trauma from a school fire, as well as the daily trauma of poverty that many students experience. The program has helped shift the culture of the school into a more positive place for students and staff with mindfulness baked into most school processes.\u003c/p>\n\u003cp>Early research on mindfulness has found that practices like focusing on one’s breath or intentionally showing gratitude can positively influence executive functioning skills that are also crucial for focusing in class, organizing work and many other cognitive functions. The importance of \u003ca href=\"https://ww2.kqed.org/mindshift/2016/01/11/research-based-strategies-to-help-children-develop-self-control/\" target=\"_blank\">self-control on life outcomes\u003c/a> has been well documented by psychologists, research that educators are now taking advantage of in classrooms.\u003c/p>\n\u003cp>\u003cstrong>DEEPENING TEACHING PRACTICE\u003c/strong>\u003c/p>\n\u003cp>Alongside discussions about how to instill character, improve school climate and motivate students to do their best work, educators are also continually trying to hone their craft, learning from research about the most effective ways to pull the best thinking out of every child. Often the articles that stimulate the most excitement and debate are not about specific curriculum or tools, but instead grapple with how to improve students’ metacognition. Researchers at Harvard have studied educators who focus on “teaching for understanding” for several years and have narrowed in on some \u003ca href=\"https://ww2.kqed.org/mindshift/2016/03/31/when-kids-have-structure-for-thinking-better-learning-emerges/\" target=\"_blank\">practices that help improve the depth of student thinking\u003c/a>.\u003c/p>\n\u003cp>In math classrooms a similar discussion is raging, with many math teachers looking for strategies to provide multiple entry points into the underlying conceptual topics in the curriculum. At the same time, most math curricula are stuffed with so many standards that teachers struggle to cover them all well. Math teachers are balancing trying to both prepare students for tests and give them the \u003ca href=\"https://ww2.kqed.org/mindshift/2016/04/13/why-kids-should-keep-using-theirs-fingers-to-do-math/\" target=\"_blank\">space and time to explore the foundations of math\u003c/a>, a key practice to future math success.\u003c/p>\n\u003cp>\u003cstrong>CAN PARENTS BE TOO INVOLVED?\u003c/strong>\u003c/p>\n\u003cp>Parents are crucial partners for teachers in the academic and social development of children. Many parents take that responsibility seriously, reading up on how they can prepare their kids for academic success \u003ca href=\"https://ww2.kqed.org/mindshift/2016/07/06/how-to-raise-brilliant-children-according-to-science/\" target=\"_blank\">through the myriad of small interactions that happen daily\u003c/a>. But the obsession with doing everything right is taking a toll on parents and may not be that great for kids either.\u003c/p>\n\u003cp>Teachers at the K-12 and university level are beginning to notice a worrying trend of overinvolvement from parents -- while well-intended, it is actually depriving kids of crucial learning experiences. Parents, too, are noticing this tendency in themselves and are \u003ca href=\"https://ww2.kqed.org/mindshift/2016/01/05/overparenting-5-recovery-steps-from-a-former-stanford-dean/\" target=\"_blank\">trying to pull back\u003c/a>, with varying levels of success.\u003c/p>\n\u003cp>\u003c/p>\n\u003cp>Reporting about education so often comes down to examining how humans interact with one another. Many of the themes that caught MindShift readers’ attention this year deal with how a bureaucratic system filled with well-intentioned people can nurture the whole child, paying attention to their academic minds, of course, but also recognizing that success in life rests on so much more. The trajectory of a life is a complicated interplay of opportunity, psychology, mentors and skills. The parents and teachers that help young people down this path have a very difficult job, but it can ultimately be one of the most rewarding ones, too.\u003c/p>\n\n",
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"content": "\u003cdiv class=\"post-body\">\u003cp>\u003cp>Every year there are some topics and conversations that grab readers’ attention more than others. In 2016, MindShift readers engaged most often and deeply with stories about the tricky job of motivating learners, especially when circumstances like poverty, learning differences and trauma complicate classroom dynamics. Educators are looking for ways to reach all facets of the complicated learners that sit in their classrooms, diving deeply into research about self-control, mindfulness programs and teaching strategies to give students structures for their thinking. And, since educating a child is a partnership between schools, families and communities, many classroom teachers and parents alike are increasingly concerned about the role parents play in nurturing and supporting students.\u003c/p>\n\u003cp>\u003cstrong>MOTIVATION AND ENGAGEMENT\u003c/strong>\u003c/p>\n\u003cp>Motivating students is a perennially difficult aspect of teaching, so it’s no wonder that there is robust interest in the \u003ca href=\"https://ww2.kqed.org/mindshift/2016/03/04/how-to-turn-on-the-part-of-your-brain-that-controls-motivation/\" target=\"_blank\">neuroscience behind motivation\u003c/a>. Researchers found that when test subjects could see how their brains were reacting to different motivational strategies on MRI images, they got better using successful approaches. But they also found it exhausting. While not yet applicable to the classroom setting, this neuroscience does offer educators insights into strategies that did and didn’t work, as well as how tiring the process can be.\u003c/p>\n\u003cp>On a more practical note, an article featuring \u003ca href=\"https://ww2.kqed.org/mindshift/2016/03/03/20-strategies-for-motivating-reluctant-learners/\" target=\"_blank\">20 tips to engage even the most seemingly reluctant\u003c/a> students also grabbed readers’ attention. No teaching approach is going to reach every student, so teachers need lots of strategies. When teachers have many ways to present information, to offer varying points of entry, and know how to demonstrate concepts from multiple viewpoints, they can better serve the different needs of their students.\u003c/p>\n\u003cp>\u003cstrong>SELF-REGULATION AND TRAUMA\u003c/strong>\u003c/p>\n\u003cp>\u003c/p>\u003c/div>",
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"content": "\u003cdiv class=\"post-body\">\u003cp>\u003c/p>\n\u003cp>Increasingly, teachers are being asked to do far more than deliver content, and that shift requires a new set of strategies and a compassionate approach to the job. Often educators are looking for guidance on how they can help kids improve self-control and behavior, as well as address their social and emotional needs.\u003c/p>\n\u003cp>Managing the behavior of 30 kids in an enclosed space is one of the most difficult aspects of teaching, so it’s no surprise that no teacher knows exactly how to respond to every situation. Yet acting out is a form of communication that can easily be misinterpreted as intentional disobedience or malice. That’s why \u003ca href=\"https://ww2.kqed.org/mindshift/2016/04/21/20-tips-to-help-de-escalate-interactions-with-anxious-or-defiant-students/\" target=\"_blank\">tips to de-escalate situations\u003c/a> with anxious or defiant students, presented by an experienced behavior analyst, was so helpful to educators.\u003c/p>\n\u003cp>Similarly, more and more educators are beginning to realize how much trauma their students have endured and how their behavior is often a symptom of those experiences. Educators are gravitating to workshops on \u003ca href=\"https://ww2.kqed.org/mindshift/2016/06/06/how-trauma-informed-teaching-builds-a-sense-of-safety-and-care/\" target=\"_blank\">how to teach with a trauma-informed lens\u003c/a>, and are seeking support as they deal with the taxing work of educating children who are suffering intensely.\u003c/p>\n\u003cp>One school turned to a \u003ca href=\"https://ww2.kqed.org/mindshift/2016/03/30/what-changes-when-a-school-embraces-mindfulness/\" target=\"_blank\">program that combines mindfulness and education about the brain\u003c/a> to deal with residual trauma from a school fire, as well as the daily trauma of poverty that many students experience. The program has helped shift the culture of the school into a more positive place for students and staff with mindfulness baked into most school processes.\u003c/p>\n\u003cp>Early research on mindfulness has found that practices like focusing on one’s breath or intentionally showing gratitude can positively influence executive functioning skills that are also crucial for focusing in class, organizing work and many other cognitive functions. The importance of \u003ca href=\"https://ww2.kqed.org/mindshift/2016/01/11/research-based-strategies-to-help-children-develop-self-control/\" target=\"_blank\">self-control on life outcomes\u003c/a> has been well documented by psychologists, research that educators are now taking advantage of in classrooms.\u003c/p>\n\u003cp>\u003cstrong>DEEPENING TEACHING PRACTICE\u003c/strong>\u003c/p>\n\u003cp>Alongside discussions about how to instill character, improve school climate and motivate students to do their best work, educators are also continually trying to hone their craft, learning from research about the most effective ways to pull the best thinking out of every child. Often the articles that stimulate the most excitement and debate are not about specific curriculum or tools, but instead grapple with how to improve students’ metacognition. Researchers at Harvard have studied educators who focus on “teaching for understanding” for several years and have narrowed in on some \u003ca href=\"https://ww2.kqed.org/mindshift/2016/03/31/when-kids-have-structure-for-thinking-better-learning-emerges/\" target=\"_blank\">practices that help improve the depth of student thinking\u003c/a>.\u003c/p>\n\u003cp>In math classrooms a similar discussion is raging, with many math teachers looking for strategies to provide multiple entry points into the underlying conceptual topics in the curriculum. At the same time, most math curricula are stuffed with so many standards that teachers struggle to cover them all well. Math teachers are balancing trying to both prepare students for tests and give them the \u003ca href=\"https://ww2.kqed.org/mindshift/2016/04/13/why-kids-should-keep-using-theirs-fingers-to-do-math/\" target=\"_blank\">space and time to explore the foundations of math\u003c/a>, a key practice to future math success.\u003c/p>\n\u003cp>\u003cstrong>CAN PARENTS BE TOO INVOLVED?\u003c/strong>\u003c/p>\n\u003cp>Parents are crucial partners for teachers in the academic and social development of children. Many parents take that responsibility seriously, reading up on how they can prepare their kids for academic success \u003ca href=\"https://ww2.kqed.org/mindshift/2016/07/06/how-to-raise-brilliant-children-according-to-science/\" target=\"_blank\">through the myriad of small interactions that happen daily\u003c/a>. But the obsession with doing everything right is taking a toll on parents and may not be that great for kids either.\u003c/p>\n\u003cp>Teachers at the K-12 and university level are beginning to notice a worrying trend of overinvolvement from parents -- while well-intended, it is actually depriving kids of crucial learning experiences. Parents, too, are noticing this tendency in themselves and are \u003ca href=\"https://ww2.kqed.org/mindshift/2016/01/05/overparenting-5-recovery-steps-from-a-former-stanford-dean/\" target=\"_blank\">trying to pull back\u003c/a>, with varying levels of success.\u003c/p>\n\u003cp>\u003c/p>\n\u003cp>Reporting about education so often comes down to examining how humans interact with one another. Many of the themes that caught MindShift readers’ attention this year deal with how a bureaucratic system filled with well-intentioned people can nurture the whole child, paying attention to their academic minds, of course, but also recognizing that success in life rests on so much more. The trajectory of a life is a complicated interplay of opportunity, psychology, mentors and skills. The parents and teachers that help young people down this path have a very difficult job, but it can ultimately be one of the most rewarding ones, too.\u003c/p>\n\n\u003c/div>\u003c/p>",
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"content": "\u003cp>How math is taught in the United States and how our students perform on international math tests continue to be areas of intense debate. The most recent \u003ca href=\"http://www.oecd.org/pisa/\" target=\"_blank\">Program for International Student Assessment (PISA)\u003c/a> results for 15-year-olds show a \u003ca href=\"https://www.washingtonpost.com/local/education/on-the-world-stage-us-students-fall-behind/2016/12/05/610e1e10-b740-11e6-a677-b608fbb3aaf6_story.html?utm_term=.057fc98521c7&wpisrc=nl_sb_smartbrief\" target=\"_blank\">significant drop in math performance \u003c/a>between 2012 and 2015 among U.S. students who now rank 40th out of the 73 countries tested. While an international comparison of this sort can never tell the whole story, PISA administrators have started including questions about how students study. The answers to these survey questions about how students approach learning math could help provide some insight into which strategies work and which do not.\u003c/p>\n\u003cp>In a \u003ca href=\"https://www.scientificamerican.com/article/why-math-education-in-the-u-s-doesn-t-add-up/\" target=\"_blank\">Scientific American article\u003c/a>, Stanford education professor Jo Boaler and Pablo Zoido, the Education Lead Specialist at the Inter-American Development Bank, explain that students reported three main strategies for learning math: memorizing algorithms, relating new topics to those already learned, and routinely evaluating learning and focusing on areas not yet learned. Boaler and Zoido draw this conclusion:\u003c/p>\n\u003cblockquote>\u003cp>In every country, the memorizers turned out to be the lowest achievers, and countries with high numbers of them—the U.S. was in the top third—also had the highest proportion of teens doing poorly on the PISA math assessment. Further analysis showed that memorizers were approximately half a year behind students who used relational and self-monitoring strategies. In no country were memorizers in the highest-achieving group, and in some high-achieving economies, the differences between memorizers and other students were substantial. In France and Japan, for example, pupils who combined self-monitoring and relational strategies outscored students using memorization by more than a year's worth of schooling.\u003c/p>\n\u003cp>The U.S. actually had more memorizers than South Korea, long thought to be the paradigm of rote learning. Why? Because American schools routinely present mathematics procedurally, as sets of steps to memorize and apply. Many teachers, faced with long lists of content to cover to satisfy state and federal requirements, worry that students do not have enough time to explore math topics in depth. Others simply teach as they were taught. And few have the opportunity to stay current with what research shows about how kids learn math best: as an open, conceptual, inquiry-based subject.\u003c/p>\u003c/blockquote>\n\u003cp>Boaler and Zoido go on to recommend that math teachers focus on presenting students with visual, engaging tasks that let students \u003ca href=\"https://ww2.kqed.org/mindshift/2015/11/30/not-a-math-person-how-to-remove-obstacles-to-learning-math/\" target=\"_blank\">grapple with the problem\u003c/a>, test out various strategies, and thus gain a deeper understanding of core concepts. They point to research showing that students who solve problems by memorizing algorithms use a completely different part of the brain than those who work out the problem with various strategies. They posit that if the U.S. wants to improve the math abilities of its young people, it must heed the research and switch approaches.\u003c/p>\n\u003cp>Countries like Canada, Estonia, Germany and Hong Kong emerged as \u003ca href=\"https://www.bloomberg.com/news/articles/2016-12-06/rich-poor-achievement-gap-is-narrowing-in-american-education\" target=\"_blank\">leaders in math education from the 2015 PISA\u003c/a> results. Not only do students in these countries score well, but the gaps between rich and poor students are much smaller.\u003c/p>\n\u003cp>https://www.scientificamerican.com/article/why-math-education-in-the-u-s-doesn-t-add-up/\u003c/p>\n\u003cp>[ad fullwidth]\u003c/p>\u003cp>\u003c/p>\n",
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"content": "\u003cdiv class=\"post-body\">\u003cp>\u003cp>How math is taught in the United States and how our students perform on international math tests continue to be areas of intense debate. The most recent \u003ca href=\"http://www.oecd.org/pisa/\" target=\"_blank\">Program for International Student Assessment (PISA)\u003c/a> results for 15-year-olds show a \u003ca href=\"https://www.washingtonpost.com/local/education/on-the-world-stage-us-students-fall-behind/2016/12/05/610e1e10-b740-11e6-a677-b608fbb3aaf6_story.html?utm_term=.057fc98521c7&wpisrc=nl_sb_smartbrief\" target=\"_blank\">significant drop in math performance \u003c/a>between 2012 and 2015 among U.S. students who now rank 40th out of the 73 countries tested. While an international comparison of this sort can never tell the whole story, PISA administrators have started including questions about how students study. The answers to these survey questions about how students approach learning math could help provide some insight into which strategies work and which do not.\u003c/p>\n\u003cp>In a \u003ca href=\"https://www.scientificamerican.com/article/why-math-education-in-the-u-s-doesn-t-add-up/\" target=\"_blank\">Scientific American article\u003c/a>, Stanford education professor Jo Boaler and Pablo Zoido, the Education Lead Specialist at the Inter-American Development Bank, explain that students reported three main strategies for learning math: memorizing algorithms, relating new topics to those already learned, and routinely evaluating learning and focusing on areas not yet learned. Boaler and Zoido draw this conclusion:\u003c/p>\n\u003cblockquote>\u003cp>In every country, the memorizers turned out to be the lowest achievers, and countries with high numbers of them—the U.S. was in the top third—also had the highest proportion of teens doing poorly on the PISA math assessment. Further analysis showed that memorizers were approximately half a year behind students who used relational and self-monitoring strategies. In no country were memorizers in the highest-achieving group, and in some high-achieving economies, the differences between memorizers and other students were substantial. In France and Japan, for example, pupils who combined self-monitoring and relational strategies outscored students using memorization by more than a year's worth of schooling.\u003c/p>\n\u003cp>The U.S. actually had more memorizers than South Korea, long thought to be the paradigm of rote learning. Why? Because American schools routinely present mathematics procedurally, as sets of steps to memorize and apply. Many teachers, faced with long lists of content to cover to satisfy state and federal requirements, worry that students do not have enough time to explore math topics in depth. Others simply teach as they were taught. And few have the opportunity to stay current with what research shows about how kids learn math best: as an open, conceptual, inquiry-based subject.\u003c/p>\u003c/blockquote>\n\u003cp>Boaler and Zoido go on to recommend that math teachers focus on presenting students with visual, engaging tasks that let students \u003ca href=\"https://ww2.kqed.org/mindshift/2015/11/30/not-a-math-person-how-to-remove-obstacles-to-learning-math/\" target=\"_blank\">grapple with the problem\u003c/a>, test out various strategies, and thus gain a deeper understanding of core concepts. They point to research showing that students who solve problems by memorizing algorithms use a completely different part of the brain than those who work out the problem with various strategies. They posit that if the U.S. wants to improve the math abilities of its young people, it must heed the research and switch approaches.\u003c/p>\n\u003cp>Countries like Canada, Estonia, Germany and Hong Kong emerged as \u003ca href=\"https://www.bloomberg.com/news/articles/2016-12-06/rich-poor-achievement-gap-is-narrowing-in-american-education\" target=\"_blank\">leaders in math education from the 2015 PISA\u003c/a> results. Not only do students in these countries score well, but the gaps between rich and poor students are much smaller.\u003c/p>\n\u003cp>https://www.scientificamerican.com/article/why-math-education-in-the-u-s-doesn-t-add-up/\u003c/p>\n\u003cp>\u003c/p>\u003c/div>",
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"content": "\u003cp>Catherine Good has experienced stereotype threat herself, although she didn’t know it at the time. She started her academic career in pure math, expecting to get a Ph.D. But somewhere along the way she started to feel like it just wasn’t for her, even though she was doing well in all her classes. Thinking that she’d just chosen the wrong application for her love of math, Good switched to math education, where she first encountered the idea of stereotype threat from a guest psychology speaker.\u003c/p>\n\u003cp>“As he talked about students feeling that they don’t really belong, I had an epiphany,” Good said. She realized the discomfort she’d felt studying mathematics had nothing to do with her ability or qualifications and everything to do with a vague sense that she didn’t belong in a field dominated by men. \u003ca href=\"http://users.nber.org/~sewp/events/2005.01.14/Bios+Links/Good-rec2-Steele_&_Aronson_95.pdf\" target=\"_blank\">Stereotype threat\u003c/a> is a term coined by psychologists Joshua Aronson and Claude Steele. They found that pervasive cultural stereotypes that marginalize groups, like “girls aren’t good at math,” create a threatening environment and affects academic achievement.\u003c/p>\n\u003cp>Good was so fascinated by how powerful psychological forces can be on learning, including her own, that she switched fields again to study social psychology, and she ended up working closely with Carol Dweck for several years when Dweck’s growth mindset work was in its early stages and not yet well-known among educators. Good now works at a psychology professor at \u003ca href=\"http://www.baruch.cuny.edu/wsas/academics/psychology/cgood.htm\" target=\"_blank\">Baruch College\u003c/a>.* Originally, Dweck and Good hypothesized that believing intelligence is flexible -- what we now call a growth mindset -- could \u003ca href=\"https://www.nsf.gov/awardsearch/showAward?AWD_ID=0813817\" target=\"_blank\">protect students from stereotype threat\u003c/a>, an inherently fixed idea.\u003c/p>\n\u003cp>“If students are first really encouraged and taught to believe in brain plasticity, our hypothesis was that they could be protected,” Good said. While that hypothesis was shown to be true, Dweck and Good also began to uncover forces that seemed to undermine individual mindsets.\u003c/p>\n\u003cp>“What we found was that students’ perception of what’s going on in their learning environments are often more important than their own beliefs,” Good said. In other words, if a classroom climate is one of fixed ability, it will override a student’s own beliefs about his brain plasticity. This effect was even more pronounced when stereotype threat was present. Students were less likely to feel belonging and were less likely to engage with content. That, in turn, led to lower achievement and lower grades.\u003c/p>\n\u003cp>[ad fullwidth]\u003c/p>\n\u003cp>“When you are looking at a long-term trajectory that’s when the culture really becomes much more important,” Good said -- especially in certain fields of study, like math and science, where stereotype threat exists and traditional classroom structures favor a laddered approach to learning that screens out the unworthy and is inherently sending fixed mindset messages.\u003c/p>\n\u003cp>\u003cstrong>APPLYING GROWTH MINDSET\u003cbr>\n\u003c/strong>\u003c/p>\n\u003cp>A recent \u003ca href=\"http://www.edweek.org/media/ewrc_mindsetintheclassroom_sept2016.pdf\" target=\"_blank\">Education Week Research Center survey\u003c/a> of 600 K-12 teachers nationwide found that over three-quarters of respondents felt “familiar” or “very familiar” with growth mindset as a concept, and nearly all reported feeling it had a positive potential for teaching and learning. A large portion of respondents also connected growth mindset with a range of positive outcomes and behaviors, but only 20 percent felt strongly that they themselves were good at cultivating a growth mindset in their students. Still fewer had confidence in their colleagues and administrators.\u003c/p>\n\u003cp>The gap between awareness of growth mindset as a good thing to incorporate into the classroom and the confidence to actually do so, especially in specific courses, may be why \u003ca href=\"http://www.edweek.org/ew/articles/2015/09/23/carol-dweck-revisits-the-growth-mindset.html\" target=\"_blank\">Carol Dweck and others are warning that growth mindset has been misinterpreted\u003c/a>, sometimes to ill effect.\u003c/p>\n\u003cp>“People are much more likely to fall back on negative stereotypes with a naive understanding [of growth mindset],” Good said. She described a study one of her graduate students recently completed that tested teachers' perspectives on student success. The graduate student gave one group of teachers an article to read that could be described as a “pop culture” understanding of growth mindset. The other group read a paper explaining that the most important way to increase student learning is for teachers to be reflective on their own pedagogical practices. Then that group reflected on new approaches they might try to help a struggling student.\u003c/p>\n\u003cp>Teachers who received only a broad brush understanding of growth mindset were less likely to reflect on their practice and more likely to shift blame back onto the struggling student for not having a growth mindset. Author, and critic of many traditional education practices like grades and standardized testing, \u003ca href=\"https://ww2.kqed.org/mindshift/2015/09/06/does-the-focus-on-student-mindsets-let-schools-off-the-hook/\" target=\"_blank\">Alfie Kohn, has also written\u003c/a> about this danger.\u003c/p>\n\u003cp>Often teachers take away two messages from growth mindset articles or trainings: Effort is important and mistakes should be celebrated. But when applied simplistically, both these takeaways can be damaging. For example, for a student who is trying hard, but not achieving success, being told to try harder could be demoralizing. And celebrating mistakes without taking time to reflect on new strategies to try again doesn’t lead to the same learning gains.\u003c/p>\n\u003cp>\u003ca href=\"https://ww2.kqed.org/mindshift/2015/11/16/growth-mindset-clearing-up-some-common-confusions/\" target=\"_blank\">Confusion about growth mindset\u003c/a> and traditional structure of many classrooms are particularly apparent in math class, and to some extent science as well. As a former mathematician turned social psychologist with a deep interest in helping marginalized groups succeed and feel welcome in science, technology, engineering and math fields, Good has some specific ideas about how growth mindset could be incorporated into the fabric of math class.\u003c/p>\n\u003cp>\u003cstrong>CULTURE\u003c/strong>\u003cbr>\nThe first big obstacle is embedded in American culture. Somehow it has become acceptable to brag about not being \u003ca href=\"https://ww2.kqed.org/mindshift/2015/11/30/not-a-math-person-how-to-remove-obstacles-to-learning-math/\" target=\"_blank\">“a math person.”\u003c/a> Good says that has to stop, especially when that type of \u003ca href=\"https://ww2.kqed.org/mindshift/2013/10/21/why-kids-take-on-adults-math-anxiety/\" target=\"_blank\">math anxiety\u003c/a> is coming from teachers and parents. “It’s almost like an infection model where the class fixates on that anxiety and is infected as well,” Good said.\u003c/p>\n\u003cp>\u003cstrong>PROBLEMS WITH ERRORS\u003c/strong>\u003cbr>\nOne concrete mathematical teaching strategy that inherently promotes a growth mindset is to present students with worked-out problems that have errors. Students follow the thinking in the problem, identify the mistakes and rework them. “Embedded in that worked example is a lovely opportunity to talk about growth mindset and mistakes and process,” Good said.\u003cbr>\n\u003cstrong>\u003cbr>\nTHINK LIKE A MATHEMATICIAN \u003c/strong>\u003cbr>\nSchool math has become almost entirely about demonstrating how to solve a problem, rather than actually engaging in \u003ca href=\"https://ww2.kqed.org/mindshift/2016/04/06/could-this-digital-math-tool-change-instruction-for-the-better/\" target=\"_blank\">the kind of problem-solving\u003c/a> that is at the heart of what professional mathematicians do. In other subject areas teachers encourage students to “think like historians” or to become writers. In those disciplines students create their own variations on expert texts and are encouraged to become practitioners. Not so in math. Good said the discussion around math should be about pushing through challenge, the same way real mathematicians do every day.\u003c/p>\n\u003cp>\u003cstrong>RETHINK ASSESSMENTS\u003c/strong>\u003cbr>\nOne of the biggest ways math teachers can embed a growth mindset into the structure and environment of class is to change the role of assessment. Rather than taking tests whose scores accumulate into a final grade, students should get credit for \u003ca href=\"https://ww2.kqed.org/mindshift/2014/12/15/how-deprogramming-kids-from-how-to-do-school-could-improve-learning/\" target=\"_blank\">returning to problems they didn’t get right\u003c/a>, recognizing their mistakes and reworking the problems. Growth over the course of the year should be rewarded. Students shouldn’t be penalized in their final grade for doing poorly at the beginning of the year if they worked hard to learn the material over time. Assessments send very clear mindset messages that are far more powerful than anything a teacher says about growth mindset.\u003c/p>\n\u003cp>“Yes, we have to give assessments,” Good said, “Yes, we have to give grades. But when teachers say this grade doesn’t mark you or indicate what you are capable of in the long term, it shifts the whole meaning of the assessment for students,” Good said.\u003c/p>\n\u003cp>She favors a \u003ca href=\"https://ww2.kqed.org/mindshift/2015/07/09/steps-to-help-schools-transform-to-competency-based-learning/\" target=\"_blank\">mastery approach\u003c/a> that allows students to go back, relearn concepts that they got wrong and earn points for that work, in part because it ensures students actually learn the material before moving on, but also because it is important for teaching a growth mindset. It shows the teacher has high expectations, but believes the students can succeed and will provide support as they work to understand.\u003c/p>\n\u003cp>“This is where assessment can drive learning, but only if you go back and look at what you did and learn from it,” Good said.\u003c/p>\n\u003cp>\u003cstrong>HELPFUL FEEDBACK\u003c/strong>\u003cbr>\nFeedback is one of the most effective ways to help a student grow, but teachers must be mindful that students will always receive critical feedback through the lens of their stereotype threat. Human brains are also wired to \u003ca href=\"https://ww2.kqed.org/mindshift/2016/04/14/how-to-get-past-negativity-bias-and-hardwire-positive-experiences/\" target=\"_blank\">pay more attention to negative inputs \u003c/a>than positive ones. When teachers couch feedback with assurances that they will continue to hold the student to high standards and that they know he can get there, it helps protect him from the stereotype.\u003c/p>\n\u003cp>On the flip side, teachers who have fixed mindsets themselves are more likely to give comforting feedback meant to make the student feel better. Comments like, “It’s OK, let’s look at where you do have strengths,” are meant well, but communicate a fixed mindset to the student. “Things we do for students to boost their self-esteem actually have these ironic effects of making students feel you don’t believe in them,” Good said.\u003cbr>\n\u003cstrong>\u003cbr>\nRETHINK ADVANCEMENT\u003c/strong>\u003cbr>\nGood sees the current practice of looking at math learning as a ladder with progressively more difficult rungs as a detrimental approach. It encourages teachers to \u003ca href=\"https://ww2.kqed.org/mindshift/2014/12/29/intro-college-science-classes-that-enable-instead-of-weed/\" target=\"_blank\">act as gatekeepers to higher- level classes\u003c/a>, funneling the “smart” kids into advanced courses and keeping out those who struggle. That in turn communicates low expectations and a fixed mindset about students’ abilities. Good said there should be multiple entry points, as opposed to a linear progression.\u003c/p>\n\u003cp>\u003cstrong>PREPARE EVERYONE\u003c/strong>\u003cbr>\nGrowth mindsets are often discussed in relationship to kids who struggle, but the concept is just as \u003ca href=\"https://ww2.kqed.org/mindshift/2014/03/17/can-focus-on-grit-work-in-school-cultures-that-reward-grades/\" target=\"_blank\">relevant to kids who breeze through the material\u003c/a>. Telling those kids they are smart is not setting them up for success later when they do struggle. For Good, that struggle didn’t come until graduate school, but she distinctly remembers feeling “not smart anymore” because she was struggling. Math teachers need to give high achievers opportunities to struggle and persevere early and often so the experience is not foreign to them.\u003c/p>\n\u003cp>Embedded in all of this growth mindset work is a general culture shift around how math is taught and who can excel at it. It’s no surprise that teachers are struggling to integrate growth mindset into their teaching practice because every child is different. When it comes to perceptions of intelligence, belonging and whether a teacher cares, many factors come into play. Most teachers were educated in math classrooms with fixed mindset messages, as were\u003ca href=\"https://ww2.kqed.org/mindshift/2016/05/08/talking-about-failure-what-parents-can-do-to-motivate-kids-in-school/\" target=\"_blank\"> most parents\u003c/a>, so \u003ca href=\"https://ww2.kqed.org/mindshift/2015/09/24/never-too-late-creating-a-climate-for-adults-to-learn-new-skills/\" target=\"_blank\">shifting the culture of classrooms and schools \u003c/a>is work that takes time and incremental changes. But when \u003ca href=\"https://ww2.kqed.org/mindshift/2016/11/02/how-one-school-changed-its-math-culture-starting-with-teachers/\" target=\"_blank\">teachers commit to that work\u003c/a>, the shift is possible.\u003c/p>\n\u003cp>[ad floatright]\u003c/p>\n\u003cp>* \u003cem>Catherine Good is currently on leave from Baruch College, serving as \u003ca href=\"http://www.turnaroundusa.org/team/catherine-good/\">Senior Research Scientist\u003c/a> at the organization Turnaround for Children.\u003c/em>\u003c/p>\n\n",
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"content": "\u003cdiv class=\"post-body\">\u003cp>\u003cp>Catherine Good has experienced stereotype threat herself, although she didn’t know it at the time. She started her academic career in pure math, expecting to get a Ph.D. But somewhere along the way she started to feel like it just wasn’t for her, even though she was doing well in all her classes. Thinking that she’d just chosen the wrong application for her love of math, Good switched to math education, where she first encountered the idea of stereotype threat from a guest psychology speaker.\u003c/p>\n\u003cp>“As he talked about students feeling that they don’t really belong, I had an epiphany,” Good said. She realized the discomfort she’d felt studying mathematics had nothing to do with her ability or qualifications and everything to do with a vague sense that she didn’t belong in a field dominated by men. \u003ca href=\"http://users.nber.org/~sewp/events/2005.01.14/Bios+Links/Good-rec2-Steele_&_Aronson_95.pdf\" target=\"_blank\">Stereotype threat\u003c/a> is a term coined by psychologists Joshua Aronson and Claude Steele. They found that pervasive cultural stereotypes that marginalize groups, like “girls aren’t good at math,” create a threatening environment and affects academic achievement.\u003c/p>\n\u003cp>Good was so fascinated by how powerful psychological forces can be on learning, including her own, that she switched fields again to study social psychology, and she ended up working closely with Carol Dweck for several years when Dweck’s growth mindset work was in its early stages and not yet well-known among educators. Good now works at a psychology professor at \u003ca href=\"http://www.baruch.cuny.edu/wsas/academics/psychology/cgood.htm\" target=\"_blank\">Baruch College\u003c/a>.* Originally, Dweck and Good hypothesized that believing intelligence is flexible -- what we now call a growth mindset -- could \u003ca href=\"https://www.nsf.gov/awardsearch/showAward?AWD_ID=0813817\" target=\"_blank\">protect students from stereotype threat\u003c/a>, an inherently fixed idea.\u003c/p>\n\u003cp>“If students are first really encouraged and taught to believe in brain plasticity, our hypothesis was that they could be protected,” Good said. While that hypothesis was shown to be true, Dweck and Good also began to uncover forces that seemed to undermine individual mindsets.\u003c/p>\n\u003cp>“What we found was that students’ perception of what’s going on in their learning environments are often more important than their own beliefs,” Good said. In other words, if a classroom climate is one of fixed ability, it will override a student’s own beliefs about his brain plasticity. This effect was even more pronounced when stereotype threat was present. Students were less likely to feel belonging and were less likely to engage with content. That, in turn, led to lower achievement and lower grades.\u003c/p>\n\u003cp>\u003c/p>\u003c/div>",
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"content": "\u003cdiv class=\"post-body\">\u003cp>\u003c/p>\n\u003cp>“When you are looking at a long-term trajectory that’s when the culture really becomes much more important,” Good said -- especially in certain fields of study, like math and science, where stereotype threat exists and traditional classroom structures favor a laddered approach to learning that screens out the unworthy and is inherently sending fixed mindset messages.\u003c/p>\n\u003cp>\u003cstrong>APPLYING GROWTH MINDSET\u003cbr>\n\u003c/strong>\u003c/p>\n\u003cp>A recent \u003ca href=\"http://www.edweek.org/media/ewrc_mindsetintheclassroom_sept2016.pdf\" target=\"_blank\">Education Week Research Center survey\u003c/a> of 600 K-12 teachers nationwide found that over three-quarters of respondents felt “familiar” or “very familiar” with growth mindset as a concept, and nearly all reported feeling it had a positive potential for teaching and learning. A large portion of respondents also connected growth mindset with a range of positive outcomes and behaviors, but only 20 percent felt strongly that they themselves were good at cultivating a growth mindset in their students. Still fewer had confidence in their colleagues and administrators.\u003c/p>\n\u003cp>The gap between awareness of growth mindset as a good thing to incorporate into the classroom and the confidence to actually do so, especially in specific courses, may be why \u003ca href=\"http://www.edweek.org/ew/articles/2015/09/23/carol-dweck-revisits-the-growth-mindset.html\" target=\"_blank\">Carol Dweck and others are warning that growth mindset has been misinterpreted\u003c/a>, sometimes to ill effect.\u003c/p>\n\u003cp>“People are much more likely to fall back on negative stereotypes with a naive understanding [of growth mindset],” Good said. She described a study one of her graduate students recently completed that tested teachers' perspectives on student success. The graduate student gave one group of teachers an article to read that could be described as a “pop culture” understanding of growth mindset. The other group read a paper explaining that the most important way to increase student learning is for teachers to be reflective on their own pedagogical practices. Then that group reflected on new approaches they might try to help a struggling student.\u003c/p>\n\u003cp>Teachers who received only a broad brush understanding of growth mindset were less likely to reflect on their practice and more likely to shift blame back onto the struggling student for not having a growth mindset. Author, and critic of many traditional education practices like grades and standardized testing, \u003ca href=\"https://ww2.kqed.org/mindshift/2015/09/06/does-the-focus-on-student-mindsets-let-schools-off-the-hook/\" target=\"_blank\">Alfie Kohn, has also written\u003c/a> about this danger.\u003c/p>\n\u003cp>Often teachers take away two messages from growth mindset articles or trainings: Effort is important and mistakes should be celebrated. But when applied simplistically, both these takeaways can be damaging. For example, for a student who is trying hard, but not achieving success, being told to try harder could be demoralizing. And celebrating mistakes without taking time to reflect on new strategies to try again doesn’t lead to the same learning gains.\u003c/p>\n\u003cp>\u003ca href=\"https://ww2.kqed.org/mindshift/2015/11/16/growth-mindset-clearing-up-some-common-confusions/\" target=\"_blank\">Confusion about growth mindset\u003c/a> and traditional structure of many classrooms are particularly apparent in math class, and to some extent science as well. As a former mathematician turned social psychologist with a deep interest in helping marginalized groups succeed and feel welcome in science, technology, engineering and math fields, Good has some specific ideas about how growth mindset could be incorporated into the fabric of math class.\u003c/p>\n\u003cp>\u003cstrong>CULTURE\u003c/strong>\u003cbr>\nThe first big obstacle is embedded in American culture. Somehow it has become acceptable to brag about not being \u003ca href=\"https://ww2.kqed.org/mindshift/2015/11/30/not-a-math-person-how-to-remove-obstacles-to-learning-math/\" target=\"_blank\">“a math person.”\u003c/a> Good says that has to stop, especially when that type of \u003ca href=\"https://ww2.kqed.org/mindshift/2013/10/21/why-kids-take-on-adults-math-anxiety/\" target=\"_blank\">math anxiety\u003c/a> is coming from teachers and parents. “It’s almost like an infection model where the class fixates on that anxiety and is infected as well,” Good said.\u003c/p>\n\u003cp>\u003cstrong>PROBLEMS WITH ERRORS\u003c/strong>\u003cbr>\nOne concrete mathematical teaching strategy that inherently promotes a growth mindset is to present students with worked-out problems that have errors. Students follow the thinking in the problem, identify the mistakes and rework them. “Embedded in that worked example is a lovely opportunity to talk about growth mindset and mistakes and process,” Good said.\u003cbr>\n\u003cstrong>\u003cbr>\nTHINK LIKE A MATHEMATICIAN \u003c/strong>\u003cbr>\nSchool math has become almost entirely about demonstrating how to solve a problem, rather than actually engaging in \u003ca href=\"https://ww2.kqed.org/mindshift/2016/04/06/could-this-digital-math-tool-change-instruction-for-the-better/\" target=\"_blank\">the kind of problem-solving\u003c/a> that is at the heart of what professional mathematicians do. In other subject areas teachers encourage students to “think like historians” or to become writers. In those disciplines students create their own variations on expert texts and are encouraged to become practitioners. Not so in math. Good said the discussion around math should be about pushing through challenge, the same way real mathematicians do every day.\u003c/p>\n\u003cp>\u003cstrong>RETHINK ASSESSMENTS\u003c/strong>\u003cbr>\nOne of the biggest ways math teachers can embed a growth mindset into the structure and environment of class is to change the role of assessment. Rather than taking tests whose scores accumulate into a final grade, students should get credit for \u003ca href=\"https://ww2.kqed.org/mindshift/2014/12/15/how-deprogramming-kids-from-how-to-do-school-could-improve-learning/\" target=\"_blank\">returning to problems they didn’t get right\u003c/a>, recognizing their mistakes and reworking the problems. Growth over the course of the year should be rewarded. Students shouldn’t be penalized in their final grade for doing poorly at the beginning of the year if they worked hard to learn the material over time. Assessments send very clear mindset messages that are far more powerful than anything a teacher says about growth mindset.\u003c/p>\n\u003cp>“Yes, we have to give assessments,” Good said, “Yes, we have to give grades. But when teachers say this grade doesn’t mark you or indicate what you are capable of in the long term, it shifts the whole meaning of the assessment for students,” Good said.\u003c/p>\n\u003cp>She favors a \u003ca href=\"https://ww2.kqed.org/mindshift/2015/07/09/steps-to-help-schools-transform-to-competency-based-learning/\" target=\"_blank\">mastery approach\u003c/a> that allows students to go back, relearn concepts that they got wrong and earn points for that work, in part because it ensures students actually learn the material before moving on, but also because it is important for teaching a growth mindset. It shows the teacher has high expectations, but believes the students can succeed and will provide support as they work to understand.\u003c/p>\n\u003cp>“This is where assessment can drive learning, but only if you go back and look at what you did and learn from it,” Good said.\u003c/p>\n\u003cp>\u003cstrong>HELPFUL FEEDBACK\u003c/strong>\u003cbr>\nFeedback is one of the most effective ways to help a student grow, but teachers must be mindful that students will always receive critical feedback through the lens of their stereotype threat. Human brains are also wired to \u003ca href=\"https://ww2.kqed.org/mindshift/2016/04/14/how-to-get-past-negativity-bias-and-hardwire-positive-experiences/\" target=\"_blank\">pay more attention to negative inputs \u003c/a>than positive ones. When teachers couch feedback with assurances that they will continue to hold the student to high standards and that they know he can get there, it helps protect him from the stereotype.\u003c/p>\n\u003cp>On the flip side, teachers who have fixed mindsets themselves are more likely to give comforting feedback meant to make the student feel better. Comments like, “It’s OK, let’s look at where you do have strengths,” are meant well, but communicate a fixed mindset to the student. “Things we do for students to boost their self-esteem actually have these ironic effects of making students feel you don’t believe in them,” Good said.\u003cbr>\n\u003cstrong>\u003cbr>\nRETHINK ADVANCEMENT\u003c/strong>\u003cbr>\nGood sees the current practice of looking at math learning as a ladder with progressively more difficult rungs as a detrimental approach. It encourages teachers to \u003ca href=\"https://ww2.kqed.org/mindshift/2014/12/29/intro-college-science-classes-that-enable-instead-of-weed/\" target=\"_blank\">act as gatekeepers to higher- level classes\u003c/a>, funneling the “smart” kids into advanced courses and keeping out those who struggle. That in turn communicates low expectations and a fixed mindset about students’ abilities. Good said there should be multiple entry points, as opposed to a linear progression.\u003c/p>\n\u003cp>\u003cstrong>PREPARE EVERYONE\u003c/strong>\u003cbr>\nGrowth mindsets are often discussed in relationship to kids who struggle, but the concept is just as \u003ca href=\"https://ww2.kqed.org/mindshift/2014/03/17/can-focus-on-grit-work-in-school-cultures-that-reward-grades/\" target=\"_blank\">relevant to kids who breeze through the material\u003c/a>. Telling those kids they are smart is not setting them up for success later when they do struggle. For Good, that struggle didn’t come until graduate school, but she distinctly remembers feeling “not smart anymore” because she was struggling. Math teachers need to give high achievers opportunities to struggle and persevere early and often so the experience is not foreign to them.\u003c/p>\n\u003cp>Embedded in all of this growth mindset work is a general culture shift around how math is taught and who can excel at it. It’s no surprise that teachers are struggling to integrate growth mindset into their teaching practice because every child is different. When it comes to perceptions of intelligence, belonging and whether a teacher cares, many factors come into play. Most teachers were educated in math classrooms with fixed mindset messages, as were\u003ca href=\"https://ww2.kqed.org/mindshift/2016/05/08/talking-about-failure-what-parents-can-do-to-motivate-kids-in-school/\" target=\"_blank\"> most parents\u003c/a>, so \u003ca href=\"https://ww2.kqed.org/mindshift/2015/09/24/never-too-late-creating-a-climate-for-adults-to-learn-new-skills/\" target=\"_blank\">shifting the culture of classrooms and schools \u003c/a>is work that takes time and incremental changes. But when \u003ca href=\"https://ww2.kqed.org/mindshift/2016/11/02/how-one-school-changed-its-math-culture-starting-with-teachers/\" target=\"_blank\">teachers commit to that work\u003c/a>, the shift is possible.\u003c/p>\n\u003cp>\u003c/p>\u003c/div>",
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"content": "\u003cdiv class=\"post-body\">\u003cp>\u003c/p>\n\u003cp>* \u003cem>Catherine Good is currently on leave from Baruch College, serving as \u003ca href=\"http://www.turnaroundusa.org/team/catherine-good/\">Senior Research Scientist\u003c/a> at the organization Turnaround for Children.\u003c/em>\u003c/p>\n\n\u003c/div>\u003c/p>",
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"disqusTitle": "What Neuroscience Can Tell Us About Making Fractions Stick",
"title": "What Neuroscience Can Tell Us About Making Fractions Stick",
"headTitle": "MindShift | KQED News",
"content": "\u003cp>Fractions are a notoriously tricky part of elementary math education for many children. Too often teachers struggle to ensure students are grasping the conceptual underpinnings of this complicated topic, resorting to “tricks” that will help them learn the procedures of adding or multiplying instead. This is particularly troubling because \u003ca href=\"http://www2.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf\" target=\"_blank\">studies have shown\u003c/a> that students’ knowledge of \u003ca href=\"http://www.psy.cmu.edu/~siegler/Siegler-etal-inpressPsySci.pdf\" target=\"_blank\">sixth grade fractions is a good predictor of their math achievement in high school\u003c/a>. This is largely because a deep understanding of fractions plays out in algebra.\u003c/p>\n\u003cp>“If a student has a solid understanding of fractions and precisely what they represent, they’re likely to perform much better with algebra,” said Valorie Salimpoor, a researcher at the Rotman Research Institute in Toronto, during an \u003ca href=\"http://home.edweb.net/\" target=\"_blank\">edWeb\u003c/a> presentation on the neuroscience of fractions. She thinks educators have an opportunity to leverage what researchers know about brain science to ensure students learn fractions well, but also admits that learning math is cognitively taxing.\u003c/p>\n\u003cfigure id=\"attachment_46867\" class=\"wp-caption aligncenter\" style=\"max-width: 957px\">\u003cimg class=\"size-full wp-image-46867\" src=\"http://ww2.kqed.org/mindshift/wp-content/uploads/sites/23/2016/11/brain-on-math.png\" alt=\"The various regions of the brain recruited when learning fractions.\" width=\"957\" height=\"716\" srcset=\"https://ww2.kqed.org/app/uploads/sites/23/2016/11/brain-on-math.png 957w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/brain-on-math-160x120.png 160w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/brain-on-math-800x599.png 800w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/brain-on-math-768x575.png 768w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/brain-on-math-240x180.png 240w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/brain-on-math-375x281.png 375w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/brain-on-math-520x389.png 520w\" sizes=\"(max-width: 957px) 100vw, 957px\">\u003cfigcaption class=\"wp-caption-text\">The various regions of the brain recruited when learning fractions. \u003ccite>(Valorie Salimpoor)\u003c/cite>\u003c/figcaption>\u003c/figure>\n\u003cp>Learning a new math concept takes a toll on the brain not only because of the new math concepts, but also because students must recruit many parts of the brain to solve any problem. For example, students need visuospatial and auditory working memory when solving a fractions problem, and they must focus attention, inhibit distractions, order tasks, recall information from long term memory and integrate new concepts into an old schema. There’s a lot of mental processing going on when learning math, so understanding how careful brain-based instruction can prime the brain for new learning becomes extra important.\u003c/p>\n\u003cp>\u003cstrong>OPTIMIZE INFORMATION PROCESSING\u003c/strong>\u003c/p>\n\u003cp>When a person learns something new she forms a connection between two previously unconnected neurons. If that connection is weak, the new learning can easily be lost through forgetting. But the stronger the pathway and the more ways a person has learned the information, the more likely it will become encoded into long term memory.\u003c/p>\n\u003cp>[ad fullwidth]\u003c/p>\n\u003cp>To improve a student’s information processing around fractions neuroscience tells us teachers should both present information and give students ways to interact with it, in a variety of ways. For example, explaining how to add fractions only uses an auditory pathway. Showing fractions on a line graph or as a pie chart are two other ways of presenting the same information. Asking students to visualize parts of the whole and show fractions using manipulatives are two more ways.\u003c/p>\n\u003cp>“Every time you are visualizing this in a different way, you are recruiting different neurons and neural connections,” Salimpoor said. And she says active learning through problem solving or manipulation is a whole different ballpark neurally than passively listening, partly because even if a student looks like she is listening she still may not be paying attention.\u003c/p>\n\u003cp>“In one case you are passively absorbing information and in the other case you have to initiate the motor sequences and carrying them through,” Salimpoor said. When students are actively learning they are using the frontal lobe to determine what needs to happen next and to organize the information, as well as the motor regions of the brain to carry it out.\u003c/p>\n\u003cfigure id=\"attachment_46869\" class=\"wp-caption aligncenter\" style=\"max-width: 670px\">\u003cimg class=\"size-full wp-image-46869\" src=\"http://ww2.kqed.org/mindshift/wp-content/uploads/sites/23/2016/11/gray-matter-over-time.png\" alt=\"In the first years of life babies are rapidly forming new connections between neurons, creating a dense network that represents deeply encoded learning.\" width=\"670\" height=\"486\" srcset=\"https://ww2.kqed.org/app/uploads/sites/23/2016/11/gray-matter-over-time.png 670w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/gray-matter-over-time-160x116.png 160w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/gray-matter-over-time-240x174.png 240w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/gray-matter-over-time-375x272.png 375w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/gray-matter-over-time-520x377.png 520w\" sizes=\"(max-width: 670px) 100vw, 670px\">\u003cfigcaption class=\"wp-caption-text\">In the first years of life babies are rapidly forming new connections between neurons, creating a dense network that represents deeply encoded learning. \u003ccite>(Source: Corel, JL. The postnatal development of the human cerebral cortex. Cambridge, MA: Harvard University Press; 1975.)\u003c/cite>\u003c/figcaption>\u003c/figure>\n\u003cp>“The more areas we recruit, the more elaborate this network becomes,” Salimpoor said. That’s important because later when the student tries to recall the new information and pings one node in the network, the entire system is primed to help retrieve the information. It’s also important for students to continue recalling that information in order to strengthen the network, even if a concept was introduced in many ways initially. Ultimately for something to be learned well it needs to be integrated into the student’s general schema, which happens through practice.\u003c/p>\n\u003cp>\u003cstrong>STRENGTHENING NEURAL NETWORKS\u003c/strong>\u003c/p>\n\u003cp>Practicing a new concept is one way good way to strengthen a neural network, but if students only practice using one problem solving strategy, that practice is leading to memorizing, not deep conceptual knowledge. To strengthen the whole network and by doing so deepening understanding, they must practice using various ways of visualizing and solving the same problem.\u003c/p>\n\u003cp>“It’s most important to create problems where students have to initiate and come up with their own solutions,” Salimpoor said. “This makes a big difference. You can never have too much of that.”\u003c/p>\n\u003cp>Neurochemicals like dopamine can also create shortcuts to encoding information. The brain releases dopamine in response to novelty or when a person is anticipating something and doesn’t know what’s going to happen. When a student is \u003ca href=\"https://ww2.kqed.org/mindshift/2016/05/31/why-emotions-are-integral-to-learning/\" target=\"_blank\">emotionally invested\u003c/a> in the learning or finds it intrinsically motivating, dopamine is also present. And dopamine helps strengthen neural networks.\u003c/p>\n\u003cp>Salimpoor uses the dopamine trick whenever she wants her toddler son to learn something new. She knows he loves trains, so if she can fold whatever new information she wants him to learn within a train narrative she knows he will be paying attention, intrinsically motivated and releasing dopamine that will help hardwire the new concepts. Rewards can also be a way to get at dopamine, but Salimpoor warns external rewards are never going to be as powerful as getting students to personally care about the learning.\u003c/p>\n\u003cp>\u003cstrong>GETTING AT CONCEPTUAL UNDERSTANDING\u003c/strong>\u003c/p>\n\u003cp>One of the trickiest things about helping students develop a deep conceptual understanding of a topic like fractions is that each student is coming into the learning experience not only with different levels of math knowledge, but also with different levels of working memory, executive functioning skills, ability to pay attention, and all the other non-content skills related to learning.\u003c/p>\n\u003cp>One way a teacher can use brain science to help students get at the deeper concepts is to relieve the pressure on students working memories as they are learning the new information. Salimpoor says working memory is a big challenge for many children. “If you can’t hold all of this new information in your mind, you can’t really process it,” she said. And if the information isn’t getting processed, it isn’t getting integrated into the large schema a child holds in his head of how things work.\u003c/p>\n\u003cp>“While some students might be very skilled at working memory, the ones who aren’t as good really suffer because they can’t take in all that information and process it, so they just tune out,” Salimpoor said. Teachers can be aware of this and try to break concepts down into the cognitive elements, always being mindful of how many pieces of new information the children need to hold in their minds at the same time to solve a problem.\u003c/p>\n\u003cp>When possible teachers can give students supports, like visuals representing the fractions, to work with as they are introduced to a concept. When the student has the visual, he doesn’t have to hold the symbolic representation of the fraction in his working memory as he figure out how to add the two together. After the concept has been introduced, teachers can slowly remove those scaffolds. Writing information down is another way student can offload some of what would be stored in working memory.\u003c/p>\n\u003cp>Working memory is one challenge, but it is very hard for teachers to identify all the specific ways students differ from one another cognitively. This is where technology can help.\u003c/p>\n\u003cp>Salimpoor has helped design a videogame focused on adding, subtracting, multiplying and dividing fractions -- the skills any student should have mastered by the end of sixth grade -- that puts all her neuroscience expertise for learning into the game mechanics. Called \u003ca href=\"http://fogstoneisle.com/\" target=\"_blank\">Fog Stone Island\u003c/a> and produced by \u003ca href=\"http://www.cignition.com/\" target=\"_blank\">Cignition\u003c/a>, the game is free and available online to teachers and students. Salimpoor said the game tracks the intangible information like working memory ability and executive functioning that a teacher would have a hard time identifying.\u003c/p>\n\u003cfigure id=\"attachment_46870\" class=\"wp-caption aligncenter\" style=\"max-width: 952px\">\u003cimg class=\"size-full wp-image-46870\" src=\"http://ww2.kqed.org/mindshift/wp-content/uploads/sites/23/2016/11/fog-island.png\" alt=\"A screen capture of Fog Stone Island juxtaposed with an image of the brain shows how multiple multiple brain regions are being stimulated while working on multiplying fractions.\" width=\"952\" height=\"540\" srcset=\"https://ww2.kqed.org/app/uploads/sites/23/2016/11/fog-island.png 952w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/fog-island-160x91.png 160w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/fog-island-800x454.png 800w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/fog-island-768x436.png 768w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/fog-island-240x136.png 240w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/fog-island-375x213.png 375w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/fog-island-520x295.png 520w\" sizes=\"(max-width: 952px) 100vw, 952px\">\u003cfigcaption class=\"wp-caption-text\">A screen capture of Fog Stone Island juxtaposed with an image of the brain shows how multiple multiple brain regions are being stimulated while working on multiplying fractions. \u003ccite>(Valorie Salimpoor)\u003c/cite>\u003c/figcaption>\u003c/figure>\n\u003cp>Salimpoor and her Cignition colleagues know full well that there are a lot of commercial games promising cognitive training that don’t deliver. There are also a lot of math games that are essentially procedural math disguised in a game form. The Fog Stone Island designers wanted to move away from both these models to develop a game that uses intrinsic and external rewards, offers multiple pathways to understand fractions, gives working memory support at the beginning of a task and slowly takes it away, and is situated within a context in which fractions would actually be used.\u003c/p>\n\u003cp>“We’ve tried to think of real life scenarios when you’d use math,” Salimpoor said. So, for example, in one corner of the immersive Fog Stone Island world, players use raw materials to build structures, a bit like \u003ca href=\"https://ww2.kqed.org/mindshift/2015/09/28/for-the-hesitant-teacher-leveraging-the-power-of-minecraft/\" target=\"_blank\">Minecraft\u003c/a>. The bricks have different lengths and the player must add like and unlike denominators to build a wall. Early in the concept the game provides a digital sketchbook -- essentially virtual working memory -- for students to use.\u003c/p>\n\u003cp>“Students can understand what they need to do in the situation without taxing their other cognitive abilities,” Salimpoor said. As they progress through the game they will gradually take over those working memory functions within the greater problem. Salimpoor finds it a little odd that she has ended up working on a videogame, but she wants good brain-based practices to be embedded in teaching and found that videogames were a far easier way to affect the many mental processes recruited when solving math problems.\u003c/p>\n\u003cp>\u003c!-- iframe plugin v.4.3 wordpress.org/plugins/iframe/ -->\u003cbr>\n\u003ciframe width=\"640\" height=\"360\" src=\"https://www.youtube.com/embed/SWKyDy78OtI\" frameborder=\"0\" scrolling=\"yes\" class=\"iframe-class\">\u003c/iframe>\u003c/p>\n\u003cp>Because Fog Stone Island is an immersive game, players can wander between the house building area, a farm plot and other zones that work on different elements of fractions. “The reason we wanted to keep it all in one game is it helps with investment,” Salimpoor said. Success in one area can lead to success in another area, and the different ways of working with fractions are integrated so students can develop a deep understanding of how the concepts are connected.\u003c/p>\n\u003cp>Salimpoor said the great thing about the game is that it adapts to the player’s level not just of math knowledge, but also working memory and executive functioning needs too. But Salimpoor is proud of how hard she and the designers worked to build those support skills into the game in a natural way that makes sense for the game. For example, rather than playing a silly side game that requires a player to repeat a sequence forward and then backwards -- a common working memory exercise -- Fog Stone Island may require a player to remember five items in the service of building part of his world. Now there’s an intrinsic motivation to build up working memory.\u003c/p>\n\u003cp>[ad floatright]\u003c/p>\n\u003cp>“In the video game we can do that so in depth,” Salimpoor said. “And really understand where each child is in each of these areas.” She says the latest iteration of the game is getting good reviews from students, who want to play even when it isn’t required for class. And very preliminary research results indicate that students are deepening their fractions knowledge outside of the game too, although those effects are still being studied.\u003c/p>\n\n",
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"content": "\u003cdiv class=\"post-body\">\u003cp>\u003cp>Fractions are a notoriously tricky part of elementary math education for many children. Too often teachers struggle to ensure students are grasping the conceptual underpinnings of this complicated topic, resorting to “tricks” that will help them learn the procedures of adding or multiplying instead. This is particularly troubling because \u003ca href=\"http://www2.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf\" target=\"_blank\">studies have shown\u003c/a> that students’ knowledge of \u003ca href=\"http://www.psy.cmu.edu/~siegler/Siegler-etal-inpressPsySci.pdf\" target=\"_blank\">sixth grade fractions is a good predictor of their math achievement in high school\u003c/a>. This is largely because a deep understanding of fractions plays out in algebra.\u003c/p>\n\u003cp>“If a student has a solid understanding of fractions and precisely what they represent, they’re likely to perform much better with algebra,” said Valorie Salimpoor, a researcher at the Rotman Research Institute in Toronto, during an \u003ca href=\"http://home.edweb.net/\" target=\"_blank\">edWeb\u003c/a> presentation on the neuroscience of fractions. She thinks educators have an opportunity to leverage what researchers know about brain science to ensure students learn fractions well, but also admits that learning math is cognitively taxing.\u003c/p>\n\u003cfigure id=\"attachment_46867\" class=\"wp-caption aligncenter\" style=\"max-width: 957px\">\u003cimg class=\"size-full wp-image-46867\" src=\"http://ww2.kqed.org/mindshift/wp-content/uploads/sites/23/2016/11/brain-on-math.png\" alt=\"The various regions of the brain recruited when learning fractions.\" width=\"957\" height=\"716\" srcset=\"https://ww2.kqed.org/app/uploads/sites/23/2016/11/brain-on-math.png 957w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/brain-on-math-160x120.png 160w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/brain-on-math-800x599.png 800w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/brain-on-math-768x575.png 768w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/brain-on-math-240x180.png 240w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/brain-on-math-375x281.png 375w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/brain-on-math-520x389.png 520w\" sizes=\"(max-width: 957px) 100vw, 957px\">\u003cfigcaption class=\"wp-caption-text\">The various regions of the brain recruited when learning fractions. \u003ccite>(Valorie Salimpoor)\u003c/cite>\u003c/figcaption>\u003c/figure>\n\u003cp>Learning a new math concept takes a toll on the brain not only because of the new math concepts, but also because students must recruit many parts of the brain to solve any problem. For example, students need visuospatial and auditory working memory when solving a fractions problem, and they must focus attention, inhibit distractions, order tasks, recall information from long term memory and integrate new concepts into an old schema. There’s a lot of mental processing going on when learning math, so understanding how careful brain-based instruction can prime the brain for new learning becomes extra important.\u003c/p>\n\u003cp>\u003cstrong>OPTIMIZE INFORMATION PROCESSING\u003c/strong>\u003c/p>\n\u003cp>When a person learns something new she forms a connection between two previously unconnected neurons. If that connection is weak, the new learning can easily be lost through forgetting. But the stronger the pathway and the more ways a person has learned the information, the more likely it will become encoded into long term memory.\u003c/p>\n\u003cp>\u003c/p>\u003c/div>",
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"content": "\u003cdiv class=\"post-body\">\u003cp>\u003c/p>\n\u003cp>To improve a student’s information processing around fractions neuroscience tells us teachers should both present information and give students ways to interact with it, in a variety of ways. For example, explaining how to add fractions only uses an auditory pathway. Showing fractions on a line graph or as a pie chart are two other ways of presenting the same information. Asking students to visualize parts of the whole and show fractions using manipulatives are two more ways.\u003c/p>\n\u003cp>“Every time you are visualizing this in a different way, you are recruiting different neurons and neural connections,” Salimpoor said. And she says active learning through problem solving or manipulation is a whole different ballpark neurally than passively listening, partly because even if a student looks like she is listening she still may not be paying attention.\u003c/p>\n\u003cp>“In one case you are passively absorbing information and in the other case you have to initiate the motor sequences and carrying them through,” Salimpoor said. When students are actively learning they are using the frontal lobe to determine what needs to happen next and to organize the information, as well as the motor regions of the brain to carry it out.\u003c/p>\n\u003cfigure id=\"attachment_46869\" class=\"wp-caption aligncenter\" style=\"max-width: 670px\">\u003cimg class=\"size-full wp-image-46869\" src=\"http://ww2.kqed.org/mindshift/wp-content/uploads/sites/23/2016/11/gray-matter-over-time.png\" alt=\"In the first years of life babies are rapidly forming new connections between neurons, creating a dense network that represents deeply encoded learning.\" width=\"670\" height=\"486\" srcset=\"https://ww2.kqed.org/app/uploads/sites/23/2016/11/gray-matter-over-time.png 670w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/gray-matter-over-time-160x116.png 160w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/gray-matter-over-time-240x174.png 240w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/gray-matter-over-time-375x272.png 375w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/gray-matter-over-time-520x377.png 520w\" sizes=\"(max-width: 670px) 100vw, 670px\">\u003cfigcaption class=\"wp-caption-text\">In the first years of life babies are rapidly forming new connections between neurons, creating a dense network that represents deeply encoded learning. \u003ccite>(Source: Corel, JL. The postnatal development of the human cerebral cortex. Cambridge, MA: Harvard University Press; 1975.)\u003c/cite>\u003c/figcaption>\u003c/figure>\n\u003cp>“The more areas we recruit, the more elaborate this network becomes,” Salimpoor said. That’s important because later when the student tries to recall the new information and pings one node in the network, the entire system is primed to help retrieve the information. It’s also important for students to continue recalling that information in order to strengthen the network, even if a concept was introduced in many ways initially. Ultimately for something to be learned well it needs to be integrated into the student’s general schema, which happens through practice.\u003c/p>\n\u003cp>\u003cstrong>STRENGTHENING NEURAL NETWORKS\u003c/strong>\u003c/p>\n\u003cp>Practicing a new concept is one way good way to strengthen a neural network, but if students only practice using one problem solving strategy, that practice is leading to memorizing, not deep conceptual knowledge. To strengthen the whole network and by doing so deepening understanding, they must practice using various ways of visualizing and solving the same problem.\u003c/p>\n\u003cp>“It’s most important to create problems where students have to initiate and come up with their own solutions,” Salimpoor said. “This makes a big difference. You can never have too much of that.”\u003c/p>\n\u003cp>Neurochemicals like dopamine can also create shortcuts to encoding information. The brain releases dopamine in response to novelty or when a person is anticipating something and doesn’t know what’s going to happen. When a student is \u003ca href=\"https://ww2.kqed.org/mindshift/2016/05/31/why-emotions-are-integral-to-learning/\" target=\"_blank\">emotionally invested\u003c/a> in the learning or finds it intrinsically motivating, dopamine is also present. And dopamine helps strengthen neural networks.\u003c/p>\n\u003cp>Salimpoor uses the dopamine trick whenever she wants her toddler son to learn something new. She knows he loves trains, so if she can fold whatever new information she wants him to learn within a train narrative she knows he will be paying attention, intrinsically motivated and releasing dopamine that will help hardwire the new concepts. Rewards can also be a way to get at dopamine, but Salimpoor warns external rewards are never going to be as powerful as getting students to personally care about the learning.\u003c/p>\n\u003cp>\u003cstrong>GETTING AT CONCEPTUAL UNDERSTANDING\u003c/strong>\u003c/p>\n\u003cp>One of the trickiest things about helping students develop a deep conceptual understanding of a topic like fractions is that each student is coming into the learning experience not only with different levels of math knowledge, but also with different levels of working memory, executive functioning skills, ability to pay attention, and all the other non-content skills related to learning.\u003c/p>\n\u003cp>One way a teacher can use brain science to help students get at the deeper concepts is to relieve the pressure on students working memories as they are learning the new information. Salimpoor says working memory is a big challenge for many children. “If you can’t hold all of this new information in your mind, you can’t really process it,” she said. And if the information isn’t getting processed, it isn’t getting integrated into the large schema a child holds in his head of how things work.\u003c/p>\n\u003cp>“While some students might be very skilled at working memory, the ones who aren’t as good really suffer because they can’t take in all that information and process it, so they just tune out,” Salimpoor said. Teachers can be aware of this and try to break concepts down into the cognitive elements, always being mindful of how many pieces of new information the children need to hold in their minds at the same time to solve a problem.\u003c/p>\n\u003cp>When possible teachers can give students supports, like visuals representing the fractions, to work with as they are introduced to a concept. When the student has the visual, he doesn’t have to hold the symbolic representation of the fraction in his working memory as he figure out how to add the two together. After the concept has been introduced, teachers can slowly remove those scaffolds. Writing information down is another way student can offload some of what would be stored in working memory.\u003c/p>\n\u003cp>Working memory is one challenge, but it is very hard for teachers to identify all the specific ways students differ from one another cognitively. This is where technology can help.\u003c/p>\n\u003cp>Salimpoor has helped design a videogame focused on adding, subtracting, multiplying and dividing fractions -- the skills any student should have mastered by the end of sixth grade -- that puts all her neuroscience expertise for learning into the game mechanics. Called \u003ca href=\"http://fogstoneisle.com/\" target=\"_blank\">Fog Stone Island\u003c/a> and produced by \u003ca href=\"http://www.cignition.com/\" target=\"_blank\">Cignition\u003c/a>, the game is free and available online to teachers and students. Salimpoor said the game tracks the intangible information like working memory ability and executive functioning that a teacher would have a hard time identifying.\u003c/p>\n\u003cfigure id=\"attachment_46870\" class=\"wp-caption aligncenter\" style=\"max-width: 952px\">\u003cimg class=\"size-full wp-image-46870\" src=\"http://ww2.kqed.org/mindshift/wp-content/uploads/sites/23/2016/11/fog-island.png\" alt=\"A screen capture of Fog Stone Island juxtaposed with an image of the brain shows how multiple multiple brain regions are being stimulated while working on multiplying fractions.\" width=\"952\" height=\"540\" srcset=\"https://ww2.kqed.org/app/uploads/sites/23/2016/11/fog-island.png 952w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/fog-island-160x91.png 160w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/fog-island-800x454.png 800w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/fog-island-768x436.png 768w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/fog-island-240x136.png 240w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/fog-island-375x213.png 375w, https://ww2.kqed.org/app/uploads/sites/23/2016/11/fog-island-520x295.png 520w\" sizes=\"(max-width: 952px) 100vw, 952px\">\u003cfigcaption class=\"wp-caption-text\">A screen capture of Fog Stone Island juxtaposed with an image of the brain shows how multiple multiple brain regions are being stimulated while working on multiplying fractions. \u003ccite>(Valorie Salimpoor)\u003c/cite>\u003c/figcaption>\u003c/figure>\n\u003cp>Salimpoor and her Cignition colleagues know full well that there are a lot of commercial games promising cognitive training that don’t deliver. There are also a lot of math games that are essentially procedural math disguised in a game form. The Fog Stone Island designers wanted to move away from both these models to develop a game that uses intrinsic and external rewards, offers multiple pathways to understand fractions, gives working memory support at the beginning of a task and slowly takes it away, and is situated within a context in which fractions would actually be used.\u003c/p>\n\u003cp>“We’ve tried to think of real life scenarios when you’d use math,” Salimpoor said. So, for example, in one corner of the immersive Fog Stone Island world, players use raw materials to build structures, a bit like \u003ca href=\"https://ww2.kqed.org/mindshift/2015/09/28/for-the-hesitant-teacher-leveraging-the-power-of-minecraft/\" target=\"_blank\">Minecraft\u003c/a>. The bricks have different lengths and the player must add like and unlike denominators to build a wall. Early in the concept the game provides a digital sketchbook -- essentially virtual working memory -- for students to use.\u003c/p>\n\u003cp>“Students can understand what they need to do in the situation without taxing their other cognitive abilities,” Salimpoor said. As they progress through the game they will gradually take over those working memory functions within the greater problem. Salimpoor finds it a little odd that she has ended up working on a videogame, but she wants good brain-based practices to be embedded in teaching and found that videogames were a far easier way to affect the many mental processes recruited when solving math problems.\u003c/p>\n\u003cp>\u003c!-- iframe plugin v.4.3 wordpress.org/plugins/iframe/ -->\u003cbr>\n\u003ciframe width=\"640\" height=\"360\" src=\"https://www.youtube.com/embed/SWKyDy78OtI\" frameborder=\"0\" scrolling=\"yes\" class=\"iframe-class\">\u003c/iframe>\u003c/p>\n\u003cp>Because Fog Stone Island is an immersive game, players can wander between the house building area, a farm plot and other zones that work on different elements of fractions. “The reason we wanted to keep it all in one game is it helps with investment,” Salimpoor said. Success in one area can lead to success in another area, and the different ways of working with fractions are integrated so students can develop a deep understanding of how the concepts are connected.\u003c/p>\n\u003cp>Salimpoor said the great thing about the game is that it adapts to the player’s level not just of math knowledge, but also working memory and executive functioning needs too. But Salimpoor is proud of how hard she and the designers worked to build those support skills into the game in a natural way that makes sense for the game. For example, rather than playing a silly side game that requires a player to repeat a sequence forward and then backwards -- a common working memory exercise -- Fog Stone Island may require a player to remember five items in the service of building part of his world. Now there’s an intrinsic motivation to build up working memory.\u003c/p>\n\u003cp>\u003c/p>\u003c/div>",
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"content": "\u003cdiv class=\"post-body\">\u003cp>\u003c/p>\n\u003cp>“In the video game we can do that so in depth,” Salimpoor said. “And really understand where each child is in each of these areas.” She says the latest iteration of the game is getting good reviews from students, who want to play even when it isn’t required for class. And very preliminary research results indicate that students are deepening their fractions knowledge outside of the game too, although those effects are still being studied.\u003c/p>\n\n\u003c/div>\u003c/p>",
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"content": "\u003cp>Many educators are aware of Carol Dweck's research on \u003ca href=\"https://ww2.kqed.org/mindshift/2015/12/29/beyond-working-hard-what-growth-mindset-teaches-us-about-our-brains/\" target=\"_blank\">growth mindset\u003c/a>. The \u003ca href=\"https://ww2.kqed.org/mindshift/2014/12/23/why-talking-about-the-brain-can-empower-learners/\" target=\"_blank\">Stanford psychologist\u003c/a> has found that the way students think about and approach challenge makes a big impact on their learning. Students who believe that they were born with a certain amount of intelligence that cannot be changed -- a condition Dweck calls a fixed mindset -- are often afraid to seek out challenging tasks and are resigned to one's perceived set of abilities. Students who see intelligence as something that can grow and change with effort -- known as a growth mindset -- tend to persist at difficult tasks, trying new strategies and ultimately performing better in school. Many \u003ca href=\"https://ww2.kqed.org/mindshift/2015/10/02/how-to-weave-growth-mindset-into-school-culture/\" target=\"_blank\">schools have begun to focus\u003c/a> on building growth mindsets in students because of this research.\u003c/p>\n\u003cp>Helping students develop growth mindsets is made even trickier because \u003ca href=\"https://ww2.kqed.org/mindshift/2015/11/16/growth-mindset-clearing-up-some-common-confusions/\" target=\"_blank\">mindsets about learning can change depending on context\u003c/a>. And unfortunately \u003ca href=\"https://ww2.kqed.org/mindshift/2015/11/30/not-a-math-person-how-to-remove-obstacles-to-learning-math/\" target=\"_blank\">math class\u003c/a> is a time when many students have preconceived notions about their abilities. Many adults, including teachers, grew up receiving negative messages about their math ability and can unintentionally pass on unhelpful messages to students through casual words or actions.\u003c/p>\n\u003cp>That's why it's impressive that educators at Two Rivers Charter School in Washington, D.C. recognized a culture of math fear among the staff and worked hard to change teachers' relationships to math as part of their broader strategy to improve math achievement. The school's Director of Curriculum and Instruction, Jeff Heyck-Williams, described their efforts in an \u003ca href=\"http://blogs.edweek.org/edweek/learning_deeply/2016/10/how_we_got_teachers_to_love_math_--_and_improved_our_math_scores.html\" target=\"_blank\">Education Week article\u003c/a>:\u003c/p>\n\u003cblockquote>\u003cp>In August of 2010, we started by listening deeply to our teachers' math stories. We recognized that if we didn't start with their learning first, we would never be able to approach the kinds of mindset shifts necessary to impact the learning of students. Teachers—even the art teacher and the pre-school teacher—wrote their math stories, sharing their deepest feelings about math and the people and experiences that led them to those beliefs. Over 65% of the stories that teachers told were negative. When they were students, our teachers had been given messages like \"girls aren't good at math,\" \"it is OK if you don't get this, you won't need it once you get out of school anyway,\" and \"math is either something you get, or you don't get.\" These messages were pervasive and came from teachers with an affinity towards math as well as teachers who couldn't stand math. By acknowledging these messages, we brought them to the surface and made teachers aware of the messages that they were explicitly and too often implicitly sending to kids about math.\u003c/p>\u003c/blockquote>\n\u003cp>By starting with the mindsets of teachers, and recognizing that each person has her own mathematical history, Two Rivers was able to empower teachers to deepen their math skills. This professional development in turn helped teachers feel capable of teaching in problem-based ways that stretch student thinking.\u003c/p>\n\u003cp>\u003c!-- iframe plugin v.4.3 wordpress.org/plugins/iframe/ -->\u003cbr>\n\u003ciframe src=\"https://player.vimeo.com/video/145866873\" width=\"640\" height=\"360\" frameborder=\"0\" scrolling=\"yes\" class=\"iframe-class\">\u003c/iframe>\u003c/p>\n\u003cp>[ad fullwidth]\u003c/p>\n\u003cp>\u003ci>\u003ca href=\"https://vimeo.com/145866873\">Learning and Loving Math: A Problem-Based Approach\u003c/a> from \u003ca href=\"https://vimeo.com/tworiverspcs\">Two Rivers Public Charter School\u003c/a> on \u003ca href=\"https://vimeo.com\">Vimeo\u003c/a>.\u003c/i>\u003c/p>\n\u003cp>The Two Rivers Charter School example is a good reminder how so often the culture of particular schools and the attitudes of \u003ca href=\"https://ww2.kqed.org/mindshift/2015/09/24/never-too-late-creating-a-climate-for-adults-to-learn-new-skills/\" target=\"_blank\">the adults in the building\u003c/a> affect efforts to improve academic outcomes. It also shows that with a concerted effort, the teachers at this school flipped the script on their own math stories, learning the math they teach more deeply, while simultaneously becoming better mentors and guides to students.\u003c/p>\n\u003cp>\u003c/p>\n\u003cp>http://blogs.edweek.org/edweek/learning_deeply/2016/10/how_we_got_teachers_to_love_math_--_and_improved_our_math_scores.html\u003c/p>\n\n",
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"content": "\u003cdiv class=\"post-body\">\u003cp>\u003cp>Many educators are aware of Carol Dweck's research on \u003ca href=\"https://ww2.kqed.org/mindshift/2015/12/29/beyond-working-hard-what-growth-mindset-teaches-us-about-our-brains/\" target=\"_blank\">growth mindset\u003c/a>. The \u003ca href=\"https://ww2.kqed.org/mindshift/2014/12/23/why-talking-about-the-brain-can-empower-learners/\" target=\"_blank\">Stanford psychologist\u003c/a> has found that the way students think about and approach challenge makes a big impact on their learning. Students who believe that they were born with a certain amount of intelligence that cannot be changed -- a condition Dweck calls a fixed mindset -- are often afraid to seek out challenging tasks and are resigned to one's perceived set of abilities. Students who see intelligence as something that can grow and change with effort -- known as a growth mindset -- tend to persist at difficult tasks, trying new strategies and ultimately performing better in school. Many \u003ca href=\"https://ww2.kqed.org/mindshift/2015/10/02/how-to-weave-growth-mindset-into-school-culture/\" target=\"_blank\">schools have begun to focus\u003c/a> on building growth mindsets in students because of this research.\u003c/p>\n\u003cp>Helping students develop growth mindsets is made even trickier because \u003ca href=\"https://ww2.kqed.org/mindshift/2015/11/16/growth-mindset-clearing-up-some-common-confusions/\" target=\"_blank\">mindsets about learning can change depending on context\u003c/a>. And unfortunately \u003ca href=\"https://ww2.kqed.org/mindshift/2015/11/30/not-a-math-person-how-to-remove-obstacles-to-learning-math/\" target=\"_blank\">math class\u003c/a> is a time when many students have preconceived notions about their abilities. Many adults, including teachers, grew up receiving negative messages about their math ability and can unintentionally pass on unhelpful messages to students through casual words or actions.\u003c/p>\n\u003cp>That's why it's impressive that educators at Two Rivers Charter School in Washington, D.C. recognized a culture of math fear among the staff and worked hard to change teachers' relationships to math as part of their broader strategy to improve math achievement. The school's Director of Curriculum and Instruction, Jeff Heyck-Williams, described their efforts in an \u003ca href=\"http://blogs.edweek.org/edweek/learning_deeply/2016/10/how_we_got_teachers_to_love_math_--_and_improved_our_math_scores.html\" target=\"_blank\">Education Week article\u003c/a>:\u003c/p>\n\u003cblockquote>\u003cp>In August of 2010, we started by listening deeply to our teachers' math stories. We recognized that if we didn't start with their learning first, we would never be able to approach the kinds of mindset shifts necessary to impact the learning of students. Teachers—even the art teacher and the pre-school teacher—wrote their math stories, sharing their deepest feelings about math and the people and experiences that led them to those beliefs. Over 65% of the stories that teachers told were negative. When they were students, our teachers had been given messages like \"girls aren't good at math,\" \"it is OK if you don't get this, you won't need it once you get out of school anyway,\" and \"math is either something you get, or you don't get.\" These messages were pervasive and came from teachers with an affinity towards math as well as teachers who couldn't stand math. By acknowledging these messages, we brought them to the surface and made teachers aware of the messages that they were explicitly and too often implicitly sending to kids about math.\u003c/p>\u003c/blockquote>\n\u003cp>By starting with the mindsets of teachers, and recognizing that each person has her own mathematical history, Two Rivers was able to empower teachers to deepen their math skills. This professional development in turn helped teachers feel capable of teaching in problem-based ways that stretch student thinking.\u003c/p>\n\u003cp>\u003c!-- iframe plugin v.4.3 wordpress.org/plugins/iframe/ -->\u003cbr>\n\u003ciframe src=\"https://player.vimeo.com/video/145866873\" width=\"640\" height=\"360\" frameborder=\"0\" scrolling=\"yes\" class=\"iframe-class\">\u003c/iframe>\u003c/p>\n\u003cp>\u003c/p>\u003c/div>",
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"content": "\u003cdiv class=\"post-body\">\u003cp>\u003c/p>\n\u003cp>\u003ci>\u003ca href=\"https://vimeo.com/145866873\">Learning and Loving Math: A Problem-Based Approach\u003c/a> from \u003ca href=\"https://vimeo.com/tworiverspcs\">Two Rivers Public Charter School\u003c/a> on \u003ca href=\"https://vimeo.com\">Vimeo\u003c/a>.\u003c/i>\u003c/p>\n\u003cp>The Two Rivers Charter School example is a good reminder how so often the culture of particular schools and the attitudes of \u003ca href=\"https://ww2.kqed.org/mindshift/2015/09/24/never-too-late-creating-a-climate-for-adults-to-learn-new-skills/\" target=\"_blank\">the adults in the building\u003c/a> affect efforts to improve academic outcomes. It also shows that with a concerted effort, the teachers at this school flipped the script on their own math stories, learning the math they teach more deeply, while simultaneously becoming better mentors and guides to students.\u003c/p>\n\u003cp>\u003c/p>\n\u003cp>http://blogs.edweek.org/edweek/learning_deeply/2016/10/how_we_got_teachers_to_love_math_--_and_improved_our_math_scores.html\u003c/p>\n\n\u003c/div>\u003c/p>",
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"content": "\u003cp>People born without sight appear to solve math problems using visual areas of the brain.\u003c/p>\n\u003cp>A \u003ca href=\"http://www.radiologyinfo.org/en/info.cfm?pg=fmribrain\">functional MRI\u003c/a> study of 17 people blind since birth found that areas of visual cortex became active when the participants were asked to solve algebra problems, a team from Johns Hopkins \u003ca href=\"http://www.pnas.org/content/early/2016/09/14/1524982113\">reports\u003c/a> in the \u003cem>Proceedings of the National Academy of Sciences\u003c/em>.\u003c/p>\n\u003cp>\"And as the equations get harder and harder, activity in these areas goes up in a blind person,\" says \u003ca href=\"http://pbs.jhu.edu/directory/marina-bedny/\">Marina Bedny\u003c/a>, an author of the study and an assistant professor in the department of psychological and brain sciences at Johns Hopkins University.\u003c/p>\n\u003cp>In 19 sighted people doing the same problems, visual areas of the brain showed no increase in activity.\u003c/p>\n\u003cp>\"That really suggests that yes, blind individuals appear to be doing math with their visual cortex,\" Bedny says.\u003c/p>\n\u003cp>[ad fullwidth]\u003c/p>\n\u003cp>The findings, published online Friday, challenge the idea that brain tissue intended for one function is limited to tasks that are closely related.\u003c/p>\n\u003cp>\"To see that this structure can be reused for something very different is very surprising,\" says \u003ca href=\"http://www.psychology.pitt.edu/person/melissa-libertus-phd\">Melissa Libertus\u003c/a>, an assistant professor of psychology at the University of Pittsburgh. \"It shows us how plastic our brain is, how flexible it is.\"\u003c/p>\n\u003cp>Earlier research found that visual cortex could be rewired to process information from other senses, like hearing and touch. But Bedny wanted to know whether this area of the brain could do something radically different, something that had nothing to do with the senses.\u003c/p>\n\u003cp>So she picked algebra.\u003c/p>\n\u003cp>During the experiment, both blind and sighted participants were asked to solve algebra problems. \"So they would hear something like: 12 minus 3 equals x, and 4 minus 2 equals x,\" Bedny says. \"And they'd have to say whether x had the same value in those two equations.\"\u003c/p>\n\u003cp>In both blind and sighted people, two brain areas associated with number processing became active. But only blind participants had increased activity in areas usually reserved for vision.\u003c/p>\n\u003cp>The result suggests the brain can rewire visual cortex to do just about anything, Bedny says. And if that's true, she says, it could lead to new treatments for people who've had a stroke or other injury that has damaged one part of the brain.\u003c/p>\n\u003cp>\u003c/p>\n\u003cp>Drugs or even mental exercises might help a patient \"use a different part of your brain to do the same function,\" Bedny says. \"And that would be really exciting.\"\u003c/p>\n\u003cdiv class=\"fullattribution\">Copyright 2016 NPR. To see more, visit http://www.npr.org/.\u003cimg src=\"http://www.google-analytics.com/__utm.gif?utmac=UA-5828686-4&utmdt=When+Blind+People+Do+Algebra%2C+The+Brain%27s+Visual+Areas+Light+Up&utme=8(APIKey)9(MDAxOTAwOTE4MDEyMTkxMDAzNjczZDljZA004)\">\u003c/div>\n\n",
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"content": "\u003cdiv class=\"post-body\">\u003cp>\u003cp>People born without sight appear to solve math problems using visual areas of the brain.\u003c/p>\n\u003cp>A \u003ca href=\"http://www.radiologyinfo.org/en/info.cfm?pg=fmribrain\">functional MRI\u003c/a> study of 17 people blind since birth found that areas of visual cortex became active when the participants were asked to solve algebra problems, a team from Johns Hopkins \u003ca href=\"http://www.pnas.org/content/early/2016/09/14/1524982113\">reports\u003c/a> in the \u003cem>Proceedings of the National Academy of Sciences\u003c/em>.\u003c/p>\n\u003cp>\"And as the equations get harder and harder, activity in these areas goes up in a blind person,\" says \u003ca href=\"http://pbs.jhu.edu/directory/marina-bedny/\">Marina Bedny\u003c/a>, an author of the study and an assistant professor in the department of psychological and brain sciences at Johns Hopkins University.\u003c/p>\n\u003cp>In 19 sighted people doing the same problems, visual areas of the brain showed no increase in activity.\u003c/p>\n\u003cp>\"That really suggests that yes, blind individuals appear to be doing math with their visual cortex,\" Bedny says.\u003c/p>\n\u003cp>\u003c/p>\u003c/div>",
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"content": "\u003cdiv class=\"post-body\">\u003cp>\u003c/p>\n\u003cp>The findings, published online Friday, challenge the idea that brain tissue intended for one function is limited to tasks that are closely related.\u003c/p>\n\u003cp>\"To see that this structure can be reused for something very different is very surprising,\" says \u003ca href=\"http://www.psychology.pitt.edu/person/melissa-libertus-phd\">Melissa Libertus\u003c/a>, an assistant professor of psychology at the University of Pittsburgh. \"It shows us how plastic our brain is, how flexible it is.\"\u003c/p>\n\u003cp>Earlier research found that visual cortex could be rewired to process information from other senses, like hearing and touch. But Bedny wanted to know whether this area of the brain could do something radically different, something that had nothing to do with the senses.\u003c/p>\n\u003cp>So she picked algebra.\u003c/p>\n\u003cp>During the experiment, both blind and sighted participants were asked to solve algebra problems. \"So they would hear something like: 12 minus 3 equals x, and 4 minus 2 equals x,\" Bedny says. \"And they'd have to say whether x had the same value in those two equations.\"\u003c/p>\n\u003cp>In both blind and sighted people, two brain areas associated with number processing became active. But only blind participants had increased activity in areas usually reserved for vision.\u003c/p>\n\u003cp>The result suggests the brain can rewire visual cortex to do just about anything, Bedny says. And if that's true, she says, it could lead to new treatments for people who've had a stroke or other injury that has damaged one part of the brain.\u003c/p>\n\u003cp>\u003c/p>\n\u003cp>Drugs or even mental exercises might help a patient \"use a different part of your brain to do the same function,\" Bedny says. \"And that would be really exciting.\"\u003c/p>\n\u003cdiv class=\"fullattribution\">Copyright 2016 NPR. To see more, visit http://www.npr.org/.\u003cimg src=\"http://www.google-analytics.com/__utm.gif?utmac=UA-5828686-4&utmdt=When+Blind+People+Do+Algebra%2C+The+Brain%27s+Visual+Areas+Light+Up&utme=8(APIKey)9(MDAxOTAwOTE4MDEyMTkxMDAzNjczZDljZA004)\">\u003c/div>\n\n\u003c/div>\u003c/p>",
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"content": "\u003cp>When Jacqui Young studied pre-calculus as a high school junior, she found the experience unexpectedly fulfilling. She didn’t consider herself a “math person,” but pre-calc came more easily to her than it did to most of her peers, and she spent a lot of time helping fellow students grasp the concepts. “It felt good to be able to understand something and then be able to walk someone else through it,” she said. “It was so gratifying, and made me want to stay on top of the subject.”\u003c/p>\n\u003cp>Satisfaction and engagement may not be the most common feelings among students studying introductory calculus. According to Jo Boaler, a professor of math education at Stanford, roughly \u003ca href=\"http://www.edweek.org/ew/articles/2012/07/03/36boaler.h31.html\">50 percent\u003c/a> of the population feels anxious about math. That emotional discomfort often begins in elementary school, lingering over students’ later encounters with algebra and geometry, and tainting the subject with apprehension—or outright loathing.\u003c/p>\n\u003cp>Professor Mary Helen Immordino-Yang, associate professor of education, psychology, and neuroscience at the University of Southern California has explored how emotions are tied to learning. “Emotions are a piece of thinking,” she told me; “we think of anything because our emotions push us that way.” Even subjects widely considered to be outside the realm of emotion, like math, evoke powerful feelings among those studying it, which can then propel or thwart further learning.\u003c/p>\n\u003cp>Is there a way to separate negative emotions from the subject, so that more students experience math with a sense of satisfaction and pleasure? Immordino-Yang believes so. “It’s not about making math ‘fun’,” she added; games and prizes tend to be quick fixes. Instead, it’s about encouraging the sense of accomplishment that comes from deep understanding of difficult concepts. “It’s about making it satisfying, interesting, and fulfilling.”\u003c/p>\n\u003cp>Adam Leaman, who teaches variations of algebra, trigonometry and calculus to high schoolers in Summit, N.J., said that a sense of awe about mathematics drew him to the subject beginning with algebra 2. “There’s something satisfying about knowing there’s an answer and knowing I have the ability to get it,” he said. Today, he sees the same pattern with his students: they are most engaged when they’re figuring out hard problems.\u003c/p>\n\u003cp>[ad fullwidth]\u003c/p>\n\u003cp>There are several ways teachers can replace student fretfulness over math with a sense of appreciation.\u003c/p>\n\u003cp>\u003cstrong>Be clear about why understanding math concepts matters\u003c/strong>. Kids who believe that they must simply endure algebra and calculus until they’re through with school—and that the actual learning is pointless because they’ll never use it again—should be reminded why understanding mathematical concepts is valuable. Most importantly, being able to comprehend a “symbolic, representative system,” Immordino-Yang says, teaches the brain how to think theoretically and logically. “Learning how to think abstractly is a useful ability in all aspects of life,” Immordino-Yang said. In fact, people who have studied complex math in high school tend to have better life outcomes, she said. Teachers who share this information may persuade reluctant math-learners to stay engaged.\u003c/p>\n\u003cp>\u003cstrong>Assign projects that help kids see math’s usefulness\u003c/strong>. Students are more apt to participate if they see a practical application to their studies. “This goes beyond learning how to balance a checkbook,” Immordino-Yang adds. By studying how fast and far vegetable oil spreads on tissue paper, for example, students can learn not only about the math concept of direct variation, but also about how oil spills are measured. Sharing stories from the news where math understanding is featured in the narrative—in a story about price fixing, say, or one on climbing interest rates—also can help students see its usefulness in the real world. \u003ca href=\"https://www.learner.org/\">Learner.org\u003c/a>, a free educational resource from the Annenberg Center, provides such practical lesson plans for math at all levels, including the \u003ca href=\"https://www.learner.org/workshops/algebra/workshop7/lessonplan1.html\">oil spill example\u003c/a>.\u003c/p>\n\u003cp>\u003cstrong>Discuss mathematical role models, and share how their ideas have changed the world.\u003c/strong> Because math is the foundation of so many other fields—physics, engineering, finance, astronomy, among others—history teems with mathematical virtuosos whose creativity and curiosity shaped the modern world. In addition to the usual suspects of math icons, including Pythagoras, Rene Descartes, and Ptolemy, more contemporary role models might spark student appreciation for the subject. There are many: \u003ca href=\"http://www.bbc.co.uk/timelines/z8bgr82\">Alan Turing\u003c/a>, who’s code-breaking during World War II helped defeat the Axis powers; \u003ca href=\"http://www.computerhistory.org/babbage/adalovelace\">Ada Lovelace\u003c/a>, who created the Analytical Engine, which presaged modern programming; even \u003ca href=\"https://www.actuary.org/files/publications/NateSilver_Bio.pdf\">Nate Silver\u003c/a>, a popular mathematician who uses statistical forecasting to predict outcomes in Major League Baseball and political elections. The humbler discoveries made by annual recipients of the \u003ca href=\"http://www.nsf.gov/news/news_summ.jsp?cntn_id=137709\">Presidential Early Career Awards for Science and Engineering\u003c/a> might also inspire.\u003c/p>\n\u003cp>\u003cstrong>Strive to minimize the sources of fear.\u003c/strong> Math anxiety stifles clear thinking. At those moments when students most need to marshal their intellectual resources—during a test, say, or when called up before class to work a problem—those who fear the subject are apt to panic and shut down. \u003ca href=\"https://news.uchicago.edu/article/2010/09/21/psychologist-shows-why-we-choke-under-pressure-and-how-avoid-it\">Sian Beilock\u003c/a>, a professor of psychology at the University of Chicago, and author of \u003cem>Choke: What the Secrets of the Brain Reveal About Getting it Right When You Have To\u003c/em>, describes the anxious over-reaction to high pressure situations as a “malfunction of the prefrontal cortex.” For nervous students who feel pressure to perform well on a test, that worry causes them to execute beneath their skill level. “Anxiety is robbing you of working memory,” Immordino-Yang explained. “You’re wasting your thought powers,” she added. Math phobic kids need help from teachers to lessen their fear.\u003c/p>\n\u003cp>Keeping the classroom “kid-centric,” Immordino-Yang said, can help. Teachers who act as facilitators, or resident experts, rather than omniscient instructors, invite students to explore without fear of messing up in front of an authority. Freeing up fellow students to explain problems also allows for more personalized instruction. During Jacqui Young’s happy year studying pre-calc, she worked with peers who struggled to keep up with the teacher’s pace. “I think it was better because I’d be working one-on-one or two-on-one with my classmates,” she said. Another way to tamp down dread is to set up class in a roundtable and encourage student-led give-and-take. Known formally as the \u003ca href=\"http://www.exeter.edu/admissions/109_1220_11688.aspx\">Harkness Method\u003c/a> of teaching, this collaborative approach to learning may be especially useful in math subjects.\u003c/p>\n\u003cp>[ad floatright]\u003c/p>\n\u003cp>Returning to older math processes and ideas when introducing new material also works to lessen anxiety about the new learning. While teaching synthetic division to his Algebra 2 class, for example, Adam Leaman reminds students that this “new” concept is a cousin of the factoring they did in Algebra 1. “They have something from the past to draw reference from when tackling this new subject,” he said. Bringing up old material this way also helps students who might have struggled when they learned it the first time. Leaman said that some kids groan when he brings up factoring, but that they often end up understanding it better when going through it a second time, and in relation to a different concept. “Even if they have the perception that they’re not good at it, when we come back to it I have students say, ‘I get it now’.”\u003c/p>\n\n",
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"content": "\u003cdiv class=\"post-body\">\u003cp>\u003cp>When Jacqui Young studied pre-calculus as a high school junior, she found the experience unexpectedly fulfilling. She didn’t consider herself a “math person,” but pre-calc came more easily to her than it did to most of her peers, and she spent a lot of time helping fellow students grasp the concepts. “It felt good to be able to understand something and then be able to walk someone else through it,” she said. “It was so gratifying, and made me want to stay on top of the subject.”\u003c/p>\n\u003cp>Satisfaction and engagement may not be the most common feelings among students studying introductory calculus. According to Jo Boaler, a professor of math education at Stanford, roughly \u003ca href=\"http://www.edweek.org/ew/articles/2012/07/03/36boaler.h31.html\">50 percent\u003c/a> of the population feels anxious about math. That emotional discomfort often begins in elementary school, lingering over students’ later encounters with algebra and geometry, and tainting the subject with apprehension—or outright loathing.\u003c/p>\n\u003cp>Professor Mary Helen Immordino-Yang, associate professor of education, psychology, and neuroscience at the University of Southern California has explored how emotions are tied to learning. “Emotions are a piece of thinking,” she told me; “we think of anything because our emotions push us that way.” Even subjects widely considered to be outside the realm of emotion, like math, evoke powerful feelings among those studying it, which can then propel or thwart further learning.\u003c/p>\n\u003cp>Is there a way to separate negative emotions from the subject, so that more students experience math with a sense of satisfaction and pleasure? Immordino-Yang believes so. “It’s not about making math ‘fun’,” she added; games and prizes tend to be quick fixes. Instead, it’s about encouraging the sense of accomplishment that comes from deep understanding of difficult concepts. “It’s about making it satisfying, interesting, and fulfilling.”\u003c/p>\n\u003cp>Adam Leaman, who teaches variations of algebra, trigonometry and calculus to high schoolers in Summit, N.J., said that a sense of awe about mathematics drew him to the subject beginning with algebra 2. “There’s something satisfying about knowing there’s an answer and knowing I have the ability to get it,” he said. Today, he sees the same pattern with his students: they are most engaged when they’re figuring out hard problems.\u003c/p>\n\u003cp>\u003c/p>\u003c/div>",
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"content": "\u003cdiv class=\"post-body\">\u003cp>\u003c/p>\n\u003cp>There are several ways teachers can replace student fretfulness over math with a sense of appreciation.\u003c/p>\n\u003cp>\u003cstrong>Be clear about why understanding math concepts matters\u003c/strong>. Kids who believe that they must simply endure algebra and calculus until they’re through with school—and that the actual learning is pointless because they’ll never use it again—should be reminded why understanding mathematical concepts is valuable. Most importantly, being able to comprehend a “symbolic, representative system,” Immordino-Yang says, teaches the brain how to think theoretically and logically. “Learning how to think abstractly is a useful ability in all aspects of life,” Immordino-Yang said. In fact, people who have studied complex math in high school tend to have better life outcomes, she said. Teachers who share this information may persuade reluctant math-learners to stay engaged.\u003c/p>\n\u003cp>\u003cstrong>Assign projects that help kids see math’s usefulness\u003c/strong>. Students are more apt to participate if they see a practical application to their studies. “This goes beyond learning how to balance a checkbook,” Immordino-Yang adds. By studying how fast and far vegetable oil spreads on tissue paper, for example, students can learn not only about the math concept of direct variation, but also about how oil spills are measured. Sharing stories from the news where math understanding is featured in the narrative—in a story about price fixing, say, or one on climbing interest rates—also can help students see its usefulness in the real world. \u003ca href=\"https://www.learner.org/\">Learner.org\u003c/a>, a free educational resource from the Annenberg Center, provides such practical lesson plans for math at all levels, including the \u003ca href=\"https://www.learner.org/workshops/algebra/workshop7/lessonplan1.html\">oil spill example\u003c/a>.\u003c/p>\n\u003cp>\u003cstrong>Discuss mathematical role models, and share how their ideas have changed the world.\u003c/strong> Because math is the foundation of so many other fields—physics, engineering, finance, astronomy, among others—history teems with mathematical virtuosos whose creativity and curiosity shaped the modern world. In addition to the usual suspects of math icons, including Pythagoras, Rene Descartes, and Ptolemy, more contemporary role models might spark student appreciation for the subject. There are many: \u003ca href=\"http://www.bbc.co.uk/timelines/z8bgr82\">Alan Turing\u003c/a>, who’s code-breaking during World War II helped defeat the Axis powers; \u003ca href=\"http://www.computerhistory.org/babbage/adalovelace\">Ada Lovelace\u003c/a>, who created the Analytical Engine, which presaged modern programming; even \u003ca href=\"https://www.actuary.org/files/publications/NateSilver_Bio.pdf\">Nate Silver\u003c/a>, a popular mathematician who uses statistical forecasting to predict outcomes in Major League Baseball and political elections. The humbler discoveries made by annual recipients of the \u003ca href=\"http://www.nsf.gov/news/news_summ.jsp?cntn_id=137709\">Presidential Early Career Awards for Science and Engineering\u003c/a> might also inspire.\u003c/p>\n\u003cp>\u003cstrong>Strive to minimize the sources of fear.\u003c/strong> Math anxiety stifles clear thinking. At those moments when students most need to marshal their intellectual resources—during a test, say, or when called up before class to work a problem—those who fear the subject are apt to panic and shut down. \u003ca href=\"https://news.uchicago.edu/article/2010/09/21/psychologist-shows-why-we-choke-under-pressure-and-how-avoid-it\">Sian Beilock\u003c/a>, a professor of psychology at the University of Chicago, and author of \u003cem>Choke: What the Secrets of the Brain Reveal About Getting it Right When You Have To\u003c/em>, describes the anxious over-reaction to high pressure situations as a “malfunction of the prefrontal cortex.” For nervous students who feel pressure to perform well on a test, that worry causes them to execute beneath their skill level. “Anxiety is robbing you of working memory,” Immordino-Yang explained. “You’re wasting your thought powers,” she added. Math phobic kids need help from teachers to lessen their fear.\u003c/p>\n\u003cp>Keeping the classroom “kid-centric,” Immordino-Yang said, can help. Teachers who act as facilitators, or resident experts, rather than omniscient instructors, invite students to explore without fear of messing up in front of an authority. Freeing up fellow students to explain problems also allows for more personalized instruction. During Jacqui Young’s happy year studying pre-calc, she worked with peers who struggled to keep up with the teacher’s pace. “I think it was better because I’d be working one-on-one or two-on-one with my classmates,” she said. Another way to tamp down dread is to set up class in a roundtable and encourage student-led give-and-take. Known formally as the \u003ca href=\"http://www.exeter.edu/admissions/109_1220_11688.aspx\">Harkness Method\u003c/a> of teaching, this collaborative approach to learning may be especially useful in math subjects.\u003c/p>\n\u003cp>\u003c/p>\u003c/div>",
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"content": "\u003cdiv class=\"post-body\">\u003cp>\u003c/p>\n\u003cp>Returning to older math processes and ideas when introducing new material also works to lessen anxiety about the new learning. While teaching synthetic division to his Algebra 2 class, for example, Adam Leaman reminds students that this “new” concept is a cousin of the factoring they did in Algebra 1. “They have something from the past to draw reference from when tackling this new subject,” he said. Bringing up old material this way also helps students who might have struggled when they learned it the first time. Leaman said that some kids groan when he brings up factoring, but that they often end up understanding it better when going through it a second time, and in relation to a different concept. “Even if they have the perception that they’re not good at it, when we come back to it I have students say, ‘I get it now’.”\u003c/p>\n\n\u003c/div>\u003c/p>",
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"disqusTitle": "Belonging and Believing: Transforming Remedial Math at Community Colleges",
"title": "Belonging and Believing: Transforming Remedial Math at Community Colleges",
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"content": "\u003cp>Developmental or remedial education forms a core service of community colleges, with a staggering 68% of all community college students\u003ca href=\"http://ccrc.tc.columbia.edu/media/k2/attachments/what-we-know-about-developmental-education-outcomes.pdf\"> taking at least one remedial course\u003c/a>, most commonly English or math. The stark number of students not prepared for college work presents a two-fold dilemma for community colleges, one that is both financial and self-defeating, eating at the very purpose of community colleges’ existence.\u003c/p>\n\u003cp class=\"p1\">First, students who are enrolled in even one remedial course have a high chance of dropping out. According to a 2006 National Education Longitudinal study, the dropout rate in remedial courses is more than 70%, with only 28% of remedial students completing a degree after 8.5 years. Second, the extra money to pay for remediation is costing states billions: the Community College Research Center (CCRC) \u003ca href=\"http://ccrc.tc.columbia.edu/media/k2/attachments/what-we-know-about-developmental-education-outcomes.pdf\">estimates\u003c/a> that the national cost of providing these courses to all students is approximately $7 billion.\u003c/p>\n\u003cp class=\"p1\">But according to Tom Bailey, who heads the CCRC at Teachers College, Columbia University, it’s the students who are paying the most. “It is students who probably have to bear the most significant costs,” he writes. “They must not only pay for the classes but also must delay their progress through college. Many students are discouraged when they find out that they are not eligible for college-level courses. This may explain the high ‘no-show’ rates among those referred to remediation.”\u003c/p>\n\u003cp class=\"p1\">It is not an exaggeration, says Rachel Beattie, director of productive persistence at Carnegie Math Pathways, to say that one developmental course can derail an entire college career, and even the future beyond it. “People will keep coming back, because they’re really persistent. We see that many of our students have been enrolled in college for five or ten, even twenty years, they’re trying to get that math credit, but no luck,” she said. “We see a lot of unproductive persistence.” Part of Beattie’s job is to help mold unproductive persistence into something more fruitful, and that involves changing both the students’ mindset and how teachers teach developmental courses.\u003c/p>\n\u003cp class=\"p1\">In 2010, Beattie and team launched the \u003ca href=\"http://www.carnegiefoundation.org/in-action/carnegie-math-pathways/\">Carnegie Math Pathways\u003c/a> at the Carnegie Foundation for the Advancement of Teaching, two developmental math courses which now operate on community college campuses in 19 states and strive to help students who need to remediate in math complete their courses but also do something more: introduce students to the “soft skills” they may be missing to help get them through college. Their primary focus: convincing students that they can learn. “About 2/3 of our students come in to the Carnegie Math Pathways with the belief that, no matter what they do, they are not ‘math people,’” Beattie said. “That there is this race of math people out there, and that they’re not one of them, and no matter how hard they try, or what strategies, it doesn’t matter because they’re never going to be one of those people.”\u003c/p>\n\u003cp>[ad fullwidth]\u003c/p>\n\u003cp class=\"p1\">After interviewing researchers and teachers, Beattie and team found that if community college math professors could instill five “high-leverage” factors into students, they had a much higher chance of completing their math courses:\u003c/p>\n\u003cp class=\"p1\">* Students believe they can learn,\u003c/p>\n\u003cp class=\"p1\">* Students have social ties to peers during the course,\u003c/p>\n\u003cp class=\"p1\">* They see the course has both short- and long-term value,\u003c/p>\n\u003cp class=\"p1\">* They have the know-how, skills and habits to succeed,\u003c/p>\n\u003cp class=\"p1\">* And finally, having faculty support students’ skills and mindset.\u003c/p>\n\u003cp class=\"p1\">So Beattie teaches professors how to teach to students who truly believe they can’t learn. The two courses, a statistics course called Statway and the quantitative math course, Quantway, are developmental in the sense of what material is covered, but \u003ci>how\u003c/i> the material is covered plays a big role. “Unlike in K-12, those of us who taught and teach in higher ed, we’re not always explicitly trained in how to teach,” Beattie said. “They know a lot about mathematics, a lot of them have PhDs in mathematics, but we show them how students learn, and how to promote mindsets and learning strategies is something that a lot of times they don’t have a bag of tricks for.”\u003c/p>\n\u003cp class=\"p1\">So the first lesson for students in both courses is how the brain learns; professors also cover the research behind \u003ca href=\"http://ww2.kqed.org/mindshift/tag/growth-mindset/\">growth mindset\u003c/a>. “You’re not wired from birth knowing how to do logarithms, that’s just not how the brain works,” said Beattie, who has a PhD in developmental psychology, and did a post-doc in cognitive neuroscience. “It’s actually quite plastic and changes based on the experiences that you have. So as you increase your knowledge you create more sophisticated connections between neurons in your brain, and you’re able to make connections to information better and in different ways, and this actually helps with experience.”\u003c/p>\n\u003cp class=\"p1\">Understanding how the brain works goes hand in hand with another success strategy, which is tackling “belonging uncertainty,” in which students believe that they don’t belong in college, or in college-level math. Beattie also shares with professors how to build trust and a sense of community in classrooms—and sometimes the strategy for belonging is almost agonizingly simple, like noticing a student who is absent and contacting them to let them know they were missed.\u003c/p>\n\u003cp class=\"p1\">In the six years since they began, Statway and Quantway have tripled the success rate of students (meaning they passed the course and could move on to college work) in half the time. While 6% of students complete traditional math pathway courses, the Carnegie classes show 50-60% of students earning their math credit in one year.\u003c/p>\n\u003cp>\u003c/p>\n\u003cp class=\"p1\">What Beattie hopes to show next is that the strategies students learn in the Math Pathways carries over to the rest of college. “There is some preliminary evidence that we are seeing from our colleges that our students are being successful in future classes,” she said. “Because many students end up transferring to four-year schools—which is really exciting—it can be kind of difficult to track them down, but we are getting some preliminary evidence of sustained success,” she said.\u003c/p>\n\n",
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"content": "\u003cdiv class=\"post-body\">\u003cp>\u003cp>Developmental or remedial education forms a core service of community colleges, with a staggering 68% of all community college students\u003ca href=\"http://ccrc.tc.columbia.edu/media/k2/attachments/what-we-know-about-developmental-education-outcomes.pdf\"> taking at least one remedial course\u003c/a>, most commonly English or math. The stark number of students not prepared for college work presents a two-fold dilemma for community colleges, one that is both financial and self-defeating, eating at the very purpose of community colleges’ existence.\u003c/p>\n\u003cp class=\"p1\">First, students who are enrolled in even one remedial course have a high chance of dropping out. According to a 2006 National Education Longitudinal study, the dropout rate in remedial courses is more than 70%, with only 28% of remedial students completing a degree after 8.5 years. Second, the extra money to pay for remediation is costing states billions: the Community College Research Center (CCRC) \u003ca href=\"http://ccrc.tc.columbia.edu/media/k2/attachments/what-we-know-about-developmental-education-outcomes.pdf\">estimates\u003c/a> that the national cost of providing these courses to all students is approximately $7 billion.\u003c/p>\n\u003cp class=\"p1\">But according to Tom Bailey, who heads the CCRC at Teachers College, Columbia University, it’s the students who are paying the most. “It is students who probably have to bear the most significant costs,” he writes. “They must not only pay for the classes but also must delay their progress through college. Many students are discouraged when they find out that they are not eligible for college-level courses. This may explain the high ‘no-show’ rates among those referred to remediation.”\u003c/p>\n\u003cp class=\"p1\">It is not an exaggeration, says Rachel Beattie, director of productive persistence at Carnegie Math Pathways, to say that one developmental course can derail an entire college career, and even the future beyond it. “People will keep coming back, because they’re really persistent. We see that many of our students have been enrolled in college for five or ten, even twenty years, they’re trying to get that math credit, but no luck,” she said. “We see a lot of unproductive persistence.” Part of Beattie’s job is to help mold unproductive persistence into something more fruitful, and that involves changing both the students’ mindset and how teachers teach developmental courses.\u003c/p>\n\u003cp class=\"p1\">In 2010, Beattie and team launched the \u003ca href=\"http://www.carnegiefoundation.org/in-action/carnegie-math-pathways/\">Carnegie Math Pathways\u003c/a> at the Carnegie Foundation for the Advancement of Teaching, two developmental math courses which now operate on community college campuses in 19 states and strive to help students who need to remediate in math complete their courses but also do something more: introduce students to the “soft skills” they may be missing to help get them through college. Their primary focus: convincing students that they can learn. “About 2/3 of our students come in to the Carnegie Math Pathways with the belief that, no matter what they do, they are not ‘math people,’” Beattie said. “That there is this race of math people out there, and that they’re not one of them, and no matter how hard they try, or what strategies, it doesn’t matter because they’re never going to be one of those people.”\u003c/p>\n\u003cp>\u003c/p>\u003c/div>",
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"content": "\u003cdiv class=\"post-body\">\u003cp>\u003c/p>\n\u003cp class=\"p1\">After interviewing researchers and teachers, Beattie and team found that if community college math professors could instill five “high-leverage” factors into students, they had a much higher chance of completing their math courses:\u003c/p>\n\u003cp class=\"p1\">* Students believe they can learn,\u003c/p>\n\u003cp class=\"p1\">* Students have social ties to peers during the course,\u003c/p>\n\u003cp class=\"p1\">* They see the course has both short- and long-term value,\u003c/p>\n\u003cp class=\"p1\">* They have the know-how, skills and habits to succeed,\u003c/p>\n\u003cp class=\"p1\">* And finally, having faculty support students’ skills and mindset.\u003c/p>\n\u003cp class=\"p1\">So Beattie teaches professors how to teach to students who truly believe they can’t learn. The two courses, a statistics course called Statway and the quantitative math course, Quantway, are developmental in the sense of what material is covered, but \u003ci>how\u003c/i> the material is covered plays a big role. “Unlike in K-12, those of us who taught and teach in higher ed, we’re not always explicitly trained in how to teach,” Beattie said. “They know a lot about mathematics, a lot of them have PhDs in mathematics, but we show them how students learn, and how to promote mindsets and learning strategies is something that a lot of times they don’t have a bag of tricks for.”\u003c/p>\n\u003cp class=\"p1\">So the first lesson for students in both courses is how the brain learns; professors also cover the research behind \u003ca href=\"http://ww2.kqed.org/mindshift/tag/growth-mindset/\">growth mindset\u003c/a>. “You’re not wired from birth knowing how to do logarithms, that’s just not how the brain works,” said Beattie, who has a PhD in developmental psychology, and did a post-doc in cognitive neuroscience. “It’s actually quite plastic and changes based on the experiences that you have. So as you increase your knowledge you create more sophisticated connections between neurons in your brain, and you’re able to make connections to information better and in different ways, and this actually helps with experience.”\u003c/p>\n\u003cp class=\"p1\">Understanding how the brain works goes hand in hand with another success strategy, which is tackling “belonging uncertainty,” in which students believe that they don’t belong in college, or in college-level math. Beattie also shares with professors how to build trust and a sense of community in classrooms—and sometimes the strategy for belonging is almost agonizingly simple, like noticing a student who is absent and contacting them to let them know they were missed.\u003c/p>\n\u003cp class=\"p1\">In the six years since they began, Statway and Quantway have tripled the success rate of students (meaning they passed the course and could move on to college work) in half the time. While 6% of students complete traditional math pathway courses, the Carnegie classes show 50-60% of students earning their math credit in one year.\u003c/p>\n\u003cp>\u003c/p>\n\u003cp class=\"p1\">What Beattie hopes to show next is that the strategies students learn in the Math Pathways carries over to the rest of college. “There is some preliminary evidence that we are seeing from our colleges that our students are being successful in future classes,” she said. “Because many students end up transferring to four-year schools—which is really exciting—it can be kind of difficult to track them down, but we are getting some preliminary evidence of sustained success,” she said.\u003c/p>\n\n\u003c/div>\u003c/p>",
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"content": "\u003cp>Fractions come up all the time in everyday life, and yet, they are often a difficult concept for elementary school-aged children to grasp. One way to help kids understand fractions as concrete things is to give them real world examples. In this \u003ca href=\"https://www.teachingchannel.org/videos/3-phase-lesson-structure-cisco\" target=\"_blank\">Teaching Channel video\u003c/a>, third grade teacher Maria Franco teaches a lesson on equivalent fractions in which she tries to give more space for student thinking and discussion.\u003c/p>\n\u003cp>Instead of asking all the questions, Franco says, \"I'm going to see how you guys work by yourselves.\" She's circling the room, listening for understanding and interesting solving problem techniques. Rather than asking the questions, she's listening to her students push one another to defend their problem solving.\u003c/p>\n\u003cp>\"You came up with you own idea. It doesn't have to be the same as mine as long as we get the same answer,\" Franco said.\u003c/p>\n\u003cp>\u003c!-- iframe plugin v.4.3 wordpress.org/plugins/iframe/ -->\u003cbr>\n\u003ciframe src=\"https://player.vimeo.com/video/163580060\" width=\"640\" height=\"360\" frameborder=\"0\" scrolling=\"yes\" class=\"iframe-class\">\u003c/iframe>\u003c/p>\n\u003cp>Understanding Fractions through Real-World Tasks from \u003ca href=\"https://vimeo.com/user11426713\">Teaching Channel\u003c/a> on \u003ca href=\"https://vimeo.com\">Vimeo\u003c/a>.\u003c/p>\n\u003cp>[ad fullwidth]\u003c/p>\u003cp>\u003c/p>\n",
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"content": "\u003cdiv class=\"post-body\">\u003cp>\u003cp>Fractions come up all the time in everyday life, and yet, they are often a difficult concept for elementary school-aged children to grasp. One way to help kids understand fractions as concrete things is to give them real world examples. In this \u003ca href=\"https://www.teachingchannel.org/videos/3-phase-lesson-structure-cisco\" target=\"_blank\">Teaching Channel video\u003c/a>, third grade teacher Maria Franco teaches a lesson on equivalent fractions in which she tries to give more space for student thinking and discussion.\u003c/p>\n\u003cp>Instead of asking all the questions, Franco says, \"I'm going to see how you guys work by yourselves.\" She's circling the room, listening for understanding and interesting solving problem techniques. Rather than asking the questions, she's listening to her students push one another to defend their problem solving.\u003c/p>\n\u003cp>\"You came up with you own idea. It doesn't have to be the same as mine as long as we get the same answer,\" Franco said.\u003c/p>\n\u003cp>\u003c!-- iframe plugin v.4.3 wordpress.org/plugins/iframe/ -->\u003cbr>\n\u003ciframe src=\"https://player.vimeo.com/video/163580060\" width=\"640\" height=\"360\" frameborder=\"0\" scrolling=\"yes\" class=\"iframe-class\">\u003c/iframe>\u003c/p>\n\u003cp>Understanding Fractions through Real-World Tasks from \u003ca href=\"https://vimeo.com/user11426713\">Teaching Channel\u003c/a> on \u003ca href=\"https://vimeo.com\">Vimeo\u003c/a>.\u003c/p>\n\u003cp>\u003c/p>\u003c/div>",
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"content": "\u003cp>Getting students excited and authentically curious about a math task takes more than presenting a word problem. Some teachers are finding that a short, high-interest video or other piece of media that raises questions in kids' minds is the best way to prime them to dive deeply into problem solving.\u003c/p>\n\u003cp>In this \u003ca href=\"https://www.teachingchannel.org/videos/teaching-subtraction-problems-nsf\" target=\"_blank\">Teaching Channel video\u003c/a> of Sarah Dietz' second grade class, she uses a video clip about cookie monster to grab her students' interest and get them questioning. The video presents a puzzle to students and Dietz makes sure to draw out their questions, honing in on the common theme (and the lesson for the day) based on their authentic questions. She's also asking them to \u003ca href=\"https://ww2.kqed.org/mindshift/2015/10/19/how-can-we-teach-math-to-encourage-patient-problem-solving/\" target=\"_blank\">decide what information they need\u003c/a> to solve their question, an important part of math in the real world that is often left out of traditional textbook problems. Then, she gives them time to work through the question they've posed using a model of the cookie package and their knowledge of various subtraction strategies.\u003c/p>\n\u003cp class=\"p1\">“To [the students], it’s not a math lesson; it’s a puzzle that needs to be solved,\" said Dietz. \"It’s a problem they want to work out.”\u003c/p>\n\u003cp>When students share their answers at the end, Dietz asks them to use their work to explain their thinking and she leaves enough time for multiple examples of different strategies. Emphasizing that there are many acceptable ways to solve a problem can help students remain open to struggle and figuring things out in the ways that make sense to them.\u003c/p>\n\u003cp>In another \u003ca href=\"https://www.teachingchannel.org/videos/kindergarten-math-addition-nsf\" target=\"_blank\">Teaching Channel video \u003c/a>about this three-step process called \"Three-Act Tasks,\" kindergarten teacher Kristin Alfonso says: \"I love that Three-Act Tasks are usually just difficult enough that even if kids can figure out really quickly on the carpet, they still have to go back to their tables and show us, and be able to prove their thinking to us.\"\u003c/p>\n\u003cp>[ad fullwidth]\u003c/p>\n\u003cp>\u003c!-- iframe plugin v.4.3 wordpress.org/plugins/iframe/ -->\u003cbr>\n\u003ciframe src=\"https://player.vimeo.com/video/168841943\" width=\"640\" height=\"360\" frameborder=\"0\" scrolling=\"yes\" class=\"iframe-class\">\u003c/iframe>\u003c/p>\n\u003cp>\u003c/p>\n\u003cp>Three-Act Tasks: Modeling Subtraction from \u003ca href=\"https://vimeo.com/user11426713\">Teaching Channel\u003c/a> on \u003ca href=\"https://vimeo.com\">Vimeo\u003c/a>.\u003c/p>\n\n",
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"content": "\u003cdiv class=\"post-body\">\u003cp>\u003cp>Getting students excited and authentically curious about a math task takes more than presenting a word problem. Some teachers are finding that a short, high-interest video or other piece of media that raises questions in kids' minds is the best way to prime them to dive deeply into problem solving.\u003c/p>\n\u003cp>In this \u003ca href=\"https://www.teachingchannel.org/videos/teaching-subtraction-problems-nsf\" target=\"_blank\">Teaching Channel video\u003c/a> of Sarah Dietz' second grade class, she uses a video clip about cookie monster to grab her students' interest and get them questioning. The video presents a puzzle to students and Dietz makes sure to draw out their questions, honing in on the common theme (and the lesson for the day) based on their authentic questions. She's also asking them to \u003ca href=\"https://ww2.kqed.org/mindshift/2015/10/19/how-can-we-teach-math-to-encourage-patient-problem-solving/\" target=\"_blank\">decide what information they need\u003c/a> to solve their question, an important part of math in the real world that is often left out of traditional textbook problems. Then, she gives them time to work through the question they've posed using a model of the cookie package and their knowledge of various subtraction strategies.\u003c/p>\n\u003cp class=\"p1\">“To [the students], it’s not a math lesson; it’s a puzzle that needs to be solved,\" said Dietz. \"It’s a problem they want to work out.”\u003c/p>\n\u003cp>When students share their answers at the end, Dietz asks them to use their work to explain their thinking and she leaves enough time for multiple examples of different strategies. Emphasizing that there are many acceptable ways to solve a problem can help students remain open to struggle and figuring things out in the ways that make sense to them.\u003c/p>\n\u003cp>In another \u003ca href=\"https://www.teachingchannel.org/videos/kindergarten-math-addition-nsf\" target=\"_blank\">Teaching Channel video \u003c/a>about this three-step process called \"Three-Act Tasks,\" kindergarten teacher Kristin Alfonso says: \"I love that Three-Act Tasks are usually just difficult enough that even if kids can figure out really quickly on the carpet, they still have to go back to their tables and show us, and be able to prove their thinking to us.\"\u003c/p>\n\u003cp>\u003c/p>\u003c/div>",
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