Extracting Data from Distractors

1. R. James Milgram Extracting Data from Distractors

2. Many Math Educators are trying to classify student errors to mechanize mathematics instruction • This has proved to be very difficult. • But using the results of large scale testing, and all the distractors, it seems to be possible to make headway.

3. Many Math Educators are trying to classify student errors to mechanize mathematics instruction • This has proved to be very difficult. • But using the results of large scale testing, and all the distractors, it seems to be possible to make headway.

4. Many Math Educators are trying to classify student errors to mechanize mathematics instruction • This has proved to be very difficult. • But using the results of large scale testing, and all the distractors, it seems to be possible to make headway.

5. We start by looking at the pattern of responses. • Then we look at some examples to get an idea of what these patterns represent.

7. We start by looking at the pattern of responses. • Then we look at some examples to get an idea of what these patterns represent.

8. Note the largest incorrect responses reverse the powers of 10. This is probably an artifact of inefficient teaching

10. This is a situation where there are two peaks, probably the result of inefficient teaching. Note that the second peak is obtained by separately adding or subtracting the whole numbers and then the tops and bottoms of the fractions.

11. And look what happens with multiplication.

12. Here is a first grade example Note that response (C) actually is higher than the correct response.

13. Here is a first grade example • Note that response (C) actually is higher than the correct response.

14. Here is a similar problem in second grade instead of first. Note that we have exactly the same effect as the original.

15. Here is a similar problem in second grade instead of first. • Note that we have exactly the same effect as the original.

16. The issue: Students are not taught that the digits in the shorthand base 10 notation describe the EXPANDED FORM. They think the shorthand notation is the name of the number. In both cases the popular answer reflects the ORDER of the digits, not their “weights.”

18. Here is such a problem: The fact that the “answer” is part of a “standard” distribution implies that the problem is with instruction! Mathematically, there is no correct answer to this question. More exactly, any continuation is possible

19. Here is such a problem: • The fact that the “answer” is part of a “standard” distribution implies that the problem is with instruction! Mathematically, there is no correct answer to this question. More exactly, any continuation is possible

21. Here are some examples: The most popular wrong response represents adding or subtracting whole numbers and separately adding tops and multiplying bottoms in the fractions.

22. A similar thing happens with converting measurements in grade 8 And this happens in successive years.

23. Comments • I do not understand the rationale the students used for distinguishing 8,496 from 84,960, or 4.65 from .465, but I suspect that these students actually multiplied by the conversion factors. • In the remaining cases 23.6 and 236, 2.79 and 27.9, most likely the students understood that one needed a conversion factor, but they seemed to believe it had to be a power of 10.

24. Comments • I do not understand the rationale the students used for distinguishing 8,496 from 84,960, or 4.65 from .465, but I suspect that these students actually multiplied by the conversion factors. • In the remaining cases 23.6 and 236, 2.79 and 27.9, most likely the students understood that one needed a conversion factor, but they seemed to believe it had to be a power of 10.