Wishes Level B21 Workbook Students Book Answers
Millicent
A prejudice that was strongly confirmed was the value of mathematical fluency. Barton says, and I agree with him (and suggested something like it in my book Mathematics, A Very Short Introduction) that it is often a good idea to teach fluency first and understanding later. More precisely, in order to decide whether it is a good idea, one should assess (i) how difficult it is to give an explanation of why some procedure works and (ii) how difficult it is to learn how to apply the procedure without understanding why it works.
As an example in the other direction, Barton gives that of solving linear equations. The danger here is that one can learn a procedure for solving equations such as , get good at it, and then be completely stuck when faced with an equation such as . Here a bit of understanding can greatly help. Barton advocates something called the balance method, where one imagines both sides of the equation on a balance, and one is required to make sure that balance is maintained the whole time. I think (but without too much confidence after reading this book) that I would go for something roughly equivalent, but not quite the same, which is to stress the rule you can do the same thing to both sides of an equation (worrying about things like squaring both sides or multiplying by zero later). Then the problem of solving linear equations would be reduced to a kind of puzzle: what can we do to both sides of this equation to make the whole thing look simpler?
That last question is related to another fascinating nugget that is mentioned in the book. Barton gives an example of a question concerning a parallelogram ABCD, where the angle at A is 105 degrees. The line BC is extended to a point E, which is then joined by an additional line segment to D, and the angle CED is 30 degrees. The question is to prove that the triangle CED is isosceles.
Apparently, this question is found hard, because one cannot achieve the goal in one step. Instead, one must observe that the angle of the parallelogram at C is also 105 degrees, from which it follows that the angle ECD is 75 degrees. And from that it follows that the angle EDC is 75 degrees as well, and the problem is solved.
Barton uses these diagnostic tests to get a much clearer picture of what his class already understands, before he launches into the discussion of some new topic, than he would by simply asking questions to the class and getting answers from a few keen students. If he diagnoses a fairly serious collective misunderstanding, then he will spend time dealing with that, rather than pointlessly trying to build on shaky foundations.
It would be an interesting experiment to ask that vector space question at some point during the IB Linear Algebra course. The results might well be surprising (or not). (Craig Barton suggests some approaches to how to ask such a diagnostic question; it may be a little harder in a lecture hall, though.)
Another couple of talks which might be of interest: Sandra Laursen on a research project on the impact of IBL (Inquiry Based Learning): and Michael Starbird asking what we want our students to get out of our math(s) courses:
I had come to more or less the same conclusion regarding short-term memory based on my undergraduate education. While tutoring my friends in the Calculus and Linear Algebra courses, I found that the most common problem they faced was being able to hold many ideas in their mind at the same time. I believe that the large number of problems that I had to solve throughout my schooling helped develop my fluency in basic manipulations. I like to think that it frees the memory by pushing those skills to an instinctive level.
on the question of diagnostic quizzes, how does the teacher predict the reasoning the student may have used to arrive at their choice by looking at the students choice? is there only way of getting to that choice?
Designing a good diagnostic test for a particular course is not easy. It requires someone who has not just a lot of experience teaching the course but has already devoted a lot of attention to diagnosing why students taking that course do poorly. Without firsthand experience, it is *impossible* to know what to look for. I say this, because I went through this myself. I taught university level math in the same naive way that Barton did. for many years. Then my department hired an instructor (Jerry Epstein, now passed away) who knew the math education research about why so many students and adults fall short in math and who had devoted a lot of effort designing and validating diagnostic tests. The questions were *below* high school level, but even at top US colleges, as many as 10% of the students (but not necessarily math majors) did poorly. He explained to us how this all happened. We of course started administering the test to our students, which confirmed everything he said. After this, we redesigned our diagnostic test (which included a careful validation process) and tried our best to change how we taught our precalculus and calculus courses.
First, we use such questions as diagnostics (in addition to straightforward arithmetic and algebraic computation problems) to see whether a student has been properly trained or not in his earlier math classes. If they do poorly on problems which require properly interpreting the meaning of the text, then they are not ready for university level math.
Perhaps at a US university, before people have chosen their majors, people might be tempted to choose another option (such as B, because vector spaces are to do with algebra and not calculus), while not noting that the obvious scalars in D do not form a field.
This is quite valuable, and definitely something more of us should really be engaging with. Mathematical pedagogy is definitely hard, but even as research mathematicians it would be useful for us all to spend some time thinking how best to communicate understanding.
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Let me point out that Mr Barton has a free podcast that covers many aspects of Math education and that is great for those of us who jog or walk dogs. Also, in his website on diagnostic questions, he asks students to comment on why they gave certain answers, so he can learn to make even better questions.
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