Tests- The Mathematical Bogeyman

I have always found assessment in the math classroom really quite difficult.  There are so many issues, but the biggest ones I can think of are the problems of mastery and depth.   Traditional math tests, like the ones that I give about every 3 weeks, and have done for 22 years, are basically a test of mastery.  Some of my colleagues strenuously object to this goal, or at least this framing of what a test is for, but in my opinion that is what a test basically is for.  Other teachers in the department would say that by giving a student another chance to solve a similar, analogous problem (sort of retesting on a problem-by-problem basis), or by showing extensive corrections, or by creating their own analogous problem along with an answer key, we allow them the space to revise their understanding and show them that instantaneous mastery is not being demanded of them.

But note the structure that all of us have accepted in the first place:  tests.  My colleagues who disagree with me give them as well, they just don’t want them to feel as punitive or “final” as they did when they were students in school.  They want them to be more of a teaching tool.  And I am sympathetic to that goal, I really am.

But I don’t do that in my classroom.  Basically, I give a test to my kids about every 3-4 weeks.  I give them past tests with answers and/or review problems with answers to study from ahead of time so that they have some idea of what will be expected of them on the test, in terms of content, difficulty, and structure.  They know that the test may look very different, or quite similar, to the kinds of problems I have given them for review, but that if they can’t at least do the review problems, they will likely have a hard time on the test.

So why do I do it this way, which evokes memories of damp English boarding schools?  Competence in mathematics means, to all the math teachers that I know, the ability to solve problems in mathematics, and not only ones that are similar to those you have seen before, but also those that are unfamiliar.  To be able to do that well, you need a good deal of exposure to a range of problems on the material before someone checks to see if you really are able to solve problems on a given topic.  Homework is one way to do that, and our text , being in my opinion much less rote than many traditional texts, has definitely helped in that matter.  But students need more practice, in more contexts, before being tested—thus the review problems and my instructions to make sure to complete a healthy number of them before a test.  What a test positively offers is a chance to know that you know; one good thing about mathematics having answers that are so specific is that a student can see if their chain of reasoning is correct at a glance, because the answer to that review problem was indeed 113.67 and not something else. 

And also for most kids, in my experience, when I did offer revision, they became much less focused before the test, and didn’t complete the review problems as thoroughly, because they knew they could “revise” later.  I don’t blame them—it is human nature.  Now I offer test revisions in unpredictable fits and starts, so that students take tests seriously the first time.

Which is not to say that I am completely happy with tests, not at all.  But they are very effective at helping a student to know what they know, and to know what they do not know.  They also are clear and are perceived by students as being fair; one thing I always tell them is that when grading tests I look to see if there is any question that basically all the students missed, because then I am likely to throw that one out as it clearly was not a fair expectation that they could get it.  So students like the finitude of tests, and they like that have a good sense of what will be on them.  And to be clear, I ask a wider range of questions on tests than I used to with previous texts, because the problems in our texts have greater range and diversity, so students naturally expect that the questions they have to answer are quite varied. 

But, ironically, the reason I’m writing this entry is because I have been thinking about the limitations of tests as assessment. What else do I want out of assessment?  Well, I want to see if students can ASK questions about extending and generalizing a problem, as well as answering them.  I want to see if students can use mathematical habits of mind more systematically and consciously when confronted with a difficult problem.  I want to see how a student handles an open-ended question, rather than one that has a specific algebraic or numerical answer.  I would like students to be able to lead a discussion of a problem, much like they do in an English class or a History class. I would like students to feel, for certain limited topics, that they have gone in more depth than their peers and have attained a real mastery of something.

Currently, the best my tests do is see if my students can handle a range of types of questions about the material at hand, maybe with a little bit of using mathematical habits of mind to make progress on the more unusual problems.    I’m not really addressing much else of what I listed in the last paragraph with tests.

So how am I going to do that?  This blog entry is already too long, so I’ll try to begin to answer that next time, or the time after if the start of school brings up a topic or two that merits an immediate response.

A critical mass problem

One of the things I like best about the way we teach math at Park is that the problems themselves serve as intrinsic motivation. Sure, not every kid is perfect about doing their homework or working as hard possible, but we’re far away from the situation where it’s the impending test that motivates a kid to do their work. Most kids are interested in the conversation that happens in class and almost can’t help but give thought to the problems before them.

Every now and then I have a class that thinks that the material is too easy, despite my feeling that most students in the class are not giving the material the thought it deserves, and sometimes even despite the fact that I know there are basic skills that most students have not mastered yet. This could happen in a geometry class, where it’s easy to trick yourself into thinking that an informal argument appealing to symmetry, say, is sufficient, when actually a proof is needed. Or, if the topic is algebra, a “which of the two quantities is bigger” question: to which savvy students often know that the answer is almost always, “they’re the same size,” even if they can’t provide the algebraic justification.

Often, it’s very smart students who have this view – they’re able to intuit their way to an answer for some problems without needing to go through the thought process that the person who wrote the problem intended. It’s great if they can do that, of course, but they may be missing a chance to generalize their method to future problems. That is, they may be missing the core content of the class. Even more importantly, in their eagerness to get the problem done, they’re robbing themselves of the opportunity to be a mathematician. If a problem seemed dumb… what do you suppose you were supposed to get out of the problem? What is its larger significance?

In these situations, if I give a test that I feel is reasonable given my expectations of the students, they don’t do very well.

Because we only give tests once a month or so, it takes too long to give students the feedback that they don’t understand everything they think they understand. Part of me has the impulse, then, to give them quizzes to hammer home the point. But this is not really what I want to do. For one thing, I don’t want students in my classes to feel that they constantly have to be completely on top of the skills and content in the course. Too often, we are in discovery mode, where we are debating the appropriateness of the very skills I’d be quizzing on. It takes time for the dust to settle. And perhaps more importantly… is a test or a quiz the only way to give feedback to a student about how they’re doing? Shouldn’t there be a way to give that feedback more naturally? In most of my classes, when the majority of the students understand the spirit of the class and the exploration, students will let each other know if their arguments are too vague. In the type of class I’m describing, where there isn’t this critical mass, it’s harder.

The way I have dealt with this issue in the past is to collect homework more often, either for a small grade or just for written feedback. Still, I’d like a way to send a message to these students that even the easiest problem contains a world of follow-up questions, generalizations, and connections to other topics. A message other than “teacher says,” of course.