## Friday, August 15, 2014

### Chapter 1 Quiz (Day 7)

This concludes Chapter 1 of the U of Chicago text. It is traditional to give some sort of test at the end of the chapter, but since we formally covered only three sections, a quiz is more appropriate here.

Now for me, designing this quiz is rather tricky. As everything else in this course, I wanted my quiz to based on the U of Chicago text. There is a Progress Self-Test included in the book. But even if I threw out the questions based on sections 1-6 through 1-8, there are some questions that I chose not to include.

For example, the first question on the Self-Test asks the students to find AB using a number line. This is very similar to some of the questions that I gave on the Wednesday and Thursday worksheets. But there is one crucial difference -- this one is the first in which both A and B have negative coefficients.

Now I know what the test writers are thinking here. The test writers want to know whether the students understand a concept. There's not enough room on the test to give both easier and harder questions. If a student gets a harder question correct, we can be sure that the student will probably get a much easier question right as well. But if the student only answers an easier question correctly, we can never be sure whether the student understands the more difficult question. Therefore, the test should contain only harder questions, since anyone who gets these right understands the simpler concepts too.

But now let's think about this from the perspective of the test taker, not the test maker. Let's consider the following sequence of hypothetical conversations:

Wednesday:
Student: The distance between 4 and 5 is 9.
Teacher: Wrong. You're supposed to subtract the coordinates, not add them. The distance is 5 - 4 = 1.
Student: Oh.

Thursday:
Student: The distance between -4 and 2 is 2.
Teacher: Wrong. When subtracting, change the sign. The distance is 2 - (-4) = 6.
Student: Oh.

Friday:
Student: The distance between -8 and -4 is 12.
Teacher: Wrong. You forgot the negative in front of the 4. The distance is -4 - (-8) = 4.
Student: Oh.

And we can see the problem here. The teacher wants the student to be able to find the distance no matter what the sign of the coordinates are -- not just when they're positive. But the problem is that the instant that a student finally understands how to solve the first problem, the teacher suddenly makes the problem slightly harder, and the student becomes confused.

Of course, you might be asking, why only give one problem on Wednesday? Why can't we give more problems to check for student understanding of the all-positive case, then move on to negatives? But you see, I'm imagining the above hypothetical conversations as occurring during, say, a warm-up given during the first few minutes of class -- and warm-ups typically contain no more than one or two questions. The student is never allowed to taste success, because each day a little something is added to the problem (like a negative sign) that's preventing the student's answer from being completely correct. The student never hears the words "You're right." And that's just with negative signs -- the U of Chicago text includes questions with decimals as well. I immediately threw all decimals out of my problems -- since decimals confuse the students even more, most notably when we draw number lines and mark only the integers.

Well, I don't want this to happen, especially not on the quiz or test where most of the points are earned. I want the student to taste success -- and this includes the student who's coming off of a tough second semester of Algebra I and is now in Geometry. Sure, if you feel that some students need to be challenged, then challenge them with all the negatives and decimals you want. But I don't want to dangle the carrot of success in front of a student (making them think that they've understood a concept and will get the next quiz question right), only to jerk it away at the last moment (by adding extra negative signs that will make the student get the next quiz question wrong), all in the name of challenging the students.

And so my quiz questions are basically review questions rewritten with different numbers. My rule of thumb is that the quiz contains exactly the same number of negative signs as the review. Some teachers may see this as spoon-feeding, but I see it as setting the students up for success. Any student who works hard to prepare for the quiz by studying the review will get the corresponding questions correct on the quiz.

Of course, some questions about the properties are hard to rewrite. I considered using the question from the U of Chicago text, to get from "3x > 11" to "3x + 6 > 17." But notice that the correct answer -- Addition Property of Inequality -- is difficult to remember and will result in many students getting it wrong. So even here I changed it to the Addition Property of Equality. After all, the whole point of learning the properties is to be able to use them in proofs. The Addition Property of Equality is much more likely to appear than the corresponding Property of Inequality. All including Inequality on the quiz accomplishes is increasing student frustration over a property that rarely even appears in proofs.

And so this is what I came up with: