Designing Multiple Answer Math Tests

I got the idea for Multiple Answer Tests originally because I wanted to prepare my kids for Common Core Tests. (I’d rather people not use that post as the primary link, as I have done a lot more work since then.)

About six months later (a little over a year ago), I gave an update, which goes extensively into the grading of these tests, if you’re curious. At that time, I was teaching Pre-Calc and Algebra 2/Trig. This past year, I’ve been teaching Trigonometry and Algebra II. I’d never taught trig before, so all my work was new. In contrast, I have a lot of Algebra 2 tests, so I often rework a multiple choice question into a multiple answer.

I thought I’d go into the work of designing a multiple answer test, as well as give some examples of my newer work.

I design my questions almost in an ad hoc basis. Some questions I really like and keep practically intact; others get tweaked each time. I build tests from a mental question database, pulling them in from tests. So when I start a new test, I take the previous unit test, evaluate it, see if I’ve covered the same information, create new questions as needed, pull in questions I didn’t use on an earlier test, whatever. I don’t know how teachers can use the same test time and again. I’d get bored.

I recently realized my questions have a typology. Realizing this has helped me construct questions more clearly, sometimes adding a free response activity just to get the students started down the right path.

The first type of question requires modeling and/or solving one equation completely. The answer choices all involve that one process.

Trigonometry:

I’m very proud of this question. My kids had learned how to graph the functions, but we hadn’t yet turned to modeling applications. So they got this cold, and did really well with it. (In the first class, anyway. We’ll see how the next group does in a month or so.) I had to design it in such a way to really telegraph the question’s simplicity, to convince the students to give it a shot.

Algebra II:

The rational expression question is incredibly flexible. I’m probably teaching pre-calc again next year and am really looking forward to beefing this question up with analysis.

Other questions are a situation or graph that can be addressed from multiple aspects. The student ends up working 2 or 3 actual calculations per question. I realized the questions look the same as the previous type, but they represent much more work and I need to start making that clear.

Trigonometry:

Algebra II:

I love the Pythagorean Ruler question, which could be used purely for plane geometry questions, or right triangle trig. Or both. The furniture question is an early draft; I needed an inverse question and wanted some linear modeling review, so I threw together something that gave me both.

I can also use this format to test fluency on basic functions very efficiently. Instead of wasting one whole question on a trig identity, I can test four or five identities at once.

Or this one, also trig, where I toss in some simplification (re-expression) coupled with an understanding of the actual ratios (cosine and secant), even though they haven’t yet done any graphing. So even if they have graphing calculators (most don’t), they wouldn’t know what to look for.

I’m not much for “math can be used in the real world” lectures, but trigonometry is the one class where I can be all, “in your FACE!” when kids complain that they’d never see this in real life.

I stole the above concept from a trig book and converted to multiple answer, but the one below I came up with all by myself, and there’s all sorts of ways to take it. (and yes, as Mark Roulo points out, it should be “the B29’s circumference is blah blah blah.” Fixed in the source.)

Some other questions for Algebra II, although they can easily be beefed up for pre-calc.

One of the last things I do in creating a test is consider the weight I give each question. Sometimes I realize that I’ve created a really tough question with only five answer choices (my minimum). So I’ll add some easier answer choices to give kids credit for knowledge, even if they aren’t up to the toughest concepts yet.

That’s something I’ve really liked about the format. I can push the kids at different levels with the same question, and create more answer choices to give more weight to important concepts.

The kids mostly hate the tests, but readily admit that the hatred is for all the right reasons. Many kids used to As in math are flummoxed by the format, which forces them to realize they don’t really know the math as well as they think they do. They’ve really trained their brains to spot the correct answer in a multiple choice format–or identify the wrong ones. (These are the same kids who have memorized certain freeform response questions, but are flattened by unusual situations that don’t fit directly into the algorithms.)

Other strong students do exceptionally well, often spotting question interpretations I didn’t think of, or asking excellent clarifications that I incorporate into later tests. This tells me that I’m on the right track, exposing and differentiating ability levels.

At the lower ability levels, students actually do pretty well, once I convince them not to randomly circle answers. So, for example, on a rational expression question, they might screw up the final answer, but they can identify factors in common. Or they might make a mistake in calculating linear velocity, but they correctly calculate the circumference, and can answer questions about it.

I’ve already written about the frustrations, as when the kids have correctly worked problems but didn’t identify the verbal description of their process. But that, too, is useful, as they can plainly see the evidence. It forces them to (ahem) attend to precision.

Of course, I’m less than precise myself, and one thing I really love about these tests is my ability to hide this personality flaw. But if you spot any ambiguities, please let me know.