Monday, January 20, 2020

Improved Math Instruction

I will be thrilled if this works. Some observations.

Brilliant new educational ideas often don't scale up. They work as long as you have motivated, specially-trained teachers who want everyone to succeed. However this seems to have allowed for some of that in its design, and may get past it.

This is not the math drills beloved by education traditionalists, but it does involve repetition and seems a cousin to drill. They are wise to not call it that. It breaks the work into even smaller components. It is emphatically not the way many newer math approaches have gone the last sixty years. Traditionalists might want to be prepared to embrace this as at least a less-bad alternative.

Farther down the article we get into some unanswered problems.  The creator of JUMP is pleased that it works from a "growth mindset."  This is very fashionable these days but remains unproven, except in the fairly obvious - well, it used to be fairly obvious anyway - idea that anyone can get better at things if they work at it. It is good to keep children, or anyone, from being discouraged too early.  Yes, it does feel bad when you compare yourself to others and aren't as accomplished.

The problem of the better students being bored is glossed over.  Supposedly, they are going to be happy because the whole class is happy and we're all in this together.  Who needs silly old grades!  We're having fun! The whole class is doing better, and the other students are happy! Uh, does this actually happen?  I remember my second son throwing himself on the floor in despair the first week of first grade, wailing "Can't we at least do the numbers up to TEN?" A friend who was a math teacher gave him the bad news that every year of his school career in math would start with review.

I also don't like that they think bell curves will go away if teachers don't expect them.  Tightening the distribution is a fun phrase, but I mistrust it. Also, what happens later in school?  Do we apply this tightening in Algebra 1, so that the class doesn't move forward until almost everyone has got it right?  Are we going to have any real mathematicians in the future, or just a lot of people who have mastered fractions and decimals? Maybe that's a better outcome, but my initial prejudice is against it.

The better students might just glaze over and self-teach, which might be better anyway.

If this does actually work better, I cynically predict it will be undermined by the education establishment who want other things to be true. They want their previous methods of "discovery" and "inquiry" to prove out.  They are sure that they will prove out. Those are just better ideas, gosh-darn it, and we are cheating children if we don't give them that experience, you fools. They will resist implementation of repetition and breaking things into smaller steps, because that is "selling students short," and not "teaching them to learn." If JUMP is brought in, they will try to relegate it to supplementary status.  Failing that, they will introduce their old methods as a supplementary status because "lots of children learn better that way," sans evidence. The supplement will gradually encroach on the mandated new program, and when there is no improvement they will claim "See?  It doesn't work.  We had better go back to the "old" way."  If you think that can't happen, that is exactly what did happen to drill.

The first part of the method might work, and even if it doesn't work totally, it might be better than what we are currently doing. Let's hold that second part at arm's length.


RichardJohnson said...

If this does actually work better, I cynically predict it will be undermined by the education establishment who want other things to be true. They want their previous methods of "discovery" and "inquiry" to prove out.

A quick skimming of the article gave me the impression this approach might work. I like the breaking down into small steps. Little by little, so students don't get overwhelmed. Second, I like the disparagement of "discovery" methods. Direct instruction is better, also because students don't get overwhelmed. It is easier when you are told how to do something. Internalize and then practice it. Why Minimal Guidance During Instruction Doesn't Work.

One problem with "discovery" is that all students are not going to independently make the same discovery at the same rate. Yet, we have a curriculum which says that ALL students will learn this. But when?

When I was in 9th grade, I independently made this discover that the various associative and distributive laws- which for New Math students meant we needed to write proofs from the beginning- can be used in estimating. But how many students in the class made that discovery? Discovery on your own is one thing. Discovery when pushed down your throat- YOU WILL DISCOVER THIS- is another matter entirely. Be spontaneous, folks, on command!

But yes, the Ed School folk will not like their "discovery" methods being disparaged.If you randomly ask a teacher how much their Ed School classes helped them, odds are the teacher will reply, "Not much." Which is a shame, because there is a need for pedagogy. It is not intuitively obvious how to best teach a given material to a given group. Ed School folk are apparently not aware that to figure that out, there are over 2,000 years of classroom instruction to peruse.

That website you pointed to had an article that had me shouting, "YES." Want to Raise Successful Boys? Science Says Do This (but Their Schools Probably Won’t)
Students—especially boys—need hours of physical activity every day and they aren’t getting enough.

Like the song says, get yer ya-yas out.

GraniteDad said...
This comment has been removed by the author.
GraniteDad said...

A better approach for gifted students would probably be to have the class continue with the JUMP program, but push them to do math online (Khan Academy or whatever) in their free time, or during class if they get bored.

james said...

Discovery is wonderful, of course. But unless you can use individual tutors--think Socrates teaching the slave about doubling the side of a square--only the top fraction are going to benefit. Another argument for homeschooling...

I'm a huge fan of Socratic teaching, but not really any good at it. One Calc teacher was great--no declarative statements except "It's time for the daily quiz" and "We have doughnuts this morning." Question, restate the question from another angle, ask someone else, repeat.

But without the background language (in this case algebra and how to do graphing), nobody would have learned anything. And that background language of math demanded, just as other foreign languages do, memorization and practice to see how it works and get used to it.

Breaking the learning down into easier bits seems like a fine thing to do, but I'm pretty sure that the faster students will still be faster.

I was given a small group of TAG second graders and an interesting hands-on lesson series for teaching manipulation of unknowns (solving equations) using a "balance scales" model. I liked it, and the kids seemed to learn the basics well enough, and if it was followed-up on it would have made algebra a lot easier for them--but this was a "talented and gifted" group to begin with, and teaching demanded at least a few minutes each of 1 on 1 with each student. The material assumed a classroom setting, but that wouldn't have worked very well. (I ended the series with a lesson illustrating {but not going into detail} that there were things that ordinary counting didn't match very well--like clocks and rotations.)

BTW, this quote is just wrong. "Mathematicians “have big egos, so they haven’t told anyone that math is easy,” he said at the World Economic Forum in Davos. “Logicians proved more than 100 years ago it can be broken into simple steps.”"

No way. Mighton is conflating the basics of abstraction and counting (elementary math) with abstraction and manipulation of unknowns (algebra) with logic (classic geometry) with research (what the math pros do). Research is irreducibly hard.

Tom Bridgeland said...

Algebra can be learned! I had to take a class for which I needed algebra as a prerequisite. Having had algebra one twice in high school, a 'C' the second time, and that 35 years earlier, I knew I didn't want to take an algebra class. So I decided to test out. I searched on line for basic algebra problems and started doing them one by one. The better websites have step-by-step breakdowns of each and every problem. Exactly what I needed and had never gotten in classes. After a month of steady practice I took the test and failed by a point or two. Studied another week and passed on the second try. After that, chemistry was a breeze. I had taken chemistry multiple times in college back in the 1980s and did very poorly, eventually passing with a 'C' after multiple tries. I had never even grasped that chemistry equations were basic algebra.

My take. Math is hard but if the steps are shown again and again and again, even a very poor math student can learn it. People who are good at math, and math teachers, whom I assume are generally good at math, never seem to understand that math is hard for other people. They show one or two examples and simply assume that anyone can see how to do the rest of them. It made me angry in school that there simply were not instructions on how to do the problems in the book, and the teachers didn't understand why I didn't find the problems obvious. Math teaching is atrocious.

dmoelling said...

I was an unwilling participant in the "New Math" of the 1960s. I remember my father was completely flummoxed and said he could not help me. I didn't need the help but years later I again saw the concepts of new math as part of a graduate mathematics course. This was part of showing how one developed different arithmetics and algebras from first assumptions. I was astonished to remember these were part of first grad arithmetic. There was no way my teachers understood this at all. This was the ultimate math "discovery" assumption.

My friend who is an Astronomer turned Music professor agreed with me that this would work if all your kids were young geniuses like Gauss. People destined to become physicists often show the ability to simply "see" the results of complex mathematics. The rest of us had to work at it do gain a modicum of that skill.

RichardJohnson said...

Brilliant new educational ideas often don't scale up. They work as long as you have motivated, specially-trained teachers who want everyone to succeed. However this seems to have allowed for some of that in its design, and may get past it.

Consider the New Math version that I took in high school: UICSM, a.k.a. Illinois Math. Max Beberman developed it at the University of Illinois lab school,teaching faculty brats. Yes, it worked for faculty brats. But offhand, one might be skeptical that what worked for faculty brats would work for more ordinary students. That was definitely my experience. I loved the proof-writing that Illinois Math required from 9th grade on. Top students- say those who were inducted into the National Honor Society or at least had that capability- liked Illinois Math. We found proofs to be fun. Not all did.

Students who were not as brilliant struggled with Illinois Math. The president of my junior class was in my math class. "No more math misery, she wrote in my yearbook. Inductive proofs were not to her taste, though they were to mine. She was at least a B student. I would estimate that Illinois Math worked for somewhere in the neighborhood of 10-20% of the student body.

Elementary school teachers as a group do not have the math aptitudes of high school math teachers. When elememtary school teachers took on New Math, things got worse. For example, many elementary school teachers teaching New Math came to the conclusion that the basic multiplication and division skills were considered no longer important. That was also one of the main beefs that the general population had with New Math. I had a geometry course in college where the professor had met Max Beberman. Max Beberman told my professor that he had never intended to have basic multiplication and division skills downgraded. But when lower-math-aptitude teachers instructed students in the New Math, that is precisely what happened- on a mass scale.