David Tall's article "Teachers as Mentors to encourage both power and simplicity in active mathematical learning" argues that mathematics teachers should mentor their students into becoming independent mathematical thinkers. He argues that teaching mathematics sequentially, as is traditionally done, leaves students doing mathematics, but not necessarily thinking about mathematics. The majority of the article is spent explaining what is meant by procedure, process, concept, and procept, and how a teacher might introduce particular topics in a manner that allows students to gain meaning from symbol manipulation.
After reading Tall's article, I was left wondering what he meant by mentor. He begins by stating that "teachers need to act as mentors to encourage their students to build thinkable concepts that link together in coherent ways," but it is unclear to me why he chose the word "mentor." Are good teachers necessarily mentors? Are mentors necessarily good teachers? There was obviously some importance for teachers as mentors, but I'm unsure as to how Tall intended to use it.
Tall's recommendations for teaching the various concepts mentioned within the article are very engaging ways of interacting with the concepts. A great deal of creativity went into some of the examples, particularly the example of a difference of squares, where he uses a rearrangement of geometric shapes. Tall is a well known mathematics education researcher with a strong background in mathematics, and although I see great benefit in utilizing the teaching practices he mentions, I question how the majority of teachers would react to non-sequential mathematics. What if the teachers themselves think of mathematics sequentially? How could one expect to teach mathematics in terms of "thinkable concepts" when they do not think of mathematics in this way? What if the teacher needs to be a mentee?
There is an enormous amount of research regarding the insufficient practices of teachers in mathematics classrooms around the world, as mentioned by Jo Boaler in the video last night. If we want to change classroom practice, it is my opinion that we need to start with teachers who have not yet entered a classroom. True, one could provide future teachers with a list of neat, embodied ways to understand factorization, but isn't this just as bad as having one way to factor it? Although there is more variety, are we not still encouraging teachers to have a toolbox of ways to do and understand things? How might we encourage teachers and students to come up with their own concept images, rather than relying on ones predetermined by someone on the outside?
In my mind, the idea of a mentor is based around guiding a student towards success, rather than holding their hand every step of the way. It seems to be a fine line however. It can be difficult to know when to step back and let the student tackle the problem on their own.
ReplyDeleteI find the idea of "sequential mathematics" to be funny. I have come to realize that everyone has a different idea as to what is actually sequential. For example, I find the math textbook that we use at our school is anything but sequential. It places converting metric units before patterns in multiplying and dividing by 10, let alone understanding what a decimal is. Ideally, for me, we would present word problems to students and have them devise ways of solving them while working cooperatively in teams. Hopefully through this team work, key mathematical concepts would be discovered and practiced. This style of teaching seems to be where we are heading with the new curriculum.
In terms of whether or not we should give new teachers teaching toolboxes, again I find myself sitting on the fence. I completely agree that teachers should allow their own intuition and desire to mold their teaching practices. A part of me (my first year teaching part of me) would have liked to have some sort of package to help guide me. It wouldn't have to be a prescribed set of lessons, but maybe some general guidance on how to get through the curriculum efficiently; it could state approximately how much time to spend on each unit. Or with the new curriculum, the toolbox could give some examples on how to convert your lessons from the old curriculum into new lessons that satisfied the objectives of the new curriculum. I like your idea of students and teachers working together to create their own concept images. Perhaps ways to do that could be in the toolbox.
I have run into this question of what a mentor is before, and I was similarly unsure just what it meant. Years ago I attended a women in math conference, it was for graduate students and folks early in their career, and I was taken aback by how frequently others referred to their `mentors'. Eventually I simply asked a few what that meant: was this another word for supervisor? Was the mentor aware of their role? What did it entail? The general consensus was that these students did have formal mentors, at least in the sense that mentor was separate/additional from supervisor, and the role had been defined using that term. Mentoring seemed to require offering help and guidance beyond math content, it addressed larger issues surrounding being a student. While Tall may not be encouraging this use, I wonder if he's hoping for more explicitness in the role of the teacher regarding larger picture mathematics?
ReplyDeleteThere is lots of discussion about sequencing in the curriculum in this thread, and I just wanted to let you know that Audrey is particularly interested in this topic (and may do her papers on this question).
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