Some years ago, Hugh Burkhardt said that “What You Test Is What You Get” – WITIWYG. So what should we be testing for in high school mathematics? There are obviously many answers, but given the ethos of our times – when everybody wants to go to college – one valid answer would be “test students so they are prepared for college”. Unfortunately too small a proportion of our high school graduates go into careers that emphasize math and science, so mathematical preparation for college must have a broad meaning.
For me, that the essential objective that math education should achieve is to create students who can think flexibly, based on logical reasoning and who are able to communicate clearly their thinking processes. By flexibility, I mean the ability to solve problems they have not seen before, problems that necessitate connections. To put it another way, our graduates not only must be VERY familiar with basic mathematical concepts and operations, but they must also be able to get away from “plug-in” problems.
I will be teaching two courses in the fall: AP Statistics and Algebra II. From previous experience, AP Statistics largely achieves the flexibility, logical reasoning and communication triad. Here I am going to concentrate more on Algebra II, especially since at our school Algebra II is a freshman course (Geometry follows it) and therefore it is a formative course in mathematics for our students.
Politics intrudes here, since for the next few years, until Common Core becomes translated into assessments, what we are still going to hear is “state standards, state standards, state standards”. I am not a great fan of these standards – they are limited to the lower levels of Bloom’s taxonomy, they have low bars for “Proficient” or “Advanced” and they are completely multiple choice.
In my opinion, our assessments should be based on what we, as professionals, know that students need to achieve. For example, from my experience in engineering school (undergrad and grad) as well in engineering practice, an Algebra course should not touch completing the square in the quadratics unit. It is more important to have students understand the meaning of roots of a polynomial function, than memorize –b/2a. Graphing and computers find the axis of symmetry and the vertex without any problem. To understand what happens to a function at its roots, why we call them “zeros” and how this is applied in practical problems – this is way more important than completing the square.
Perhaps even more important than content, is the type of problems we should use in our assessments. If we are going to educate critical thinkers, multiple choice problems can have only a limited role. Here are some types of problems that I think should be part our assessments in Algebra and that I plan to use in the quizzes for SBG.
(a) Function generation from patterns.
(1) How many circles (black and white) are in Box 9?
(2) Write a formula to find the number of circles in Box n. Explain your results.
(3) The total number of circles in a box is 265. What is the box number? Show work.
The mathematics here are not very difficult. It is rather the form of the question and the fact that we connect the mathematical idea of a function with a pattern that is more important. A more challenging variation is to just to have number (3) above as the problem.
(b) Conceptual understanding
Example: In each of the four equations below, the solution depends on the constant a. If a > 0, what is the effect of increasing a on the solution of each equation – does the solution increases, decreases or stay the same?
(1) x – a = 0
(2) ax = 1
(3) ax = a
(4) x/a = 1
Again, solving these equations is pretty simple, but how often do we check for student understanding of what we mean by a solution? How often do we check that our students understand how and why solutions can vary?
(c) “Realistic” problems
Some educators assume that if we put in homeworks or in exams “real world” problems, the kids will understand why math is important and they will be eager to solve such problems. Unfortunately, research (Willingham, Boaler) shows this is not necessarily the case. This research shows that “problem context [can distract students] from [the problem’s] deeper mathematical structure”. (Shannon). Therefore we should NOT trivialize the math, just to make up a “real world” problem.
Example: For health reasons people should limit their efforts, for instance during sports, in order not to exceed a certain heartbeat frequency. For years the relationship between a person’s maximum heart rate and the person’s age was described by the formula:
Recommended maximum heart rate = 220 – age
Recent research showed that this formula should be modified slightly as follows:
Recommended maximum heart rate = 208 – (0.7 * age)
A newspaper article stated “A result of using the new formula instead of the old one is that the recommended maximum heart rate (heartbeats per minute) for young people decreases slightly and for old people it increases slightly”.
(1) From which age onwards does the recommended maximum heart rate increase as the result of the introduction of the new formula?
(2) The new formula helps determine when physical training is most effective: this is assumed to be when the heartbeat is 80% of the recommended maximum heart rate. Construct a formula for calculating the heart rate for most effective physical training, expressed in terms of age.
Realistic? Yes! Trivial? NO!
We can use examples such as the last one in groups, for small projects – whatever. The important thing is to expose students to such problems.