Most of the time I find it pretty useless to talk about kids with their old teachers before the school year starts. It’s not that the teachers don’t have important things to say. They do, I just have no idea who they’re talking about. Let’s talk in a week!

Yesterday, though, I had a really nice chat about my class roster with a colleague. She’s teaching kids I’ve taught; I’m teaching a lot of her kids from last year. At first we fumbled for something useful to say, and we landed on the usual: this kid’s great, this kid loves to gab, this kid can fall through the cracks, etc.

The breakthrough was when we narrowed things down. Who do I need to form a relationship with first? She had some ideas, and then I shared mine.

My class roster has three names marked. I will get to know them first.

A few years ago, some math teachers were discussing a book on twitter. The book had made the case that the existence of “learning styles” for kids is a myth. To some of the teachers in this discussion, this was very surprising.

Shouldn’t it be? We’ve all seen kids that seem stuck on an activity…until we present the material in some new way. Note-taking leads to learning for some kids, but not others. Other kids seem to lose track of an explanation halfway through, but thrive when given a chance to read it instead. And we’ve also all taught kids who seem to think through movement — these are kids who seem to be intellectually confined when physically constrained.

This is a tricky question, and I’m familiar with the anti-learning styles studies.

I don’t think the big issue with learning styles is that there’s no evidence for it, though. On its own, that’s only a bit troubling to me. Instead, I think there’s a risk that our learning styles work will go against our efforts to promote a growth mindset about intelligence.

The growth mindset literature encourages us to help kids see intelligence as plastic; their smarts can grow with effort. We would never tell kids that they just aren’t smart enough to understand a verbal explanation. How much different is it to tell them they’re a “visual learner,” and therefore less likely to understand that same verbal explanation?

This all comes from the best of intentions, of course. We want to make sure that different kids get their different needs met. But we have to be very careful not to do this in a way that encourages kids to identify with what they’re naturally better or worse at. We need to give that individual help in a way that sends the message that through hard work and the aid of teachers, learning will happen.

As I was discussing this on twitter, another educator mentioned that we also risk lowering expectations for students, either implicitly or explicitly, when we start designing tasks that seem to avoid areas where they’re perceived to be weak. That’s rough for a kid, and probably not what’s best for the class.

That’s my case against learning styles.

This is an area of my 3rd/4th Grade teaching that I want to do a much better job with this year.

Why does this stress me out?

When I first started teaching 3rd/4th, I was at a very fluffy moment in my teaching. I was just going to make sure that students conceptually understood multiplication, and fluency would just happen when it happened.

This worked for some kids, but I remember a number of kids that made very little progress in their multiplication fluency over the course of our time together. At the same time, I was seeing the benefits of the conceptual work because they could find a strategy for tackling bigger multiplication (like 31 x 12) but it would TAKE FOREVER. Because of the fluency.

Around this time I also got hired to do some research reading on multiplication fact fluency. I’m not going to do the citation thing, because no one cares, but that experience along with my teaching helped me clarify my views about how learning multiplication facts happens.

What’s the difference between conceptual understanding, fluency and having facts memorized?

There are lots of terms that we try to use. There are people who have “standard” definitions of all these things. Here is my current understanding of how to promote fluency, and how it relates to practice, timed stuff and conceptual understanding:

• Consider a multiplication problem like 6 x 7. There are a lot of strategies that kids could use to figure out this answer. If a kid can find out the answer using any mathematically valid strategy, then I’m happy to say that this student has conceptual understanding of multiplication. By this I mean the multiplication operation, roughly “what multiplication means.”
• To find 6 x 7 a kid might count by 7s, or by 6s. They might find 5 x 7 and then add a 7. They might find 3 x 7 and double that. And so on. All of these are informal strategies for finding 6 x 7. I have a lot of informal strategies that I use for mental multiplication in my own mathematical life. For example, 12 x 7 I can figure out by doubling 6 x 7. It’s not that I know 12 x 7, it’s that I can quickly figure it out using informal strategies. This is what I think a lot of us mean by “fluency” with multiplication — the ability to efficiently use informal strategies to derive an answer.
• On the other hand, I basically never have to use informal strategies for single-digit multiplication because I’m the kind of kid that had no trouble committing these facts to memory in my elementary years. TERC Investigations calls this “just knowing” what 6 x 7 is, and this is what a lot of us call “having the facts memorized.”

This isn’t news to any of you, but I just wanted to clarify my terms before I used them a bunch.

But how do all of these things — conceptual understanding of multiplication, fluency with informal strategies, and having facts memorized — fit together in kids’ learning?

How do multiplication facts get memorized?

Let’s start with having facts memorized. From experience and my reading, I think the way you come to have facts memorized is by holding a fact in your head and trying to remember it. This is sort of like if I told you my middle name (I’m not gonna) and then it’s rattling around in your memory and a few minutes later you had to remember my name and you’re like….uh, oh yeah, it’s ______. Those moments, piled together, eventually commit my middle name to your long-term memory. It’s memorized.

I’m saying a lot of things and this isn’t really “writing” as much as barfing thoughts. But this is an important point to me: to have facts memorized, you need to try to remember them.

These moments where you’re trying to remember a fact that you’re holding in your head can occur either artificially or naturally. Artificially, through practice. But a lot of practice isn’t actually designed to create moments where you’ve got a fact in your head and you’re trying to remember it. Consider a test that just gives a child a multiplication problem to solve. A student could complete this task by just using some strategy to derive the fact, and they haven’t practiced remembering the fact at all.  Or consider those ubiquitous Mad Minute things. They can’t possibly help you practice remembering a multiplication that you don’t already have pretty available to you through fluent derivation with informal strategies.

A lot of harm has been done to kids in the service of creating remembering practice for kids. So why not avoid them entirely?

Kids can get these opportunities to remember facts in more natural settings. Here is a way that this can happen:

• Kids have a conceptual understanding of the multiplication operation.
• You then teach kids a bunch of informal mental strategies for deriving multiplications.
• Kids get fluent at these informal strategies.
• As kids do other multiplication work, they frequently find themselves deriving multiplication through these informal strategies. Then, later in the problem, they have to remember what they derived. This creates remembering practice for kids.
• And if they don’t remember it? They just rederive the multiplication using an informal strategy, and then they’re like “oh yeah!” Another remembering opportunity.
• BUT THE KEY THING IS THAT IT’S NOT THE DERIVING ITSELF THAT LEADS TO THEM GETTING REMEMBERING PRACTICE AND THUS COMMITTING THEM TO MEMORY.

The above, I think, is how some writers, PD people and academics envision how kids could naturally come to have their multiplication facts memorized.

…but the natural approach wasn’t working for enough of my students. So I’ve adopted a more balanced approach that mixes the natural and artificial pathways towards getting remembering practice.

What do I think would be best for kids in my context?

In my 3rd and 4th Grade classes, I never want to ask kids to do something that isn’t meaningful to them. At the same time, I worry a lot about how numeracy impacts the math kids feel confident with in my high school classes. For this reason, I’m not satisfied with just fluency with informal strategies for my elementary students — I also want to help them come to have these multiplications memorized.

My ideal approach goes something like this:

• Conceptual Understanding: Make sure kids have some strategy for solving a single-digit multiplication problem, either on paper or in their heads. I want them to have conceptual understanding of the multiplication operations.
• Using Number Talks to Develop Informal Strategies: Figure out some small group of multiplication facts, and make sure that my kids become fluent in using informal strategies to derive these multiplications.
• Create Practice Cards: I give students index cards and ask them to write that small group of multiplication facts on those cards. I ask them to mix those in with a bunch of cards they’ve mastered. (Flash card key #1: keep the success rate high.) Write informal strategy hints (e.g. “double 3 x 7”) at the bottom of the card, if they want. (Flash card key #2: it’s your deck, you do what makes sense.)
• Practice with Cards: Practice your deck, alone or with a friend. Go through the whole deck at least twice. (Flash card key #3: try to keep it low stress).
• Follow-Up With Quizzes to Assess: Figure out if we’re ready for a new informal strategy, a new set of facts, a new type of talk, a new set of cards to add to our decks, or whatever.

I was doing this in 4th Grade last year, but I was unsystematic and sloppy. I think I know how to do this so that kids don’t get stressed out and that it’s enjoyable and fun. Honest to god the kids loved the cards and often asked if we could work on them. The kids liked them so much that I started putting non-multiplication stuff into their decks. My sales pitch for this sort of practice — some things that we figure out are worth trying to remember — applied to lots of things. By the end of the year I was asking kids to put in a few division problems, some fraction addition and subtraction and multiplication into their decks.

What is wrong with this approach?

I am a sloppy, unsystematic teacher. Partly this is an artifact of my teaching context — I teach 4 grade levels in 4 different rooms, don’t have much wall space, don’t have a consistent stock of supplies in my rooms, etc. — but I know I can do better.

Part of how my sloppiness expressed itself was that practice with cards was whenever-I-remembered. I think it would be better if I scheduled this in to the week somehow.

The bigger issue was that I don’t have a good plan for how to break up the multiplication facts. I’m sure there are resources that I could use here, but what I need is something like Set 1: multiplication involving 2s and stuff you can figure out by using doubling strategies on those 2 facts. Set 2: multiplication by 10s(?) Set 3: Using halving and multiplication by 10s to figure out other stuff, etc. I want to make sure that the informal work with strategies is structured over the course of the school year to cover all the facts. I want the strategy work to precede the remembering practice work. Do you know of a resource for this? I’d be interested in it!

I think those are the major issues that I want to focus on this year.

What do you think?

This isn’t Michael’s Take On Multiplication. The above was “Michael’s response to a tweet.” This is a rough shot at describing how I’ve been handling this in my thinking and my classes lately.

The seed of social reconstructionism in education that Counts had planted in the mid- and late 1920s flowered. By the early 1930s, his sentiments were being echoed by educational leaders throughout the country. Even some of those who had championed the child-centered movement, like Kilpatrick, were drawn wholeheartedly into the new orbit. Somehow the long unemployment lines and the soup kitchens dampened the spirit of optimism that had earlier prevailed, not only about the future of capitalism, but with respect to romantic ideas about the natural development of the child in the school setting. Both social efficiency–fitting the individual into the right niche in the existing social order–and developmentalism, with its emphasis on freedom and individuality for children and adolescents, gave ground to the feeling that the schools had to address ongoing social and economic problems by raising up a new generation critically attuned to the defects of the social system and prepared to do something about it.

Kliebard, The Struggle for the American Curriculum p. 157

There is a lot of talk about open curricular materials, but I hear significantly less talk about open curricular frameworks.

Consider this free-but-sadly-not-open 7th Grade curriculum from the charter network Match. I am sure that it’s going to be helpful for some people, and I expect that it will be helpful for me in my 8th Grade class at certain points of the year. But what would be REALLY helpful is if I could just delete all the content and links from that curriculum and use it to share curricular resources with the other high school geometry teachers at my school.

Or consider Julie Wright’s Math and Social Justice site. It’s great, but there are limits to the way you can organize curricular materials on Google Sites — links pile up, it’s hard to tag things by grade, and there’s the danger that with further accumulation of resources things get harder and harder to find.

Imagine if there was something like the Mathalicious framework that Julie could use to organize all those resources. That would make it much easier to collectively develop open curricular resources online, even without any of the baked-in collaborative technology that others have imagined.

A dream I have: that just as there is a team of organizers for the Global Math Department and Twitter Math Camp there could be a team of benevolent web programmers who work on projects of communal importance. They would work on developing the web with the needs of math (fine and other) teachers in mind.

I lack any of the programming capacity to manage such a group, but I did just guzzle a cup of coffee and in my caffeine high let me say that if you’re interested in being part of such a team drop a comment or let me know in some other way.

A writer is, mostly, a professional amateur. – Ta-Nehisi Coates

Public amateurs can have exceptional social value, not least because they dare to question experts who want to remain unquestioned simply by virtue of accredited expertise; public amateurs don’t take “Trust me, I know what I’m doing” as an adequate self-justification. But perhaps the greatest contribution public amateurs make to society arises from their insistence — it’s a kind of compulsion for them — on putting together ideas and experiences that the atomizing, specializing forces of our culture try to keep in neatly demarcated compartments. – Alan Jacobs

The point is not to replace specialists, but to open the hermetic quarters of specialized knowledge to public forms of interrogation. So it is almost an anarchist position: people should be entitled to learn what they need to learn and to contribute to the decisions that affect them. It’s a question of cognitive sovereignty. These positions and methods: amateur research, self experimentation, collective experimentation, unregulated discourse, exposure of interest and transparency, collaboration with non-conforming scientists or experts, are meant to invade specialties with questions of value, questions that most specialization is designed to eliminate. Since art is a realm where values are debated, one of the points of the artist as public amateur is to perform learning publicly to bring the question of value to the production of knowledge. – Claire Pentecost

It doesn’t take long for a conversation between teachers to include something sarcastic about the fad du jour. By being sarcastic, we put up an umbrella to try protect our sanity from the ideas raining on us from administrators, academics, and yes, even colleagues. I will go further, and boldly say to the proponents of the current pedagogical panacea: I’m sorry, but whatever “evidence-based” product you’re selling today, I’m not buying. The research it is based on is flawed. The anecdotes that support it only apply to specific circumstances which are not easy to replicate. In short, as I have written before:nothing works. – Henri Picciotto

Looking for ways to expand your career while staying in the classroom is a real trick. What I’m wondering is if part of the mismatch has to do with a tension between amateurism and professionalism. A teacher is a professional amateur and every other role in education calls for expertise. Teacher-researcher, teacher-speaker, teacher-leader, all of these are amateur-expert pairings, and maybe for that reason they make a mismatch.

But if a writer is a professional amateur, and if that’s what teachers are too, well?

Goodnight nobody, goodnight mush.