Chapter 8, page 180
Table 8.7: Examples from math
Students have learned to solve problems such as this one:
Solve for a.
(2a + 1) (a - 1) = (3a - 2) (2a - 4)
Solution: 2a^2 - 2a + a - 1 = 6a^2 -12a - 4a + 8
15a - 9 = 4a^2
4a^2 - 15a + 9 = 0
(4a - 3) (a - 3) = 0
Obviously, you can ask many questions of the same type. But also ask questions that “tweak” the
process in various ways.
For example:
- Give a problem with a denominator, such as a - 3 as a denominator.
- Give a problem where it is not necessary to multiply out, such as (2a + 1) (a - 1) = (3a - 2) (a -
1).
In general, at the end of a lesson, you should give problems in which it is appropriate to use the
procedure taught in the lesson, and you also should give problems in which it is inappropriate to
use the procedure taught in the lesson. Otherwise, students will not learn when to use the
procedure, and when not to use the procedure. - Give a more complex problem with a cube in it.
- Give a problem that cannot be factored.
- Give a problem in which the numbers don’t turn out to be whole numbers. (In real life, unlike
most math problems, the answers don’t come out to nice round numbers!) - Add a complexity such as: 2a(a-3) - 6a = 16a^2 - 3a + 9.
- Have students write a problem that can be solved using this method.
- Have students write a problem that looks as if it can be solved using this method but actually
cannot.
Finally, just a couple of examples from literature.
Who do you find more appealing as a Jane Austen protagonist, Emma of Emma or Elizabeth of Pride
and Prejudice? Give specifics to justify your view.
Judging from these five poems, what kind of person do you think Emily Dickinson might have been?
Provide evidence for your answer.
To recap, the purpose of the previous sections was to illustrate questions that are successful at
assessing understanding. Students cannot answer these questions using strictly rote memory. As a teacher,
it is exceedingly useful for you to construct a large pool of questions of this sort. The questions have three
uses:
- You can use the questions to define understanding for this topic. By constructing novel, challenging
questions and problems for a topic, you define for yourself what it means to understand that topic. Once
you have this understanding, you can design your instruction to promote this understanding.