Five Classroom Dimensions That Show Deep Math Learning Is Happening

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Mia Buljan remembers the specific moment eight years ago when she realized she had to give kids more space to grapple with a problem on their own. She was filming a student working on a math problem with her iPhone (something she does regularly so she can review her strategies and plan next steps). “At that time I thought my job was to be super helpful,” she said, “like ask some pointed questions, or give some suggestions of where he might go next.”

But before she could help the student, her attention was called away by a disturbance on the other side of the room among her 34 students. When she turned back to the struggling student, he had solved his issue. Buljan hadn’t moved the camera the whole time, so she captured him figuring it out on his own.

“The whole time I was distracted and not talking to him at all, he was thinking and redesigning his problem,” Buljan said. At that moment she knew she needed to stop talking so much. She still provides support, but she’s changed the kinds of questions she asks. She used to ask what she calls “funneling questions,” prompts that lead the student where she wanted them to go, like, “what comes next.” Now she tries to ask focusing questions like, “how do we get started?” Or, “who do you know who’s already good at this that you could ask?”

“I have watched so much wrong counting it hurts,” Buljan said. “The urge to fix their thinking is so strong, but I just changed my own mindset for creating space for them to think.” She’s learned to think of this approach as “going at the pace of the learning,” a phrase she heard from Akiko Takahashi, an expert in lesson study.

Buljan no longer rushes to cover everything in the pacing guide. Instead, she spends as long as is necessary on the most fundamental structures of math, making sure students know those really well. By her logic, going slower at the beginning is more efficient because her students learn concepts like subtraction well once, rather than having to learn it again in third, fourth and fifth grade.


She applies the Teaching For Robust Understanding of Mathematics (TRU) framework in her classroom. TRU could feel like another “new” math program, but it’s actually a simple way to remember the things many good teachers already know. TRU is five basic dimensions that will sound familiar to most teachers: content, cognitive demand, equitable access to content, agency, authority and identity, and uses of assessment.

“I’ve organized things that the whole field knows, so that there’s a small enough number of things to keep in mind,” said Alan Schoenfeld, University of California Berkeley professor of education and mathematics, and the person behind the TRU framework. “The main virtue of TRU is not that I’m telling you anything new.”

The five dimensions of the TRU math framework.
The five dimensions of the TRU math framework. (Mathematics Assessment Project)

The TRU Framework focuses on how students experience the math, not on what the teacher is doing. “Our framework says you should really be focusing on, 'What does it feel like to be a student in that classroom?” Schoenfeld said. “What’s the experience from the point of view of the student? Because that’s what shapes who the student becomes.” And how a student feels about him or herself has everything to do with motivation, persistence and agency.

When Schoenfeld introduces the TRU Framework to teachers, he often shows three classroom videos and asks educators to make a list of all the behaviors they see happening. As a group they then categorize those observations into the five framework dimensions. In this way, teachers co-construct an outline of important elements in a classroom and can see that when they are present rigorous learning is happening.

“They provide a straightforward way to focus on and reflect on practice in a way that will really make a difference,” Schoenfeld said.


Buljan teaches second grade at Glassbrook Elementary school in Hayward, California. She’s been using the TRU framework for several years and finds it particularly helpful when her students are having difficulty with a concept. Thinking through the TRU framework lens helps her step back and focus on aspects like agency and authority.

For example, if two students get different answers, she might talk about the idea of proof, asking them to convince one another of their right answer. “That’s some of the richest growth, when they’re able to have that conversation,” Buljan said. Approaching their learning impasse through the TRU dimension of authority helped her to structure students’ conversations differently. Focusing on authority and agency in that moment led to student growth.

“You don’t ever just use one [element of the framework],” Buljan said. She has found the idea of cognitive demand particularly helpful in her diverse classroom, where students speak 11 different languages, and 90 percent qualify for free or reduced priced lunch. Buljan says teachers have a tendency to provide too much support to English language learners in their attempt to help them access the content.

“The danger we get into is you can scaffold so much that you pull the thinking out of it,” Buljan said. “So the trick is, how do we create access into certain kinds of problems, but still make it about the kids doing the proof and doing the thinking themselves?”

The TRU framework has kept Buljan focused on creating cognitive demand for all her students, regardless of language barriers or prior knowledge. She does that largely by focusing on the structure of problems, as opposed to one specific standard in the second grade curriculum. She wants her students to deeply understand fundamental mathematical structures like place value, or how to group and ungroup numbers, so she tries not to give students “rules” that will help with one kind of problem, but later could lead to confusion.

Subtraction is a good example. Some teachers tell students the “rule” is to always subtract the smaller number from the larger number. But that rule gets subverted when students start doing multidigit subtraction and they see each column of numbers as a free floating problem detached from the idea of place value.

Instead, Buljan gives students a lot of thinking time. “It is brutal,” she said. “Sticking with second graders long enough for them to push through all that confusion and get to a place where all those underlying structures are part of who they are is tough.” But, it’s also rewarding. When her students hit challenging problems, she tells them if it was easy, they would already know it. “They totally get it," Buljan said. "They will say things like, ‘I’m really learning now,’”

Teaching with the TRU framework has also prompted Buljan to think creatively about how structures she uses for English instruction could be applied to math. She joked that in second grade, no one really cares about math, it’s all about reading, so there’s a lot of professional development around reading strategies. Buljan has adapted many of them for math.

For example, she’s applied writers and readers workshop to math workshop. She usually introduces a topic quickly, gives kids 30 minutes to think through the problem together, and then she does a quick wrap-up. There’s very little direct instruction in her math teaching. She also has adapted the idea of a “mentor text,” or in math a “mentor problem.”

“When you teach reading there’s this idea that I’m not teaching you to read this book, I’m using this text to teach you a strategy that you can use to read any book,” Buljan said. She’s used that same idea to teach kids math problem solving. She’s not teaching them how to do a certain type of problem, she’s trying to teach how this problem can help her students solve any problem. Instead of teaching rules of adding and subtracting, as a class students focus on describing the parts of the problem, what’s happening in the problem and how to talk about patterns in math.

Buljan said her second graders work on just six problems for half the year. The quantities will change and the items being added or subtracted change, but essentially the problem is the same. Within that familiarity, students are identifying parts of the number sentence, using location and quantity to describe patterns, determining what type of problem it is and moving forward with various strategies, but it all feels safe. And for second graders, small changes in the problem feel big - going from adding apples to adding stickers makes it a whole new problem, Buljan said.

To meet the equitable access to content part of the TRU framework for her many English language learners,  Buljan has modified the common practice of using sentence frames. She felt a typical sentence frame like “ I thought __ because __” funneled student thinking too much. So now she just gives sentence stems like, “I noticed,”  to help model how students can have an academic conversation. She then monitors how they are using those stems and gives immediate feedback.


David Foster from the Silicon Valley Mathematics Initiative shot videos from the first several weeks of Buljan’s class and the last several weeks of class to show how she works to develop her students’ language, conversation ability, and classroom culture. Foster has been coaching math teachers for decades and he likes the TRU framework because it distills all the research into five easy-to-understand and recognize dimensions. He finds teachers often struggle to get kids to work productively in groups because they haven’t spent time at the beginning of the year developing a class culture of trust and collaboration. And they often aren’t giving students a worthwhile task that’s worth discussing in the first place. He says if teachers pull a problem out of the textbook only one in 90 times will it be a worthwhile problem.

Foster says many of the structures Buljan used intuitively are great strategies to home in on questions of equity, agency, authority and cognitive demand. Sentence frames help students get into the habit of defending their thinking with evidence. Number talks help teachers pinpoint exactly where student misperceptions lie and are a venue for students to practice talking about math. Roles and norms can help ensure every group member has an equitable role.

Good teachers are doing these things already. Foster has seen very effective math classrooms in almost every school he enters. His work is to help all math teachers improve the quality of their teaching, something everyone needs. He finds the teachers who are most effective in this process are the ones who are never completely satisfied. They are the ones who leave at the end of the day worrying about how to reach the one kid who is still struggling. That hunger to improve is a huge part of becoming more effective.


Alan Schoenfeld, in collaboration with colleagues at the University of Michigan, recently received a National Science Foundation grant to develop tools to coach math teachers in effective classroom practices. Schoenfeld is focusing his side of the work on high school teachers in Oakland using TRU along with lesson study. The central office math coaches have found the framework useful as a point of departure for conversations with teachers.

“It creates a structure for someone to give feedback and engage in reflecting with that teacher on something the teacher is interested in working on,” said Barbara Shreve, Oakland Unified’s Secondary Math Coordinator. She’s also found it helps administrators on classroom rounds focus in on what they’re seeing.

“There have not been a huge number of spaces where teachers get to talk together about the meat of what happens day-to-day in the classroom,” Shreve said. She hopes conversations centered around TRU will give everyone the same point of departure and a useful language to move towards solutions. “Success is going to look like having a much more common language for talking about the successes and challenges we’re experiencing as educators,” she said.

Schoenfeld, for his part, hopes to use the research period to develop a set of tools that could help other districts conduct coaching and professional development around the five dimensions of TRU. In previous research funded by the Bill & Melinda Gates Foundation and verified by independent evaluators at UCLA’s National Center for Research on Evaluation, Standards, and Student Testing (CRESST), Schoenfeld found that teachers who were trained on the TRU framework and used it in their classrooms saw on average an improvement in student understanding that correlates to 4.6 months of additional learning.


"We documented changes in the teacher's behavior over time because of the lessons and the support,” Schoenfeld said. Teachers stopped telling students what to do and instead got students to work the problems out for themselves. The structure of lessons forced teachers to teach differently. Schoenfeld hopes that if he can develop an effective toolkit, more districts can easily scale up their work on TRU.