Math Education

My outlook on teaching math

One of the best pieces on this blog, reprinted in advance of writing more about teaching. Read, critique, praise. Your call.

• Best mathematics teaching inspires. At all levels and all ages it is possible to communicate some of the elegance, power and beauty of this most abstract subject.

• Mathematics encapsulates abstraction from the real world. A child learns to count spoonfuls, learns to count people, learns to count fingers, learns to just plain count, and in the process acquires the abstract concept of, for example, “two.” The child takes ownership of this concept, and can reapply it freely. As adults we may take “two” for granted, but we have never met it, never touched it, never tasted it. It is one of the first completely abstract concepts that we ever owned.

• Learning mathematics involves skill acquisition, drilling, repetition, and instruction by an authority. It also involves independent construction of knowledge, connection to physical or real world situations, reflection by the learner, and independent reapplication to new situations. Traditional instruction has been overwhelmingly weighted to the former list, standards based instruction to the latter. Neither by itself gives learners adequate opportunity to take ownership of the abstract concepts that make mathematics beautiful and powerful. Best mathematics instruction carefully blends traditional and standards-based techniques.

• I strongly believe that instruction should be adjusted or modified to meet the needs of the current students. This entails a constant process of carefully planned experimentation, reflection, adjustment, and evaluation. Further, I have found it valuable to share with students information about modifications (pacing, depth, styles of instruction, balance of traditional/non-traditional work), and to solicit additional feedback from them.

• Concept ownership takes place more readily when the learner considers him or herself a stakeholder in the process. To this end it is desirable to foster a sense of control or ownership of other aspects of the classroom, including, as appropriate, involving the students in some decision making (see above). It is also possible to make students part of the subject itself, whether through data studies of the class or students’ families, or the creation of geometric figures based on the students’ own birthdays.

• An effective instructor is also a learner. I continue to take courses in mathematics and to study on my own. I am an avid problem-solver. I have never stopped trying new techniques in the classroom, and modifying, or rejecting them based on actual experience. As a role model it is necessary to share this love of learning with students. I freely admit when I do not know, and gladly share with students how I intend to search for “the answer.”

• A teacher of mathematics must be able to distinguish between right and wrong answers. A teacher evaluates alternate approaches, and distinguishes between minor and conceptual errors. The teacher can place a topic into a broader mathematical context, and answer the questions “Where is this topic next applied within mathematics?” and “Where is this topic applied outside of mathematics?” (if there is an answer) Grade level curricula are a subset of mathematics as a whole. It is the teacher’s responsibility to ensure that what is being taught not only leads to a correct answer, but is mathematically valid and will not need to be corrected at subsequent levels. It is not acceptable to know just a bit more than the students. An effective teacher’s knowledge of mathematics must be extensive.

Do kids get most units?

I don’t think so. I don’t think they have a very good sense of distance, of rate, of volume, of area, as expressed in most standard units (traditional or metric).

Can we do anything about this in math class? Does it matter?

Yes, and yes.

It matters in the real world. There’s a piece of literacy that’s missing if I kid knows 5 miles is far, but can’t get more specific. How much is 2 gallons of soup? Adults don’t get square feet. And move to metric in the US, and lots of people are lost. We encounter the terms daily. We should understand how much, how far, how long, how fast, in terms that make sense.

And it matters in math class. All those annoying word problems, in context, with answers in cubic feet or meters per second. Shouldn’t kiddies know if their answers make sense? What good is the context if they don’t sense the scale?

Teach them to convert

So there’s two pieces here. There are a number of ways to convert. I like what I call factor-label. Example. I want to know how fast 100 meters in 9.69 seconds is in miles per hour. Look at this:

$\frac{100 meters}{9.69 seconds} \times \frac{60 seconds}{1 minute} \times \frac{60 minutes}{1 hour} \times \frac{1 kilometer}{1000 meters} \times \frac{5 miles}{8 kilometers}$

Now, ‘cancel’ the units (it’s not really math, but it works) as if we were canceling common factors in fractions, multiply across, and presto: 23 point something miles per hour (the bad 8:5 conversion limits my significant digits, but no matter. I can’t perceive the difference between 23 and 24 mph. Does 23.2 vs 23.3 really matter to the kids?)

There’s other ways to convert units, but the kids must be armed with some tool.

Teach them human-scale reference units

Miles per hour. Sounds so natural. Rolls off the tongue. But I am fairly confident that most of my students don’t have a good grasp of how fast 2 mph, 10 mph, 20 mph, 50 mph, 100 mph, etc, really are.

Time is okay, but it is worth teaching them to count out seconds. Really.

Distance is tougher. Little distances? Put rulers in front of them. Ask kids to show with their fingers, for example, 3 inches, 2 centimeters, one foot, 5 centimeters, one inch. Drill it a little here and there. They will get better, but they need practice. From feet, once they are down, get some estimates of heights of ceilings, widths of classrooms, lengths of hallways. Estimate, measure, estimate again. They will get better.

Bigger distances? In New York I use blocks (I specify short Manhattan blocks). Twenty blocks (approximately) make a mile. Reexpress them in meters, in kilometers, in feet, in yards. But let “block” be a good unit, one that they can refer back to.

Area? Estimate, measure and multiply, estimate more. Classrooms. Desktops. Sheet of paper. The classroom makes a good standard, human-scale unit.

Volume. You know, this is tough. I fall back on liters (thank you Coke!), but I don’t work much with it. Do the volume of the teacher’s desk, shock them with the answer, and that’s pretty much it. It helps if they have an inch cube or a foot cube in front of them. The centimeter cube is too small and they don’t ‘feel’ the relationship. I haven’t seen a meter cube, but I think it would be too big. Textbook volume wouldn’t be bad, but it’s different for each book, and the cover can throw things off.

Rate. That’s the big one. Miles per hour is foreign. I start with seconds per block. We estimate normal walk, slow walk, brisk walk, run, bicycle/skates/skateboard, slow car, fast car, and then take the reciprocal and convert to miles per hour. Can do kph, too. And meters per second. Seconds and parts of minutes they get. Blocks they learn. And that gives them something to hang their hats on for the harder (but more common) units.

Who knows how multiplication is taught? Not me, not Keith.

This is about an argument about nothing.

A respected math columnist went after teachers for saying that multiplication is repeated addition, but it turns out that he doesn’t know if many teachers do this. I called him on it. And his response came up short.

Background

Yup. One more Devlin post. Synopsis so far for those of you who weren’t watching the whole multiplication vs repeated addition follies.

Keith Devlin, back last Fall, wishes that he could stop teachers from saying multiplication is repeated addition. He elaborates, big time, in “It Ain’t No Repeated Addition” in July. Denise, who teaches math, thinks about it, and asks, then how should we teach multiplication? That’s when the comments get a bit out of hand. Denise posts again. Some other people post. Even I post.

Mostly the posters and commenters were yelling and screaming about whether or not multiplication is repeated addition. In all of this, the question that matters - how should we teach, was pretty much buried.

Question pops up

Fast forward a few days. I am in New Orleans, setting up classrooms. And I stop to skim a variety of elementary and middle school math texts. And I don’t find the error Devlin is chasing. Instead I find books discussing and introducing multiple meanings of mathematics.

Could there be some texts that say Repeated Addition = Multiplication? Sure. But my unscientific sample didn’t find them. Could some teachers ignore the texts and teach Repeated Addition = Multiplication. I know that some do. But I don’t really know if it is very many. So I wondered out loud if Devlin was jousting with a straw man.

Devlin’s rebuttal

His recent column, he’s making one more go of it, attempts to rebut 6 arguments. It is longer because he will “be quoting from some of the leading mathematics education scholars of the twentieth and twenty-first centuries…”

But when he comes to my arguments, um, no. He provides next to nothing. There is one British ed journal article that says teaching multiplication as repeated addition is a problem (from ten years ago, directed to British national policy, looks like the research was a small study in London.)

And his coup de grace? Studies (one British, one Canadian) that show adults, when asked to define multiplication, respond with repeated addition.

(To look for yourself, find the heading “The Problem Is WIdespread” about three quarters of the way down)

Now, think for a moment. Of the various models we may use in teaching multiplication, isn’t repeated addition the strongest? Isn’t that exactly what you would expect an adult, 15 or 30 years removed from grade school to recall first? They remembered what we should expect them to remember - but that doesn’t tell us what they were taught.

Could he have cited something else? Yup. If he found state or national standards telling teachers to teach RA = M, but I don’t think they exist. If he had found studies that said, “teachers do this a lot”… If he could show us texts that do the same… maybe they are there. Josh at TextSavvy might know?

Two things went wrong here.

Like the engineer who comes to a school knowing math but not knowing how to teach it, Keith Devlin arrived to a topic (math ed) that he remembers. He was a student. And he probably remembers better than most. But we are talking memories, not current knowledge here.

And second. Something I recognize. Stubbornness. Look how well he writes. Pick any other column. Pick his recent interview. There’s intellect, there’s quality of expression. He hasn’t poorly defended his position because he argues poorly; it’s just stubbornness without facts supporting it.

I’d be interested in recommendations about multiplication should be taught, but as for this topic, I think this will be my last post.

Thinking about a Carnival of Mathematics

That’s #39 - hosted by It’s the Thought that Counts. Clever name, huh?

It’s a new blog to me. The authors are A (computer science) and Z (physics). Subtitle: “critical analysis and interesting ideas.” Self-description of content: “commentary on all manner of topics — politics, society, science, morality, religion, and whatever else comes to mind.” So, the writing’s good. While your looking at the Carnival, and trying the puzzle, you might take an extra peek around.

Puzzle? Yup. Just for fun, he (she? they?) leads off with a combinatorial poser (39 people sitting around a circular table, none in the right place…) (Looks combinatorial, but might bend quicker to algebra).

Reminds me of this old problem, that appears never to have gotten a general solution on this blog. I’ll repost, soon.

New math blog: f(t)

A nice addition to the links here: f(t) is a new (from July) high school math teacher blog. Kate teaches in Syracuse, New York. (I’ve been there!)

So far she’s posted problems, lesson ideas, and a little bit about her work. Nicely written, easy to read.

Best of luck!

Once more on Devlin (but not on teaching)

Keith Devlin, mathematics commentator, wrote: that “Stopping teachers saying that multiplication is repeated addition” would be a good thing. He sharpened it (”I wished schoolteachers would stop telling pupils that multiplication is repeated addition“), and repeated it (”a plea to mathematics teachers to stop telling students that multiplication is repeated addition.”)

Now Josh has an interview with Devlin, and has opened up the comments section.

Denise at Let’s Play Math took it seriously, and wrote. The subsequent storm got covered by Denise, Mark, and many others, including me. Josh at TextSavvy wrote a bunch, but without allowing comments.

Now Josh has an interview with Devlin, and has opened up the comments section. I think the discussion there may become interesting.

For the record, I don’t question Devlin’s math (although there is wiggle room there), but his approach, once engaged with teachers, was irresponsibly inflammatory. And, for the record, the ‘error’ he picks on is certainly not universal.

Devlin’s strawman

These last two weeks I have been in and out of elementary and middle schools. Great chance to peek at math texts. And you know what? I am not finding the books that say that multiplication is repeated addition.

Today I look at some (Scott Foresman?) whatevers, maybe 3rd, 4th and 5th grade. The book discussed “meanings” with an “s” of multiplication, and ran through several sections that covered repeated addition, skip counting, arrays, and cartesian product.

The arrays, by the way, were of relatively fat dots, meaning, I hope, that half rows or quarter rows could be introduced at some point in the future.

This whole brouhaha about “stop telling kids multiplication is repeated addition” — do we know that this happens everywhere and all the time? Are there textbooks that get it wrong?

Or was Devlin just jousting with imaginary opponents?

Multiplication? Addition? Comments?

Kenneth Devlin wrote “stop telling your pupils that multiplication is repeated addition” and all hell broke loose.

There was a storm at Let’s Play Math, and then Denise wrote a second post. There is still a storm raging at Good Math Bad Math. And a bunch of places I don’t normally go. And then Josh at Text Savvy has written 11 posts (they start here) - but he disabled commenting, which turns the conversation into an echo chamber. (Josh, it looks bad if you complain about your comments not being published if you run a site where comments are not allowed)

So, my two cents.

Multiplication is not repeated addition.

Multiplication can represent repeated addition.

Devlin’s point was directed to how we teach little kids math, and he blew it. So we stop telling kids that x = + (rep) and we tell them what exactly instead? Don’t ask Devlin. He devoted a second column to the issue, and never got there.

Doubling back, what was his objection? That we say “math is repeated addition” and that somehow this ruins kids’ ability to handle arithmetic: “Multiplication simply is not repeated addition, and telling young pupils it is inevitably leads to problems when they subsequently learn that it is not.” He’s wrong.

What should we do?

Teach multiplication through repeated addition, skip counting, counting arrays, finding unit areas of rectangles, Cartesian product, scaling… That’s too many to start, isn’t it? Pick one, then one more. And tell the kids that multiplication will manifest itself in many other ways as well… Add more… and remind them… No big deal here.

By the way, congratulations to Denise for handling this in an intelligent way. And for, ever so briefly, becoming the central blog in a math ed tempest.

Carnival of Mathematics returns

#36 over at Rigorous Trivialities, after a break. (The previous #35 was at Catsynth.com, back a month ago…)

This edition is short (15 links to 14 blogs), punchy, leans to the advanced, but has some school math, and starts with me. Check out the Math and Logic Play puzzle, and the provocative Out in Left Field post about how math is taught.

The Carnival of Mathematics homepage has a list of all the previous carnivals. I think they need volunteers for future weeks.