The Time I Dropped Out of Math Class in the Fourth Grade

I confused social convention for human brilliance.

This is probably overly confessional for a newsletter, but a memory crashed back into my head last week that’s extremely pedagogically formative for me and also relevant to our current moment where teachers are working with students with a really wide range of understanding from the last year of remote teaching.

I was homeschooled from kindergarten through eighth grade with a five-hour break in the fourth grade where I begged to be enrolled in the neighborhood public school and then begged to be unenrolled.

My mom and I visited my new teacher after school. Everything checked out. The classroom looked like a classroom, not a dining room, and I was kind of excited about that part maybe. The teacher gave me the day’s homework to take home and complete.

I got home and I got very stuck and then I didn’t go back to school.

I still remember the assignment, decades later. We were supposed to cut out plastic straws and tape them to a sheet in the shape of scalene, isosceles, and equilateral triangles.

These were names I had never seen before, through no one’s fault except the arbitrary ordering of mathematical ideas in two different curricula, but that was that.

What is (mathematical) truth but a lie agreed upon?

So much of mathematics rests on the foundation of social convention. Why is the horizontal axis labeled “x” instead of “m.” Why do we plot the horizontal coordinate first. Why do we evaluate arithmetic operations in the order we do.

You can’t take the square root of a negative number until you can. The angles of a triangle always add up to 180° until they don’t. Polygons always have an integer number of sides until they don’t.

It seems very easy, as a teacher, to ask questions that ensure only students who understand a social convention look mathematically smart—like wearing glasses that filter out every color but one. 

It seems very easy, as a student, to try to answer those questions, run aground on the social convention, and assume that I am not mathematically smart.

Some #educatorgoals then:

  1. Help students understand that their question “Why is it that way? Why couldn’t it have been another way” is an important and necessary one and has been, in fact, a primary engine for the advancement of the field of mathematics throughout history.

  2. Help students understand other conventions were possible. “Yeah, we could have decided the first coordinate was the vertical coordinate. That would have been just fine. We had to decide on something though and you and I weren’t in the room.”

  3. Ask the questions that gave rise to the need for a social convention.

We developed a coordinate system to describe location precisely, so consider giving students the opportunity to try to describe location precisely without one.

We developed names for triangles because certain triangles exhibited certain interesting characteristics we wanted to group together. So consider asking students to look at a bunch of triangles and group them in ways they find interesting.

We created social conventions to serve our needs, not the other way around. We created social conventions—scalene, isosceles, etc.—to facilitate our brilliance. But social conventions aren’t a substitute for human brilliance, nor should we ever let social conventions undermine a student’s appreciation for human brilliance or their sense of themselves as brilliant.

➕ Updates

Great recommendations from Mylène DiPenta.

🎁 What Else?

📬 Mailbag

In my last newsletter, I asked you to share, “​​How did your journey with math and numbers begin? Where does your enthusiasm, your passion come from?” Some fantastic responses follow.

Linda L.

My HS Calculus teacher (1971-1972 school year) turned math class on its head for me. Each day he posed an interesting problem and put us to work in groups to grapple with it, followed by a lively whole-class discussion. I absolutely loved his class - first time I wasn't bored stiff in a math class. I decided to major in math and later to become a math teacher, because I fell in love with math that year and wanted to do for other kids what he did for me. Almost 50 years later and newly retired, I hope I had some success at that.

50 years! The impact education has on macro-level factors like GDP and life expectancy is questionable, but it’s undeniable that a single teacher in a single year can have a massive impact on the course of a kid’s life.

Sarah N. loves math precisely because of the opportunities it gave her to challenge social convention.

As a young child I would "sneak" my mom's puzzle books, erase her answers, and then redo the puzzles. I have always enjoyed a challenging puzzle, and I think that just flowed over into math. I'm fascinated by the idea of math concepts growing and changing. Such as the idea that we "can't" divide by zero... why not?  It was once believed that we "can't" take the square root of a negative, yet we do! I like the creative and rebellious side of mathematics.

Ashley T describes #familygoals for me.

My mom was (and still is) a math teacher and so math was a natural part of my daily routine (we actually used to do mental math at the dinner table before eating together) so relationships with numbers began early. But as I grew, I actually began to struggle at random times in math throughout my high school and my college career as a math major. Along the way there were always specific teachers encouraging me, supporting and growing me. Realizing this, and how important persistence was to a math mindset, my passion for both teaching and social emotional learning grew. As a teacher that is what I founded our classroom culture on and now it has greatly impacted my work as a math specialist. My mom, sister (also a math teacher), and I have a group text message that is almost like our family PLC where we chat math and teaching and I just love how we can all be so enthusiastic and bond together over what we do.

Mark V’s love for math and numbers evolved and hopped across careers.

I remember in 4th grade crying in the classroom when I raced to take my answers to the teacher to get them checked and having to return, more than once, to redo them. So much for timed math tasks. She didn't do that much after that. I remember competing against John Wambaugh and Barry Moat to see who could do better at the work given us or on tests. We were good at it. But, competing to get some kind of prize, such as a grade, isn't what gave me a love for math. I think the first real eye opener came in college when in Elementary Analysis we derived the Fournier Series, and for the first time, I made the connection how the strain you could measure at an oil well at the surface related to what happens downhole at the pump.  Basically, it is differential equations in action. I was an engineer 20 years, then came to my passion, teaching.

Back-to-School Resource Round-Up

Hometown teachers here in Oakland, CA, started teaching again this week, kicking off a school year that promises to be at least 2% less challenging than the last. I’m over here at Desmos, worried about all of them, gnawing on my paw, trying to stay level by reading interesting people reflecting on the start of their school year.

In case you don’t have the same time or inclination to refresh twitter dot com as often as I do, here are some of those interesting back to school reflections and resources.

Fawn Nguyen wrote a note to new teachers.

You can measure the students’ enthusiasm for your class quantitatively: time how fast they arrive at your door the next day. :) What teachers do is really hard, and I hope you’ll reach out to people who have your back. Always make family and yourself a priority. The students and the lessons should bring you joy. If they don’t, I hope you’ve saved up to start a vineyard. 

Howie Hua posted a thread of back-to-school activities.

Sarah Carter is one of the craftiest math teachers I follow and was probably more bummed than most by the turn to remote learning last year. She’s clearly back in her element now, thinking about a yearly “Statistics Scrapbook” and back-to-school experiences that are tuned up for in-person learning.

The first week is one of my absolute favorite parts of the school year. I love crafting a fun and engaging experience for students that usually involves a little math and problem solving, too!

Chanea Bond wrote What I Wish My White Colleagues Knew, which includes lots of useful advice for White teachers working in multi-racial departments this next school year.

Alex Shevrin Venet invites us to reframe the common icebreaker question, “What do you wish your teacher knew?” to “What do you wish your teacher would do?”

When we ask students in the first days or weeks of school, “What do you wish your teacher knew?” we’re essentially asking for a disclosure. The question itself implies a secret. It acknowledges that there are things that are hidden between student and teacher, things that aren’t shared for some reason or another. The question asks students to vault over those barriers and share anyway.

Sarah Strong turned her Dear Math icebreaker into a Desmos activity as well.

What Else

Mailbag

Author Nihshanka Debroy reached out to interview me about my path in math education and as I answered I found myself pretty curious how the thoughtful folks who subscribe to this newsletter would answer as well. Here’s one of his questions:

How did your journey with math and numbers begin? Where does your enthusiasm, your passion come from?

Hit reply, send me your answer, and I’ll send you mine.

Updates

Chanea Bond reached out on Twitter to mention that teachers in predominantly white institutions should also benefit from her post.

Surviving this next school year with three infinite resources.

When it comes to student learning this next year, imagine that students missed an episode of a good T.V. show.

Teachers are apprehensive about teaching this next school year, and rightly so. Even setting aside concerns about infection and reinfection, it is unclear to many teachers what it looks like to reboot student learning after this last school year which—for worse and occasionally for better—was unlike any previous year.

The good news is that this next school year will be, in several crucial ways, exactly like any other year.

More good news is that while many of the most helpful resources for teachers are finite and nonrenewable, teachers have several resources at their disposal that are both infinite and infinitely renewable.

Time is a resource that is very helpful but finite and nonrenewable, for example. Many teachers had less instructional time with students last year and many of those teachers will try to teach grade-level standards this next year in spite of time lost in the previous grade. 

I asked teachers in my last newsletter how they were planning for that task, and teachers came through, suggesting several infinite and renewable resources, each of which can make teaching easier next year.

Infinite Resource #1: Your understanding of this challenge.

Last year, student learning conditions shifted categorically—from in-person to remote, synchronous to asynchronous. That required categorically new solutions.

But if you’re all back in the classroom this next year, learning in the same room at the same time, it’s worth wondering if your challenges are in a different category from other years or just at a different point on a continuum.

Michael Pershan takes the latter perspective and it puts him more at ease:

When I taught at a school with fairly strict tracking for a place of its size, I would always end up with a few of the lowest track classes in my schedule. These were extraordinarily difficult classes to teach, for just a whole host of reasons: their expectation of failure, the challenging behavior, the unaddressed learning accommodations, and shaky understanding of the previous curriculum.

The case I'd make for why we shouldn't freak out about learning loss is because from the perspective of a classroom teacher, we already are handling situations like it all the time.

Last year’s categorical shift asked teachers to devise new technological and pedagogical innovations. This year’s continuous shift asks you to do the same work of differentiating your instruction as always, just differentiating across a wider band of student experiences than you have in past years.

Infinite Resource #2: Your affinity for students.

Teachers who believe that a positive and strong relationship with students is a medium for social, emotional, and mathematical learning will have an easier year than teachers who believe the work of teaching is only to support mathematical learning.

Here’s Jodi Donald in the comments:

What helps my students "catch up" is not so much about the math but my commitment to establish rapport between the students and me and create a safe learning space where kids can talk math, learn from each other, and use mistakes as learning opportunities.

Additionally, your belief that students have lots of mental images and early ideas that you can recruit to help them learn math is going to be even more helpful this year than it has been in previous years. It will require more effort to start with abstract and formal knowledge (like formulas and procedures that students didn’t experience last year) than to start with the sensory and intuitive ideas students were developing all throughout the last school year and then use those ideas to help them develop abstract and formal knowledge.

For example, starting with “Which is steeper?” offers students an entry point and offers you many more resources for instruction than starting with “Calculate the slope.”

Infinite Resource #3: Your understanding of math.

A teacher who believes that math comprises a lot of small ideas—each one carrying the same mathematical load as any other, each one an essential prerequisite for some other idea—will experience a lot of unnecessary burden this next year. If math is a house of cards, then every individual card needs your full attention.

However, teachers who understand that math is about a small number of very large ideas will find it much easier to help students connect their previous learning to learning they may have missed last year. In particular, many teachers should be happy to have a) standards that treat math as a story, b) an understanding of the progressions of large ideas across grades, c) a curriculum that emphasizes different storylines in math proportional to their size and importance.

When it comes to student learning this next year, imagine that students missed an episode of a good T.V. show.

This hypothetical T.V. show is a good T.V. show, which means that the characters are well-developed. They behave in ways that are interesting, dynamic, sometimes surprising, but ultimately true to their nature. 

This also means the story makes sense from one episode to the next. There is an “A” plotline that runs through the entire season. Then “B” plotlines that run across smaller arcs of episodes, and “C” plotlines that often start and end in a single episode. The T.V. show has spent much more time on “A” plotlines than “C” plotlines.

The characters and story are so well-developed that someone can miss an episode, jump into the next, and learn everything they need to know—especially for the “A” plot—from a short “Previously on Math Class” preface to the next episode.

For example, one of the big plotlines in our middle school math curriculum is the different ways we can represent equivalent ratios, which eventually gives rise to a new storyline with linear relationships.

In the Grade 6 episode of our curriculum, a student generates equivalent ratios in Pizza Maker.

In the Grade 7 episode, the student sees what those equivalent and nonequivalent ratios look like on a graph in DinoPops.

In the Grade 8 episode, the student sees graphs that are linear as well as proportional in Turtle Time Trials.

If a student missed any of those episodes, teachers can open up a screen from that lesson and offer a short “Previously in Math Class” preface that catches students up on the main characters and their “A” plotlines from the missed episode. That’s only possible if we understand math as a story and understand whether a particular plotline is an A, B, or C plotline.

I won’t trivialize the challenges of this next year. Teachers and students deserve many resources they may not have, time and political leadership perhaps chief among them. Those resources are finite and nonrenewable.

But beliefs are ideas you can cultivate without respect to time. I have developed some of my ideas about math, students, and learning much slower than my peers, and other ideas much faster. You can develop them on your own in a room while reflecting on your day, while reading something like this, while chatting with your colleagues.

Those beliefs are infinite and infinitely renewable, and each one will support your work during this next school year, a year that, in many important ways, will be just like any other.

🎁 What Else

  • Achievement for Good is a new project that seeks to create assessments that are culturally affirming, particularly of Black and Latino students. Their leadership team and advisory board make it a project to watch IMO.

  • RIP Robert (Bob) Moses.

Correction: Bad Graph

Numbers and graphs are hard even for numbers and graphs professionals.

I posted a graph of Desmos usage in my last newsletter that was incorrect. It expressed curriculum usage as a fraction of lessons run divided by total possible lessons that could have been run, and it’s come to my attention that my calculation of total possible lessons was too large by an amount I’m still sorting out. Please accept my apology for a careless error with numbers and graphs.

While you’re here, I hope you’ll take a look at our commenters who are writing thoughtfully about their approaches to “opportunity loss.” Leeanne Branham describes a common theme of “weaving” old and new ideas together that we’ll return to.

Having used the [Desmos] curriculum for the first time myself this summer with students who had struggled during the school year, the readiness checks were a great guide to what previous topics needed to be woven in. The curriculum is so inviting that there was 100% zero need to teach a unit of missed material. They could all find a way into that first lesson.

Learning Loss vs. Opportunity Loss

🛠️ Correction

I posted a graph of Desmos usage in this newsletter that was incorrect. It expressed curriculum usage as a fraction of lessons run divided by total possible lessons that could have been run, and it’s come to my attention that my calculation of total possible lessons was too large by an amount I’m still sorting out. Please accept my apology for a careless error with numbers and graphs.


At this time last year, teachers were making plans for a full year of remote teaching rather than a single emergency quarter. Now, those same teachers are making plans for how they’ll teach in the wake of that year of remote teaching. 

All kinds of debris bob around in that wake right now, all of it complicating the work of teaching. Students may have experienced the death of family or friends. They may have experienced health crises of their own. They may have experienced poverty in a new or deeper way. 

Every teacher I talk to is very stressed about that debris from their year of remote teaching. They are particularly stressed about “learning loss,” something which they are frequently told (by people who I observe are often not classroom teachers) does not exist.

In one sense, the critics of “learning loss” are correct here. Learning is the sort of thing that can become inert and not easily used, but we can take deliberate measures to reconstitute that learning later. So “learning loss” isn’t a particularly accurate description of cognition. Neither does the term do justice to all the learning students gained while at home.

But my opinion is that those critics are working much harder to decry the term “learning loss” than they are working to name and address a real phenomenon that teachers do have every reason to worry about.

Students did lose something last year. We can argue over the value of that thing and its value relative to other things they gained. But every anecdotal source I have tells me that teachers were able to teach many fewer lessons this year than last year.

Yet many school districts are still asking teachers to teach next year as though the students they’re receiving had the same opportunity to learn as they would have in years that weren’t consumed by a world-historic pandemic.

Even if “learning loss” is a misnomer, teachers have every reason to be anxious about that “opportunity loss.” Students lost a lot of opportunities to learn math this last year and teachers would likely benefit from some concrete, specific, actionable suggestions for their preparation.

So let’s offer and trade some ideas together.

How should teachers teach ideas this year knowing that many of their students will not have experienced related ideas from the previous year?

Smash the reply button or leave a comment. In a future email, I’ll share some of your answers, some answers from my colleagues at Desmos, and some of my own.

Leave a comment

🎁 What Else

  • I am obsessed with the question Jenna Laib asked in a recent post, “Do students need to finish every problem?” One key to teaching with problems is knowing when the class has had sufficient experience with the problem to benefit from a conversation about it, which is generally earlier than when the entire class has finished the problem.

  • If you’re a parent of small kids, check out this thread of children’s math books. I impulse bought a bunch on Amazon and my kids are loving Baby Goes to Market and Countablock, in particular.

  • What’s your favorite line from Math Person? High schooler Julia Schanen’s ode to mathematics is by turns heartbreaking and actual-lol-level funny. Personally, “gnawing on a bone” put me on the floor.

  • “Two lines are 2 apart.” Love an interesting tweet-sized problem.

  • Press the “Anonymize” button in Desmos and your students’ names will change to names of mathematicians. A small team at Desmos recently made large changes to the names of mathematicians that appear. Check out the what and the why.

Loading more posts…