Let’s Talk about High-Quality Mathematics Instruction – Part 3
Jason Gauthier, Ph.D.
Mathematics Education Consultant
Allegan Area Educational Service Agency
Welcome back to our discussion about the how and what of High-Quality Mathematics Instruction (HQMI)! In this post, I’m returning to a comment I made in an earlier post: mathematics is done by people—it is a fundamentally human endeavor (Jacobs, 1982; Su, 2020). My last two posts, I think, have shown that mathematics instruction is complex; and mathematics is as well. This post is concerned with the complexities of the individuals who do mathematics, whether in school, in life, or anywhere. This layered, compounded complexity is what makes teaching mathematics effectively so challenging—and so rewarding when we do it well.
Mathematical Agency, Identity, Authority in the Classroom
Have you ever tried to do something fun and happy while you were in a really bad mood? Everything about that normally enjoyable activity is different and sometimes harder, based on your mood. What about trying to do something you “just know” you’re bad at? Not only is the activity more difficult, it is also less enjoyable, and you are more likely to miss any progress you make at getting better at it. My point in these two little instances of imagining is that our mood and our perceptions of our ability deeply affect how we engage with an activity. The same is true for students in our mathematics classes. All of them are growing, changing people with emotions and moods, all of which affect the way they engage with our attempts at facilitation (and we are humans too, by the way). The moral of the story here is that our emotional state and our identity as a learner of mathematics play vital roles in how we engage in learning and doing mathematics (Martin, Aguirre, & Mayfield-Ingram, 2013; Middleton, Jansen, Goldin, 2017).
High-Quality Mathematics Instruction must attend to students’ identities as people--children, even--as explorers, learners, and doers of mathematics. Teachers must be aware of and attentive to students’ mathematical identities and should engage students in ways explicitly designed to foster positive relationships with mathematics and positive self-images as learners (see, for example the Essential Instructional Practices in Early Mathematics: Prekindergarten to Grade 3 (MAISA GELN Early Math Task Force, 2019)). Whenever I ask a room full of teachers what they remember about school mathematics, some inevitably recount the days they “learned” they were bad at math. For some it was timed tests of basic facts; for others it was an abstract alphabetical avalanche in algebra class. Those experiences taught them that they were bad at math, either implicitly (the algebra example) or explicitly (the timed facts tests). Those are the kinds of experiences we absolutely must prevent.
We need students to grow comfortable with mathematics--with sense-making--and with using it in their lives. I’m not saying everyone needs to love math, but everyone should leave our care able to engage with at least a moderate level of confidence in mathematical problem-solving. We don’t want people to continue to react with strong negative emotions to the simple appearance of numbers in a context. So how do we accomplish these goals?
Discussions in this space most often deal with three interrelated ideas: identity, agency, and authority in mathematics. The Teaching for Robust Understanding in Mathematics (TRU Math) Framework (Schoenfeld et al., 2016) discusses these three ideas as a domain of classroom instruction:
The extent to which students are provided opportunities to “walk the walk and talk the talk” – to contribute to conversations about disciplinary ideas, to build on others’ ideas and have others build on theirs – in ways that contribute to their development of agency (the willingness to engage), their ownership over the content, and the development of positive identities as thinkers and learners (Schoenfeld et al., 2016, p. 3).
So, we need to provide opportunities for students to contribute their own ideas about mathematics and to engage in collective idea sharing and building. These activities must explicitly encourage students to engage and build a desire to do so. They must allow students to have some sense of being the mathematical authority of their own ideas—this might be thought of as a de-centering of the teacher as the mathematical authority. The teacher does not give up total authority, but rather pushes students to collectively make sense of problematic contexts rather than being the primary (or even the only) arbiter of what is mathematically correct. That’s a tall order, to be sure. However, there are some concrete, actionable steps we can take to move in that direction.
Knowing Our Students and Valuing Their Ideas
First, I would argue that in mathematics teaching there is no more important imperative than “know your students and value their ideas.” Get to know your students as people as you engage with mathematics. This knowledge will serve you well in forming positive relationships with them and supporting their learning. Further, this knowledge allows us to create much warmer, emotionally safe (and hence more inclusive) environments in our classrooms--through encouraging mathematical risk taking and even small things like connecting students to context by bringing tidbits of them into your math problems and such. Second, strive to create a classroom in which you value all student ideas, whatever their source. Look for the good or the correct in each idea and push students to individually and collectively build upon that. One way to do this is through routines such asStronger and Clearer Each Time (Zwiers, 2017) or engaging students in rough draft thinking (Jansen, 2020) where the goal is to get ideas out and revise them as understanding grows. Relatedly, students should also have many opportunities to share their thinking with others, whether small or whole group. Structuring class time to make this possible is a vital step in building students’ identities. Consider how the launch-explore-summary lesson structure of the TTLP (from Part 2 of this series) offers students these opportunities.
Building robust mathematical identities is long-term work that should be part and parcel of HQMI. There are many facets to this work, many paths a teacher might take in achieving success in it. However,f: leveraging students’ ideas as learning opportunities, positioning students as mathematical authorities, and attributing importance to students’ ideas. The first of these is best thought of as a lens for a teacher to use when observing students and as an integral part of the classroom culture. Again, consider how facilitating a problem-centered lesson inherently affords teachers the opportunity to see students’ work and from there to choose to bring those ideas forward as opportunities for the whole class to benefit from discussing and thinking about them. Even approaches that did not result in a correct answer are potentially useful, as they can be built upon by the class so that the whole group has collectively made sense of a way of thinking about a problem. This is challenging work for a teacher, but when done well, it results in deep discussion and sense-making of mathematics, a goal we outlined as part of HQMI previously.
Positioning and Attribution in Mathematics Teaching
Related to using students’ ideas as opportunities for learning are two specific practices (mentioned above) that should be part of that process and part of HQMI writ large: positioning and attribution. Let’s take positioning first. This is the practice of providing students opportunities to be experts on their own thinking. For example, if a student has an idea that the class should consider, it should be that student who presents the idea to the class and explains their thinking. Any rephrasing or other work done at that point is directed toward the student who had the idea. In particular, the other students should be the ones doing the rephrasing and questioning. This positions that particular student as having valuable knowledge, thereby building confidence (agency) in sense-making. Further work by the class, whether clarification or refinement should also involve the student whose idea is under consideration. Teachers might use talk moves (Chapin, O’Connor, O’Connor, & Anderson, 2009) to allow the class to process (think time) or to clarify (rephrasing), but ultimately those discussions should center on the student who had the idea. This will increase student-to-student dialogue and de-center the teacher as the primary authority on all mathematical ideas in the classroom. The teacher and students might be considered to be co-constructing the knowledge at that point.
One important caveat to consider when implementing this practice is to pay close attention to which students’ ideas are forwarded most often in your classroom. If it is always the same small number of students, then you are only building the identities of those students and you might be implicitly telling all other students that their ideas are not worthy of consideration. In particular, if you know your students and know which of them have more fragile or less robust mathematical identities, you can use positioning to deliberately strengthen those identities over time by choosing to center particular ideas brought by those students. There is a deep connection between student agency and student identity--one might even consider agency as identity in action--and both of these must be developed together.
Along with positioning students as experts on their own thinking, publicly and privately attributing importance to students’ ideas is also a vital practice in building mathematical identities. Students need to know that their thoughts are important and their ideas valid. Most of what we do sends implicit messages about this and many students will miss those messages. To remedy this, teachers must be explicit in identifying important ideas and attributing them to the student or students who devised them. I am not advocating for boundless and random praise of students’ ideas (both correct and incorrect). Rather, it is important to use what you know about students’ current mathematical identities to plan your attributions intentionally. This can be difficult during the hectic ebb and flow of discourse in your classroom; however, it is vital work. Important ideas can come up in small group or in whole group conversation and in both cases it is important for teachers to identify those ideas and ensure that they attribute them appropriately. You might try formulating attributions in ways similar to those below:
“Jaime just had an important thought. Jamie, would you say more about your thinking here?”
“What do the rest of you think about Allie’s important mathematical idea?”
“I just heard Terance say something that I think we all need to think about a bit more. . .”
“Group 4 used Brent’s idea about decomposing 26 to add here. Brent, would you talk about what you were thinking here? It’s an important idea and I want us all to think deeply about it.”
Another option might be to adopt a strengths-based approach to teaching mathematics (e.g., Kobett & Karp, 2020) in which we focus on the strengths (both mathematical and otherwise) that students bring to the classroom and leverage them to enrich the entire classroom experience. This kind of approach inherently builds agency, identity, and engagement in mathematics classrooms. And the work can begin with concrete ideas like in Kristin Frang’s 8th grade mathematics classroom (Frang, 2021), where she maintains a jamboard of student strengths that she and the class have noticed.
Doing this work is, perhaps, not overly complex; however, remembering to do this work and incorporating it into your practice can be challenging. If you’d care to see an example of this work in action, take a look at this video from Achieve the Core. It is a Kindergarten classroom working on numbers that make ten and there are excellent examples of positioning and attribution within the first 10 minutes of the video.
In closing, I can’t stress enough that just getting out and trying some of these things is a vital first step. It’s true that new routines and practices can feel awkward at times--but stick with it. Keep trying new things, keep finding new ideas and listening to new people. Watch for those moments where you notice you could have done something differently and instead of beating yourself up, simply recommit to being intentional and support yourself in doing that work. Even something as simple as a post-it note reminder in your notes and the clipboard you carry with you: “Leverage. Position. Acknowledge.” And above all, don’t forget to celebrate your successes by yourself and with your class. Being intentional about knowing your students is the key. You’ll know which of them need to be positioned as experts more often. You’ll know which need to be told that they have important ideas. All students need these things, but some need it more and having a knowledge of your students’ identities will help you build them up when they need it, showing them that they do have good ideas and that they can succeed in mathematics, no matter what their past experiences have been.
Chapin, S. H., O'Connor, C., O'Connor, M. C., & Anderson, N. C. (2009). Classroom discussions: Using math talk to help students learn, Grades K-6. Math Solutions.
Frang, K. [@KristinFrang]. (2021, March 10). Great question. I start in my 8th GR class talking about the strengths we bring to class already. Every time . . . [Tweet]. Twitter. https://twitter.com/KristinFrang/status/1369801128435650563?s=09
Jacobs, H. R. (1982). Mathematics, a Human Endeavor: A Textbook for Those Who Think They Don't Like the Subject. W. H. Freeman & Co.
Jansen, A. (2020). Rough Draft Math: Revising to Learn. Portsmouth, NH: Stenhouse.
Kobett, B. M. & Karp, K. H. (2020). Strengths-based Teaching and Learning in Mathematics: Five Teaching Turnarounds for Grades K-6. Corwin.
Martin, D. A., Aguirre, J., & Mayfield-Ingram, K. (2013). The Impact of Identity in K-8 Mathematics Teaching and Learning: Rethinking Equity-based Practices. Reston, VA: NCTM.
Michigan Association of Intermediate School Administrators General Education Leadership Network Early Mathematics Task Force. (2019). Essential Instructional Practices in Early Mathematics: Prekindergarten to Grade 3. Lansing, MI: Authors.
Middleton, J., Jansen, A., & Goldin, G. A. (2017). The complexities of mathematical engagement: Motivation, affect, and social interactions. In J. Cai (Ed.) Compendium for Research in Mathematics Education (pp. 667-699). Reston, VA: NCTM.
Schoenfeld, A. H., & the Teaching for Robust Understanding Project. (2016). An Introduction to the Teaching for Robust Understanding (TRU) Framework. Berkeley, CA: Graduate School of Education. Retrieved from http://map.mathshell.org/trumath.php orhttp://tru.berkeley.edu.
Zwiers, J. (2017). Stronger and Clearer Each Time. Retrieved from:https://jeffzwiers.org/tools