# Focus on Math

## Helping children become mathematicians!

### Math Assessment: There’s More Than Just One Way to GradeDecember 6, 2013

In the world of education, not all assessment is created equally. In the area of mathematics, folks seem to think marking or assessing student work is very easy to do: things are either right or wrong – full stop. This could not be farther from the truth.

Most of us grew up in a system that was based on that kind of marking, the kind that looked for a particular math problem to be only right or wrong. Tests and quizzes were comprised of many examples of problems, and we endeavoured to get as many as we could correct, generally using the same procedures over and over to solve the items.

That, however, is a very shallow level of assessing. Students can produce an answer to a problem without knowing or understanding the underlying mathematical concepts – and it is there, in the conceptual knowledge, that the “real math” lives. So a student who has done a page of problems (say, long division or quadratic equations) may be able to follow memorized rules and procedures to come to a correct answer without having the faintest notions of the “why” behind the rules and procedures. There was no assessment of the “real math”.

My friend Katie Wagner, who teachers a number of math courses at McMath Secondary School (isn’t that a cool place to be teaching math!), has written in her blog about two different kinds of math assessment:

Standards-Based Assessment — a grading method where students’ proficiency on a specific outcome is assessed based on a set of pre-established standards.

Outcome-Based Assessment – a grading method that assesses students discretely against the particular outcomes that they are to learn in a course, a unit, or such.

I encourage you to read Katie’s posts and think about the assessments you are doing with your students in your classroom (or, if you are a parent, think about the assessments your children are experiencing). Hopefully you will have some points of discussion and/or questions after you read about the other possibilities.

We need to be helping students understand the “why’s” of mathematics, to understand the deep concepts that underlie all of the rules and procedures, and we need to ASSESS students for those “why’s” and understandings.

Thanks for being a “guest speaker”, Ms Wagner!

Mathematically yours,

Carollee

### Using a Rubric in Math Problem SolvingMarch 3, 2012

If we want students to become better problem solvers, not only must we provide situations where they can practice their problem solving skills, but we also need to make sure they are thinking meta-cognitively about the problem solving skills they are developing. One way to do that is to use a rubric with students.

Sandra Cushway, another teacher in my district, and I are presently in our 5th year of an action research project concerning teaching the whole curriculum through problem solving. As part of that project we developed a problem-solving rubric modeled somewhat after the rubrics developed by the BC Ministry of Education. Thus we chose to evaluate the same 4 aspects of mathematical thinking as those of the Ministry’s rubrics (namely, Concepts & Applications; Strategies and Approaches; Accuracy; and Representation & Communication), but we wrote the descriptors as “I statements” so students could self-assess. (Download our rubric here.)

The ministry’s rubrics were developed for specific grade levels (link here to see those) but Sandra and I and chose to make one rubric that was applicable to many grades.
For instance, if a student might choose this statement in the Strategies & Approaches aspect: “I chose a strategy that worked. It allowed me to get an answer but it took a long time, and was confusing in places.” That statement can apply to a primary student using strategies for double-digit addition as well as for a high school student looking to solve trig problems.

If you are new at using rubrics, may I suggest this: it is much better to begin the self-assessment process with a single aspect. In other words, choose one line from the rubric like “Strategies and Approaches” and only use that “strip” across the page with students. Talk about what the different levels mean and show samples of problem-solving work at each of the four levels. Have students work together to assess some work so they can get a feel for evaluating what good (and poor) strategies look like.

Let me know how the rubric works for you and your students! I know it can make a difference in the quality of work they do.

Mathematically yours,
Carollee

### We assess what we believe is important.February 21, 2012

Why are some things perceived as a “waste of time”?

I visited a high school math class today where the ‘Foundations of Mathematics and Pre-calculus’ students were beginning a unit on measurement, including conversions between imperial and metric systems. The teacher spent quite a bit of time facilitating a discussion with the students about the concept of measuring: the various aspects of things that we measure, how we might measure things in “difficult” situations (e.g., finding the surface area of an irregularly shaped puddle), and when in life we use particular imperial units of measurement.

It was a great discussion, with the students putting forth many ideas, some wonderfully “out of the box”. As it got to the point that the teacher was setting the students to work doing some of the questions from the text book, he came over to me and made the comment, “I think the kids think this discussion was a waste of time. I know that when I was in school, I would have felt a discussion like today’s was a waste of time.”

I have no doubt that both statements are true – that the teacher remembered feeling discussions about the “big ideas” were a waste of time, and that his students, although they found it interesting, also felt it was a waste of time. The question, however, is why are such discussions perceived as such?

Personally, I think it comes down to the fact discussions about conceptual understanding are not given any honour, any value. Students look at how marks are derived and figure out pretty quickly that those things which are important are what show up on tests, quizzes, and such. If nothing about the big ideas, about the conceptual understanding, about the meaning behind the mathematics is asked when it “counts”, then it seems clear that those things are not valued. They are not important.

We, as teachers, have to decide what is really important in mathematics, and if that includes conceptual understanding, then we must include questions and/or tasks that get to the heart of that conceptual understanding in our assessments. It is not always easy to do – such items will likely not fit into a multiple-choice kind of test to be marked quickly on a Scantron. But again, we must ask ourselves what we value, what we believe is important.

Our assessments will reflect our true beliefs.

Mathematically yours,
Carollee