First Weeks of School (part 2)

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“Dear Mr. Husmann, I love match. But I done no it. Please help with it.”

During the first week of school I gave pre-tests every day of class. After their last pretest, I asked students to write me a quick note to tell me something about themselves, and received the above letter from one of my 8th grade students. If that doesn’t make every math teacher in the world a little bit weepy, I don’t know what will.

Sadly, this is a typical situation for students here in Liberia. My students are no exception. Here’s the results of the pretests I gave that first week:

Day 1:

Answers basic multiplication facts perfectly
8th: 1 out of 11 students (11%)
9th: 2 out of 6 students (33%)

Sucessfully multiplies two 3-digit numbers:
8th: 3 out of 11 students (27%)
9th: 1 out 6 students (17%)

Day 2:

Solves at least one of three long division problems correctly:
8th: 0 out of 14 students (0%)
9th: 2 out of 8 students (25%)

Able to add fractions with different denominators:
8th: (no time to test)
9th: 1 out of 8 students (13%)

Day 3:

Solves at least one of three long division problems correctly:
8th: 4 out of 17 students (24%)
9th: 4 out of 10 students (40%)

Able to add fractions with different denominators:
8th: 1 out of 17 students (6%)
9th: 2 out of 10 students (20%)

As you can see, these students are crying out for help.

While administering these pre-tests, I tried out a new anti-cheating procedure. I would write a small problem set on the board (say, three long-division problems), and then give them limited time to complete the section (say, 6 minutes). When the time was up, they would all fold their papers over, and then I would write a new problem-set on the board. This new set would be solved on the folded paper. They were not allowed to unfold their paper to go back and review their problems, and we would continue folding and doing new sets until the paper was too small to do work on.

Since most cheating I’ve observed happens when students finish a long test early and have time to share their answers, breaking a test up into smaller, timed bites makes this a lot more difficult. If a student finishes a 20 minute test in 12 minutes, it leaves 8 minutes for the student to sit back in their desk, disrupt the class, and let a neighbor copy their answers. But if that same test is given in 4 sections of 5 minutes, that same student will only have 2 minutes free per section – the constant movement and transition gives them a lot less time to get bored and cause trouble. For the rest of the class, it starts to scaffold the idea of pacing oneself during a test.

Some may complain that this testing strategy is unfair to the students that work slow. What about the student that needs 10 minutes for a 5 minute section? Well, I would argue that for subjects like arithmetic, speed is an essential component of mastery. And there’s no reason to work slow: one does not need to think creatively or have flashes of insight to solve a long-division problem. It’s completely mechanical.

Furthermore, to be sucessful in higher math (especially if you don’t have access to a calculator), arithmetic skills need to be second-nature, automatic — if you have to put any thought into your long division while you are solving a calculus problem, you are not going to have the brain space to maintain your train of thought and get the flashes of insight you need in calculus. To see the big picture of the problem that you’re working on, you can’t get bogged down in arithmetic.

Knowing about my pre-test procedure now, you might argue that its low success rate was due to the limited time or nervousness of my students. But if you looked at the types of errors being made, I think you would agree that even ideal emotional conditions wouldn’t have changed the results much: A student that is adding fractions by adding both the numerator AND denominator doesn’t need more time to think about their decision… they simply didn’t know how to add fractions. All the students that seemed to know the proper procedure got at least one question of four without mistake, and are reflected as the “successes” in my statistics above.

From the pre-test results, I think it is clear that I am going to start by teaching basic arithmetic to my students. Specifically, we’re going to start by getting our times tables down cold, and then move to other skills like multiplying multi-digit numbers and long division. While I am drilling these skills, I’ll be trying to awaken a general number sense and abstract thinking ability (you’ll get an idea for how I intend to do this as I post my lessons).

As I teach and test these skills, I am going to emphasize the following two principles I’ve distilled from the discussion above:

1) speed + accuracy = mastery
2) mastering rote mechanics creates space for higher-level thought and creative inspiration

After stating those two principles in such a general and barebones sort of way, I couldn’t help but laugh. This is how you learn how to play the violin. You must build your technical ability (speed and accuracy) if you want to have a greater capacity to express yourself with the instrument (higher-level thought and creative inspiration).

But it’s bigger than that: I would argue that these principles apply to every sort of human endeavour. The arts, sports, relationships… In clown school we used the image of a river to describe the interplay between these two dynamics, with the “form” of the river being rote technical ability, and the “flow” of the river being the creative impulse. If you’re interested, I’d love to talk more about this in a different post.

In the meantime, I’ve got some lessons to plan.

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2 Responses to First Weeks of School (part 2)

  1. Anthony Lenz says:

    I love this post. I really like to learn different teaching philosophies like this. It really makes you really dig deep and meta-think.

    ‘speed + accuracy = mastery’ – Agreed!! sounds like practice = perfect; like developing muscle memory for your brain… brain/memory memory????

    Keep up the good work. 😀

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