I was at a day panel that was organized by Walter Whiteley in relation to the CMESG meeting that is being held at York University this weekend when I wrote this. The first talk was about the competing goals of mathematics education and mathematicians.
The speaker was Peter Liljedahl, a professor of Mathematics education at SFU who teaches mathematics to future teachers. He emphasized that you cannot truly separate math and math ed and that there are common goals but he also talked about the differences between the goals of math and math education.
Some interesting ideas that came up in this talk:
1. The first is that sometimes people who are interested in learning math education are interested in learning how to teach mathematics and not about learning the mathematics content. This can be a conflicting goal with mathematicians who tend to believe that teaching mathematics well requires that the teacher understand the subject with a real depth.
2. The other idea that I thought was interesting was he addressed how teaching courses for math teachers (for example in this program) is different than other mathematics classes. The undergraduate classes that he gives in a 'for Teachers' program do not go into depth. The courses I have taught in this MA program always keep in mind two ideas (1) the students taking the course do not have the same background as each other or even a core basis that I can assume everyone knows and (2) the content that we do teach is not required for other courses. In other graduate and undergraduate courses the purpose is to teach a subject with a depth that can only be reached by assuming the students have understood and remember the material that came before.
3. He also touched on some of the skills that we need to provide students with for them to succeed in mathematics. For him it was less about what content we teach students, but what skills we provide them with. Some examples of skills that he mentioned are: how to read a math textbook, how to recognize when a teacher is skipping steps (and why), how to approach a problem that cannot be solved in 5 minutes, etc.
From my iPhone so this may be edited later.
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1) The more you know, the better teacher you will be. For example, if you have only studied up to the level you are teaching, how will you be able to cope with student questions that extend beyond the curriculum? How arrogant (and unrealistic) to assume that you ever know all that you need to know about your subject.
ReplyDeleteThat being said, there is also a time and place for learning effective teaching strategies. Faculty of Ed. programs should do that.
A good undergrad program for future math teachers should incorporate both.
2) I can only hope that the required math courses given by other instructors in the SFU UNDERGRAD ‘for Teachers’ program do go into depth. It’s super important at that level ‘cause it’s really, really hard to go back and get that depth later on in life :)
On the other hand, York’s MA for Teachers program is designed to meet the needs of teachers at different stages of their careers. You are taking the right approach with the 2 ideas you mention. Most other undergrad or grad math courses have prerequisites. It’s hard to go into much depth if you have to start from scratch in each course.
[It’s interesting to look at this idea of depth in light of the previous blog posting that included the reference list for the advanced mathematics course and the article from Taylor, Whiteley, and Barbeau. Seems that the current approach in high school is to favour teaching depth in understanding a few (and very few!) key concepts, while sacrificing exposure to other topics. Clearly depth v. breadth is an issue at this level also.]
3) Yup. The most important things we can instil in our math students are confidence and perseverance. If we can help them see that math is intrinsically beautiful and useful, and give them the confidence to persist when facing a problem, it will carry them so much farther than just teaching them a bunch of formulas and algorithms. How true the saying, “whether you think you can or think you can’t, you’re right” (and that one IS on the Internet!). When it comes to math, our students need to believe that they CAN.
And once they have this ...
Many of our students don’t even realize that there are examples and explanations in their textbooks – teaching them how to effectively read the book is a great, and often overlooked idea.
Hi Alice. Maybe we can get others interested in this topic, so I think I should clarify (1) a bit and perhaps introduce a few controversial ideas.
ReplyDeleteMy take is that mathematics education wants to answer questions like "how do we teach mathematics more effectively?" This is a goal for which mathematicians and mathematics education have common ambitions. Generally speaking, mathematicians are just too busy doing mathematics to have an interest in studying how people learn in order to answer that question.
Ideally one would like an answer to this question which can be applied to mathematics at any level to any mathematics classroom. Maybe this is not possible, but more realistically one could develop some general techniques that can at least be applied at (say for instance) the junior/intermediate level so that it is not so important that you know the mathematics, just that a teacher can learn it better than the students and so is able to teach it.
You can see how this idea might grate on someone like me because I believe that having a deeper understanding of mathematics helps you to be a better teacher and I would like to see even elementary school teachers much better trained in mathematics than they currently are.
I have heard a study cited (by Walter Whiteley) which showed that teachers who take several more advanced undergraduate math courses are less effective teachers than those who take just a few introductory math undergraduate courses. I am sure that the study had a more precise hypothesis because I have to reject out of hand the statement as to general to be justified in that form.
It is hard to make general statements about how to teach mathematics at the junior/intermediate level when there are elementary school teachers who have a fear of mathematics.
I want to clarify my comment on (1) too. I said, "the more you know, the better teacher you will be". Implied in this is the phrase 'relative to your original level of teaching competence'.
ReplyDeleteFor example, I do not believe that a person with a PhD in math (or an MA) is automatically a better teacher just because they have that piece of paper. With reference to the study Whiteley cited, there's not enough info. here to properly discuss it, eg. what's the definition of 'effective', how's it measured, what's the sample size, and so on.
But in general, teachers need to be able to communicate the ideas at an appropriate level for the audience. This is where the PhD types may be at a disadvantage for teaching public school. They are functioning at such a higher level mathematically that it might be challenging for them to come up with good explanations for things they take for granted, like operations with integers and fractions.
It is helpful if a teacher understands the struggle that some of their students are facing. This is perhaps easier for someone who had to work hard all the way through to get 70's or 80's, rather than for the 95% student. Are these the students who only took math courses up to the second year level? Possibly.
Here is an interesting article about the nature of mathematics and math research and math teaching. Check it out.
http://mail.baylorschool.org/~dkennedy/treeofmath
Let me clarify what Peter Liljedahl said about his courses for teachers at SFU (e.g. point (2)).
ReplyDeletePeter emphasized that even his math education courses are 60-90% math. SFU offers a sequence of undergraduate courses designed for teachers to get math credits to meet a 'teachable' requirement. He says that he likes teaching these courses because he can choose from topics that have 5000 hours worth of material and pick and choose what he likes to teach in 39 hours (3 hours x 13 weeks). The important thing for him was not the subject, but the skills and techniques and ideas which are independent of the material which can uses in these courses.
Note this is very different from a course which picks a subject and goes into depth, say for instance, complex analysis or algebra, where one learns some main concepts within a subject that often come up over and over in mathematics.
This is different yet again from courses which are very specialized in a subdomains like operations research or cryptography or numerical analysis. The courses get more specialized the further along you go.
This is why I loved the essay that Alice posted a link to. The analogy for the examples I am listing above are 'fruit of the tree,' 'trunk' and 'branches.'
I think the essay makes a lot of sense too.
ReplyDeleteAnd, for those that are encouraging us to expose our students to more interesting and challenging problems from a variety of 'branches' of math (rather than just calculus), this essay provides a nice rationale why teachers need to have taken math courses beyond the first and second year 'trunk' level. How can you give your students a glimpse of what's out there in the branches, using a technology ladder or not, if you have never been there yourself? Furthermore, you need to have progressed far enough out and spent long enough on the branches in order to be comfortable taking your students there.
Reminds me of Mike's comment: "Having a deeper understanding of mathematics helps you to be a better teacher". Case closed.