The Math Teacher’s Job is Neither to Teach the Lesson, Nor to Help Individual Students Who are Struggling!

….but rather, to prepare a lesson from which ALL the students can learn!

… according to the way that Japanese math teachers are taught their craft, as described below. You will find that these methods, which include Lesson Study, are pretty much the exact opposite of American “Direct Instruction” or “Teaching Like A Champion.”  Given that nobody claims that Japanese students lag behind American ones in math or science, perhaps we in the US could profit from examining how other nations’ teachers do it. Note also that this description is of mathematics lessons in elementary school, not middle or high school.

Please read the following description and leave comments on what you think.

From Tom McDougal. Lesson Study Alliance, Chicago [and brought to my attention by Jerry Becker. – GFB]
It’s not the teacher’s job to teach the students!

By Tom McDougal

What?? You might be thinking. What else could the teacher’s job be but to teach?

The teacher’s job is to ensure that students learn, all of them, we hope, though we know we will usually fall short.

In Japan, most (elementary) math lessons are designed as  “teaching through problem solving” lessons (TtP). A teaching through problem solving lesson typically includes the following parts:

1.  introduce the problem
2.  explicitly pose the task for students
3.  students work on the task (5-10 minutes)
4.  share student ideas
5.  compare and discuss the ideas for the purpose of learning new mathematics
6.  summarize major points from the lesson
7.  student reflections

(There is sometimes overlap, and a back-and-forth between some of these, e.g. #4 & #5 may be combined.)

While students are working on the task (#3), the teacher walks around the room, monitoring their progress. Japanese educators have a term for this, kikkan shido, or  “providing] guidance between the desks.” They recognize that there are different ways to do kikkan shido, and it is often a subject of discussion in Lesson Study. During planning, for example, a team will usually discuss how – or whether  – the teacher should respond to a student who exhibits a particular misconception; during the post-lesson discussion, there may be argument about whether the kikkan shido was effective. And, it is considered a skill that new teachers need to develop.

Teachers who are inexperienced with TtP lessons often make an unfortunate error while doing kikkan shido: they see a student who is struggling, or who has done something wrong, and they stop and help that student. After several minutes the teacher moves on, encounters another student who is having trouble, helps that student, and so on. Then, suddenly, time is up, and the lesson ends.

There are at least four important drawbacks to this type of kikkan shido. First, as my description suggests, it uses up a lot of time. The teacher may never get around to all of the students, and other students who need help may never get it. Second, by addressing misconceptions privately rather than publicly, the teacher deprives other students of the opportunity to analyze those misconceptions and learn why they are incorrect. Any experienced teacher knows that certain misconceptions are very common, so when one student makes an error that stems from a common misconception, that offers an opportunity to “inoculate” other students against making the same error sometime later.

The third problem with tutoring students individually is that it conflicts with the whole premise of teaching through problem solving. You expect that some, or even all, of the students will have difficulty with the task; that’s why it’s called “problem solving” and not “practice.” Teaching through problem solving involves an expectation that students will have difficulty, but that the comparison and discussion phase will address their difficulties and that, by the end of the lesson, all (or almost all) of the students will have learned what they need to know.

And fourth, we want to help students learn to give viable arguments and to critique the reasoning of others, the third Standard for Mathematical Practice in the Common Core State Standards. To accomplish this, we need for students to share and discuss different, perhaps conflicting solutions. Students need to do the critiquing, not the teacher.

Of course, some errors are simply the result of sloppiness, or otherwise unrelated to the main learning goals of the lesson. So when the teacher sees an error while conducting kikkan shido, he or she has to decide: should this be addressed privately or publicly? What should I say to this student? Do I expect that, by the end of the lesson, this student will understand what he or she has done wrong? This is a tricky decision, and an important part of lesson planning is anticipating different student responses, correct and incorrect, and deciding ahead of time how to handle them.

Caring teachers naturally feel drawn to help struggling students: they feel like it is their duty to help those students right now. To counteract that impulse, I say, bluntly:

It is not the teacher’s job to teach the students. It’s the teacher’s job to create a lesson that teaches the students.


‘Discovery Math’ is Weird but a Good Idea Nonetheless

This was brought to my attention by Jerry Becker
From, Saturday, September 3, 2016. SEE
No, teaching math the “old-fashioned way” won’t work: Paul Wells
In response to the latest EQAO report, many parents insist that “discovery math” is the cause of low test scores in Ontario.
By Paul Wells  (National Affairs)
According to the latest EQAO report, half of Ontario Grade 6 students don’t meet the curriculum standard in math. That’s a problem. But it’s not the only one.

What worries me is that only 13 per cent of students who didn’t meet the provincial standard when they were in Grade 3 manage to catch up so they meet the standard for Grade 6. That’s the lowest number on that indicator in five years.

If you fall behind in math you stay behind. That’s why it’s important to get it right, not just at some vague moment in the future, but for kids who are in Ontario schools right now.

Fortunately, every parent in Ontario is sure they know how to teach math. Many parents want to get rid of “discovery math,” broadly defined as “doing it weird.” If only that loopy Liberal government would teach math the way we learned it when we were kids, the theory goes, there’d be no problem.

Sure, great, except for one thing. Very few parents I’ve met can perform more than the most rudimentary arithmetic for themselves. If you all learned math so well, why do you inch toward Junior’s algebra homework with a cross and a bulb of garlic?

Discovery math, to the extent it means anything, is an attempt to apply in a formal setting the insights about numbers that good mathematicians use routinely. People who are comfortable with numbers use all sorts of strategies to work with them. Confidently, through a kind of learned intuition.

So subtracting 272 from 836 is an altogether different proposition from subtracting 998 from

1,002. In the first case, you’re likelier to write it all out, solve the ones column first, carry 100 to the 10s column so you’re subtracting seven from 13, and so on. In the second case, I’d count up four from the lower number to the higher. It’s a really big drag on a kid to make her do the second problems the same way as the first. And parents who read “add to subtract” on a homework sheet, chuckle and roll their eyes, are committing malpractice.

This summer I made my stepson spend some time on Khan Academy, an educational website, to brush up his math before he enters Grade 8. He was briefly baffled by questions that asked, say,

6 1/4 – 3 3/4. One way to do it is to convert both sides to improper fractions. But it’s easier if you simply recognize that 6 1/4 is the same as 5 5/4. You can do the differences in your head in about two seconds.

The question is, how do you produce the kind of students who will make that insightful leap? All I know for sure is that you don’t do it by teaching a bunch of rules students will learn by rote – the beloved “old-fashioned way.” That may work for basic math facts. I did make our son practice his basic addition, subtraction and times tables one summer until he knew them from memory. I wish schools would take more time to nail those basic facts down. Since our school wouldn’t take the time, I did.

But very quickly, math becomes so complex you can’t have a rule for everything. Khan Academy teaches and tests 111 different skills at the fifth-grade level alone. You’d go crazy learning a rule for each skill. You must be able to intuit a useful method for each situation.

Modern curricula recognize, and try to teach, that flexibility. I refuse to say that’s a mistake. There is even empirical evidence it’s not. A March report from PISA, the international testing organization, found that in countries where students say they rely heavily on memorization, they scored starkly lower on complex advanced math questions than students who memorize less. “To perform at the very top,” the report concludes, students must learn to do math “in a more reflective, ambitious and creative way.”

What’s to be done about those declining EQAO scores? First, Ontario should support teachers by sharing best teaching practices more widely. In some countries, like Japan, teachers spend far more time mentoring younger newcomers to the profession, and sharing techniques among colleagues. Ontario schools should follow suit.

Second, support students by giving them more practice time. The only way to learn how numbers work together is by tackling incrementally more difficult questions, lots of them, over time. Kids need to practice insight just as their parents practiced times tables. If they do, they may just grow up knowing how to do math, not just how to complain about math teachers.

Paul Wells is a national affairs writer. His column appears Wednesday, Friday and Saturday.
It never appears to occur to either journalists or educational conservatives (or political ones) or to those deeply invested in undermining public education in the interest of turning it into a for-profit investment that curricula come and go due to fluctuations in standardized test scores, but the one sacred cow that is NEVER seriously interrogated is the testing process or its concomitant methods. Give me control of the tests and how they are scored and I ABSOLUTELY GUARANTEE that I can make results fluctuate to suit any political agenda and outcomes one might wish to see.
Mathematics itself has changed almost not at all when it comes to the content of K-12 curricula in most countries (and certainly in the US and Canada). Blaming decreasing test scores entirely on a teaching approach to math that is politically unpopular misses almost entirely that if assessments are skewed away from the kinds of thinking that teachers are trying to help students develop, it’s a slam dunk that scores wlll go down. And when assessments are developed to reflect more conceptual understanding (and scores go up), the conservatives and nay-sayers scream that the tests are “fuzzy.”
Once this sort of politicization of education is allowed to dominate the conversation, as it clearly is doing in this article and in many of the accompanying comments, there’s no chance for thoughtfu educators to pursue anything but lock-step, computation-dominated “math” teaching. Only that’s not math, and my Smart Phone does all of that vastly quicker, more accurately and more easily than nearly every human who has ever lived or ever will. If you want
 kids to be adept at replicating donkey arithmetic, so be it, but no one I teach will be encouraged to limit herself in that way.
Published in: on September 15, 2016 at 10:20 pm  Comments (1)  
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Is Math Necessary?

This is worth reading. It’s a fact that we do NOT have a shortage of trained STEM grads, and it’s also true that very, very few people will ever use any concepts from advanced math in their work or in their day to day lives.
(As a former math teacher, I rejoice when I find a way to use relatively advanced math, eg algebra 2 or above, in the real world – which shows you that it doesn’t happen every day, even for someone who’s actively looking for it.)
So why do we require every single HS grad to master whatever the current Algebra 2 curriculum consists of?
via Mike Simpson  (remove)
In his new book The Math Myth: And Other STEM Delusions, political scientist Andrew Hacker proposes replacing algebra II and calculus in the high schoo …
Published in: on March 3, 2016 at 3:33 pm  Comments (2)  
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On the War in Vietnam, and its Memorial

Vietnam Veterans Memorial

This is a starkly geometrical monument to the tens of thousands of American fighting men and women who died in the Southeast Asian theater of war, from 1957 to 1975.

At first, it just looks like a long, black, polished granite V or chevron half-buried in the ground. When you get closer, you begin to see that each of the stone panels has names that were carefully carved into the stone, five names per line, in no immediately obvious order.

But the names are in fact carefully catalogued in chronological order by date of death (or initial injury/wound/disappearance). The order starts in panel 1E, just to the right of the center bend, and continues to the right as you face the wall, up until the single line of names on panel 70E, for May 25, 1968, and then continuing with another single line of names on panel 70W, also for 5/25/68, and continuing from left to right up to panel 1W.

If you want to look up a person who died in that war, you can look their name up alphabetically online or in a printed directory that is the size of a large old-fashioned telephone book. There you can find their branch of service, rank, casualty date, state of origin, and a code giving the panel and line number on which his name is engraved.

Math connection:

  1. How many names are carved on the wall?
  2. What was the midpoint of the war, as far as US combat deaths are concerned?

The second question is by far the easiest to answer: it was May 25, 1968, because the panels on the left of the vertex are virtually identical to the ones to the right of the vertex in terms of numbers of names.

The first question is a bit harder.

It was suggested by an elementary-school child that one could simply think of each half of the V as a triangle; if we were to use our imagination and flip, slide, or rotate one side above the other one, we would get a rectangle. Then, count the number of lines of names in the tallest panel, multiply by the number of panels, and then by five (because there appear to be five names per line, no matter how long or short the names) and we should have a pretty good estimate.

Here’s the idea:

triangles of Vietnam memorial

Using that rule, the tallest panel that is all names (no extra inscriptions) has 137 lines of names. The shortest panel has a single line of names. There are 70 panels to the east, and 70 to the west. Re-arrange them to make a rectangle, and we have 70 panels with about 138 lines of names, each with 5 names per line. Multiply all that and we get 48,300 names.vietnam war mem 1Unfortunately, this number is much too low: more than 55,000 American service personnel died in Vietnam.

What went wrong?

Let’s look at the wall again, more carefully.

bend in bottom of wall

Each side is in fact NOT a triangle. They are quadrilaterals. The bottom edge makes a bend at about panel 20, a little bit to the right of the head of the man in blue looking at the wall. It’s easy to miss.

If you spend some time counting and recording the number of lines on a sample of panels, you will notice that the number of lines does not go up uniformly.

I carefully counted and recorded the number of lines of names for about a few dozen panels, and found that my data, when graphed, looked more like this:

triangle rect vietnamvietnam war mem 2

(By the way, I didn’t count every single line in all of those panels, except for the ones to the right of panel 30. I found patterns that allowed me to figure out how many names there were in panels 3 through 17. How did I do it?)

But how do we calculate the total number of names now?

My original triangle was easy enough, but the new, green triangle was pretty difficult to measure the area of, so I decided to cut the figure up differently, as shown here:

vietnam war mem 3

We have a light blue trapezoid (or close enough) on the left, which will be matched with another trapezoid rotated from the west side, producing a rectangle whose base is 20 panels long and that is about 137+128 lines tall; so it has 20 times 265 lines of names, or 5300 lines of names.

The darker blue triangle will be matched with its twin, also rotated from the west side, to produce a 50 by 128 rectangle with 6400 lines of names.

That’s a total of 11,700 lines of names.

Five names per line give 58,500 American military deaths from the Vietnam War.

Which is quite close to the official figure of 58,272 names.


Obviously there was more than one country involved in this war. If you want to see memorials to the Vietnamese, Cambodians, Laotians, South Koreans, Australians, and other nationalities, both fighters and civilians, who were killed in that same conflict, you will need to look elsewhere, overseas.

Estimates of the number of Vietnamese, Laotian, and Cambodian killed during and right after the war vary widely and are not very accurate. It is safe to say that several million Southeast Asians were killed during the war and its aftermath.

If one were to build a memorial to all those Southeast Asian casualties with the names spaced in the same way as in this one, how large would that memorial need to be?

For scale: If we begin counting at 1960 and end at 1980, that’s about 20 years, or 240 months. If three million Southeast Asian combatants and civilians were killed during that period, then that means that about 12,500 Vietnamese, Cambodians and Laotians were killed by bombs, bayonet, napalm, bullets, knives, drowning, and so on, each and every single month. In four months, that’s about 50,000 dead, which is pretty close to the total number of American combat deaths for the entire war.

Two hundred forty months divided by 4 months gives 60, which means that one would need approximately sixty memorials the size of this one to honor the memories of the millions of Southeast Asian civilians and fighters killed during and after the Vietnam War.


Post script:

While touring this memorial on Saturday, November 9, 2013 (i.e. two days before Veterans Day / Armistice Day) with four other math teachers, I was shown a couple of features I hadn’t noticed before:

  1. Every ten rows, in every second panel, there is a dot; this makes it much easier to count the total number rows in each panel.
  2. A few lines of names actually do have six names. It does not happen very often, but is definitely enough to throw the count off by a bit.


Some grim calculations in honor of today being Veterans Day (actually, the 11th day of the 11th month of 1918 was the day that an armistice (a cease-fire) was proclaimed to stop the fighting on the Franco-German front during World War One.

58,000 Rough estimate of US war dead in Vietnam theater
         1,218,000 square feet at 3 fet by 7 ft per coffin
               43,560 one acre (in square feet)
                       28 acres neded to bury all American Vietnam War dead
2,000,000 rough number of Vietnamese, Cambodian and Laotian war dead
       42,000,000 square feet at 3 fet by 7 ft per coffin
               43,560 one acre (in square feet)
                     964 acres neded to bury all Southeast Asian Vietnam War dead
                    1.51 number of square miles needed


And here is what I just wrote on the time line of a colleague, who had the bad fortune to get drafted during Vietnam and served over there:

“As someone who lived through the Vietnam War era dreading the day when I would be ordered by my draft to report for induction in a war that I knew for a fact was completely unjustified, I am sorry that you and your buddies were drafted and had to fight and risk or lose [your] lives over there and fight against one side in a civil war. (Some wars are justified, many wars are not.) I am glad that my lottery # ended up being a little too high to get called up. I did my best to try to stop the war… I suspect that our efforts, along with the [much more important] actions of many soldiers etc who saw no point in the war, may have saved a lot of lives in getting the US out of there. In any case, I’m glad you made it back safely, Ben.”

Published in: on November 11, 2013 at 7:19 pm  Comments (4)  
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A quick look at some of the Common Core math standards, grades 7 & 8

I was prepared to be appalled by the Common Core math standards, but I’m not.

The CC math standards — at first glance, anyway — actually look quite a lot better than the old middle-school math standards we used to have in DC, which had interminable lists of many minute details kids were supposed to know — and which lists repeated themselves over and over again in grades 4, 5, 6, 7 and 8.

Mile wide, centimeter deep it was.

So far, I don’t see any sign of that ridiculous nonsense in the new CC standards.

However, I can just see teachers requiring their students to copy and recite turgid prose like this, which is a direct quote from page 56 of the PDF. It means something to me, but to how many other adults?

“Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.

For example, collect  data from students in your class on whether or not they have a curfew on  school nights and whether or not they have assigned chores at home. Is  there evidence that those who have a curfew also tend to have chores?

A Page For Folks Who Might Find that they Like Math

A web page with very fun explorations that show what real mathematicians do:×17-4-coloring-with-no-monochromatic-rectangles/

(Hint: long division, balancing checkbooks, and adding fractions aren’t such a big part. I suspect that if we taught some of this stuff in class, kids would see that, in fact, math has some fun aspects — it’s not all drudgery. There is a lot of “gee-whiz” stuff that is accessible to the average layperson or young’un.

Published in: on February 9, 2012 at 4:49 pm  Comments (1)  
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More Problems With DCPS Curriculum and DC-CAS

Upon taking a closer look at the DCPS standards and the DC-CAS, I submit that they should probably both be ignored by any teacher who actually wants to do right by students. If you are doing a good job teaching the things that students should actually know, it won’t make much difference on their DC-CAS scores. Conversely, if you teach to the DC-CAS, you are short-changing your students.

Case in point: Standards in Geometry and Algebra 1 ostensibly covered on the 10th grade DC-CAS. Recall that all 10th graders at this point in DCPS have supposedly finished and passed Algebra 1, and are enrolled in at least Geometry by 10th grade.

I have prepared a little chart giving the standards (or learning objectives) for Geometry: the ones listed in the DCPS list of learning standards, and the number of questions that I found on the page of released DC-CAS questions that supposedly address that standard. There is almost no correlation at all. In fact, if you threw a dart at the topics and chose them randomly, you would do a better job than the test-writing company did.

Published in: on March 23, 2011 at 12:37 pm  Leave a Comment  
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Why Does EVERYBODY need to learn Advanced Algebra?

An interesting article by Dennis Redovich:

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