Auntie Ann makes a great point on my last post:

Giving many problems and demanding wordy answers on a test are mutually exclusive. In the time it takes to explain in words one problem, a student could demonstrate their proficiency on several problems with different mathematical concepts. Writing wordy explanations is much slower than giving a student a variety of different questions.Assuming that the point of making students explain their answers is to distinguish those who really don't understand the math from those who've simply made stupid mistakes, then there are two possible approaches.

1. Assign a smaller number of problems so that students spend time explaining their answers.

2. Assign a larger number of problems.

Back in the day, we got perhaps ten times as many problems per session as students do today.

A student who is prone to stupid mistakes won't get nearly every answer wrong; a student who doesn't understand the math will. The type of answer generated by stupid mistakes often looks different from the type of answer generated by conceptual misunderstandings. Assign enough math problems, and a competent teacher can easily distinguish between the two types of student. Include harder problems that involve more mathematical steps than today's problems do, such that more students will naturally write down their mathematical steps, and it's even easier to distinguish those who understand from those who don't.

Doing lots of math problems (and getting timely feedback on them) is probably also a better way for students to overcome conceptual misunderstandings than explaining a much smaller number of problems is.

And its a great way for everyone to get better (especially more fluent) at math.

## 2 comments:

But does doing a lot of problems increase the probability of stupid mistakes? Students could get sloppy and/or bored, especially if the teacher then assesses the homework so that conceptual understanding is prioritized over the "stupid" mistakes in assessment.

In some long problems, as well, stupid mistakes can turn into significant conceptual errors. My classic example, because i used to do this, is z's that would turn into twos through sloppy handwriting -- a problem gets way easier (or unsolvable) when you remove one of the variables.

And, in the end, an answer can be just as wrong when a stupid error (i.e. arithmetic miscalculation, etc.) has been made.

bj

Not an either/or issue. If children cannot explain the concept, when they practice the problems, they may reinforce their errors. With a new concept, I prefer to have fewer practice problems (maybe only 5), and have them explain their reasoning. Then I give a quiz. If I see errors, I can address them before they are etched in cement. Finally we practice for fluency.

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