Stop Thinking About Math Education Politically

A colleague passed along an article from The Economist, "America's Maths Wars" (https://www.economist.com/united-states/2021/11/06/americas-maths-wars) that discussed how teaching math in the U.S. has become a political issue: "Conservatives typically campaign for classical maths: a focus on algorithms (a set of rules to be followed), memorising (of times tables and algorithmic processes) and teacher-led instruction. ... Progressives typically favour a conceptual approach to maths based on problem-solving and gaining number-sense, with less emphasis on algorithms and memorizing."

The more I read and think about how we learn math, the more annoyed I get by articles like this (and there have been many!), and those that espouse the "best practices" of one side or the other.  The argument that learning math is about the basics versus about conceptual thinking is an old and dusty one, and at the moment, I believe neither side is valid.

People argue that math, like reading, is fundamental, so we need to make sure every student learns it. I don’t disagree with this, but how we teach reading is also hotly debated (https://www.washingtonpost.com/education/2019/03/27/case-why-both-sides-reading-wars-debate-are-wrong-proposed-solution/). I wonder if the perceived importance of math and reading inclines people to think of them as part of the zero-sum "pool of power" and less thoughtful about how to teach them?

I think that math (and perhaps reading?) should be taught more like we teach art, music, or creative writing. In these subjects, we encourage students to get messy and draw/play/write what they know, provide tips and critique on their work, teach basic principles of structure, suggest edits based on historical/cultural norms (hopefully a variety of them), and how to break those norms effectively. And many students find joy in these subjects, and many students who don’t pursue them as careers still continue to draw, play, or write into adulthood.

Teaching math as SOMETHING YOU MUST KNOW AND HERE’S HOW TO LEARN IT misses the point. It loses sight of the fact that there is joy in math, and helping students to get messy and use what they know, providing feedback, and teaching the history and basic structures of the subject are all important parts of the learning process.

The “conservative” idea about “teach the basics” is good: learning a basic addition algorithm is good, but needs to be connected to students’ experiences instead of valued as an isolated skill. If we only focus on the “basics” we suck the joy out of and ignore the students’ connections to the subject.

Likewise, the “progressive” idea about “teach concepts” is good: seeing how the basic addition algorithm and all sorts of methods work are good, but need to be connected to students’ experiences and thinking instead of scatter-shotted without providing focus. If we only focus on the “concepts” we ignore any procedural fluency.

Art, music, and creative writing teachers know how to teach the basics of their fields and provide feedback that values the student’s contribution. They teach some basic techniques and encourage their students to practice those and to indulge their curiosity and imagination. They critique work (including their own), identifying good and interesting ideas and suggesting changes to make the work better. They recognize that not all students will pursue careers in these fields, but help students appreciate and find joy in the work.

The back-and-forth arguments about "best practices" for math education miss the mark, in that they both make assumptions about how ALL students learn math. I learned many years ago that successful teachers build up a repertoire of strategies that they can match to their students, content, and context.* It's not about one way being right and the other wrong. Teaching math is about figuring out when to teach the algorithm and when to let students explore; who needs clear and actionable feedback and who needs to just know if the work is right or wrong; and how to demonstrate perseverance and express joy in the messy, creative, and sometimes straightforward work of doing math.

I agree that math is something you should know, but ignoring the students’ and our own ideas and experiences in order to adhere to a “conservative” or “progressive” curriculum is not the way to help students know it. Do we need to teach basic techniques and have students practice them? Of course. Do we need to encourage students to indulge their curiosity and imagination? Certainly. Let’s also learn to critique their work (and our own) effectively, to identify the good and interesting ideas, and to suggest ways to make improvements. We need to help students appreciate and find joy in math, even if it’s not part of their career plans.


(*I was lucky enough to learn from Jon Saphier and his book The Successful Teacher about repertoire and matching many years ago. I still refer to the book for guidance when I start to feel frustrated about some aspect of my teaching.)

It's Been Another Rough Week

What's a math teacher to do?

The headlines this week have been about anti-vaxxers, climate deniers, partisan fighting, and a missing white woman.  So many of the problems stem from a lack of critical thinking (about information, about race, about consequences ...) and critical thinking is what I'm supposed to be teaching!  Except I'm not in the classroom this year, and the stuff I have typically taught has been so far removed from the lived experiences of my students, that I've been finding it difficult to even think about how I can have any impact.  And while I've felt disconnected in the past, being a middle-aged, cis-gendered, white male, I really don't like this feeling of powerlessness.

I have been trying to keep up with reading articles and books and attending webinars about issues and practices around diversity and inclusion and about promoting student agency through instruction and assessment. At the moment, I'm feeling really overwhelmed by the depth and breadth of the information available and the apparent lack of effectiveness my learning is having on the world around me.  

With so much information out there, why is it that states and school boards still think it's okay to bury uncomfortable truths?  How come law enforcement can rally to find a missing white woman in a state where hundreds of indigenous women are unaccounted for?  And what can be the motivation to avoid providing vaccines and other aid to communities and even countries who are struggling (often because of policies and actions of more powerful groups)?

Even as a middle-aged, cis-gendered, white male, brought up in a religious household, I know the answer to those questions.  Too many people who look like me are too comfortable with their own power to look beyond themselves.  And aside from trying to be aware of the impact I have on others and learning from my mistakes, I'm not sure at the moment what I can do.

My response today is to write about this.  And try to figure out what else I can do.

Games and Education

A couple weeks ago, I attended almost every session at the GENeration Analog tabletop games and education conference, sponsored by Analog Game Studies.  The first day focused on board games and the second day on role-playing games (the tabletop kind, not the video game kind).

So many good speakers and interesting topics, and I ended up doubling my articles/website reading list, and following more folks on Twitter.  I want to capture some of my thoughts about the conference before other shiny objects distract my attention more than they already have.

Here are some take-aways I want to remember:

• Jorge Moya-Higueras presented some research on gamification versus playing games that indicated a lower engagement rate for gamification, possibly because of the extrinsic rewards of earning badges and points, versus the intrinsic reward of enjoyment that simply playing games produces.
• I want to pick up a copy of Critical Play by Mary Flanagan.  Her talk on "Values and 'Enculturation' in Tabletop Game" was really intriguing. 
• Steven Dashiell's talk on "Gamer Stores and Gilded Doors: Narrative analysis of minority gamers' experiences at analog game spaces" included a couple ideas I need to think more about.  In particular, he pointed out that understanding the rules is not the same thing as understanding the culture, and that immersion requires socialization.  It makes me think about how I "find my tribe" in role-playing and board games, because of the shared language I typically experience outside of the game, like Monty Python jokes and references to books like Lord of the Rings or TV shows like Firefly.  Dashiell talked about his experiences as a kid enjoying RPGs at his local game store, but not feeling part of the crowd because he did not understand that shared language.  This reminds me that I need to be cognizant in all spaces about how I help or hinder others from being welcome; signaling my own geek culture is okay, but do I also value other cultures present around me?  Also, just as Dashiell spent time on his own as a kid going back to watch and read what the others were talking about, it is up to me to do the same when I am in spaces where I am an outsider.  It also reminds me of the importance of spaces created specifically for marginalized populations, and makes me appreciate people who are willing to share those spaces with wider audiences so that me, as a white, straight, cis-gendered, mostly neuro-typical male can get a glimpse inside the thinking, emotions, and relationships of people who are different from me.  I am grateful for people like Tanya DePass, who hosts Into the Motherlands, an actual-play game, on her YouTube channel for opportunities like this.
• The keynote presentation from B. Dave Walters on "Diversity and Inclusion" was outstanding!  I loved his quote "Storytelling is sacred.  It makes us human."  And I appreciated his advice to "tell the stories you want to see in the world."  These are definitely ideas I want to remember when I return to the classroom and tell the story of math, and when I write stories and adventures.

There were other talks that included information on specific games or research about using games in classrooms.  I especially enjoyed the presentations by Catherine Croft of Catlilli Games, Scott Nicholson on game design, and Rebecca Y. Bayeck on Historical African Board Games.  I definitely plan to attend GENeration Analog next year, hopefully in person!

Test Questions

 There's been lots of online discussion and articles lately about holding students accountable, while still showing grace and compassion, especially during the pandemic and as we move out of it.  My school has chosen to eliminate semester exams entirely, even after we are back in school full time next year.  We're also moving to block scheduling.

I've been thinking about alternatives to testing, because what I really want my students to learn is much more than some math facts and procedures they can quickly produce on a timed test.  So I'm wondering if portfolios might be the way to go, for at least part of the grade.  But what do I want my kids to learn, and what goes into a portfolio?

First, I do want them to know some math when they leave my class.  It would be horribly unfair of me to send them on to the next level without making sure they can perform some of those procedures.  (And I'm also thinking about how I can put some of those into a historical or cultural context; students will be far more likely to buy into procedural stuff if they understand its context.  I'm not talking about "making math useful" - I find the useful math becomes really boring really quickly, especially since we can always whip out our pocket computing devices and find answers.)  The other things I want my students to be comfortable and confident with are perseverance in the face of uncertainty and imagination in using their backgrounds and skills.  These last two would be really hard to show or evaluate on a timed test.

Here's what I would want to see in a portfolio problem:

• A significant problem.  Not just an exercise of a math procedure, but something the student had struggled with, revised, and thought about.
• Evidence of their problem solving: persistence, creativity, use of prior knowledge, verifying their results, finding further questions.
• Evidence of metacognition: What were their main struggles?  What are the key learnings they take from this problem?
• Evidence that they understand some of the important math of the class.  For this, I'm thinking that each portfolio problem must address a different topic, and the work must demonstrate an understanding of that topic.

I would want to have each student solve and write about maybe three portfolio problems per quarter?  I want the students to have opportunities to get feedback and make revisions.  But how do I describe the bullet points above on a rubric that the students and I (and their parents) understand?  What's the criteria for success?  And is it even possible for me to provide quality feedback on so many portfolios in a timely manner?

And more questions ... By using portfolios, am I actually grading a student's writing abilities?  What about students who have trouble keeping their work organized?  Might there be an opportunity for an "interview" version where I have the students talk to me about their work?  I'll need to give some sort of homework or some way for the students to practice the skills and procedures; should I assign points to that as well?  And should there still be small quizzes to check progress and give the students feedback on that aspect of the work?

I'm glad I have extra time to think about this, rather than turning it in at the end of the period.

Thank You, and Things to Think About

Last week, I finished teaching several Exploding Dots lessons to a group of seventh and eighth graders.  I was remote, and their classes were hybrid; their teacher, Ms. Anna, and a few of the students were in the classroom with masks, and the rest of the students at home.  Most of the kids did not have their cameras on, and several were active in the chat.  When I asked them to think about a problem, I could hear nothing because everyone was muted, but I could see the students in the classroom having some animated discussions.  I was really glad to be talking to students again, and recognized that while my experience this year is different (and much less difficult) than Anna's or other teachers, I still learned a few things.

First, the students were amazing!  I'm not surprised by this, but I think I was starting to forget how exciting it is to be with kids.  I was new to their class and only coming once a week, so the novelty of the experience probably had something to do with their engagement.  Even so, learning their names and a few of their faces, and getting to hear some of their thinking was the highlight of my weeks with them.

I was also reminded how difficult teaching remotely really is.  I missed seeing the students who were not on camera, and I wasn't able to see what was on anyone's papers and whiteboards, or hearing their discussions.  Some colleagues have shared stories about their remote and hybrid experiences, and I know my few times in this classroom are nothing compared to what they've been going through.  Many kudos to Anna, Quinton, Sara, and Matt, along with all the teachers this year; you are all incredible professionals making a very difficult situation work.

Building relationships with students takes practice, patience, and perseverance, and is much more necessary and difficult in a remote situation.  So thank you to James Tanton, who trained a group of us on Exploding Dots, for demonstrating how building relationships can look, even when you can't see or hear some of the students.  I was gratified that one of the students would say "Hello, Mr. Peter!" the minute he joined the zoom call, another would make sure to appear on camera at the beginning to say "Hi!" and others would share their favorite movies, or how they were feeling when we did a quick ice-breaker question at the beginning of class.

It is likely that because I wasn't grading the students and I was teaching them something outside the ordinary curriculum, they showed a great willingness to participate.  Which has me thinking again about how I use grades and feedback in my own classroom, and what ideas I choose to teach and focus on.  The first few lessons of Exploding Dots are more about seeing something familiar in a new light and learning how mathematics works, than about one more procedure or fact.  Along with that, it brings some humanness and history to math, which I supplemented with things I've found at the Library.  I think the most important bit is that it also encourages students to play with math ideas and connect some things they already know in new ways.

During one lesson, we played the game NIM, and the students quickly figured out that there must be some sort of trick that allowed me to win the game.  In their breakout rooms, they played the game against each other and discussed strategy.  One student, thinking about her next move, said, "Wait while I look into the future."  By the end of the lesson, some students had figured out some strategy, and they were excited to show me what else they learned about the game the next time we met.  So, how do I incorporate games in my classes on a regular basis?

In the last lessons, I showed the students how Exploding Dots related to multiplying polynomials, which they were working on during the days I wasn't with them.  I took a poll in one class about how well they were understanding what I was talking about, and found very mixed results.  So I put three different problems, of varying difficulty on screen, and asked the students to solve whichever one looked challenging but doable.  On camera, the students in the classroom raced for their whiteboards, and after a few minutes of working, I asked the students if they were ready to discuss their thinking.  Almost everyone, including those off camera gave a "wait" sign, and I realized that the students who started with the problem that looked the easiest were trying out the other problems as well.  By the end, everyone felt comfortable with the middle level problem, and a bunch more were satisfied with the harder version.  How will I build this kind of differentiated activity into my lessons when I return to the classroom?

One of the best moments for me was when a student who was mostly quiet and off camera responded in the chat with "I'm not sure about this, but I think ...".  His thinking was excellent, even though he was unsure where to go with it.  I asked him if he would mind turning on his mic while I asked him some questions.  He agreed, and as he explained and clarified his own conceptions, other students started typing ideas in the chat.  For me, it was an amazing example of trust and flexibility, and both Anna and I praised his participation, pointing to all the ideas he expressed and helped other students generate.  And this happened with lots of students as we went along.  So, what was it that Anna and I were doing that helped the students be willing to publicly wrestle with new ideas?

I want that kind of productive struggle and student engagement in my classroom all the time, and I need to think about how to bring that to life even more.  In the meantime, a big "Thank You!" to Ms. Anna and all the students in 701, 801, and 802!  You were wonderful to work with, and have given me lots to think about!

Use Your Imagination!

“Imagination is more important than knowledge. For knowledge is limited to all we now know and understand, while imagination embraces the entire world, and all there ever will be to know and understand.” ~ Albert Einstein

I recently had a conversation with colleagues about the need for some families to have their kids accelerated in math.  The idea that a district would de-track their math classes was met online with cries of "socialism" and "liberal equity agenda!"  The less extreme messages were about "providing the best possible experience to allow my child to excel" or "making sure my child gets into a good college."

Whether these are knee-jerk political reactions, NIMBYism, or honest feelings of wanting the best for our students, I think they all display a fundamental misunderstanding about the nature of mathematics.  Unfortunately, the topics and ways we teach and the ways we assess math contribute to this misunderstanding.  Too often, math is seen as a set of skills to be checked-off.  The more you can check off the list, the better college you can get into.  Or, if you're feeling math-phobic, what's the minimum you have to check off to graduate?

But mathematics is far richer than a set of skills or a method for tracking success.  There is lots to be noticed, and lots to be wondered about.  Whether you're in grade 5 math, 9th grade Algebra 1, or 12th grade Calculus, there are so many interesting, useful, beautiful, amazing, and curious paths to explore that everyone can find something to enjoy, and no one should be bored.  True mathematics takes imagination, and every answer can prompt a new question.  Many of the skills are important, but mathematical thinking goes far beyond being able to memorize multiplication tables or solve equations.  And imagination is not built or nurtured by remaining in homogeneous groups with the same set of classmates on the same track, whether it's the "higher" or the "lower" track.  To be good mathematicians, to be good thinkers, to be good people, we need to surround ourselves with thoughts and ideas that are not always like our own.  Detracking is not an attempt to "dumb it down", but a way of expanding everyone's understanding and imagination.  And people with imagination are never bored.

Here's a math problem to get your imagination going: Roll six standard dice.  Find a way to separate them into two groups so each group has the same sum, if possible.*  Can you predict when it's possible and when it's not?  Can you create an algorithm to solve this problem?  What if you used only four dice, or used seven dice, or eight dice, or N dice?  What if you roll seven dice and have to discard one; can you predict which die to discard?  Back to the original problem, what's the probability that the six dice can be separated into equal groups?  Can a set of dice be separated into three equal groups and how many dice would you need to guarantee this is possible?  Can you create a game out of this, and would different colored dice or differently sized dice add some new twistsand questions?  There are a lot of skills built into this problem, from addition and division to conditional probability, to computer science.  With some imagination, math can be a far richer experience for everyone, and having students who think about it differently all in the same class will only enrich the experience for everyone. 

If you want to see more examples of imaginative math check out James Tanton (@jamestanton) or Numberphile (@numberphile) or Sunil Singh (@mathgarden).  And there are many others!

*I first saw the dice problem here: https://twitter.com/davidporas/status/1386378256476254221

Choose Your Open Middle Adventure

 A couple weeks ago, I participated in a workshop with Robert Kaplinsky.  He was talking about "Open Middle" problems.  These were basically a typical math problem, like 8-3x=12, but with blanks instead of numbers.  The questions come in three parts that ask the students to fill in each blank with a number between one and nine, with some condition attached.  Conditions range from "so that the solution is equal to zero" to "so that the solution is a small as possible" with other conditions between.  Check out his website for more details, https://robertkaplinsky.com/  He can explain it far better than I can.

I'm thinking about these type of problems, because they remind me a bit of those old "Choose Your Own Adventure" books.  The reader takes on the role of the protagonist and starts with the first chapter, but at the end of that, you, the character, are offered a choice of actions.  Depending on which choice you pick, the book tells which chapter to read next.  There's another choice at the end of that chapter, and so on until you find the treasure, or rescue the prince, or die a horrible death at the bottom of a spiked pit.  The great thing was, even if you reached the horrible end of the story, you could back up and try again.  I read some of those books over and over, making different choices each time, just to see what would happen.  (And isn't that what problem solving involves?)

The Open Middle problems are like that because once you get past the first part, the problem opens up with lots of choices in the middle.  Students can engage at a number of levels by guessing and checking, making a table, thinking about a general rule, or whatever approach they think of.  Then, the students get to tell their part of the story in class!  My job as the teacher is to see which paths everyone is taking and get a conversation going about those choices.  Kind of like when my brother read the same adventure book I did, and we talked about the different choices we made and the endings we found.  The other nice thing about Kaplinsky's problems are that it's okay if students don't get to the third part.  It's designed to be a harder question, with multiple ways to solve, but unlike the middle, does not have many correct answers.  (At least for the ones I've seen so far.)  I'm thinking that after the students work on the first two parts, and discuss their methods,  I would help solidify the math idea, and assign the third part for homework.  I certainly wouldn't grade it on correctness, as some of these problems are hard.  But I would try to provide feedback to the students about whatever they write or tell me about their process.  

I'm looking forward to digging into Kaplinsky's work some more and trying these problems in class.  It probably won't go well the first time, but the fun thing is, I can make a different choice and try again the next day.  Math is nowhere near as dangerous as that spiked pit.

Choose Your Own Adventure Books
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My Work

 So much has happened in the last three months (not to mention the last two weeks!) and I have been struggling to sit down and write a blog post.  I'm not sure why I've not been able to write; I've had plenty of ideas.  And I've been writing other blog posts for the Library as well as journal articles and presentations.  And I've really enjoyed writing those, as well as bits of fiction for myself.  But sitting down to write here seemed like too much work.  At the same time, the Fellowship has been talking about being on "a national stage", which I am finding a bit difficult to deal with.  Even writing some twitter posts has started giving me a cramp in the neck.  I think I'm feeling pressure to write something amazing and profound with a catchy title and important point.

Since I'm enjoying writing some bits of fiction on the side, last week I decided to join the Storytelling Collective, and write something every day in February.  Since then, I've been in a bit of a panic about my commitment to write 500 words a day.  How can I possibly accomplish that when I can't even find time to take a daily walk?!  You'd think that with a pandemic going on and me sheltering at home that missing things like a daily commute would open up some extra time, but truthfully, it's often hard to get myself out of bed in the mornings.

Today, I'm meeting some other Fellows for an outdoor and socially distanced lunch, and I have to finally mail the thank you notes to family members who sent me Christmas gifts.  So I had to get up.  And I looked at my computer and thought about all the things I've been wanting to write here and felt that panic welling up.  Then I thought about my blog's title: Show The Work.  Sometimes, getting to the mailbox is the work.  Sometimes, getting up in the morning is the work.  Sometimes, just opening the computer and writing anything is the work.

So there you go.  This is my work right now.  And I'm going to continue to show it.