learning math is hard, part deux

(Part one of this post can be found here.)

In the first part of this post on the math program, Connecting Math Concepts, we were discussing about how the program is field tested and how error diagnosing and correction is built into the program. I needed to describe those two aspects of the program briefly to get to the aspect of the program that I intended to discuss -- practice.

Student practice is built right into CMC. That's one of the reasons why the program is field tested beforehand; to determine how much practice students need to retain the material taught. Unlike in most math programs, in CMC material is not just taught, tested and then permitted to lay fallow whereupon it is quickly forgotten by the student. Do you remember the threat of the dreaded end of year cumulative test back in K-12? You dreaded it because you knew that you had forgotten most of the material presented in the first half of the year. You don't have such a luxury in CMC, all tests are cumulative. The only time a skill isn't practiced or tested is because it's been incorporated into a more difficult skill. At the end of the year, students are expected to have retained all the material presented during the year. This is exactly what is needed when learning math.

Since CMC has been field tested with lower performing students and since CMC is designed to accelerate student learning as quickly as possible, you can get an idea for how much practice is needed for a lower performing student to retain the material. The program is designed to provide sufficient practice with a little bit extra to account for things like student absences, but not too much since that would hinder the acceleration. So, the practice provided in the program should turn out to be about what is necessary for a lower performing student to master the material at about the fastest rate he can handle. Cutting to the chase, the amount of practice that a lower performing student requires t o learn math is simply enormous if CMC is an accurate guide.

There is way too much practice for my son. I routinely cut out about every other practice lesson for each topic because I don't want him to get bored and our time for lessons is limited afterschool. Plus, I want to keep the ball rolling and stay far ahead of the wildly inappropriate nonsense that gets taught in his Everyday Math class.

So, you might be thinking that I'm only cutting out about half the material. Nope. I'm cutting out far more than that. I'm cutting out all the "extra practice" lessons that are scheduled for students after they fail a proficiency test. Since he's never failed any portion of any test so far, I haven't had to go back and reteach any lesson. At most he'll get a few problems wrong due to his desire not to being math work at night when he could be playing Lego Star Wars II on his PS2, but so far he's always stayed in the proficiency range no matter how fast I go.

In addition, I've never given him any worksheets from the extra practice workbooks or the blackline master worksheets. And, i skip all the games that sneak in more extra practice since there's no one to play against since he's the only student. Occasionaly, I'll play against him to give him an idea how fast I can work the problems so he has any idea how fast he's going to be expected to work the problems. He's not as fast as I am yet, but he routinely does his problems in half the time allotted in the timed exercises. So, he's starting to approach automaticity on some of the stuff he's learned so far.

Lastly, I've been known to skip the last 30 or lessons at the end of the year since most of this material will be quickly reviewed at the beginning at the next level.

I'd estimate that I cut out about 2/3 to 3/4 of the total practice provided in CMC which accords pretty closely with Engelmann's estimation that higher performers can be accelerated at about 3-4 times the rate of lower performers. And, it doesn't surprise me at all that lower performers need every last bit of all that practice I'm cutting out. Math is all about learning abstract concepts and our brains are not wired to learn abstract concepts easily. It also doesn't surprise me that in most math program, with the exception of Saxon, lower performing kids aren't getting close to the amount of practice they need to retain the math they've been taught.

Hence the widespread failure we see in math education.

And, this assessment doesn't even get into the messy area of the initial presentation of the material enabling the student to understand the concepts in the first place. I'll cover that aspect of CMC in future posts since we're now just starting to get into the interesting areas of math instruction. I'll leave you with this. CMC presents the material so clearly and concisely that I only have to "teach" for about five minutes each lesson. The rest of the time he's working problems using the skills I just taught or practicing previously taught skills. Ironically, that's probably far less teaching that goes on in your typical discovery learning/constructivist heavy math class. I'll show you who's the real guide on the side.

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.