# Micro-math vs. the bell curve

There’s A Better Way to Teach Math, writes David Bornstein on the New York Times’ Opinionator Blog. Jump Math, a K-8 curriculum used in Canada and England, might even “eliminate the bell curve” in math class, Bornstein suggests.

John Mighton, once a Toronto math tutor, developed Jump.

” . . . very early in school many kids get the idea that they’re not in the smart group, especially in math. We kind of force a choice on them: to decide that either they’re dumb or math is dumb.”

Children’s working memory is limited, Mighton says. Many children have trouble “remembering math facts, handling word problems and doing multi-step arithmetic.”

Despite the widespread support for “problem-based” or “discovery-based” learning, studies indicate that current teaching approaches underestimate the amount of explicit guidance, “scaffolding” and practice children need to consolidate new concepts. Asking children to make their own discoveries before they solidify the basics is like asking them to compose songs on guitar before they can form a C chord.

Jump breaks down math problems into “micro-level” concepts to ensure complete understanding. “No step is too small to ignore,” Mighton says. “Math is like a ladder. If you miss a step, sometimes you can’t go on.” Students who succeed in small steps build the confidence needed to “jump” forward, he believes.

In a controlled study in rural Ontario, the Jump fifth graders “achieved more than double the academic growth in core mathematical competencies.”

1. I went to the “Jump Math” site and looked at their sample problems for the beginning of third grade. (My daughter has just finished Grade 1 Saxon.) In the first 44 sample pages, there was nothing she couldn’t do.

SO my question is:

Is the bell curve disappearing because all students are now learning, or because the curriculum has just gotten easier, so that the tests aren’t measuring students’ abilities as accurately, especially at the high end?

(My daughter’s not a genius, and Saxon is not super hard, so it seems going from first to third grade math should pose a challenge– as opposed to a ‘correct answers shouted from the other room as she does something else.)

2. Roger Sweeny says:

If each student had a personable teacher with infinite patience guiding him through Jump, I’m sure there would be wonderful progress.

Whether an ordinary teacher with 20 heterogeneous students can do it is a lot less certain.

3. Mark Roulo says:

I fear that Jump Math solves the problem of some kids doing poorly by … um … going slower:

Example problem from their site for 4th grade:

The snow is 19 cm deep at 3 pm. 5 cm of snow falls every hour. How deep is the snow at 7 pm?

Example problem from Singapore Math 4A workbook:

David bought 2 computers at \$3569 each. He had \$2907 left. How much money did he have at first?

Having said this, there are at least two things in favor of Jump Math:
(a) The kids seem to be doing better. Jump Math may well be quite an improvement over what they had been doing, so comparing it to something that I think is even better may be making the perfect the enemy of the good.
(b) The kids are doing two part word problems. This is very good.

What I don’t like is that the kids can do these problems by repeated adding rather than by using multiplication. The kids will do that if they can, which does not reinforce/practice the long multiplication that they should know by now. I also prefer that the kids start using larger numbers in problem by 4th grade so that the large numbers become less scary.

Still, this could well be an improvement over what they had before. If there is explicit instruction in how to do math this could be quite a step up.

4. SuperSub says:

I don’t think this is a fad as much as rediscovery by some of effective centuries-old mathematical education… though I do agree that the difficulty might be a little low for what could be taught. That being said, when you have 4th graders doing real 1st grade math, it seems an entirely appropriate step to catch them up.

5. Mark Roulo says:

Hmmm … I find an explanation of how to do long multiplication (section “NS7-8 Long Multiplication”) in the grade seven example pages. The problem they are working in the example is: 3 × 42.

I’d accept that this might just be review (one problem with only seeing the first pages of a given year is that the problems might be review to deal with forgetting over summer), but by 7th grade I’d kinda expect that one wouldn’t be reviewing 3 × 42.

This would concern me.

I wish I knew when they started teaching fractions formally (e.g. adding fractions with different denominators). It appears that fractions arrive quite late.

6. We have been teaching math for literally thousands of years, how could we not know how to do it?

7. Michael E. Lopez says:

We have been teaching math to aristocrats with tons of time and resources, or to slaves with no choice but to learn math or die for thousands of years. (Math was, in ancient Greece, for example, not one of the “liberal” arts because it was a skill that was associated with manual labor. After the fall of Rome, Math moved into Capella’s seven liberal arts (arithmetic and geometry) and became something that was taught to the wealthy and those who eventually entered monasteries and then universities.

Teaching math to everyone is a concept less than 300 years old, and only in a small part of the world.

It’s OK to rely on history for our points, but let’s be honest about what we’re talking about. Circumstances and motivations matter.

8. SuperSub says:

Rob-
Because the people given the responsibility of teaching it (primarily elementary teachers) don’t value mathematics and often don’t know mathematics, let alone how to teach it properly.
Our society, from the poorest to the wealthiest and most powerful, has collectively forgotten the value of focus and hard work, constantly seeking the quickest and easiest path.

9. cranberry says:

I looked at the sample worksheets. The 8th grade worksheet covered multiples. The company chose the pages to post, so I’m not unfairly picking on an exceptionally easy sheet.

That isn’t easy, it’s remedial. If you set the requirements low enough, of course the achievement gap disappears, but not in a good way.

There may be an argument to make against heterogeneous classes. The teachers may get better results on the whole, because the instruction (as described) tries to discover the very basic concepts a student didn’t understand. If you don’t understand addition, you won’t understand multiplication, and you won’t understand algebra. It makes sense not to rush through a curriculum with students who don’t have a firm grasp of the concepts. On the other hand, for some students this would be much too slow.

This is an argument for different speeds of teaching. It is not a curriculum which should be used for all students. There are some students who are ready for Algebra (real Algebra) in 7th or 8th grade.

Others discuss the curriculum here: http://mathforum.org/kb/message.jspa?messageID=7436719&tstart=0

10. Mark Roulo says:

“The company chose the pages to post, so I’m not unfairly picking on an exceptionally easy sheet. “

Yes, but the company *also* consistently posted the first 20-40 pages of worksheet for each year. It looks pretty far behind where a student for any given grade level “should” be, but maybe a bunch of this is refresher to deal with summer learning loss?

I’m *still* worried about things like the first exercises in the grade 4 workbook showing how to count on your fingers (with pictures!), though …

The pacing does look pretty slow.

11. Michael E. Lopez says:

Cranberry –

I’m not sure “remedial” is the right word. I might try something stronger, like “insulting” or “pathetic.”

I’m also vaguely disturbed by the questions that asked “Is 12 a multiple of 4? How do you know?”

I suspect that the “how do you know” question has some specific right answer. That troubles me. I’d rather think that the following would all be good answers:

* “Because I memorized my times tables and 12 is in the four row.”

* “Because 4×3=12”

* “Because 4 is a factor of 12” (Which, let us be honest, is exactly the same thing as saying that 12 is a multiple of 4.)

* “Because 4×2 is 8, and 20=12+8, and 20 is a multiple of 4.”

My favourite (read least favourite) question though is when we’re told that:

Alana wants to find all pairs of numbers that multiply to give 10. She lists each number from 1 to 10 in a chart. She looks for a second number that multiplies with the first to give 10.

a) Why didn’t Alana list any number greater than 10 in the first column of her table?

b) Why didn’t Alana list 0 in the first column of her table?

This is eighth grade math we’re talking about (supposedly… bear with the fiction for a moment). How are these as answers?

a) Alana didn’t list any number greater than 10 because, like us, she was apparently never taught about fractions, or perhaps was mistakenly told that fractions don’t count as “numbers.” Alternatively, she might not have been serious about finding all number pairs that multiply to give 10 because she wanted to finish her little experiment before she died.

b) Alana didn’t list 0 in the first column on her table because she didn’t want to be tempted to have to double her work load and include negative numbers, which she should have learned about before 8th grade but apparently, like us, hasn’t. Also, as discussed above, she wanted to finish her experiment before she died and doubling her work load didn’t facilitate that. Finally, it might be the case that Alana was actually performing some real practical task rather than just acting on some strange, disembodied desire to “find all the pairs of numbers that multiply to give 10”, and while we were not given information about that task, and so cannot dutifully report it here on this worksheet, there may be practical issues that inform her decisions to only dwell on integers between 0 and 11, exclusive.

That’s the sort of smart-ass answer I would have given when I was in 8th grade (I might not have used the expression “strange, disembodied” but close enough.)

12. Mark Roulo says:

Michael,

The phrasing of the 8th grade example question strongly suggests that fractions are covered very late and/or are pretty much an afterthought.

So … 8 or 9 years to nail down math with the Natural numbers.

Sigh. Might be an improvement over what the kids are doing now, but … man …

13. SuperSub says:

I wonder if the designers took the schedule from current curricula to make it more marketable… my 7th graders constantly ask me for help on their math homework – fractions.

14. Cal says:

It’s extremely obvious that none of you work with students on the lower half of the ability spectrum. And I very much doubt that Michael Lopez would have given that answer.

Obviously, what Jump Math does (as some have pointed out) is slow math down to a crawl. That’s what many students need. But it’s not ignoring the bell curve or flattening it if it takes some kids five years longer to get to algebra (assuming they can).

15. Cardinal Fang says:

Mighton says many students struggle with “remembering math facts, handling word problems and doing multi-step arithmetic.”

I’m always bemused by people who imply that word problems are some kind of advanced exercise that only the best math students can handle. (I don’t mean Mighton thinks word problems are abstruse. His quote was merely a jumping-off point.)

In reality, word problems are the whole point. If you can’t figure how much 16 oranges cost if oranges are five for a dollar, why bother even learning the times table? What are math facts for, if not for using in real world situations?

Outside the classroom, nobody is going to come up to you and say, “Quick, what’s 16 times 100/5?.” If they did, you could just whip out a calculator if you couldn’t compute it in your head. But very likely you will need to figure out if you can afford to buy oranges.

If you can understand how to set up math problems, but are bad at mental arithmetic, you are inconvenienced. If you’re great at mental arithmetic but can’t set up problems, your arithmetic skill is useless.

16. Roger Sweeny says:

Michael E. Lopez,

I feel fairly sure they were asking, “What are the factors of ten?”–and I’ll bet the students knew that. Listing the positive integers from 1 to a number and then asking for each, “can I divide this evenly into the number?” is the basic way to find the factors of a number. (Of course, you only have to go halfway through.) It takes some understanding to explain why you don’t have to list any number past ten (multiplying eleven–or more–by some number to get ten is only possible if the “some number” is less than one, and one is the smallest positive integer). The reason you don’t list zero isn”t trivial. Zero times anything is zero, so you can’t multiply zero by anything to get ten.

No doubt this is obvious to you, and it’s obvious to me (it had better be; I teach high school physics and chemistry). However, it is not at all obvious to a large number of my ninth graders.

We have math curricula which say students should have been taught a whole lot of things. And most of them can memorize well enough to get a passing grade on a unit test. But then they forget, and when they come to me, I find an astounding number have remarkably little in the way of math skills. It wouldn’t surprise me if half of them got that question wrong.

17. Remember that it is only half way through Mathematica Principia that 1+1=2 occurs. Math is actually extremely, extremely subtle. Smart kids skip steps; until they don’t get it. Weaker students really need every simple step and its reason, in simple language, explained. This is very, very hard.

Ask yourself: what really is subtraction? Explain all of the ideas to a first grader. Good luck.

18. georgelarson says:

Dennis Ashendorf

I think your point is true, but I know you are not recommending that anyone use Principia Mathematica to teach elementary math. The steps may be small but they are abstract and pointless to many. They were to most minds before Russell and Whitehead.

“Explain all of the ideas to a first grader.” My mother did for me and i was a hard headed 6 year old.

19. North of 49th says:

The Times article really doesn’t give a clear idea of Mighton’s work. For one thing, although JUMP is used in some school systems it is not widely accepted in Ontario. My district forbids its use because it is considered too focused on rote learning, algorithms, arithmetic etc. and not enough on discovery and concepts. JUMP is not on the approved list of curricula for the province of Ontario, or to my knmowledge of any other province.

However, to get an idea of what Mighton is all about, you need to read his books (which are quite good and thought-provoking), or hear him speak. The workbooks are NOT the JUMP “method” or “program,” they were developed in order to assist parents or teachers wanting to use JUMP in the classroom or at home (for a decade or more JUMP was exclusively an out-of-school tutoring program, targeting low-income children in inner city schools). The workbooks are designed to follow the Ontario math curriculum, which has some distinctive characteristics that are different from many USA expectations.

— math facts are not expected to be learned (memorized) at any point. Teachers are not allowed to spend much time teaching them. This gives Kumon a biiiiiiig market hereabouts.
— specific algorithms are also not required to be learned or mastered. Calculators are allowed from day 1 even in the provincial tests for grades 3 and 6.
— the emphasis is on using manipulatives and generating your own problems
–math textbooks and “programs” are strongly discouraged (no Everyday Math around here)
–some algebra is taught in middle school — it is one of the five strands that are taught — but USA-style Algebra is for Grade 9 and 10.

I’ve been to one of Mighton’s two-day seminars and I was quite impressed by him. He does NOT “dumb down” the math (quite the opposite — he had a spectacular ability to ramp up even the simplest instroductory exercise into a challenging one in a very short time; unfortunately I can’t explain how he did it — but he did, consistently, with topics such as fractions, integers, binary numbers, and several others. However, he does strongly believe in starting where the students are — which in many cases is in a state of learned helplessness with no idea what to do even to begin. He teaches the finger-counting stuff to those who need it — always some in every class, since we have full inclusion here, and some kids with real developmental delays will be in every class. He was brilliant at getting every student (these were elementary kids — Grade 5 and 6 when I observed) involved and challenged, all at the same time, on the same topic, but at multiple levels of difficulty. I think most teachers would have trouble doing this consistently (I know I would).

We have a big problem in Ontario with math achievement in secondary school, which is IMO due to the weak and fragmented elementary math curriculum. I have graded our provincial math tests and students can get a top mark for wrong answers as long as they “explain their thinking” in a prolix and terminology-heavy manner. Not a good omen.

But people should not blame Mighton or JUMP for “dumbing down” the math — he is trying to deal with the results of poor curricula and has had some pretty spectacular successes with student so far behind that it was thought they could never get high school math credits or were considered cognitively handicapped. Some of these have since completed advanced secondary math and gone on to university math.

I suspect Mighton’s success comes in part from his being outside of the education scene entirely. He is an award-winning playwright and got his PhD in math after achieving reknown in the arts. He’s a low-key, self-effacing individual but a real out-of-the-box thinker and so totally unwelcomed by the ed school crowd. One thing that struck me from his books was his recognition, from experience, of the non-linear nature of learning where in some cases, consolidation of what appear to be low-level skills precipitates a huge jump in conceptual development and achievement. The late Ogden Libndsley and Eric Haughton have written about this and I have observed it myself, more in the language area than in math.

20. About a decade ago I went back to my old elementary school and came upon my old 3rd grade teacher, Mrs. Barton, who was set to retire after 39 1/2 years teaching.

To make a long story short, she dropped everything and had her students show off for me. Among other things every single one of them knew every single one of their multiplication tables–just like we did over 25 years before.

We need more Mrs. Bartons and less “creativity” and fads.