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.

## 2 Comments

There’s much to discuss in this post. I hope we can discuss it both on- and offline.

I don’t usually offer test revisions either, unless there’s some situation where many people do badly and the class really wants a second chance. What appeals to me about the policy that some of our colleagues have (everyone always has a chance to take a second test, on their own time, with problems that closely correspond to the original, for the opportunity to make up half the points) actually doesn’t have that much to do with pedagogy. It’s just that students only wind up taking about eight tests during the year, and so it’s not until March or so that their average starts to become really meaningful. Before then, their grades could be flukey. Sometimes, I can tell that students’ performances on, say, the first three tests don’t really reflect their understanding. Other times, I’m pretty confident that the grades _are_ reflective of their understanding, but the student or their parents don’t believe this, and I don’t have much evidence to back it up. Either way, it would be nice to have more test grades _without_ having to have the class in “test mode” all the time.

More broadly, I’ve come to think of assessment in math class as kind of a sad thing. In other subjects, you can “make” something — you can write a paper containing your original ideas, or you can design an experiment, or you can build a functional model of something. On a math test, you just get to solve problems that everyone in the class is solving. Because the test is supposed to be doable, they’re not even going to be problems that require huge amounts of creativity (we save those for when we’re not assessing). But I’ve yet to find some other kind of assessment that I think does as good a job of showing how good a mathematical thinker a student is — let alone whether or not they know the content.

Problem posing requires greater cognitive demand than problem solving. Why not give the solution and challenge the students to create a problem? Also use more formative assessments in your class, observations, presemtations by students, to ensure mastery of concepts before the test. Formative assessments will allow you to understand any misconceptions well before test day and a good range question each day will provide you with a better understanding of what students know and are able to do before your lesson begins. Use exit cards to assess the key understanding that your students leave class understanding to help you prepare the next lesson. These steps should alleviate your and your student’s angst about the test