The Chinese Remainder Theorem by Example

I learned some math today. What happened was during lunch I was trying problems on Alcumus and I came across a problem I absolutely hated. It went like this:

The marching band has more than 100 members but fewer than 200 members. When they line up in rows of 4 there is one extra person; when they line up in rows of 5 there are two extra people; and when they line up in rows of 7 there are three extra people.

How many members are in the marching band?

I sighed loudly to no one in particular and started working it out. I wrote down a lot of numbers that fit the “two extra people when there are rows of 5” constraint (which I know as “2 mod 5”). I wrote 102, 112, 122, 132, 142, 152, 162, 172, 182, 192, 107, 117, 127, 137, 147, 157, 167, 187, and — wait for it — 197. And then I started crossing numbers off the list that didn’t work for the “rows of 7” constraint (the first ten numbers) and then searched one by one to find the last one.

I showed it to my friend who I was sitting next to, he said that there has got to be a better way to do this.

Anyway, I find the solution and punch it into the website. I got it right (woohoo) and start trying to make sense of the solution, which is a truly awful Wall of Text. See?:

Screen Shot 2019-01-23 at 1.17.05 PM.png

OK, but it mentions the Chinese Remainder Theorem, which I know as “that thing from group theory that I didn’t understand.”

I wanted to learn about the Chinese Remainder Theorem so I went to wikipedia, which is useless for this sort of thing, and then I said to my friend (who was still sitting nearby): I’ve got to find an example.

So I search, looking for something I can make sense of without already understanding it, and then I find this. It’s not great but it’s thankfully mostly-free of symbols, so I invest some effort into making sense of it. I work this through, step-by-step on paper.

The+Chinese+Remainder+Theorem.jpg

My friend and I keep talking about it, and I think I figured out how it worked. And then I thought that I’d write the sort of examples that I wish I had been able to find in my initial search. So, while I’m no fan of Search Engine Optimization, I want to help this post make its way into Google Image search so I’m going to (Chinese Remainder Theorem Example) try my best (Chinese Remainder Theorem Example) to (Chinese Remainder Theorem Example) help Google (Chinese Remainder Theorem Example) find it.

This is not a complete explanation — there are definitely some gaps, and I’m sure it could be improved — but hopefully it’ll give you a start.

So here’s the explanation, with examples in the style of my favorite example resource, which is also the style of example I mostly use in my teaching.


This all begins with linear congruences. For example, there are lots of numbers that are 2 mod 5 (i.e. they have a remainder of 2 when you divide them by 5). 12 is congruent to 2 mod 5. So is 22. So is 1002, 10000002, etc., etc., there are a lot such numbers.

There are likewise numbers that have a remainder of 1 when you divide them by 3. Like 4, which is congruent to 1 mod 3. So is 34, 667, 333333333334, etc., etc.

So are there numbers that are both? And how do you find them? The Chinese Remainder Theorem says that there is a process that works for finding numbers like these. Here is an example of that process in action:

chinese remainder theorem examples - google docs-page-001 (2)

There’s probably no way to understand this without working through each step of the example — sorry! — but part of what I think is cool here is that this is a constructive process. What you’re doing is building this number, and the number is pieced together from parts that are designed to work. The reason this has to work comes in two parts:

  1. Ingredient A: Anything times 7 is 0 mod 7. Anything 5 times anything is 0 mod 5. Anything times 13 is 0 mod 13. 156 x 45 x 23 x 8 is 0 mod 45.
  2. Ingredient B: You can multiply numbers and get back to 1. So if you start with 3 mod 4, you can get back to 1 by doing 3 x 3 mod 4. If you start with 2 mod 5, you can get back to 1 by doing 2 x 3 mod 5. Can you always do this? Good question, it had better work for us here though.

These are the two ingredients that we use in that example. We make a number that is a sum of a multiple of 7 and a multiple of 3, which I imagine in this case to be little self-destruct signals waiting to be activated. When you want the number to be 2 mod 3, the multiple of 3 part is activated and it blows up, counting for 0, leaving just the multiple of 7 that you multiply by the thing that sets it back to 1 (thanks Ingredient B) and is then scaled by 2. Tada: 2 mod 3.

And you can do it again to get 1 mod 7. This time the mod 7 part explodes, counting for 0. You’ve multiplied 3 by the number that makes it 1, mod 7, and that’s perfect. You get 1 mod 7.

There are subtleties here. To test yourself, you might want to see if you can spot the mistake in this example:

chinese remainder theorem examples-page-003

One other little twist is that so far, we’ve just discussed how to find any number that fits both conditions. But what if you’re looking for the smallest positive number that fits the bill? In short, just count backwards, though at first you may be surprised by what you have to count backwards by…

Chinese Remainder Theorem Examples-page-002 - Edited.jpg

A beautiful thing is that this process works exactly the same way when there are three conditions instead of two. That is, it almost exactly works the same way. Because we now want to plant even more little self-destruct signals in each part of the number. We want to rig things so that there are three terms, two of which self destruct every time we consider one of those congruence conditions.

Here we go:

chinese remainder theorem examples-page-006

And that’s that! The procedure just keeps going for more and more terms.


One thing I haven’t thought a great deal about yet is where to go from here, mathematically. What would be fun ways to extend or apply this idea? What are some good problems to try next?

I don’t know, I just started understanding this a few hours ago. But here are a few ideas:

  • If you can do it, do it backwards. (I learned this from Kate Nowak.) But it’s not so much fun to start with a number and find some congruency conditions. Like, yes, divide 46 by 5 and you get remainder 1 and divide it by 7 you get 4, big whoop. But what’s kind of cool is that each of these uses of the Chinese Remainder Theorem end up partitioning the number. So can you know how to partition the number without going all the way through? Does each partition work for some initial conditions?
  • The process always works for producing a number that fits the congruency conditions, but it doesn’t always produce the smallest. What’s up with that? How much bigger do they end up? Is there a way to know how many times bigger your number will be than the smallest possible integer?
  • Does any of this relate to a procedure for solving systems of linear equations? Could you solve this graphically? Is there a nice graphical way of representing any of this?

I’ll keep thinking!

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How to make worked examples work

I’ve written a few times about how I use worked-out examples (and mistake analysis) in my teaching:

The resource I’ve returned to again and again is Algebra By Example, which is really well-done. (And get excited for Math by Example, their expansion pack for 4th and 5th Grade!)

I realize that there are two things that I haven’t yet shared: some research on how to design these things well, and my routine for using these materials in my teaching.

The Research

Whenever you get a bunch of people talking about research in education, somebody eventually mentions how sad it is that research in education usually happens without the input of educators. Wouldn’t it be grand if researchers and educators were more collaborative?

Yes! This indeed would be grand, and it should happen more often. But sometimes it does happen, and those partnerships can produce really wonderful things. Algebra by Example is the result of one of these partnerships.

The project was led by Julie Booth of Temple University, and Booth has written a very nice piece in an NCTM journal about the design process her team uses and the research behind it:

McGinn, K. M., Lange, K. E., & Booth, J. L. (2015). A worked example for creating worked examplesMathematics Teaching in the Middle School21(1), 26-33.

Booth’s piece is very nice. It cites a more research-y piece that more carefully trawls and summarizes the worked example literature, and that is also an interesting (though more research-y) read:

Atkinson, R.K., et al. (2000). Learning from examples: Instructional principles from the worked examples research. Review of educational research, 70(2), 181-214.

Though it’s tough to summarize these pieces, here are a few important principles when designing example tasks:

  1. Try to put everything in one place…
  2. …but also try to visibly separate different steps.
  3. Switch up correct examples with incorrect work.
  4. Ask kids to explain what’s going on.
  5. Ask kids to try it on their own.

Try to put everything in one place. That’s a NO for splitting the problem and the solution with a not-so-useful line of text. Keep words at a minimum; ideally, you can see the whole example at once.

Worse:

Screen Shot 2018-11-28 at 3.00.13 PM

Better:

Screen Shot 2018-11-28 at 2.59.49 PM

It’s useful (according to Booth’s research and also me) to switch up correct with incorrect examples.

Can you spot the error?

Screen Shot 2018-11-28 at 3.03.50 PM.png

I like the way this incorrect work has some good thinking (I like the regrouping idea) and that it also clearly distinguishes each step.

From experience, I’ll say that people new to writing examples often try to do too much with the space they have. The goal really is to eliminate all but the most important thinking from the example, and it’s OK to use multiple examples to get at different aspects of the thinking. Overstuffing one of those little boxes can be a sign that it’s time to break this up into two smaller problems.

So much for the visual design of these materials, which is really a huge subject in of itself.

Some of the most interesting bits of those reviews are about self-explanation. Here’s a bit from the more research-y review:

Research on explanation effects suggests that self-explanations are an important learning activity during the study of worked examples. Unfortunately, the present research suggests that most learners self-explain in a passive or superficial way.

“Passive or superficial”: YES! All too often this is how students interact with an explanation (or with feedback).

So how do examples help at all, given the tendency of students to just passively say “OK fine yep I get it” when they see an example? I use a routine to really make example activities hum.

The Routine

Whenever I use an example activity with students, here is my routine.

  1. Get ready: I show the problem, covering up everything besides the problem itself. I usually use a document camera, so I mean I literally cover it up with my hand. Sometimes I ask students to begin thinking about how they’d approach the problem before I reveal the student work.
  2. Read: I ask everyone to read the student work to themselves. I ask students to offer a quiet thumb to let us know that they finished reading. I usually tell students to put a thumb up if they’ve read each line, even if they don’t understand everything fully yet.
  3. Discuss: I assign partners (really I’ve already done this) and ask students to discuss the example until each person understands every line of the student work. (I focus my class on spreading understanding, not on solving problems.)
  4. Explain: Every Algebra by Example problem has prompts for student explanation. When I make my own materials, I always include such prompts. It’s at this stage in the routine that I make sure everyone tries to answer the prompts. Then I usually call on a student to explain.
  5. You Try: This is where we help students try it on their own. There’s a whole art to picking good problems for students to try here — they should change the surface-details in some way, while keeping the underlying ideas constant. This is where we try to keep kids from making false generalizations, and it’s another level of protection against superficial understanding.

Using the Algebra by Example materials, the routine flows from left to right:

Screen Shot 2018-11-28 at 3.23.04 PM.png

It’s not so easy to design these prompts, either! Here are some suggestions, from Booth’s piece:

unnamed.png

If you’re in the mood for a bit of practice, you might think about how you’d fill in the rest of this example task:

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Example analysis works especially well as a feedback routine. There was a problem that kids especially had trouble with? Write an example that focuses on how to improve, and then follow the routine with some time to revise and improve the quizzes (or a re-quiz, or etc.).

All of this — the examples, the prompts, etc. — is worth your time, I think. It’s a lot of fun for kids to puzzle out how someone else is thinking, and the format allows us to really focus on a whole idea. It’s a bit of a myth, I think, that these types of problems are boring.

It’s also a myth that this kind of math is boring. It’s not, because there’s something to be understood here. And fundamentally, understanding is interesting.

Worked Examples and Loop-De-Loops

This is a Loop-de-Loop:

1-3-6-loop.jpg

At camp, I taught a session on Loop-de-Loops, a mathematical object from Anna Weltman’s book. I had about 30 people in the room (15 counselors, 15 kids) looking for patterns and trying to figure out a bit about what Loop-de-Loops are all about.

I introduced Loop-de-Loops using Chris Lusto’s fantastic site, asking everyone what they noticed and were wondering after I hit “Show Me.” (Go try it; it’s a blast.)

Screenshot 2018-08-01 at 2.45.21 PM.png

This was perfect for us, as I wanted the focus of the session to be about asking questions. (Lord knows that the kids spend enough time solving problems in this camp. Asking questions is at least half of a full mathematical picture.) Lusto’s site makes it easy to quickly explore many different lists of numbers, generate theories and see if those theories hold up.

In fact, it’s too easy to generate these Loops with the software. For the sake of understanding, it’s always good to construct stuff by hand a couple times. Often we notice different things when we construct objects by hand than we do on the computer.

My first time running this session, though, I realized something: it’s hard to make a Loop-de-Loop!

IMG_1136.JPG

Students and counselors struggled to draw these things — though, when they saw these objects on the computer, they were extremely confident that they understood how to make them. Surprise!

Today, I ran the session for another group of kids and counselors. This time, I came prepared. I wrote a worked example activity that aimed to help everyone better learn how to make these cool Loopy things:

example 2

example1

I quickly made a handout right before class today. (Hence the marker.) I knew that I needed to include the three things I always include on one of these things:

  1. The example, clearly distinguishing between the “task” and the “student work,” and trying to make sure to only include marks that contribute to understanding. (See those little arrow heads? I noticed that people had a hard time keeping track of directions while drawing these.) I tried to remove any distracting text or clutter — reading about the split attention effect helped me learn how to do this.
  2. Prompts for noticing the most important stuff. Research suggests that students often don’t explain things to themselves, or do so superficially. (And experience totally confirms this.) Prompts, along with a clear call for students to spend a minute responding to the prompts, helps a lot with this I find.
  3. A chance to try it out on your own, with the model nearby to help, if you get stuck.

I first read about this structure while reading about Cognitive Load Theory, but things didn’t click for me until I also saw the Algebra by Example project. Two other pieces have helped me better understand this bit of my teaching:

Here’s how I do these things. First, I put the “student work” up on the board and ask everybody to silently study it. (People need time to think before talking!) Then, I ask everyone to check in with a neighbor and to take turns making sure each can explain what’s happening. (This is usually where there are “ah!”s and “oh wait!”s.) Then, we talk about the prompts. After that, I tell everyone to try it on their own.

This part of the session went so much better today than my first pass. Everyone was still challenged by drawing these, and there were still a lot of mistakes. But the difference between these two sessions was precisely the difference between productive and unproductive struggle. Instead of flailing around when they got lost in the construction, everyone had something to go back to. Ah, OK, so this is how it’s done.

Part of my job is also to help the counselors support the students in their math work. A lot of the counselors tell me they struggle with knowing about how much to give away to a student when they are stuck. And while I totally know what they mean, I always tell them that their job isn’t to give away stuff or to avoid giving stuff away. Their job is to get the ball rolling for the student, get thinking happening, as quickly as possible — and then to step back and let that thinking happen.

Today I needed to get the ball rolling. It was a session about posing problems, and I started with asking everyone to notice and ask questions. But an important part of getting the ball rolling was a worked example.

Study an example, see the world

I’ve been a math teacher in New York City since 2010, a few months after I graduated from college. It’s the only job I’ve ever had, besides for little things over the summer when I was a teen. (In order: babysitter, camp counselor, Pepsi vendor at Wrigley Field, tutor. All kind of relevant to teaching, come to think of it.)

Though I teach math, math didn’t feel easy for me as a student. It was never where I shined. An exception was geometry, with its heavy emphasis on proof. Proof felt natural for me in a way that algebra didn’t.

When I began teaching, I realized that for many students the situation is reversed — it’s proof that feels unnatural and cumbersome. Writing a proof involves combining statements in ways that seemed to mystify many students. This was especially true early in my career.

After a few years of hitting my head against the wall, I started to understand what made this such a difficult skill to teach. Proof is the closest that mathematics comes to writing, and writing itself is impossible without reading. How can a student who has never read an essay possibly write one? I concluded that my students needed to read more proofs.

It took me a few more years to understand how exactly to pull this off in class. My big frustration was that my students wouldn’t devote enough attention to the proof examples I shared. I would distribute a completed proof and ask the class to read it with care. Very often, it seemed that they missed the whole point of the proof. They couldn’t read it carefully yet — they didn’t know how.

Now, things go better when I share proofs in class. One big difference is I have a much better understanding of all the subtle conceptual understandings that go into a proof, many which were invisible to me at first. (In teaching, it can be trouble when a topic comes naturally to you.) There are many aspects of a proof that I need to help them uncover.

Besides for a better understanding of the subtleties of proof, I’ve learned to structure my activities in sturdier ways. I’ve learned to design these activities so that they have three parts:

  • The proof example
  • Comprehension questions about the example
  • Proof-writing practice, with the example as a model

I didn’t come to this structure on my own, by the way. I came to it through reading about Cognitive Load Theory (where these are sometimes called “example-problem pairs”) and especially from seeing it in some especially well-designed curricular materials:

Screenshot 2017-11-30 at 6.49.21 PM.png

Also:

Screenshot 2017-11-30 at 6.50.45 PM.png

 

(In fact, I didn’t really understand how to make my own example activities until I saw many models in these curricular materials. I needed examples, myself.)

So, for instance, I created this proof example for my students this year:

DM8ANAWWsAACjSC.jpgLooking back, the example isn’t perfect. It ended up being a bit visually crowded, and it might have been better to eliminate some of the letter-abbreviations. In class, I actually covered up each stage of the proof to focus their attention on each part.

In any event, this activity shows a lot of what I’ve learned about teaching proof. I knew I wanted to make explicit the complicated two-stage structure of some congruence arguments, so I worked hard to create a pretty clear example for my students. I then called on students to answer a trio of analysis questions about the proof — there’s a lot to notice, and students don’t yet know how to notice the underlying structure of this kind of proof all on their own. Finally, I ask students to use what they’ve noticed on a related pair of problems, so that students see that there’s something here that’s generalizable to many different kinds of diagrams.

Even when my proof activities aren’t structured so rigidly, I try to include variety and a chance to practice. Here is a simpler activity, but I still call for students to do a bit of proof-completion in the second prompt:

DM8ANArXkAItunW.jpgSometimes when I talk to other teachers about examples, they tell me they’re worried that kids will just try to unthinkingly copy the model. I do know what they mean, but it’s not what I see with my kids. I think that part of the reason is that I reserve example-analysis for when I worry that the math is going to be difficult, even overwhelming for many students. There is certainly a way to misuse these activities, and perhaps if I used these sorts of tasks on less complex material I would see unthinking imitation.

One of my jobs is to help students see things that they can’t yet see — things like the logical structure of a good mathematical argument, or the way just a tiny bit of information about a shape can guarantee a whole lot more. When things don’t come naturally to my students, what I’m learning to do is to design an activity that opens up a little window into the mathematics so they can learn to see new things.

Addendum (1/6/18): I just came across this lovely line from Paul Halmos:

A good stack of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one.

Teaching, in General

I. 

If you give a quiz covering lots of different topics, you’re going to get a lot of different mistakes. Which leaves you with a dilemma: how do you address those mistakes?

Yesterday’s quiz in geometry was a review quiz, so the topics were from all over the place:

  • angles in isosceles triangles
  • inscribed angles in a circle
  • area of triangles, parallelograms and trapezoids
  • congruence proofs

As expected, kids distributed their not-quite-there work fairly evenly across these topics. (OK so that’s not true, there were a lot of issues with the congruence proofs. There always are and always will be. Sigh.)

Here were two bad options for returning the quiz:

  • Try to address all the issues with individual comments. First, it’s a game of whack-a-mole that is guaranteed to drive me insane. Second, what should I do? Try to leave perfect hints? Say nothing, and let kids figure out on their own what they did wrong? Show them the correct way to answer the question, and thereby eliminate anything for the kids to actually think about when I return the quizzes?
  • Pick just one thing to focus on. Reteach that one thing in a careful way, then return the quizzes and ask kids to revise.

The second of the two options is great when there the mistakes are in the same galaxy. (I wrote about this in a post, Feedbackless Feedback.) But, I’m realizing now, this isn’t a terrific move when the mistakes are distributed across many topics. Because on what basis should I pick something to focus on reteaching? Any choice would be equally bad.

II. 

While reviewing the class’ quizzes, I found myself falling into written comments, at least until I figured out what else to do with the quizzes.

I used to write long, wordy comments that were essentially hints on the margins of the page. (“Great start! Have you tried multiplying both sides of the equation by 3?”) I came to dislike those sort of comments, as they just focus focus focus attention all on THIS problem. But I don’t particularly care about whether a students gets this problem correct; I care about the generalization.

What I’ve fallen into is, whenever possible, writing a quick example that’s related (but not identical) to the trouble-problem (the problem-problem) on the page. I do this below on the second question:

image-3.jpg

Then, I ask kids to revise the original on the basis of the example (or anything else they realized).

After writing a few of these example-comments, I realized I was taking a lot of time doing this, and repeating myself somewhat. I also realized that I don’t know if I could repeat this on every page for the congruence proofs, as the problem itself was reasonably complex:

image-2.jpg
Hard to read. I use the highlighter to flag errors, but this student highlighted the triangle himself.

I wasn’t sure what to do. Then, I remembered something I had read from Dylan Wiliam — I think it’s in Embedded Formative Assessment. His idea there was that you can give all the class’ comments to everyone, and then kids have to decide which comments apply to them.

I thought, OK, I can work with this. So I quickly (quickly!) made a page of examples, one for every mistake I saw on the quiz:

ums-copier@saintannsny.org_20171115_095927 (1)-page-001.jpg

My routine in class went like this:

  • Hand out the examples for revision.
  • Hand back the quizzes with comments.
  • Search for an example that’s relevant to your mistake.
  • Call for revision on the basis of the examples. Work with friends, neighbors. Of course, I’m available to help.
  • Then, try the extension task.

This was my first time trying this, but I thought it went well. Solid engagement, really good questions, no unproductively stuck students.

When you do something good in teaching, you never really know if it’ll work again, but I’ve got a good feeling about this one. It feels like a lot of what has already worked for me, but in a better order.

III.

Harry Fletcher-Wood is very nice and has a lot of interesting thoughts about feedback. As such, Harry and I very nicely disagree about a pretty interesting question about feedback: how can you teach people how to give better feeedback?

The usual caveats apply: I am not a teacher teacher, but Harry is involved in teacher education, and I have no idea if I’m right on this.

In any event, Harry recently published a really cool post where he tried to synthesize a lot of the research on feedback into a decision tree:

Feedback-decision-tree-1.6

Now, this is awesome as a synthesis. But just because something is a good description of feedback doesn’t mean that it’s useful prescriptive advice. My favorite example of this comes from Pólya’s strategies for mathematical problem solving. Alan Schoenfeld has a nice way of putting it in Learning to Think Mathematically — the strategies have descriptive, but not prescriptive validity:

In short, the critique of the strategies listed in How to Solve It and its successors is that the characterizations of them were descriptive rather than prescriptive. That is, the characterizations allowed one to recognize the strategies when they were being used. However, Pólya’s characterizations did not provide the amount of detail that would enable people who were not already familiar with the strategies to be able to implement them.

In other words, just because a heuristic is a good description of practice doesn’t mean that it is an effective pedagogical tool. And that’s precisely my concern with Harry’s decision tree.

Feedback is a high-level concept that describes a TON of what happens in teaching. And any guidelines for how to give feedback effectively are also going to be high-level in a way that reminds me of Pólya’s moves like “find a simpler problem” or “draw a picture.”

And just as Pólya’s moves struggle because they aim to guide problem solving in geometry, algebra, topology, etc., all areas of math, Harry’s decision tree seems to me an attempt to guide feedback in all areas of teaching — math, history, medical school, etc.

Of course, Harry doesn’t intend for this to be the only thing guiding students, but neither did Pólya. My question is whether these generalizations themselves are helpful, beyond whatever ways that teacher educators can make them concrete and specific for teachers.

But what’s the alternative?

I don’t know yet. I can say a few things now that I couldn’t a few years ago:

  • I think domain-specific — math-specific, history-specific — generalizations will be more useful than domain-general ones.
  • I think that the generalizations can productively come in the form of instructional routines.

And, with this post and the other one, I now have two generalizations I can make about giving feedback in math class.

First: if there’s a problem that a lot of students have trouble with, consider a reteaching/revising cycle like the one in this image:

Screenshot 2017-11-15 at 1.24.03 PM

Second: if mistakes are sprinkled across too many topics, consider something like the revision routine I described in this post.

IV.

My bet is that a lot of knowledge about teaching looks like this. It’s not that there isn’t knowledge about teaching that accrues, but that we look for ways to scale things out of their contexts. Then we call those things myths and talk about how we have to kill ’em.

In general, generalizations about teaching are hard to come by. But nobody teaches in general. All teaching is intensely particular. These kids. These schools. This idea.

Some people are skeptical of the possibility of making generalizations about teaching, and the vast majority of people are cheery about making sky-high generalizations that cross every context. There’s a middle position that I want to find. There’s a sweet spot for knowledge about teaching, though I don’t know if we’ve all found it yet.

Feedbackless Feedback

I.

Not all my geometry students bombed the trig quiz. Some students knew exactly what they were doing:

Screenshot 2017-05-26 at 3.12.57 PM

A lot of my students, however, multiplied the tangent ratio by the height of their triangle:

Screenshot 2017-05-26 at 3.19.05 PM.png

In essence, it’s a corresponding parts mistake — the ’20’ corresponds to the ‘0.574’. The situation calls for division.

Half my class made this mistake on the quiz. What to do?

II.

Pretty much everyone agrees that feedback is important for learning, but pretty much nobody is sure what effective feedback looks like. Sure, you can find articles that promise 5 Research-Based Tips for great feedback, but there’s less there than meets the eye. You get guidelines like ‘be as specific as possible,’ which is the sort of goldilocks non-advice that education seems deeply committed to providing. Other advice is too vague to serve as anything but a gentle reminder of what we already know: ‘present feedback carefully,’ etc. You’ve heard this from me before.

As far as I can tell, this vagueness and confusion accurately reflects the state of research on feedback. The best, most current review of  feedback research (Valerie Schute’s) begins by observing that psychologists have been studying this stuff for over 50 years. And yet: “Within this large body of feedback research, there are many conflicting findings and no consistent pattern of results.”

Should feedback be immediate or delayed? Should you give lots of info, or not very much at all? Written or oral? Hints or explanations? If you’re hoping for guidance, you won’t find it here. (And let’s not forget that the vast majority of this research takes place in environments that are quite different from where we teach.)

Here’s how bad things are: Dylan Wilam, the guy who wrote the book on formative assessment, has suggested that the entire concept of feedback might be unhelpful in education.

It’s not looking like I’m going to get any clarity from research on what to do with this trig quiz.

III.

I’m usually the guy in the room who says that reductionist models are bad. I like messy models of reality. I get annoyed by overly-simplistic ideas about what science is or does. I don’t like simple models of teaching — it’s all about discovery — because I rarely find that things are simple. Messy, messy, (Messi!), messy.

Here’s the deal, though: a reductionist model of learning has been really clarifying for me.

The most helpful things I’ve read about feedback have been coldly reductive. Feedback doesn’t cause learning . Paying attention, thinking about new things — that leads to learning. Feedback either gets someone to think about something valuable, or it does nothing at all. (Meaning: it’s affecting either motivation or attention.)

Dylan Wiliam was helpful for me here too. He writes,

“If I had to reduce all of the research on feedback into one simple overarching idea, at least for academic subjects in school, it would be this: feedback should cause thinking.”

When is a reductive theory helpful, and when is it bad to reduce complexity? I wonder if reductive theories are maybe especially useful in teaching because the work has so much surface-level stuff to keep track of: the planning, the meetings, all those names. It’s hard to hold on to any sort of guideline during the flurry of a teaching day. Simple, powerful guidelines (heuristics?) might be especially useful to us.

Maybe, if the research on feedback was less of a random assortment of inconsistent results it would be possible to scrap together a non-reductive theory of it.

Anyway this is getting pretty far afield. What happened to those trig students?

IV. 

I’m a believer that the easiest way to understand why something is wrong is usually to understand why something else is right. (It’s another of the little overly-reductive theories I use in my teaching.)

The natural thing to do, I felt, would be to mark my students’ papers and offer some sort of explanation — written, verbal, whatever — about why what they did was incorrect, why they should have done 20/tan(30) rather than 20*tan(30). This seems to me the most feedbacky feedback possible.

But would that help kids learn how to accurately solve this problem? And would it get them to think about the difference between cases that call for each of these oh-so-similar calculations? I didn’t think it would.

So I didn’t bother marking their quizzes, at least right away. Instead I made a little example-based activity. I assigned the activity to my students in class the next day.

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I’m not saying ‘here’s this great resource that you can use.’ This is an incredibly sloppy version of what I’m trying to describe — count the typos, if you can. And the explanation in my example is kind of…mushy. Could’ve been better.

What excites me is that this activity is replacing what was for me a far worse activity. Handing back these quizzes focuses their attention completely on what they did and what they could done to get the question right. There’s a time for that too, but this wasn’t a time for tinkering, it was a time for thinking about an important distinction between two different problem types. This activity focused attention (more or less) where it belonged.

So I think, for now, this is what feedback comes down to. Trying to figure out, as specifically as possible, what kids could learn, and then trying to figure out how to help them learn it.

It can be a whole-class activity; it can be an explanation; it can be practice; it can be an example; it can be a new lesson. It doesn’t need to be a comment. It doesn’t need to be personalized for every student. It just needs to do that one thing, the only thing feedback ever can do, which is help kids think about something.

The term ‘feedback’ comes with some unhelpful associations — comments, personalization, a conversation. It’s best, I think, to ignore these associations. Sometimes, it’s helpful to ignore complexity.

Beyond “Beyond Explaining”

There’s a good reason why educators often talk about the need to move beyond explanations. People who don’t know much about teaching think all the action in teaching is about the clarity of the explanation. (That, and getting kids to listen to your ultra-clear explanation.)

There’s much more to the job than that, of course. Michael Fenton puts this nicely in a recent post:

In my first few years in the classroom, I held the notion that the best way to improve as a teacher was to hone my explaining skills. I figured that if I could explain things more clearly, then my students would learn more. […]

The best way to grow as a teacher is to develop my capacity to listen, to hear, to understand. […] This doesn’t mean that I’ll stop working on those other skills. But it does mean I have a new passion for learning about listening—really listening—to students and their thinking.

I think this focus on listening is wonderful, and Michael did say that he’s going to keep working on his other skills, which is a nuanced take. But what about the title of the post, “Beyond Explaining, Beyond Engaging”? Philosopher Eric Schwitzgebel has a helpful distinction between a piece’s headline view and its nuanced view:

Here’s what I think the typical reader — including the typical academic reader — recalls from their reading, two weeks later: one sentence. […] As an author, you are responsible for both the headline view and the nuanced view. Likewise, as a critic, I believe it’s fair to target the headline view as long as one also acknowledges the nuance beneath.

So let’s take on that headline: should we go “beyond explaining”? If we’re trying to improve our teaching, it could be that getting better at listening has a higher payoff than getting better at explaining. But my experience has been that there isn’t any strict hierarchy of payoffs in teaching. Teaching evolves in funny ways. Last year I taught an 8th Grade class that pushed on my classroom management. This year I’ve spent a good deal of time learning how to tutor students with learning disabilities. I’d hate to say that explaining is some sort of basic teaching skill, the sort of thing novices focus on but more experienced teachers don’t need. Teaching is weirder, more cyclical, more web-like than that.

Maybe, though, we should move beyond explaining because it’s easy — or because pretty much everybody knows how to do it well after their first two years on the job.

That might be true, for all I know. If I doubt it, it’s only because it’s only over the past year that I’ve really started to understand some of the things that make a good explanation hum and lead to great student thinking, instead of slack-eyed drooling from the back rows of the classroom.

Besides, a lot of what I’ve learned about explaining comes from outside math education writers and speakers. Which started me thinking that maybe this knowledge (if it’s even true) isn’t as well known to math teachers as it could be.

Here’s what I think I know about giving good explanations to kids:

1. Study Complete Mathematical Thoughts; Don’t “Roll Them Out”

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This isn’t really an explanation, but it’s a complete example. Source: Algebra by Example

The first students I taught were subject to an especially painful type of instruction. I don’t know exactly how to describe it. Maybe an analogy would help. Imagine a magician (Ze Great Pershan-o) who is waaaay too detail-oriented: OK AND NOW CHECK IT OUT: I PUT MY HAND IN THE HAT! AND NEXT, I CLENCH MY FIST! HERE COMES THAT HAND SLOWLY COMING OUT OF THE HAT! ETC!

This is basically also how my explanations worked. HERE IS AN EQUATION! (WRITES AN EQUATION.) WE WANT TO SOLVE FOR X! (WRITES ‘X = ?’) HOW SHOULD WE DO THIS, CLASS? WELL, WE COULD DIVIDE BY 2. BUT THAT WOULDN’T HELP US VERY MUCH. LET’S INSTEAD SUBTRACT 2, ETC!

The phony enthusiasm was a problem. Another problem was that I was feeding the math one mini-idea at a time rather than presenting them the complete mathematical thought. I’ve come to think that when we do this — when we roll out the explanation, line by line — we lose a lot of kids.

Do you know that thing you do when you’re trying to understand something hard in a math textbook? How you put one finger at the top of the explanation, and then go line-by-line to make sure you understand each piece? But then you go back and try to make sense of the whole? That sort of self-explanation is where the learning can come from in an explanation, I think, and if we roll the explanation out, we’re making it harder for kids to look at how the pieces fit together.

So, when we’re ready to explain an idea to students, we ought to be offering them a complete mathematical thought. No need to dice it down to the atomic level, like Dumb Houdini or whatever.

Practically, this means that if I intend to “show the steps” in an explanation, I make a real effort to reveal them all at once. I project or photocopy artifacts like the ones that Algebra by Example create. If I’m working one-on-one, I’ll scribble a full example down on the page, rather than coaching a kid through that example step-by-step.

This idea, of course, isn’t my own. I came to understand this from reading about cognitive science, and especially cognitive load theory. There’s more to math than explaining, but there are ways of explaining things that preserve the math and others that make it harder for students to make meaning. Fully worked-out examples can help kids make meaning from our explanations, I think.

2. Use Arrows to Emphasize Process, Change and Action

This was an aha moment for me. I came to think people across education were saying similar things about how it’s easier for people to think about actions, rather than properties. You can hear this idea bouncing around research on how kids solve word problems, how our minds especially remember narratives, and what constitutes good writing.

Nobody has told me that these ideas are related. I’m a bit worried that I’m connecting totally unrelated people and ideas. Still, here’s the idea:

An interesting result of Cognitively Guided Instruction is that numerically equivalent word problems are often handled very differently by children. Consider these two problems:

Problem 1: Jill has 8 jellies, but a raccoon eats 5 of them. How many does she have left?

Problem 2: Jill has 8 jellies. A raccoon has 5 jellies. How many more jellies does Jill have?

Numerically equivalent problems — 8 minus 5 — but the first problem contains an action that is easy to represent. It would occur to a lot of kids that they could solve the first problem by dealing out 8 counters (or whatever) and then removing 5 off them from their pile. That’s directly modeling the action of Problem 1.

Problem 2 doesn’t contain an easy-to-notice action, so direct modeling would be less likely to occur to children. Carpenter and pals found that, in fact, kids didn’t use direct modeling strategies for Problem 2, and as a result Problem 2 was a bit trickier for kids to handle.

(This is 100% true, in my experience.)

Word problems with actions, essentially, contain stories that are easy for us to represent and understand. And stories are the sort of thing that our minds most easily grasp and symbolically represent. As Dan Willingham notes:

Research from the last 30 years shows that stories are indeed special. Stories are easy to comprehend and easy to remember, and that’s true not just because people pay close attention to stories; there is something inherent in the story format that makes them easy to understand and remember. Teachers can consider using the basic elements of story structure to organize lessons and introduce complicated material, even if they don’t plan to tell a story in class.

Incidentally, Tom Newkirk makes a similar observation about what constitutes good writing in his book Minds Made for Stories:

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Later, Newkirk argues that part of what makes good non-fiction writers good is they find subjects and actions to metaphorically represent abstract structures. They turn “evolution is a process whereby genes are randomly mutated” into “mutagens are constantly attacking our genetic material, altering it in ways that have the power to change the direction of an entire species” or whatever. Action, action, action.

(I’m also pretty sure this connects to Anna Sfard’s work on the way we tend to turn mathematical processes into mathematical objects but I’m not sure I have all the pieces put together yet.)

Back to Planet Classroom: What does this mean for my teaching? Practically, a lot of annotated arrows.

Instead of an example of simplifying expressions that looks like this

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…I try to recreate it with annotated arrows, to emphasize the actions involved:

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Our minds privilege stories, which means that they privilege change over inaction. If our explanations can include more doing things to things, this can help kids see what we mean a bit more easily.

For a lot of fun use of arrows to represent the actions in otherwise static representations, check out David Wees’ new tumblr for mathematical annotations. A prime example:

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Arrows — especially annotated arrows — can help transform examples (i.e. what correct work might look like) into explanations that help someone understand the examples.

3. Describe Mathematical Pictures, Though This is Harder Than You Might Expect

The What Works Clearinghouse (WWC) is this big federal initiative to try to sort through the evidence for various educational claims and give clear recommendations. The thing is, there is a ton of dissatisfaction with their standards for recommendations. Some people think their standards of evidence are weirdly strict. Others say they privilege large experimental or quasi-experimental studies over other forms of evidence.

Anyway, they have this report on helping students who struggle with math, and I like it. Their fifth recommendation is all about visuals:

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Why are visuals important? At least partly because words don’t distract you from pictures — you can pay attention to both at once. (Unlike reading a slide and hearing it explained to you, where the words interfere with each other. You might find yourself doubly distracted in that situation.)

This relates to dual-coding theory, a theory from cognitive science that deserves to be better know in math education. Like worked-out examples or the privileged role of narrative, it’s a legitimately useful bit of cognitive science to know.

If you’re looking to teach a strategy, describing a (complete!) mathematical picture (with arrows!) can help.

“So to figure out 7 minus 2, I started with 2 and counted up until I got to 7 and that was 5, so I figured out that 7 – 2 is 5.”

The thing, though, is that it’s very easy to mess this up. A “mathematical picture” is not “a bunch of written numbers or words.” That’s not describing a picture with words — that’s just using spoken words to describe written words. I don’t think that helps as much, according to dual-coding. Words can distract you from words.

I’m not saying that board work in the above tweet is bad at all. My point is just that these equations are going to compete for attention with any spoken explanation in a way that (dual-coding says) a picture wouldn’t. (Though check out those annotated arrows!)

A problem: when I think about it, there are a lot of mathematical topics that I can’t think of a good picture for. And even for ones where I can (e.g. the connection between area and multiplication) those visual representations aren’t obviously connected to their numerical ones to kids. Those connections need to be carefully taught. Ideally, they’re built into a curriculum.

I mean, it’s obvious to me that you can carve up a rectangle into 4 quarters and this represents multiplying (x + 3)(x +7), but anyone who’s tried knows that this isn’t obvious to high school math students.

So while it’s great to aim for “describing a mathematical picture” as an ideal for explanation, we’re limited by the mathematical pictures that kids understand and that we know.

I love learning new pictures for mathematical ideas. I feel like this year I really realized the power of the visual representation of the Pythagorean Theorem to help my kids understand its meaning and use:

But there are a lot of topics where I don’t know good visuals to go with the numbers, equations or words. I’d love if we could find more of them.

What Beyond Explanations Shouldn’t Mean

I could be wrong, but I don’t see a lot of people writing or thinking about how to give good mathematical explanations. This is despite the fact that the vast majority of teachers I talk to say they give explanations often, even if they are a bit embarrassed by this. (They shouldn’t be, I think.)

And the vast majority of thinkers and writing about teaching would tell me that they aren’t anti-explanation, just against a mono-focus on explaining as the core of teaching.

So why doesn’t anyone write about giving good mathematical explanations? Three possibilities I can think of.

  1. Unlike me, pretty much already knows how to give good mathematical explanations.
  2. People don’t think that improving our explanations is worth the ink. It’s a low pay-off instructional improvement.
  3. Anti-explanation ideals make it trickier to talk about improving our explanations.

I’m pretty sure it’s not Possibility 1. I think Possibility 2 sounds good, and Possibility 3 is a solid maybe and is anyway related to 2.

Either way, now you know what I’ve recently figured out about explaining stuff to kids:

  • Represent complete mathematical thoughts
  • Use annotated arrows to emphasize action and change in those representations
  • If possible, describe mathematical pictures

So, what’s next? Just last week I tried out a new representation of solving trig problems with my geometry students. It flopped:

But then I made a little tweak, and it went better. Which got me thinking: we’ve got this whole internet thing. Why aren’t people sharing more of these images? Is it less fun for us to share pictures of our own work? Does it seem self-promoting in a way that sharing other things (e.g. activities) doesn’t?

I’m not sure. But I think that this work is valuable, and is worth sharing. Explanations are nitty gritty, but it’s important nitty gritty.

A Worked Example Workaround?

When you decide to explain some math to a kid, how should you explain it? Step-by-step, or all at once?

There’s an issue with step-by-step explanations: kids have to remember what you’ve already said in order to understand what comes next. This means that there’s often a lot to hold in their head!

There’s an issue with fully worked-out examples: by not developing math slowly, in full view of the student, you make it seem as if the solution was dropped out of the sky. It can present a false picture of math: as constituted entirely of an encyclopedia of procedures that mathematicians memorize, look-up, and employ on canned problems.

I was thinking, today, about whether there is a way to get the best of both worlds. My mind wandered towards my 8th Grade class. We’re studying slope, so I launched class by putting two right triangles under the document camera. Which is steeper?

Students debated, thought some more, offered good and better approaches. I kept a record of their ideas, which I scanned after class:

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I’m wondering, what if I started tomorrow’s class by projecting this image back on the board? I’d say, Here’s what we figured out yesterday. I’d like to give you two new triangles to look at today. I’m going to keep yesterday’s work up, in case it’s helpful.

Would that give a distorted view of mathematics and its development? Would that give students the benefits of worked-out examples?

Too Much of A Good Thing

Greg Ashman:

If more guidance makes minimally guided approaches more effective then why not use a fully guided approach? Won’t that be still more effective? It is an argument that plays out again in the book and one that offers little comfort to proponents of open-ended problem solving in high school maths classes.

But, Jordan Ellenberg:

The difference between the two pictures is the difference between linearity and nonlinearity, one of the central distinctions in mathematics…Mitchell’s reasoning is an example of false linearity—he’s assuming, without coming right out and saying so, that the course of prosperity is described by the line segment in the first picture, in which case Sweden stripping down its social infrastructure means we should do the same.

But as long as you believe there’s such a thing as too much welfare state and such a thing as too little, you know the linear picture is wrong. Some principle more complicated than “More government bad, less government good” is in effect. The generals who consulted Abraham Wald faced the same kind of situation: too little armor meant planes got shot down, too much meant the planes couldn’t fly. It’s not a question of whether adding more armor is good or bad; it could be either, depending on how heavily armored the planes are to start with. If there’s an optimal answer, it’s somewhere in the middle, and deviating from it in either direction is bad news.

Also, John Sweller:

That is not to say that there are no disadvantages to the use of worked examples. A lack of training with genuine problem-solving tasks may have negative effects on learners’ motivation. A heavy use of worked examples can provide learners with stereotyped solution patterns that may inhibit the generation of new, creative solutions to problems.

Greg’s argument is, “If a bit is good, isn’t a lot better?” But this sort of falsely linear thinking isn’t compelling, no matter what you think about direct instruction.

 

How did 69 turn into 29?

Last year, while reading and writing about cognitive load theory, I came across something weird that I couldn’t explain. A paragraph from Greg Ashman’s latest reminds me of this puzzle. It’s really small and inconsequential, but it’s been bugging me. Maybe you can figure it out.

He writes:

One of my PhD supervisors did an experiment in the 1980s. Undergraduates were given as series of problems. Each problem involved a starting number and a goal number. The participants had to get from the first number to the second using only two moves which they could repeat: multiply by three or subtract 29. The problems were designed so that each one was solved by alternating the steps. Although the students could generally solve the problems, very few ever worked out the rule.

Great. Multiply by three, or subtract 29.

Except you go back to that paper, and it’s actually subtract 69.

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Where did Greg get the “subtract 29” from? I don’t know, but it could be from this piece by Sweller in 2016.

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Anyway, totally unimportant. Completely uninteresting. But. Did he forget? Was it a typo? Did he decide — as so many before — that 69 is a funny number to talk about in classes?

If you see me and I’m looking pensive, this is probably what I’m thinking about.