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Why Times Tables?

Lots of teachers disagree about whether students need to know their times tables. I’m teaching 7th grade for the first time and I’m thinking a lot about times tables, because some of my students aren’t fluent and I want to understand whether or not that matters.

Context: I’m teaching proportions right now. Here’s a problem I recently gave my students:

I think a reasonable person does this with a calculator. Sure you could do it by hand, but why? Real humans who want to avoid making a mistake definitely use calculators.

Here’s another problem:

Sure, this one is totally reasonable with mental math. But if it’s ok to allow a calculator for the problem above, why not for this one?

Here’s my take:

Times tables aren’t useful for solving problems, they’re useful for learning to solve problems.

Let’s look at the two problems again. The first problem I wouldn’t want to use when introducing a new concept. There’s too much going on. But the second problem is a great introduction to proportions. If students can see that each number on the right is three times the number on the left, they will have a better understanding of the patterns in the table and the idea of a proportional relationship. They could use a calculator to see that pattern — but all the effort involved in using a calculator saps valuable working memory that could focus on making other connections.

Ok so takeaway one: times tables maybe aren’t useful for solving problems, they’re useful for learning to solve problems. Here’s takeaway two: the traditional 12×12 times table is silly and arbitrary. What’s so special about 12? After teaching 7th grade for the last six weeks, here are the times tables I wish my students knew:

Ok the tens look a little awkward, but the idea is simple. I can’t use the same numbers in every problem, so I would love students to be fluent in a decent variety of multiplication facts. But I don’t care about 9×7, or 12×8, or 6×11. It’s been rare in the last few weeks of my class that those facts come up.

Final point. It’s not only about multiplication. If a students knows that 4×6=24, that’s awesome. But it is just as important that they know that 24 / 6 = 4. And one piece of knowledge that keeps coming up, and that I think is another distinct skill, is knowing that if 4 x something = 24, that something is 6. Each of these facts creates opportunities to make connections and better understand new ideas because of the working memory they make available while looking at a new problem.

To summarize: times tables are important, but it’s all about quality over quantity. I don’t need students to know every fact, I need them to know a smaller subset forwards, backwards, and sideways to free up space in working memory. And the point of freeing up that space isn’t to solve problems, it’s to notice other patterns while solving problems that help students learn new ideas.

I Think “Meet Students Where They Are” Is Sometimes Bad Advice

I think “meet students where they are” is sometimes bad advice.

My goal this year is to teach my 7th graders 7th grade math. Plenty of my students are “behind.” It would be easy to say, “well they aren’t ready for 7th grade math so we’ll start with something they are ready for.”

I’m trying all sorts of things to provide support. Adding scaffolding to help students access what we’re learning. Taking time each week to practice foundational skills that students will need soon. Giving students chances for retakes and lots of time on assessments to make success attainable. I’m constantly trying to find new ways to give more students access. What I won’t do is go backwards and say, “ok I have some students on a 3rd grade level so we’ll start by teaching 3rd grade standards.” That’s demoralizing, it lowers the bar for everyone, and I can fit in mini-lessons and chances for extra practice while working on grade-level math.

I don’t mean that I’m just going full steam ahead with a normal 7th grade math curriculum. I’ll have to cut a few less essential things from some units to have the time to do the important stuff well and give students support they need. Part of providing scaffolding is creating a bridge from where students are to where I want us to get. But I worry that the advice “meet them where they are” can lower expectations and lead to a never-ending spiral of remediation.

When I’m at my best we are making steady progress while also putting a few minutes each day toward review of essential foundational skills. I try to simultaneously scaffold students to access the next topic while identifying things they need to work on from previous years and building in time for mini-lessons and practice. I’m not perfect and I often fall short. But I do what I can to keep us moving forward. My goal isn’t to figure out all the 3rd grade standards students don’t know. It’s to figure out what I need to teach them to access 7th grade standards.

Observations in a New Job

(some mostly disconnected thoughts about transitioning schools)

Observation number one:

I’ve been a retake skeptic for a while. I’ve offered retakes at two different schools, a charter school and a private school. In both cases the students who did retakes were mostly the ones who needed it least, trying to increase their grade from a B to an A rather than finally understanding something they struggled with the first time around. There were exceptions, and I could hassle the students who would benefit more from retakes, but it didn’t work as well as I wished.

I still offered them — I’m not against retakes. I just didn’t see them as the most effective tool I had.

But in my new school tons of students are taking me up on retakes. It’s awesome.

There are three pieces I think might be contributing. First, there is a school-wide policy that requires teachers to grade formative assessments at least every two weeks and offer retakes for these, so students are familiar with the idea that they can retake at least some assessments. Second, I’ve been giving these formative assessments in Desmos and offering the retake as an extra screen that I unlock after they take it the first time around. Students can take them any time, anywhere, and resubmit via Google Classroom to let me know I should grade it. That workflow has kept reassessments simple and straightforward for me and students, and avoids the mess of having students ask for retakes and me chasing them around with the correct piece of paper. Finally, I’m setting aside time each week for students to use a digital practice tool. This time is both great and terrible, a topic for another post, but it has given students a bit of time they can also use for retakes, or check in with me for clarification, which has increased uptake.

I should add that I have a great department of thoughtful folks who I’ve learned from. I’ve inherited the ideas above in different forms from past practices, which has helped my transition. I can’t take all the credit! I’m learning here.

Second observation: when I talk with students about retakes, it’s clear that the largest motivating factor is grades. Students can check their grades online, and they want good grades. There’s a lot of talk about ungrading and similar ideas online right now. I agree with many of those folks that grading is flawed and harmful for many students. But I also have to share a dirty little secret: as a new teacher at a school, leaning on grades as a motivator is huge. It’s not ideal, but students already care.

I’m not going around saying “hey students make sure you retake so you get a good grade,” I’m trying to frame things around learning and all my nobler goals, but that’s not often why students are doing it. And maybe that’s fine. Because something else I’ve observed is that, while the complicated assessment systems that experienced teachers design can be great, they are almost impossible for new teachers to use well. I see ungrading as something that has a lot of potential in the right hands. But typical grading practices are often a good option for new teachers because there are so many other things teachers need to deal with. Weird grading and feedback systems are not at the top of my list of what makes the biggest difference while adjusting to a new school.

This whole thing has been a great lesson in how specific school culture can be. Right now I would say that things are going fine in the new job. A lot of things could be better, but plenty could be worse as well. And the knowledge that has served me best has mostly been about adapting to a new culture and a new student community, rather than bringing in my own prior experience. Of course my years teaching are important, but I’ve had to adjust over and over again, and a lot of my best choices have been doing what other people suggest.

Learning is Not Binary

I’m two weeks into a new teaching job, switching from high school to 7th grade math. Lots of things are different.

My students have a hard time with foundational skills we’ll need this year, particularly with fractions. I’ve been trying to figure out how to support them. I don’t want to wade into the learning loss debate, I find that whole thing pedantic and unhelpful. But I do have one relevant observation about learning that I think gets missed sometimes:

Learning is not binary. There is middle ground between knowing something and not knowing it. To offer an alternative conception, there is a continuum from recall — I know it without being prompted — to recognition — I can do it when I am reminded — to relearning — I knew it before and I can relearn it quickly.

It would be easy for me to say, wow most of these students don’t know how to multiply a fraction by a whole number hold the phones we’re doing fractions boot camp for the first few weeks of the year. Instead I could try teaching a ten minute mini-lesson and see how much students are able to do after that. My experience the last two weeks is that more comes back than I would have thought.

To me this is an example of deficit vs asset thinking. I’ve been trying to learn more about deficit and asset thinking for a while but I often find the distinction fuzzy and hard to pin down. In this case, deficit thinking means seeing that a student can’t do something and assuming they know nothing about it and need to be retaught from scratch, lowering expectations and taking time away from other things we could be learning. Asset thinking means seeking out strengths, even when they’re hidden below the surface, and building off of them.

Learning Distributions Redux

I still think sometimes about the ideas behind this old post of mine. I was wondering which distribution I should strive for in my teaching:

I’m still not sure I have a clear answer. But another layer I’m thinking about is how a linear measure of “student learning” oversimplifies how I want to support students. Content knowledge is one thing, but students who have struggled in math class in the past often have negative beliefs about themselves and about their learning. Those beliefs do a lot to undermine learning in the long term. If students show up to class believing that they’re bad at math and math class is pointless they’re likely to continue having a hard time.

One distribution to think about is which students learn the most — is it the students who have done well in math classes in the past, or students who have had a hard time? A second distribution is how I influence student beliefs and dispositions about their own learning. This is an area where I want to prioritize students who have struggled in the past.

I can try to facilitate those beliefs a few different ways. One is having a constant barometer of how successful students feel. If students feel like they are constantly getting things wrong they are likely to internalize that as a part of their identity in math class. If I can start class with tasks that are accessible to every student and scaffold success in challenging work I can help more students to develop positive beliefs. I want to have a barometer on success for all students but pay particular attention to students whose beliefs I want to influence the most.

A second strategy is assigning competence. When I ask a student to share a great idea or a new connection I am publicly signaling that student’s competence. It’s often only a few high-achieving students in a class who regularly have their ideas highlighted. I can keep a special eye out for high-quality work from students who often struggle. It’s important not to tokenize or compromise; students can see through me when I’m celebrating someone without good reason. But when students do great work I want to celebrate it, and I want to prioritize celebrating students who I know often struggle.

I’d love to think more about other dimensions of what I hope students take away from my class. It’s easy to generalize about a group of students — “they learned this” or “they didn’t learn that” — but there’s a wide range elided by those statements. I want to get better at thinking about and acting on that complexity in ways that support all students.

Never Do Anything a Kid Can Do

We must not fool ourselves, as for years I fooled myself, into thinking that guiding children to answers by carefully chosen leading questions is in any important respect different from just telling them the answers in the first place.

-John Holt, How Children Fail

I think “never say anything a kid can say” is bad advice. I realize that it’s a common refrain in some education circles and there was that popular NCTM article. But some of my worst teaching has come from a desire to get kids to say things so I don’t have to. It’s not a terrible idea in the abstract, but it often leads to games of “guess what’s in my head” that don’t lead anywhere. Having students talk in class is great. Having students talk so that they can say the magic words that I decide are important is a waste of time.

Here’s a try at better advice: never do any math a kid can do. Math class should be about students doing math. If they spend all their time watching me do math they’re not learning very much. I think that first advice, never say anything a kid can say, leads to students playing fill in the blanks bit by bit, which is a hollow shell of doing math. Instead I should ask myself, “what is the least I can do to set students up for success doing this math?” When I’m doing this problem, could they be doing it instead? If I try to make students do everything they won’t have the tools to access it they won’t do anything at all. But I want to be deliberate in putting all of the work on students that they can manage. Then I can have them talk with each other about the math they did. They’ll say lots of great stuff, they’ll do math, and we won’t spend any time playing “guess what’s in my head.”

Maybe for some people, “never say anything a kid can say” means “say as little as you can to set students up to do some math then have students talk about that math, wash, rinse, repeat.” That’s not what the linked article above says, and I don’t think that’s the most common interpretation of the phrase. But that’s the advice I wish I’d gotten years ago, and it’s advice that sets students up to learn math and not just to learn whatever I have in my head for them to guess.

Memory

I enjoy learning about the human mind and human memory. It’s a huge rabbit hole. A few times I’ve started and stopped writing a longer piece summarizing what I understand about memory. But each time I end up making it really complicated. Memory is complicated, but the lessons I take from it as a teacher are pretty simple. This is my attempt to distill those ideas into a few principles I use in the classroom.

  • Working memory is where thinking happens
  • Thinking about things puts them into long-term memory
  • We can only think about a few things at a time
  • Overloading working memory with too many things at once makes it less likely anything will end up in long-term memory
  • We make mistakes when working memory is overloaded, and also when we mistake two similar things
  • Long-term memory is what we think with
  • Long-term memory doesn’t have a limit
  • Having more stuff in long-term memory makes thinking easier
  • Long-term memory isn’t just facts and figures — it’s also skills, strategies, and feelings and literally everything else we think about and do
  • The more often we use something the more easily we can retrieve it from long-term memory, and this works best when we space out those retrievals
  • Stuff tends to go out the way it comes in: we remember things best in the context that we learned it
  • It’s hard to transfer knowledge from one context to another, but this becomes more likely when we have experience thinking about that knowledge in different ways and different contexts

12 bullets. Not simple, but not the most complex thing in the world either. I’m fascinated by memory and I’m sure I’ll keep trying to learn more about it. There’s lots more to memory than these bullets, but the more I read the more I come back to the same few ideas that feel relevant to teaching and learning.

Taking Hands

I have largely stopped asking students question and calling on a raised hand. And for the most part when I do it it’s because I didn’t plan that part of class very well. I’d be happy if I never took hands for questions I ask again.

One caveat — I’m not saying I don’t let students raise a hand to ask a question. I’m talking about students raising hands to answer questions I have.

My basic issue is that calling on raised hands warps everyone’s perception of learning. Let’s say a student gives the “right” answer. I have no idea what that means. I’m calling on someone who has volunteered their answer, so it’s a biased sample of the class and I have no idea what other students know or don’t know. Let’s say a student gives the “wrong” answer. I could interpret that as a sign that more students are confused but I don’t really know. What’s worse, giving a wrong answer in math class is pretty fraught socially. Now I need to take some time to help that student understand where the confusion came from. Maybe that’s helpful for other students but maybe not, I don’t know.

Here are a bunch of things I do instead of taking hands:

  • Give students a problem or several problems to solve, on their own, with a partner, or with a group. Circulate and pick out one or several students to share with the class.
  • Same situation, but pick out a common mistake to address. I can choose to share it anonymously or have a student share depending on where we are with classroom culture and how socially safe that feels.
  • Same situation, but just offer some summarizing thoughts myself and give students a chance to check their answers.
  • Have students study some worked examples with the full solution written out. Discuss with a partner what they notice, take questions, then have them try one on their own.
  • Spend less time modeling and doing “guided practice” up front. Instead, break instruction up into small chunks where I give explicit instruction and student try problems on their own, and debrief those problems using one of the strategies above.
  • Recognize when an activity isn’t doing well — usually because I underestimated the knowledge students need to be successful — and stop the activity entirely to change course rather than trying to rescue it with lots of questioning.

I could go on. But moral of the story is I find that classic of the math classroom, working through a problem at the board while asking students to raise hands to help me out along the way, a relic of my past teaching and something to avoid whenever I can.

What Gets Measured Gets Valued

One thing I’ve learned about teaching during the pandemic will stay with me: what gets measured gets valued.

Where I live, and in much of the rest of the country, the official criteria for exposure to Covid-19 are simple: if you were within 6 feet of someone for more than 15 minutes in the two days before they test positive for Covid-19, you are considered a close contact and have to go into quarantine.

But the coronavirus doesn’t care exactly how many feet apart you are or exactly how long two people are close together. Distance and time are two important variables in transmission of respiratory viruses, but they’re just two of many. It also matters whether people are wearing masks, the quality of those masks, how many people are present, how confined the space is, whether they are indoors or outdoors, the quality or lack of ventilation, whether the infected individual is coughing or sneezing, exactly how close people are and how long they are together beyond a binary 6 feet/15 minutes, and more.

But once we decide to measure something, that something becomes the focus above whatever we don’t measure. A video came out over the summer of a school board member in Georgia suggesting that their schools can be in compliance with that rule by having students change seats every 14 minutes.

That’s the example that got public attention, but those same conversations have been happening behind closed doors all year. The 6 feet/15 minutes rule has narrowed attention on those two variables, and has distracted from other factors worth thinking about.

The close contact rule takes a complex, messy, uncertain situation and tries to make it objective. And I understand why the rule is what it is! I understand that the rule needs to be simple to enforce. I understand that masks aren’t a factor so that indoor dining to continue with distancing to help keep restaurants afloat. But regardless of anyone’s good intentions, the rule causes people to believe that you can only get Covid-19 if you are within six feet of someone else.

That same thing happens in education. Here’s an example. Teachers often grade for completeness and take points off for lateness. We do so with the best intentions — we don’t have time to look through every answer of every assignment, and late work is a pain to manage. But when we focus on measuring completeness and late work, we send a message that completeness and late work are what we value.

This phenomenon is everywhere. It would be easy to tee off here on standardized tests. Right now the AP Calculus test is on my mind. I need to make sure students can answer those contrived Mean Value Theorem problems and have a ton of practice interpreting a function written as an integral of a piecewise function. But standardized testing is too easy of a target. What we measure becomes what we value in ways large and small, every time we try to take some complex situation and distill it into a simple metric. Learning is always complex and schools are full of incentives; every measurement distorts those incentives and creates new ways to be successful that don’t actually involve learning.

There’s no easy solution, but it’s important to be honest about the ways that measurement distorts values. When we pretend that measurements don’t influence what we intend to measure we amplify the distortions. Measurements are often mathematical, and math is often held up as a higher truth. Those measurements can be taken at face value when they shouldn’t. It’s important to strip away that veneer of objectivity and be honest about what measurements are actually measuring.

For All/There Exists

Two different statements come up often in mathematical proofs: “for all” and “there exists.” “For all” makes a statement that is true in every case. For instance, for all numbers divisible by both 2 and 3, they are also divisible by 6. “There exists” makes a statement that is true in at least one case, but need not be true in every case. For instance, there exist whole numbers not divisible by any whole number except for 1 and itself.

I think a similar distinction is useful when talking about teaching. There are some things in the “for all” category — things that should always be true, that I should strive to do every class, every day. Academic safety falls into that category. Every student should feel like they can take risks, share ideas, and be wrong, all with unconditional support. I should strive never to compromise on academic safety. I fall short on this all the time, but I need to set “for all” as my goal and work to help every student feel safe every day.

There are other things that fall in the “there exists” category — things that students should experience but don’t need to happen every day. Learning through discovery falls here for me. I think every student should experience mathematical discovery. And it’s hard to get discovery right, so this can’t be a one-off every few months. At the same time, I don’t believe students need to discover something themselves to understand it, or need to experience mathematical discovery every class.

There are other examples. Practice is a “for all” — I value practice, and I want students to practice every mathematical concept they encounter. Representation is “there exists” — I can’t show every student a mathematician who looks like them every day, but I can strive to share mathematicians who represent each of my students several times over the course of our time together.

Discourse around teaching can get lost when we confuse “for all” with “there exists.” I need to hold myself to a high standard around academic safety, every day and for every student. But it would be easy to get defensive and say, “but this other student feels safe, so x student should feel safe too!” Teachers face a constant onslaught of decisions and information; I have to avoid cherry-picking examples to fit my narrative. And it’s easy to make the opposite mistake, to take something that should exist somewhere and assume it has to exist everywhere. The value I place on discovery doesn’t mean that every lesson has to be a discovery lesson, and doing so risks losing sight of my true goals. One of the biggest challenges of teaching is how many decisions I have to make a day, and how quickly I have to make them. I hope distinction can help me to better live out my values and avoid lazy shortcuts.