## Monday, June 1, 2015

### Semester 2 Final Exam (Day 178)

This is finals week at the school district whose schedule I'm following on the blog. And so today I am posting my version of the second semester final exam.

As usual, let me give my rationale for choosing these particular questions. When I wrote this final, I wanted it to serve not only as an in-classroom final, but what my vision of an ideal Common Core test, like PARCC or SBAC, should look like.

I've talked several times about the traditionalists who prefer that test questions focus more on content and less on labels. The questions at the end of each chapter of the U of Chicago are divided into four sections, Skills, Properties, Uses, and Representations (SPUR). So we conclude that the traditionalists prefer tests that are heavy on Skills (where most of the content is), and light on Properties (where most of the labels are).

I don't agree completely with the traditionalists here -- especially not in Geometry class. Geometry, after all, is all about proofs, and the reasons that appear in proofs are labels and properties. So if one isn't learning about labels and properties, then one isn't really doing Geometry.

A test that selects from the questions in the U of Chicago text would naturally have mostly Properties and Representation questions, and this is what I started to write. But one traditionalist argument for having more Skills than Properties questions is that with a Skills-based test, students who have the necessary Skills can take the test cold, without having to study for a long time, and still get an excellent grade. But a test that contains many labels and properties would require even the smartest students to spend time learning the particular names of the labels and properties. This is significant considering that one major argument against standardized tests like the Common Core tests is that they require so much time for test prep.

I spent lots of time on this blog preparing for the PARCC test -- not my final exam. Yet I didn't want my test to be just PARCC problems. And so I took questions from the U of Chicago text -- and since I didn't post test review for these question on the blog, they ended up being Skills questions, just as the traditionalists desire.

So here's how I wrote the test. This is a cumulative exam covering the whole text. But it was hard for me to find some good problems for Chapters 1 and 2, and I did just post a review sheet last week for some of the angle theorems from Chapter 3, so I began with Chapter 3. I decided to include numbered questions from the text that were multiples of five, starting with Question 5 and stopping at the end of the Skills section. For Chapter 3, there are six questions that would be included, Questions 5, 10, 15, 20, 25, and 30. But I had to drop Questions 10 and 30 because the particular skill for those questions involve drawing, which isn't easy to do on either a multiple-choice final or a computerized Common Core test.

Here is the chapter breakdown: for Chapter 3, I included four questions, but for Chapter 4, I included just one question. For Chapter 5, I included four questions, but for Chapter 6, I included just one question again. We notice that Chapters 4 and 6, where the transformations are taught, have very few Skills questions, since the main skill in both chapters is drawing the images, and I've already decided to drop all drawing questions. This is in accord with the traditionalist distaste for the Common Core transformations like reflections and translations. But Chapter 7 has just two included questions -- for questions on SSS, SAS, and ASA are also just Property questions.

Chapter 8 has the most included questions, with a whopping ten of them. Seven of these questions are from the Skills section. But after I wrote this test, I've having second thoughts about these. Many of these questions are not straightforward. For example, students are asked to find the perimeter of a square given its area or vice versa, as opposed to finding either of them given the side length. Now traditionalists like these types of problems because they require students to think deeply about the problem -- and I agree, but only up to a point. I have no problem with some of the questions requiring students to think outside the box, but when every question is this difficult, students will eventually become frustrated. But unfortunately, I ended up choosing the multiples of five, and these just happen to be the more difficult problems.

No matter what anyone else says, I want to include some problems from the Uses section, since I still want to demonstrate how math can be applied to the real world. So this means that I include questions 50, 55 and 60 from the Uses section of Chapter 8.

Chapter 9 is a tough chapter, since we covered Chapter 9 only briefly so we could get to Chapter 10. I included three questions (one from Uses) for Chapter 9. Chapter 10 is, of course, a big chapter, and so I included six questions (two from Uses) for this chapter.

Chapter 11, on coordinate geometry contains no Skills problems at all. Dr. David Joyce criticizes coordinate geometry, and so I include only two Uses questions from this chapter. From Chapter 12 I included five questions, with two of them from the Uses section.

Chapter 13 contains very few Skills or Uses questions -- it's a chapter focusing mainly on Properties, just like Chapter 2. So I included no questions from this chapter -- and recall that Chapter 13 will be broken up for my curriculum next year. From Chapter 14 I included six questions, with one of them from the Uses section. Since I covered Chapter 15 only briefly, I was only able to include one question from this chapter -- otherwise Chapter 15 would be a great Skills-based chapter.

This leaves five questions from the PARCC Practice test. I decided to continue the pattern and stick to multiples of five, so I included questions 10, 15, 20, 25, and 30. As we expect for PARCC questions, of course these are mostly Properties questions. These are already set up to be multiple choice -- still I had to set up the questions from the U of Chicago text so that they could be multiple choice as well.

Of course, I set up the questions to be multiple-choice for the purposes of the final. If this really were a computer-based exam, then I would have more free-response questions -- especially those requiring students to enter only a numerical answer.

Notice that Representations has been completely shut out of this test -- and Representations includes graphs and coordinate geometry. One problem with graphing questions on the computer is that they often require students to drag the graph to the correct location -- and this confuses them. I believe that there should be more graphing questions, but it's not clear to me how to make them so that more students can draw them on the computer easily. The only question that involves a graph is officially a Skills-based question from Chapter 8 -- students are to estimate the area of an irregular region.

I still like the idea of a computer-based test, though. Many people say that they oppose Common Core because it's "one size fits all." But the whole point of a computer-adaptive test like the SBAC is to avoid being "one size fits all" -- the same test for every student. I can easily imagine a computerized test asking a question such as to find the area of a square given its perimeter. A student who answers this incorrectly (say, by simply squaring the perimeter). can get an easier question such as to find the area of a square given its side length. Those who answer correctly, on the other hand, can get more difficult questions such as to find the area of a circle given its diameter or circumference, then move on to difficult volume questions, and so on.

Indeed, students who get many questions right could move on to some above-grade-level questions, if time allows. Unfortunately, I doubt that the actual SBAC does this. So SBAC fails to use the full power of having a computer-adaptive test. I wonder whether more traditionalists would be in favor of a computer-adaptive test like the SBAC if students could jump to above-grade-level questions.

Here are the answers to my posted final exam:

Once again, I don't post a Form B for this exam.