For students to develop a deeper understanding of statistics, they need to be actively involved in reasoning and communicating their conceptual understanding. Come experience several instructional strategies based on the AP Statistics Cours and Exam Description, including building the model solution, stand & talk, error analysis and quick write.
How many of our students truly enjoy math class? Is math class only enjoyable for students who perform well on it? Are you sure? In this session, we will explore some ideas in how to facilitate #MathPlay in the classroom creating a more enjoyable math experience for all our learners. Learn how effective #MathPlay can lead to deeper understanding and students taking ownership over their learning.
Centering the Standards for Mathematical Practice I’m a huge believer that students should use mathematics to understand, analyze, and critique the world around them. To this end, I believe that every student should take courses that teach linear and exponential growth, statistics, and basic geometry before graduating high school (more thoughts on statistical literacy coming in future writings). This content is the main focus of most math classes until right around Algebra 2, where the practicality of some topics starts to become much more debatable. I’m not sure I can rationally argue that ALL students need to be able to factor a polynomial, find the asymptotes on a graph of a rational expression, or solve a radical equation in order to be deemed college and career ready and using Algebra 2 as a gatekeeper to opportunity seems to be a capricious choice. More thoughts by Christoper Edley Jr. here.
Any time I wade into a conversation about the actual content students should be required to take before leaving high school, it inevitably turns to “When would a student ever use this math?” If you’ve been in math education long enough, you’ve probably been unable to avoid this type of query whether from students, parents, or fellow educators. Typically, when it comes to a topic that may have questionable practical value, I look for opportunities for students to work on their critical thinking, most times through examining the content by using the Standards for Mathematical Practice (SMPs). These standards help to clarify exactly what critical thinking skills that we, as math educators, have an opportunity to sharpen.
For a specific example of what this can look like, I will use a recently taught example of studying quadratic equations. I’m not sure the daily usefulness of possessing some of the related algebraic skills surrounding quadratics, so I tried to examine the content from the lens of the SMPs, specifically SMP #7: Look for and make use of structure and SMP #8: Look for and express regularity in repeated reasoning. I wanted my students to use their prior knowledge to explore some quadratic structures and draw some generalizations. We started by talking about minimizing. “Why is minimizing something a person would want to do?” Answers included saving time or money, using resources wisely, reducing pollution and waste, among others.
We took this conversation to then examine some expressions. The task was to find the minimum value for some given expressions, starting with (x–8)2. Students used Desmos, while I encouraged them to use a guess and check strategy to see if they could find the minimum value of the expression. After a few minutes, a few students shared their conclusion that 8 is the number that makes this expression have a minimum. We discussed why this was, landing on the idea that any time an expression is squared, the minimum value that could be produced would be 0 because all other values would be positive, and therefore larger. We tried a second round with (x+5)2, then moved to several more examples of the form: (x–h)2.
Next, the wrinkle of a value outside of the parentheses was introduced: (x–2)2–1. Highlighting the structure again, some students asked questions about how a minimum value could be negative, given the conversation we just had about the minimum value of squaring an expression being 0. An opportune time to discuss the order of operations! Some students were still hung up on how the expressions were being evaluated because signed arithmetic was not their favorite topic. In reflection, next time I will offer more opportunities to work with the evaluation of an expression first, before asking for guessing and checking on the minimizing, but overall, students were able to, by the end of class, identify from an expression of the form (x–h)2+k what x-value would create a minimum value, and what that minimum value would be, but most importantly WHY that x-value created a minimum based on the structure of the expression.
I have taught vertex form for quadratics many times before, often with an approach from translating the graph, but this approach allowed me to center the standards for mathematical practice AND connect to something that students were able to see the value in, the need to minimize a value. In the future, I would definitely want to dive deeper into their conversations about what in society should be minimized, how we might represent those quantities, deciding if there is consensus on minimalizing these things, and how this might impact their lives before diving into the abstract mathematics. I also would revise some of my approach in the math itself, and here is a Desmos activity that I created based on my reflections from class.
What are some topics you have upcoming that might benefit from an SMP first model? How can you create a structure that will help students focus on the conceptual thinking versus the procedure? I’d love to hear your ideas!
Maps In the last GMD newsletter, @Idil_A_ introduced #GMDReflects as part of a reflective journey into our own teaching practices. If you haven’t read her piece, you can read it here. ??
I’d like to engage with #GMDReflects by thinking with and about maps. I have to admit, I’ve always taken maps for granted. To me, they’ve always just been the thing that hung in a classroom wall. They’re where different bodies of land and water are located and, in the case of Google Maps, they tell us how to get to places we want to go.
But in recent years, I’ve begun to go down these rabbit holes that have pushed my thinking about maps and the practice of mapping.
To put my conclusion up front: Maps are a big deal. Math/math ed twitter, let’s talk about maps more!
To be fair, lots of people have been talking about maps. I just wasn’t tuned into the conversation. Here are two important conversations that I missed the first time around.
Projections: Maybe some of us learned that because the Earth is spherical-ish but a map is flat, then you get different maps based on the types of projection you use. But I certainly didn’t learn that maps projections also bear political implications. Back in the day, Eric Gutstein talked about this in his middle school lessons on the Mercator projection, which is the traditional map in the U.S. and was developed during European colonial expansion. Under the Mercator projection, countries closer to the equator appear smaller, thereby diminishing their importance in the eyes of the mapmaker and map consumer. READ HERE
Mathematics for Spatial Justice: I heard about teaching math for social justice but didn’t know about teaching math for spatial justice. People like @Laurie_Rubel have done some amazing research in this area to combine teaching mathematics for social justice with critical perspectives on space and place informed by geography: READ HERE
Even deeper down the rabbit hole, there are people who ask: what even is a map? For example, this CityLab project shares homemade maps of people’s stories about how their lives have been shaped by the pandemic.
Maps aren’t just bodies of land and water or even a network of streets. They can be tools for reflection and personal meaning making. And how you design a map can say a lot about your values and your relationship to the world.
“Maps have done a lot to confuse things for people and I think more lands have been lost to native peoples probably through mapping than through physical conflict.”
Instead, he has created different kinds of maps that center indigenous voices and world views through the Zuni Map Art Project. His maps defy conventional notions of where things should be on a map, how maps should be scaled, and what maps are meant to do. The Counter Mapping video showcases some of his beautiful maps, so check it out.
———————— When I was teaching in NYC, I mostly biked to work. And as almost any bike commuter in any U.S. city knows, you think a lot about road infrastructure. I made this map to think about that time when biking and teaching were such big parts of my life they were practically intertwined.
Pedestrian and cycling paths in NYC (with some red highways). Data collected from the OpenStreetMap project and drawn in R using the WGS84 projection used in GPS satellite navigation. It would be great to do more maps, map making, and counter mapping in the classroom. Even beyond this, how many of us are asked to tell our own stories through maps? Until recently, I had never been to a PD or event where I was asked to make a personal map or to expand my mind about what a map could be. If there’s no space for map making in PD, let’s make a space for it on Twitter.
I’ll close with one more quote from Jim Enote:
“North does not have to be at the top…scale is unnecessary. What’s more important is these stories of the history described in these vignettes of experience.”
Culturally Sustaining Teaching – A Working Group, A Community When Math for America Los Angeles gets together for our monthly meetings, teachers break out into self-selected working groups for part of the day. This year (and last) our working group is titled Culturally Sustaining Teaching (CST). During our time together, facilitated by Sammie Marshall, we have tackled questions like “What can culturally sustaining teaching look like in secondary math classrooms?” For those of us who have been part of this group for multiple years it has become a place where we can come to figure out together what culturally sustaining teaching might mean for our particular students. As we’ve worked through a series of tasks, and shared stories, lessons, reflections, and concerns we’ve not only become better equipped to serve our students but we become a community. This year we’re hoping to share a bit of our community with the readers of the GMD Newsletter.
During our first meeting in September we welcomed new members, and the group was asked to share what they would like to get from our time together this year. Some responses included: “I want to continue to use this group to help me reflect on how I interact with my students;” “Strong examples of CST in the classroom where students actually have positive feedback about it (avoiding feel-good teacher centering);” “Address different aspects of social justice in education; explore ways to empower students to seek justice and fight for what they believe in;” “Learn what CST is;” and “Learn with trusted colleagues about how to make my class a more affirming space.” While thinking about Culturally Sustaining Teaching leads us to reflect on our teaching practice and think about how to bring about greater equity and justice in our classrooms, it’s not a list of teaching practices that one can implement and check off one by one. Instead, we spend our time in these working group meetings sharing our struggles, better understanding how our students might be experiencing our classrooms, thinking about things that we might want to try to do, and learning how to be resources for each other.
In our first meeting, we worked through a Desmos activity looking at four dimensions of Culturally Sustaining (math) Teaching. We have linked it here if readers would also like to see what we discussed and we encourage you to share your own thoughts!
In our most recent meeting, we focused on the theme of power and justice: how can we use STEM to understand oppression and combat it? Much of our conversation was around the use of “social justice lessons” to help students read and write the world with STEM (Gutstein, 2006). We split into two groups allowing teachers to choose their own adventure.
In the gallery walk, teachers reflected deeply about Harper’s findings. For example, Harper, in describing how racism was found to be relevant both to students of Color and white students, wrote, “teachers viewed TMSJ [Teaching Math for Social Justice] as a way for students of Color to use mathematics to understand, critically and deeply, the educational inequities impacting their lives,” (p. 283). One in our group responded, “I want to get to a place like this one day as a teacher,” but several expressed anxiety about racism coming up in curriculum. When reading Harper’s finding that some teachers avoided discussing race and racism, even when the data they examined clearly showed racism, one teacher reflected that planning was key: “breaking down the underlying SJ content (even if the lesson is not intentionally a SJ focused lesson) is just as important as planning for the [mathematics] content.” Another noted that “The path of least resistance is to avoid uncomfortable conversations and focus on content,” but by knowing that that’s a potential pitfall, we can plan for and hopefully avoid it. Several commented on how powerful SJ math could be, but also a sense that it is not to be waded into lightly.
The second group came together to think about what might be helpful to increase adoption of social justice mathematics lessons. There are a number of useful compendia out there already (Math and Social Justice Collaborative MTBoS site, RacialMath, Skew the Script). One of us observed that sometimes when these links get shared, our colleagues find these compedia overwhelming–there is just too much stuff to sift through. What might be more helpful are some personal recommendations from people that we trust. For example, what if in our departments or PLCs we kept a running list of social justice mathematics lessons that we personally tried? The list would include a name and contact information so that others in the group could contact that person and ask for specifics, or the list could include some personal reflections right after the lesson. These shorter lists of personalized recommendations might be more helpful than a massive list of resources, and hearing what went well and what didn’t could help us better prepare to teach it.