Transcript

INSTRUCTOR:

We are interested in learner responses to the question-- if a is 4, what is 2a plus 1? We know that 2a plus 1 means 2 times a, add 1.

The most common learner response is likely to be 9, in which case, they're working out 2 times 4-- 4 instead of a-- then plus 1. And the answer is 9. And that's the most common response.

But you're also likely to get a range of incorrect responses. One of these is the answer 3a. In this case, the learner's probably looking at 2a plus 1 and ignoring the letter. So they see 2a plus 1 and write the answer 3. In some cases, they then add the a back in, again, just to do something with it. So that's likely to be an incorrect response you'd get, and we call it letter not used or letter ignored.

This is also the case with the incorrect response 7. Here, we have 2. The a is replaced by 4. And we have plus 1. That's read as an instruction to add up the numbers 2, 4, and 1, which gives 7. Sometimes learners, aware that this is algebra, will add an a in. Those three answers are the ones that you're most likely to find.

Sometimes you'll also find an answer such as 25. Here, the learner looks at the 2a next to each other and thinks about this as a two-digit number. So 2 with a 4 is 24, giving the answer 24 plus 1 is 25. This is quite a reasonable view, as it is how we write two-digit numbers. But in algebra, 2 followed by a has to be recognised as 2 times a.

Finally, you might find you get strange numbers, such as 17. What probably happened here is a confusion between 2a and a squared. 2a is a plus a or 2 times a. But confusing the addition and the multiplication can give us a times a. When a equals 4, that is 4 times 4 plus 1 equals 17.

So those were a range of correct and incorrect responses to the question-- if a is 4, what is 2a plus 1?

You have the correct response-- 9. You have responses such as 3a and 7, where the letter is ignored. And the calculation is done on the numbers available, perhaps with the letter added in afterwards. And you have a sensible, but incorrect interpretation of 2a as a two-digit number.

In your class of, say, 30 learners. Most will give the correct answer. But you will have a few who show these misconceptions. If you see these responses, you will know that they are based on some reasoning, but it includes a misunderstanding of algebraic conventions.