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Providing public voice and leadership to support and advance high quality teaching and learning of mathematics for all students.
In Jason Gauthier's final installment of "Let's Talk About High Quality Math Instruction", he ties all of the previous posts together by breaking down the notion that, "a classroom math learning community is a group of individuals who show up expecting to explore unfamiliar problems and make sense of unfamiliar situations using mathematics" and ways in which we make that happen. Click here to read this latest post and join us in thanking Jason for sharing his team's wisdom through these last few months in the comments! In what ways has this series helped shape your thinking around high quality math instruction?
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
Let’s Talk about High-Quality Mathematics Instruction – Part 2
Engaging in Flexible Thinking and Problem-Solving
By: Jason Gauthier, Ph.D.
Allegan Area Educational Service Agency
In my last post, I summarized the process a group of us went through to develop a vision of high-quality mathematics instruction (HQMI) that reflected what we believed as mathematics educators, what we wanted for our students, and what we could support with research. We developed a vision that allowed us to be clear and succinct in discussing the what of HQMI. We never intended to stop with the “what.” Rather, we also spent a significant amount of time puzzling through, debating, and generally thinking deeply about the how of HQMI as well. And that is where I’ll pick up in this post.
The How of HQMI
So, let’s talk about the how of HQMI—particular practices and mindsets that allow teachers to do the work we’ve described. As part of the process of creating this vision of HQMI, the group developed our initial work in considering student and teacher actions through examining our own beliefs about mathematics, teaching, and identity. This was an important step because, as Hiebert (1999) reminded us, research can do many things, but it can’t tell you what you value. This post is concerned with diving a bit deeper into how teachers can create the conditions for students to effectively “Engage in Flexible Thinking and Problem-Solving” in mathematics classrooms. Along the way, I’ll do my best to connect to research via both citations and brief explanations of important findings.
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Let’s Talk about High-Quality Mathematics Instruction
It all started with a question—one that, on the surface, seemed easy to answer. For years, the mathematics education community has been maintaining that the best preparation for assessments of any kind—including high-stakes assessments such as SAT—is high-quality mathematics instruction. Let’s call it HQMI for short. When provided with the aforementioned advice regarding preparation for SAT, an administrator asked, “So what is good math instruction anyway?”
And there it was. The “simple” question. There are no shortage of answers to it, many based on research and many based on gut feeling or familiarity. However, none of those answers was easily provided in the moment. None of them were short, memorable messages that balanced brevity with intelligible information. In rooms full of teachers, consultants, and college professors, if you ask this question you will not get identical answers. And those answers will likely be neither short nor non-technical. And to be clear, they shouldn’t be. Good math teaching is complex and highly technical. It might be one of the most complex activities a human being can undertake. And so, no attempt to summarize it will be simple. But a group of us tried anyway, realizing that it might be helpful to have such an answer to hand when asked by those who are not steeped in the work and research (policymakers come to mind readily).
[NOTE: The work I share in this post is not solely my own. It is the result of a collaboration of expert mathematics educators from across the state. It is also the result of a process which I will only summarize here.*]
The process began with examining what we as mathematics educators want for our students when they leave our collective care at graduation (with regard to mathematics, specifically). We interrogated and debated what that might look like as students engage in learning. Then we considered, for each of those ideas, what teacher moves might produce conditions ripe for that kind of student activity. After we collected these ideas, we examined them for themes that might allow us to begin the process of summarizing and condensing those ideas into a more manageable form. We wanted the final product to serve a three-fold purpose:
Create a message that summarized research-informed mathematics instruction.
Create a message that could be used effectively with very brief time to provide explanation.
Create a message that could be used effectively with more expansive time allotments.
The themes helped us collapse the message into what we called an “elevator speech”—a message that could be delivered and understood in under two minutes. Careful expansion on those themes provided us with a message that was longer, but also more detailed. Personally, I always envisioned this message as a way to unite the mathematics education community and support teachers in their incredibly complex work. The message had to go further than NCTM’s Principles to Actions which, while fantastic and a go-to resource, spends a good deal of time talking about what HQMI is but much less time on how to actually do it in great detail. We aimed to do both.
The What of HQMI
So, going back to the question asked by the administrator in the beginning of this post, what is HQMI? As with the process I described above, let’s start with what we want for our students upon graduation. What habits of mind do we want them to have? What do we want their relationships with mathematics to look like? What kind of ways of knowing and doing do we want them enculturated into?
HQMI engages students daily in making sense of problematic situations, building flexible thinking processes over time which leverage tools, patterns, visualizations, and representations to develop creative solution pathways.
The essence of doing mathematics is sense-making—that is, engaging with a problematic mathematical situation that is (relatively) unfamiliar and attempting to make sense of it in order to find a resolution to the problem. Because this is the essence of doing mathematics, then students must engage in this kind of activity on a regular basis, preferably daily. The ultimate goal of this work, supported by effective pedagogical moves (discussed in subsequent posts), is to give students a coherent set of experiences that allow them to become experienced at solving problems. Gaining this experience involves the creation of knowledge, skills, and habits of mind over an extended period of time. And that is a key point to remember: learning takes time. Developing expertise takes even longer. Developing this expertise involves a good many mistakes and missteps as well as an equal number of recoveries and ah-ha moments. It involves playing around with mathematics and seeking to understand rather than driving as quickly as possible for an answer. This flexibility in the use of strategies and ideas is a hallmark of problem-solving expertise. Further, understanding comes from exploring the different ways one might conceptualize and attack a problem. It also involves being strategic in the use of those different ways. Studying problems in this way also makes explicit for students the repertoire of strategies, tools, and representations they have and can use to attack whole classes of problems. Throughout their journey through school mathematics, we want students to become problem-solvers and sense-makers. We want this, yes, because these skills will be valuable to them in life after school mathematics. But more than that, we want them to have these skills because doing mathematics is something that humans do and our lives, both individually and in aggregate, are better for it.
Developing Positive Mathematical Identities
HQMI attends regularly and explicitly to students’ developing mathematical identities, fostering belief systems that promote student agency through the development of a growth mindset and empower students as mathematical authorities.
Because mathematics is something that humans do, we ought to keep one critical fact in our minds as we think about this idea: mathematics is done by people. And people bring all sorts of knowledge, emotions, ideas, and pre-conceptions to the table when they do anything. Those factors always have an impact, either large or small, on what we do and how we do it (e.g., how confidently we perform a task or how creative we are within the confines of an activity). Mathematics is no different. The reality we as math educators face is that teaching mathematics is identity work. We have to help students build identities as sense-makers and problem-solvers in mathematics. This is not tacit work. On the contrary, we have to take great pains to help students grow into confident problem-solvers. Students must come to see themselves as capable of doing mathematics, to trust in their abilities to make sense in all the ways discussed above, even in the face of mistakes and missteps in the process. Students also must have some confidence in their own reasoning and not rely solely on an outside authority to judge the quality and correctness of their work. This confidence shouldn’t be boundless, however, as students must also know that even experts make mistakes and try things that don’t work—that is part and parcel of doing mathematics. Student should also be able to bring themselves into the work of doing mathematics. By this I mean that math class should value all students’ ideas, contributions, and kinds of knowledge. Ultimately, we want students to see themselves as people who are capable of successfully doing mathematics and using it to make their lives better.
Becoming Part of Classroom Mathematical Learning Communities
HQMI intentionally creates and continuously attends to a classroom community of mathematical thinkers, engaging students both individually and collaboratively in communicating about students’ mathematical ideas and mathematical thinking.
Lastly, despite the popular belief, mathematics is not something that is done only in isolation or solely within a person’s mind. People are social creatures and people do mathematics, therefore there is no reason we might expect mathematics to be merely an individual endeavor. Consider that school mathematics encompasses thousands of years of human mathematical thought and development. The idea that we might expect a student to sit quietly in a desk for 13 years and gain a perfect understanding of that great quantity of ideas is, quite frankly, outrageous. And so, mathematics is something we do together, as a community in search of understanding. And we are better off for it. We learn more, we generate and try more creative approaches, we make more and better sense of situations, and we push each other in a classroom community focused on making sense of big mathematical ideas. We do all this through communication with each other, both teacher and students. And so, mathematics classrooms might be conceptualized as a group of people engaged in communicating about and developing a common understanding of mathematical ideas. It’s important to remember, though, that this common endeavor also involves a good deal of individual thought, oftentimes prior to community engagement. The false tension between individual and social learning is one we do not acknowledge—doing mathematics and making sense necessarily involve both types of activity. And those types of activity must be focused on the ideas and mathematical thinking of the students in that community.
While I’ve elaborated a bit on the what of HQMI here, it’s helpful to see the three main pieces all at once:
That’s the elevator speech. HQMI is about those three things. These are not the only possible pillars, but they are research-informed and tested in classrooms. Further, each of them can be connected to research and resources that elaborate in much greater detail on the actual processes and activities of mathematics classrooms consistent with this vision. I’ll talk more about those in the next in this series of posts.
This ending may seem unsatisfying. I can’t say that I disagree; however, I’d love you to come back to read the next post in this series so my choice to end here is at least somewhat deliberate—an academic cliffhanger, so to speak. In the next entries in this series of posts, I’ll explore in more detail exactly what each of the pillars mean and how we might go about engaging in mathematics this way.
* The group of mathematics educators who worked on this document would not, perhaps call themselves experts, but I would. This work would not have been possible without their vision and dedication. Thank you to, in no particular order, Denise Brady, Brad Thornburgh, Danielle Seabold, Debbie Ferry, Gerri Devine, Joe Elsheikhi, Jodi Redman, Kathy Berry, Kathy Dole, Kim Hubert, Karen Reister, Marie Smerigan, Mary Christensen-Cooper, Mary Starr, Jamie McClintic, Megan Coonan, Mike Klavon, Anne Marie Nicoll-Turner, and Michelle Tatrow. Any omissions are mine and are entirely unintentional. Lastly, I must thank the Michigan Mathematics and Science Leadership Network for hosting our meetings.
Jason Gauthier is a Mathematics Education Consultant for the Allegan Area Educational Service Agency.
Perseverance in Problem Solving
By Chelsea Ridge, GVSU
“Make sense of problems and persevere in solving them” This has always been my favorite mathematical practice. I think because in a mathematical sense, it is a skill that underlines all content areas, but on a personal level it reminds me of what is most important. During this time of the COVID-19 pandemic, Stay Home Stay Safe Orders, and Continuity of Learning Plans, we are all called to persevere.
As I work to understand various problems, both personally and professionally, I strive for perseverance. As I dig through the overwhelming amounts of data, I persevere in understanding the situation and am inspired and appreciative of the statisticians, data analysts and graphic designers that are working to make the data accessible to everyone. In a public health crisis, high quality data is required to drive the decision making process.
I recently read Teaching Through a Pandemic: A Mindset for This Moment and one line of that article really stuck with me: “Everyone thinks they can’t before they can.” During this time especially, personal health and wellness is imperative in order to continue to persevere. In order to serve all students, a level of commitment greater than we have ever seen is required. I am inspired by all the educators that are doing what would have been classified as impossible a few weeks ago.
As I continue to work on my mindset during this challenging time while also working to accomplish daily tasks in new and unique ways, I am called to analyze problems and solve them. This has required perseverance greater than what was required during my 10 mile race training regimen of 2015 (running is hard). As I consider meeting the needs of the diverse groups of people we serve, it can be overwhelming. To guide me through this process, I have found success with the Innovators’ Compass.
A couple of years ago, I was connected with the Design Thinking Academy at Grand Valley State University. We were doing some new work on student programming and were looking for assistance both in program development and creating some Design Thinking opportunities for middle school students. Over and over, as I have used the tools and refined my skills in these types of problem solving situations, I have been impressed by the creative solutions teams of individuals of any age have been able to accomplish.
As I have encountered challenging problems, I have turned to the Innovators’ Compass to organize thoughts and guide group conversations. Their homepage states: “People finding better ways forward in anything we do,” and I think that summarizes the strength of the compass. As complex problems require perseverance in solving them, the Innovators’ Compass provides a platform that encourages perseverance in all situations. In the past, I have used the compass to solve problems ranging from finding a cross-cutting theme for a group project to working with middle school students to discover how they might live more sustainably. During the unique challenges presented by the COVID-19 situation, I have used the compass to consider how the Regional Math and Science Center might enhance our social media strategy to address the changing needs of our constituents as well as considering how we might create a summer camp experience that does not happen face to face. None of these prompts have a single, clear solution pathway and the Innovators’ Compass supports the continued work toward a rich solution.
The uses of the Innovators’ Compass are definitely vast and countless examples can be found on their website or on Twitter with #innovatorscompass. One thing I have found to be true is that utilizing this structure has broadened my mindset, increased my perseverance, and provided far more creative solution pathways. I appreciate that the structure is intended to be cyclical, so as things change it is easy to continue on embracing what is learned into a new solution pathway that may be more viable.
As we continue to persevere in this rapidly changing environment, I want to praise everyone for all the work that has been done. Keeping relationships first as we move forward educating the whole child is paramount to our success. I encourage you to continue to persevere and address challenging problems in all settings with the same vigor we dig into our favorite flavor of mathematics. In all settings, be well and keep your wellness a priority.
**The resource lists for remote learning, SEL, personal health and wellness, and supporting students are vast, so I won’t try to link them all here. If there is something specific you are looking for that you can’t seem to find, please contact me and I will help to connect you with the best resource.**
Chelsea Ridge is the Math Program Coordinator at the Grand Valley State University Regional Math and Science Center. firstname.lastname@example.org @GVSURMSC
A Positive Relationship with Math
By: Jessica Tufnell
Teachers are at the forefront of promoting positive mathematical relationships to learners. As the elementary school years progress, the math concepts become more abstract and can begin separating student feelings about the subject by achievement. As a 5th grade teacher, I see too many students associating themselves as “bad” at math and therefore plateauing their engagement in the subject simply because they don’t innately believe that they are capable of having success. As my math block rolled around in the beginning of the year, I saw a common trend: raised shoulders, sullen eyes, fewer hands being raised, and more judgement than love being spread when learners engaged in dialogue, all signs of stress associated with math. It didn’t matter how many pep talks or individual and group affirmations I gave, I realized my whole community of learners needed to believe they were capable of tackling the white elephant in the room themselves, math.
Engaging student learning around mathematics in my classroom is always overarched with the same purpose: Students are able to identify as mathematicians that are capable of perseverance in problem solving. The two key words being identifying and capable. If I don’t have a full community of learners that see themselves as being capable of solving a problem or using a strategy then what good is my best prepared lesson? I decided to shift my focus to emphasizing the “how” rather than the end product. But unfortunately you can’t just tell a ten year old that you care more about the process rather than the product!
I decided to bring mindfulness into my math mini-lessons to give students coping strategies that are not directly aligned to the math concepts being taught, but welcome the acknowledgement of discomfort in the learning process.
The first strategy is to “Take a Breath.” This can look and sound however the learner wants it to. Maybe they push their chairs back, maybe they put their pencil down...I don’t really care how it looks. This gives my students the opportunity to notice whatever they were feeling towards a math task and also gives them the grace to take a step back. Some students never take a breath, some do every lesson. I began to see fewer raised shoulders during my teaching time because students knew they were able to step away during their individual practice if they were feeling overwhelmed. It is also a great visual cue to drive me to individual consultations.
The second strategy is to “Give Yourself an Affirmation.” Because my affirmations were well intended but didn’t fully change a student’s feelings towards math, I wondered if a self-affirmation would? When we see ourselves as something, we become it. Giving my students a space to tell themselves something positive as a mathematician or anything else attempts to give them a space to live in the math lesson or again, fully step out of it. I began to see doodles of affirmations on math notebooks, or hear students say, “I am good at math even if I am struggling with this strategy.”
Cultivating an environment where these strategies were safe and welcomed took time and trust. I consistently use terms like, “This is new, we aren’t aiming to master it by the end of today,” “Whatever you are feeling right now is okay,” and “Even if we got that answer wrong it doesn’t mean we aren’t a mathematician.” The more my students see me step back from a math lesson and acknowledge their feelings as people, the more they individually cope. The more affirmations they hear as options during our morning meetings, the more I see them lifting themselves up. The more I model, show, and sit beside them when we get to a hard concept, the more willing they become in tackling it. Acknowledging student struggle opened the door to conversations about students overcoming fears during math.
Without addressing the stress that some students feel towards math, I feel like I am leaving them behind. As much as it is my job as the educator to make sure that learners can efficiently and effectively use a strategy when they move onto the 6th grade, I personally feel that it is just as much my role to cultivate an environment where risk taking, grace giving, and a love for math occurs. The better my learners feel they are at math, the better they do.
Jessica Tufnell is a 5th grade teacher in Kent City.
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Overcoming Professional Isolation through StoryCircles
By Chandler Brown
Due to the COVID-19 crisis, teaching and education as we know it are going to look very different this fall. Many school districts are moving forward with remote instruction, which may result in teachers having limited opportunities to interact not only with their students, but also with their colleagues. It is important now more than ever to ensure that professional learning opportunities offered to teachers focus on opportunities to create professional learning communities where educators can be supported and learn alongside one another.
StoryCircles, a new professional learning opportunity for Algebra 1 and Geometry teachers across the State of Michigan, has brought together groups of educators in online professional cohorts to focus collectively on representing lessons and discussing how to facilitate productive mathematical discussions through problem-based lessons. Using web-based tools developed by the GRIP lab at the University of Michigan, teachers are able to represent various ways different lessons could go - considering collectively common classroom situations and challenges using lesson storyboards and maps. From there, teachers have discussions around decision points and student conceptions where they debate and examine alternative instructional moves and probable outcomes. These larger discussions around the different approaches of instruction allow teachers not only to expand their repertoire of practices, but also to build a community that supports one another in an otherwise seemingly isolated field. Our research has demonstrated the potential for StoryCircles to support teachers in gaining valuable knowledge needed in teaching secondary algebra and geometry. In our most recent work, we have been using StoryCircles to support teachers in gaining the skills necessary to facilitate productive mathematical discussions of problems-based lessons.
The opportunity to join StoryCircles’ next cycle couldn’t come at a better time. We as educators are now facing a new normal where it is our responsibility to engage students in meaningful lessons, both synchronously and asynchronously. The StoryCircles community is solely based online so that teachers from all over Michigan can join in on the discussions, help expand the visualizations described above, and find new and innovative ways to give instruction during these unprecedented times. Accessibility, support, and professional development are the foundational pillars that hold this group together.
To learn more about the program please fill out this interest form and register for a live webinar on September 21st at 5:00 pm EST. This free webinar will be hosted by Dr. Amanda Milewski (University of Michigan), who will briefly describe the StoryCircles program. After that, we will hear from practitioners who will share how they learned from each others’ knowledge and experience through discussion of alternative instructional actions. Finally, ample time will be devoted to respond to questions from the webinar participants.
Fostering a Positive Math Identity, Within and Outside of the Classroom
By Morgan Denyes
Math has always been one of my absolute favorite subjects. As a student, I enjoyed the opportunities to think outside of the box and solve “problems” in creative and thoughtful ways. As a scholar, mathematics provided explanations for several real life situations that I desired to have a justification for. As an educator, math is fascinating to teach, yet can also be a struggle to help students realize that being “good” or “bad” at math is a fallacy. As I began my first year of teaching, I was shocked to see how many students presumed the identity of “I’m not good at math”. As I reflected on my schooling in mathematics at Michigan State University, and my experience as a child, I saw this as an opportunity, and a challenge, to foster an environment where each and every one of my third graders would leave my classroom with the understanding that they are mathematicians.
Identity is described as the belief that a child has of themselves. When we take a deeper look at mathematics, we see that this identity has to do with the belief that a child has of their personal ability in mathematics. This identity is built on several grounds. First, personal experience with math provides an opportunity for children to identify themselves as “good” or “bad”. Secondly, what I like to call outside factors, is the environment in which the child is exposed to. For example, if they have a parent who consistently mentions that “I am not good at math”, the child will likely develop these same beliefs because that is the message that is being portrayed to them. While in my teacher preparation program, I was taught how to develop a positive math identity within my future students, but this was only one sided. That is, I learned what I can do within my classroom, phrases to say, but not much beyond that. I quickly realized that this would not be enough to influence a child’s belief of their ability in mathematics. I then asked myself “How can I help my students, beyond our classroom?” I came up with many possible solutions, but decided to continue to build positive math talk within the classroom but also take it a step further and provide education for parents in mathematics. Not the same education that their child is learning, but rather education on the power of their words and their perspective on their child's development in mathematics. By focusing on these two factors, I have noticed immense improvement in my students personal mathematics identity, but also developing a growth mindset.
Within the Classroom
At the beginning of the year, I quickly learned that students need several supports to guide them in the right direction. I wanted my students to hit the ground running and begin with some fantastic mathematical explanations. What I quickly realized was that my students were not quite ready for what I was expecting them to do. For example, when I asked my students to provide an explanation or a noticing of a pattern, they were stumped on what to say. When I pushed them to explain more, they were very confused. As many teachers have experienced, I needed to do some ground work before I could expect them to communicate like mathematicians. As a trusted colleague once told me, you must go slow to go fast. I then took a step back and gave my students the support that they needed to be successful. To support math talk, I provided sentence stems to structure their thinking. For example, when I want them to predict or make a noticing, I ask them to respond starting with “I notice” or “I predict”. As they became more comfortable and began using these on their own, I pushed them a little more and added the “why”. As I modeled, my third graders began to understand that mathematicians don’t just notice or predict, but they give good evidence for why they came to that conclusion. Moving forward, when I wanted students to model their thinking, I had to first model what I expected of them. Throughout all of this, my high expectations remained. I firmly believe that high expectations breed high results. If my students knew that I would take work that was not their best, they would not strive to do better. To prepare students to be successful on their own, I made sure that my expectations were clear. For example, when asking students to show their thinking with words, pictures and an equation, I modeled what this would look like. If a student were to bring me their work missing one of these items, I politely ask them to return to their seat and make sure that they are meeting my expectations. I will be honest with you, this happened quite a few times at the beginning of the year, but after winter break I noticed I had to do this less and less. It was in those moments that I knew that my hard work was paying off.
Providing support for my students is important to their success. Another support that I find more important than academics is our classroom community. When a child feels safe and secure within our classroom, they are more likely to take risks, have a growth mindset, and be vulnerable when they are struggling. In my classroom we have a marble jar. We use our marble jar to capture three things. The three ways that we earn marbles are when 1) We are kind, 2) We take a risk, and 3) We are vulnerable. I do not believe in a behavior management system that rewards my expectations. I expect my students to be ready to learn, to do their assignments, and to show their work. When a student goes above and beyond, or exceeds my expectations, I will provide praise. Just like everything else, I cannot expect students to take risks and be vulnerable without modeling this first. As a child, I remember thinking that my teachers knew everything in the world, and now that I am a teacher I know that is not true. Within my classroom, I tell my students when I make a mistake, take risks and am vulnerable. It is through this communication that they realize that taking a risk is not a bad thing. By modeling this behavior in myself, my students feel safe and secure because their teacher has made mistakes and has taken risks as well. When they portray these behaviors, I target and praise them. For example, If I were to call on two students to answer or explain a question. One student may give me the correct answer and explain how they got it. I would thank them for their work and move on because this student met my expectations. However, if the other student began their response with “I am a little confused, but I am going to take a risk” and attempted to explain their work, even if it was not correct, I would praise this behavior. I want my students to try and take risks. Math is much more than numbers and correct answers. It is investigating, noticing patterns, problem solving and analysing the reasonableness of your work and so much more. I want my students to question, investigate, fail, and try again. The only way to do this is to target and praise that behavior. As the school year has progressed, I have witnessed the fruits of my labor. My students are willing to take the risk and fail, and they do this because I have built a safe space for them within our classroom.
Positive math talk does not remain in the classroom, but rather follows the child wherever they go. At home I want my students to continue to take risks, to be vulnerable and have a positive math identity. Once a child leaves my classroom, I cannot control what they are experiencing, but what I can do is inform the parents of the hard work that we are doing and provide explanations as to why it is important. As I have witnessed in my own family, my parents are confused and afraid of this “new math” that my younger sisters are working on. As a teacher and a daughter, I have had many conversations with my mother to explain to her that the math is the same, but the way we are approaching it is quite different. We are no longer dictating how students tackle math, but rather providing them with all of the tools and allowing them to choose which strategy makes the most sense to them. This is the same exact message that I have communicated to the parents of my students over and over again. Due to this, I have noticed several parents changing their narrative when it comes to math and as a result, their child’s math identity has blossomed.
In addition to open communication and education for parents, I have altered the math homework that I send home. At the start of the year I was not quite ready to give my attention to homework and so I just copied and sent home the pages that my curriculum provided. What I quickly noticed was that only a handful of students were returning their work. When I dug deeper, I learned that several students were confused and their parents did not know how to help them. I then asked myself “What is the purpose here?”. Research has proven that pages and pages of math homework does not help a child progress, or even understand the content more. If sending math pages home was not working, and instead was hurting my students math identities, then why would I continue to do it. After this realization, I shifted my ideas on homework towards fluency practice. At my school we are lucky to have a math curriculum that has a constant rotation of math games that reinforce the skills that we are teaching. My students love to play these games with their peers and are very familiar with the rules. Why not send these games home for students to play with their parents and practice the skills they are learning in the classroom! So I did it, and the results were amazing. After creating math game folders for each of my students, they took them home and began playing them with their parents. Each week, my students were expected to play three games of their choice from the folder, tell me who won each game and identify efficient strategies that they used while playing. What was originally a handful of students turning in their homework shifted to practically my entire class. Many parents told me that they loved the change in homework. My students were eager to tell me that they beat their parents, and I was thrilled to see that they were getting the skill practice without even realizing that they were doing “Math homework”. By providing simple and fun math games, my students grew much more than if they had to complete a math page. As for my parents, they are able to have positive math conversations while witnessing the skills that their child has been working on and may need practice with. As a teacher, I couldn't imagine anything better.
In conclusion, I know that I do not have all the answers on how to foster the identity of my students. What I do have is experience on what works and what does not. As I continue my career beyond this first year, I will learn more and more. I will continue to communicate the importance of positive math talk with my students' parents, send math games as homework, provide several supports within my classroom, have high expectations, and create a safe space for my students to take risks. Just as my students take risks and try new things, I will do the same. As Robert Jean Meehan once said:
“Teachers who love teaching, teach children to love learning.”
That is my purpose, and I will strive to do that each and every day.
Morgan Denyes is a teacher in Kent City, Michigan
Cartoon Avatars and Storyboards Give Teachers the Chance to Explore Their Practice
By: Brenna Dugan (firstname.lastname@example.org)
The average classroom teacher makes more than 1,500 decisions each day.
In the field of teacher education, there is growing interest in creating opportunities for teachers to examine and improve these daily decision-making processes, since they are the building blocks of an instructor’s teaching practice. This is usually done by studying practices of others and observing their own, then attempting them in a new way. The overall goal with this type of professional development is to create opportunities for educators to apply effective practices in their classrooms with each decision they make throughout a school day.
Dr. Pat Herbst and Dr. Amanda Milewski, along with University of Michigan GRIP Lab, have taken this concept a step further, knowing that technology can support engagement with practice-based pedagogies. For 15 years, Herbst and his associates have been refining digital tools to study teaching and support teacher growth as they reflect on, and experiment with, teaching practices.
With the goal of advancing mathematics education in particular, Herbst and Milewski developed a process called StoryCircles to build on and honor what teaching practitioners already know. The StoryCircles process allows instructors to represent classroom scenarios by creating storyboards with cartoon avatars. Teachers use the cartoons to script, visualize, and discuss mathematics lessons in a shared digital format. The frame-by-frame storyboards they create through StoryCircles make it possible to show interactive details and event timelines in a classroom. This can include the physical locations of events taking place in a room as well as the language they use in class.
The StoryCircles process relies on a cast of cartoon characters to “play out” what happened or could happen in a classroom scenario. The granularity of the storyboarding process—that is, the ability to lay out a classroom event scene by scene—allows for valuable specificity. “Distinct from a lesson plan, where teachers say what they would do but cannot show it happening,” Milewski says, “the StoryCircles process allows teachers to demonstrate exactly how they would do something. The granularity of the storyboard isn’t so detailed, though, that it would make an instructor feel exposed.”
Usually, discussions of teaching practices privilege more abstract descriptions of practice or overly detailed video recordings of practice. Herbst, Milewski, and their associates have found that the combination of visual and verbal representations in storyboards can accurately expose tacit or nonverbal issues for the sake of shared discussion, whether the conversation is about actual happenings or predicted ones. The creation of the storyboards—and the group conversation that follows—leads to a larger discussion about the varied approaches to teaching a given topic. In all cases, a StoryCircles group creates a common artifact through this social process. Laid out on a storyboard, a scenario is easily analyzed by teachers who can consider multiple pedagogical moves and outcomes.
Herbst and Milewski note that the digital format of the storyboards allows practitioners to share knowledge with one another and helps them to focus specifically on incremental changes. Few other teacher professional development processes take into account these incremental improvements, tending to focus on larger predicaments of practice that make practical changes difficult.
American practitioners rarely experience professional development in which teachers discuss the meaning of a piece of student work or consider the actions that could be taken in class. “StoryCircles tries to break this isolation,” says Herbst, noting that the storyboards represent knowledge that can be “shared, challenged, bolstered, refuted, and possibly also learned.” He believes that, by increasing the chances for colleagues to summon what they know about instruction while they are together, they can expand their communal knowledge. With practitioners at the center of this model, their professional knowledge and shared learning is elevated. “We value what they know as opposed to valuing what they can learn from us,” adds Herbst.
A mathematics consultant for Macomb Intermediate School District, Deborah Ferry asserts that these lessons are applicable across a range of scenarios. “No matter what their situation is, urban or rural, teachers share many of the same struggles and need the support of others to find solutions,” says Ferry. “They learn new ways to engage their students with the concepts. The StoryCircles process provides them with a safe environment.” Ferry also guided facilitators to collaborate with Milewski in engaging teachers who were implementing EMATHS curricula to represent EMATHS lessons using StoryCircles.
Current support from a James S. McDonnell Foundation grant is enabling the study of the ways in which StoryCircles supports teacher learning. StoryCircles particularly triumphs at blending professional development efforts from outside the classroom with teachers’ own experience and perspectives. It brings forth a middle-ground approach in which teachers actively experiment with new practices as they scrutinize and share their own classroom practices.
This story was adapted from its original version inMichigan Education magazine, available at soe.umich.edu/magazine.
Rumbling with Vulnerability
By Andrew Smith
Kent ISD Math Specialist
At a time where there is so much uncertainty in our world, many are being asked to perform in extraordinary ways. Nurses are being asked to risk their own safety and health in caring for patients diagnosed with Covid-19, restaurant employees are either without a job or transforming to curbside servers and teachers are being asked to teach in a way that our system has never been asked to with overnight notice. During this time of change, we can experience a variety of emotions and we can also choose to lean into the pause and find joy.
Over the last month, we have heard from two teachers, Holly Vostad and Kelly Compher, from Godwin Heights Public Schools. In their blog posts, they shared narratives of the positive changes in practice that have occurred in their classrooms over the past year. Although they varied in years of service, grade levels taught and major/minor areas of study, what emerged as common themes in both classrooms were an eagerness to professionally learn and a willingness to be vulnerable.
“What teachers are doing in classes with students on a daily basis has the greatest potential to influence the academic outcome of students, and the more challenged students are in social capital terms, the more true this is (Darling-Hammond, 2000; Hattie, 2009; Marzano, Pickering & Pollock, 2001; Nye, Konstantanopooulos, & Hedges, 2004).” As Holly and Kelly prepared for the 2019-2020 school year, they both embarked upon eight full days of Add+Vantage Math Recovery Training. Holly and Kelly were both thankful for the opportunity to increase their content knowledge around mathematics and also wondered how to transfer this new knowledge into their classrooms with students. “True learning (as a permanent change in thinking or behavior) happens when the learner is an active participant in constructing knowledge and in constantly thinking about how new information confirms or challenges previously existing thinking (Dack & Katz, 2013, p. 27).” Holly and Kelly were both rumbling with the question, “What might this change look and sound like with our students?”
This school year, I had the opportunity to walk alongside Holly, Kelly, about 20 of their colleagues and the students of North and West Godwin Elementary Schools as a mathematics specialist. Every Monday, teachers would open up their classrooms for observations, modeling, co-teaching, co-planning and Cognitive Coaching cycles. Amidst all of the initiatives and number of items on both of their plates, Kelly and Holly were both able to make the learning around mathematics priorities in their classrooms. In our Mondays together, we were able to learn lots about each other, about the teaching and learning of mathematics and about the students in their classrooms. We planned lessons, stumbled through facilitation and also had those “lightbulb” moments that all teachers dream about.
During this pause of Covid-19, there has been plenty of time to reflect on Mondays at Godwin Heights Public Schools. As I reflect upon the time spent there, what emerges most are not the strategies learned or the content we were able to explore, but the vulnerability and leadership of Kelly and Holly. “The definition of vulnerability as the emotion that we experience during times of uncertainty, risk and emotional exposure (Brown, 2018, p. 18)” and both Holly and Kelly were able to model this for and with their students on a weekly basis as they tried new opening routines, altered the structure of their math blocks and critically examined how their actions and beliefs aligned or misaligned from their identities, values and beliefs. The change of practices that happened in these classrooms impacted students and it occurred because Holly and Kelly were willing to rumble with the unknown.
As you prepare to transition into the summer, what might it look and sound like to examine your own identity, values, beliefs, actions and behaviors? How might you be modeling vulnerability for your colleagues and your students? Who might you invite into your learning space(s) to walk alongside you as you rumble with new learning? What might actions and behaviors look and sound like to create positive change in your classroom, building or district?
Andrew Smith is a Math Specialist for Kent ISD.
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