kitchen table math, the sequel: learning math is hard

Wednesday, January 3, 2007

learning math is hard

Update: For more one error detection and correction, take a look at this video (quicktime) starting at about 5:30. He's talks about error correction with reference to reading instruction. It continues on into this clip up until about 7:00 and math gets discussed for the last 3 minutes or so.

That's my current position based on teaching my six year old son math for the past year and a half.

Actually, that observation isn't based on my son having difficulty learning math. So far he hasn't. It's based on the the material we've skipped. It is that differential that separates the higher preforming math students from the lower performing math students. That differential represents an enormous amount of practice.

Unlike most parents who use Saxon to teach math, I'm using Connecting Math Concepts. Both programs are scripted, both use a mastery learning "basic skills" approach, and both have lots of practice built into the program. Both are complete programs which don't require parents to know how to teach math; knowing elementary math is sufficient. For most kids there is not much difference between the two. Contrast this with Singapore Math which does require some teaching skill to present and requires practice to be supplemented. That's not meant to be a knock against Singapore Math, each program has its strengths and weaknesses. I actually think that the ideal K-6 elementary math curriculum would be some combination of all three programs, capitalizing on the strengths of each.

For the purposes of this post, however, I want to focus on the practice aspect of learning math. To master elementary math a student needs to practice what's been learned until it is automatic. Unfortunately, most math programs do not provide sufficient practice to safeguard against the ravages of forgetfulness.

Most parents do not take control of the educational process until there the need to remediate becomes evident. At this point, there is a tension between the need to devote time for practice and the need to reteach the child to get him back on track as quickly as possible. Practice tends to get the short end of the stick at this point. It shouldn't.

One aspect I like about CMC is that it's been field tested so you can be certain that if the student has the math skills to enter a level of the program, the program will teach clearly enough and provide enough practice for the student to reliably master all the material presented in this level within one school year, about 120 lessons.

The most important aspect of CMC, however, is that error diagnosing and correcting are built right into the program, unlike almost every other math program. Let's face it, if students didn't make any errors while learning math, a trained monkey could teach math using almost any commercially available math program. It is in the diagnosing and correcting of student errors where most math programs fail. When students derail, many teachers are unable to get them back on the track. Math, being brutally cumulative is not forgiving at all when students derail.

This is CMC's greatest strength.

CMC is designed to minimize students errors in the first place by providing clear instruction in small instructional steps. Students are then tested frequently (workbooks are checked after every lesson and tests are given every two weeks) to check student errors. based on the ten unit tests, student errors are evaluated and a built-in remedy is provided to the student based on the errors the student made. The student is then retested to see if the remedy worked before the student is permitted to advance. If the student were permitted to advance without mastering the material, then the diagnosing and correction of errors would be become much more difficult come the next ten unit test because now the teacher doesn't know where the student went astray. Was it one of the new skills taught in the past ten lessons of was it one of the previously taught skills? Now extrapolate out 80 more lessons and try to figure out where the problem is for a newly taught skill that the student can't do. Forget about it.

Contrary to popular belief, the greatest shortcoming of the "constructivist" math programs is not the less than clear presentation of new skills, though this is certainly a problem; it is that error detection becomes virtually impossible. This is not so much a problem in a class full of higher performers, but it is deadly in a class where students make errors.

I see this post is getting a bit longish and I still haven't touched on the main point -- practice. So, I'm going to break it up into two posts since there's already much to chew on in this post. More to come.

Part two here
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14 comments:

Catherine Johnson said...

Built-in diagnosis of errors - incredible.

I'm very eager to see how CMC does this (and what is then done to remediate errors).

Fantastic.

Catherine Johnson said...

Thinking these issues through, I've come to believe that not only do standard curricula and pedagogy penalize slower learners, who don't get enough distributed practice; standard curricula and pedagogy, in one way, richly reward fast learners who get more practice than they need, and hence end up having "overlearned" practically everything taught regardless of how often they paid attention in class, did their homework, or stayed home sick.

Overlearning is the key to automaticity, and when a fast learner attends a class aimed at average learners, he or she is virtually guaranteed to reach overlearning.

This isn't an efficient way to do it, but I'm positive it works.

Anonymous said...

"Most parents do not take control of the educational process until there the need to remediate becomes evident."

Exactly. That's what happened here. I wish I had put my daughter in KUMON in preschool. But I really wasn't thinking about math back then. I'd heard so many horror stories about whole language that I focused on reading and phonics. In hindsight, I should have been pre-teaching math also. Live and learn.

Anonymous said...

You have an interesting point about how high achievers overlearn in classrooms aimed at average kids, Catherine. It's ironic because most parents don't understand the need for overlearning. They complain to teachers that there's too much repetition and their bright kids aren't being challenged.

Instructivist said...

I am back from a long vacation and am thrilled to see KTM back in business again. It's as lively as ever.

It's good to know that there is another quality math program out there. I looked at the CMC offering and was wondering what the difference between Level D and E is. Perhaps Ken can fill us in.

KDeRosa said...

I've just begun level C which is the second grade program. That means level D is third grade and level E is fourth grade. I'll have to find the scope and sequence at home to tell you the exact differenes between the two.

Catherine Johnson said...

hi, instructivist!

You should come do some posts here!

(You should have an invitation or two stacked up in your email.)

Robyn

It's ironic because most parents don't understand the need for overlearning. They complain to teachers that there's too much repetition and their bright kids aren't being challenged.

This idea just hit me the other day!

It's true that parents of fast learners always complain that the pace is too slow, etc. - and they're probably right.

But given the fact that our curricula and pedagogy don't have built-in error diagnosis & remediation, there's an obvious upside that we just haven't noticed.

Ever since I started relearning math I've marveled at how solid my math education - what there was of it - is.

Now I know why.

I was a fast learner, and I had SOOOO much practice that I way overlearned.

e.g., now that I've finished Saxon Algebra 1, I realize that the two years of algebra I took pretty much corresponded to just one year of algebra in Saxon.

Actually, Saxon Algebra 1 goes further than my two years of algebra did.

My high school taught algebra 1, geometry, and algebra "2" in sequence, which means that I had essentially 3 years of practice of one year's worth of material.

No wonder I remember how to set up an equation or two or three and solve for 'x'!

SteveH said...

The student is then retested to see if the remedy worked before the student is permitted to advance."

The goal of a math curriculum should be that everyone gets an 'A'. If you let them go on (especially in K-6), then it's hard to imagine (without outside help) a 'C' going to an 'A'.

Anonymous said...

Catherine,

I'm also amazed at how solid my math education was, and I was not a strong math student. But it comes back to me pretty quickly (although I have to admit that I can't remember a thing from trigonometry except sign, cosign and ... that other thing).

That's why this reform math movement is so ironic to me. I learned the old way, with lots of drill. And I'm pretty proficient with everyday math, although don't think I could have graduated from engineering school or programmed computers.

With these reform programs, they're trying to make everyone into a "mathbrain" and they're failing miserably. For the average and below kids, they're not even teaching proficiency in basic arithmetic.

Keith Devlin, NPR's "math guy", writes:

"The human brain evolved into its present state long before mathematics came onto the scene, and did so primarily to negotiate and survive in the physical world. Our brain does not find it easy to understand mathematical concepts, which are completely abstract."

My brain especially. But with lots of practice and drill, I learned math. Today, I'd be classified.

LynnG said...

Ken -- CMC sounds interesting. My worries with EM in the schools is that it is so easy for teachers to never notice gaps and errors, due to the very limited practice in the math journals and homelinks. Afterschooling has helped me discover my kids' errors, because it is immediate. When you only have 1 or 2 or 3 kids in front of you, you can keep track of weaknesses. It is my sense that kids in heterogeneous classrooms would make errors all over the place. Even with frequent tests to diagnose errors, can a teacher deal with all of the various problems that might crop up in a whole class? Do you know any schools successfully using CMC? How much depends upon skilled teachers paying attention to the results of the diagnostic exam?

Barry G. said...

I'm also amazed at how solid my math education was, and I was not a strong math student. But it comes back to me pretty quickly (although I have to admit that I can't remember a thing from trigonometry except sign, cosign and ... that other thing).

What amazes me are articles like the one in the Baltimore Sun (at http://www.baltimoresun.com/news/local/bal-te.md.math02jan02,0,2408740.story?coll=bal-local-headlines) that Ken linked to, which has the sentence that has become pat garbage for edu-journalists: "In the 1960s, students didn't have to understand why a formula worked. It was enough to memorize the facts and do the problems."

I have my 60's math books; elementary and high school. There was sufficient explanation given as to why things worked. Perhaps the only exception was the invert and multiply rule, but they at least showed the pattern; i.e., 6/4 is the same as 6 x 1/4; 1/4 divided by 2 is the same as 1/4 x 1/2, etc. Singapore does this same approach. Show the pattern enough (division is the same as multiplying the dividend by the reciprocal of the divisor)and then go into dividing by fractions. Only Saxon makes an attempt at an explanation.

But I'm digressing. The rule wasn't simply given; it was built up to. Multiplication was explained as repeated addition; the multiplication facts weren't simply presented with no explanation of what multiplication is. And we had plenty of exercises in which we had to say what to use to solve a problem--multiplication or division.

The reform texts feel you need "real world" connections, so they have students generate their own data; e.g., take the ratio of length of hand to length of arm for a sampling of students and graph the data; what is line of best fit? Nice way to kill a day when you can be going over basic skills.

Anonymous said...

I agree, Barry. The old way wasn't just rote. Math was taught in a logical sequence.

Anonymous said...

I'm also very proficient with calculators, even though I never had calculator lessons.

rightwingprof said...

I'm not using any particular book with my tutee. The initial plan was for him to bring home his worksheets and we'd work through them. Well, that didn't work, since he didn't know the most basic arithmetic. Then when he got that, the initial plan didn't work because by then, I'd seen how disorganized and superficial the course was. So I'm trying to pull the course together for him by pulling things he's learned into what we're doing, and spending a lot of time on problem-solving logic (and I mean that in the most literal, traditional sense), so he can dissemble a problem and attack it on his own.

I did a math minor as an undergrad and tutored for the math department. Although this is a lot of work, it's a great deal more rewarding than trying to tutor somebody in analytic geometry who hasn't been to class in three weeks and has a test coming up in two days. That was the norm tutoring for the math department.