Taking a discussion of what’s good for prospective math teachers (explanation, representation, translation, and conversation of/about math) and seeing that’s it’s really good for all math students.
As part of our conversation with Dr. Ball (see the previous post), the question came up about how we should be preparing future math teachers. Because we wanted to consider teachers at all levels (K-12) and didn’t really want to get into specific mathematical topics, Dr. Ball presented 3 ideas supported by research and one that was a current interest. I’ll go through the ideas below, but what struck me as I thought about the ideas was that they could and should be part of every math class because they were key in what it means to be a mathematician, and thus good for all students.
The ideas (with my clarifications, comments and examples) (educators = prospective math teachers in math classes):
Being able to explain mathematical ideas accurately – For educators this means explaining concepts, practices, etc. in both a variety of ways and at different levels, and doing so with mathematical accuracy; this involves more than just mathematical knowledge but the basis does come from mathematics. For the rest of the students, as we desire them to not just be able to do the mathematics, but also they should be able to understand the concepts behind the processes, and there’s probably no better way to develop and assess this type of learning is to explain it, especially using different methods. It would be very natural to include activities involving these type of explanations by all students in most of our math classes.
Being able to use a variety of representations of mathematical ideas appropriately – For educators this is the essential skill to address the diversity of learning styles in a typical class; it forms the basis for the Rule of 4 (or 5) (i.e. symbolic, in spoken and written words, in tables, and in graphics) and the focus in most curriculum on developing student fluency with data in these variety of forms. For the rest of the students, being able to convert ideas from one form to another is a necessary skill for learning in different teaching environments, is useful for making sense of new ideas, and is valuable for ones who will work in applied fields with non-mathematicians. Presentations in math classes should already be using a variety of representations and it would not take much to add opportunities for students to work at expressing ideas in different ways.
Being able to translate mathematical ideas from one context to another – This is more about the nature of mathematics as it focuses on learning the essential nature of a mathematical idea (sometimes called the invariances) and then seeing how that idea is expressed in different areas of mathematics. For the educators, the focus might be on ideas that are more prevalent in K-12 mathematics, like proportionality and equivalence, that also are present in higher mathematics. But, it also would extend to looking at how ideas in the higher math classes exist as extensions of K-12 topics, and how the ideas of translation, equivalence, etc. are directly manipulated to develop new ideas or new areas of mathematics and to develop deeper understanding of those ideas. For the rest of the students, this is an essential part of mathematics and should be a part of every math course (and not just some sort of capstone experience). All students should see all of mathematics as both a distinct set of areas, but, more importantly, as a strongly connected set of ideas, with higher level ideas held in common. This is (or should be) part of the instruction in each course, and is probably part of the student work, but, due to it’s primacy in mathematics, should be increased in all classes.
Being able to have an accurate, comprehensible conversation about mathematics – For all this is just the ability to express ideas correctly. So often we only ask students to write mathematics and often accept some pretty incoherent scribbles as ‘justification’. For the educators, this would just be the baseline of being able to speak mathematics and to make good logical arguments. For the rest of the students, we ‘talk’ to our students all the time, and this would just ask them to be able to do the same. Including student oral communication in a class can be challenging both in scheduling and assessing, but could be done more informally as part of other assessments, like having a student orally present the results of a problem, or using online resources to record themselves, and then use some peer evaluation for assessment.
Overall any efforts to modify math courses with a focus on improving the preparation of future math teachers, would inherently improve the quality of the course for all math majors.