I have been away for awhile, so I'm glad to be writing again. In my last post I talked about how I think that the order we present the basic math/pre algebra topics in developmental math classes is not the best for adult students.
If we are going to change the order in which we teach the material, I feel it also important that we change HOW we are teaching the material.
I
remember one student who I was tutoring back in the 1980's. He was struggling in his Elementary Algebra class, and he
asked his instructor if he could use his calculator in class to make things
easier for him. The teacher said no, as is standard in that class, and he
challenged her by saying, “I’m majoring in Computer Science, so when I graduate
my computer will do this for me, why should I learn all this?”
The teacher responded with what I would classify as a
standard ‘pre-college’ response, “If all your computers go down and you need to
figure out something, then what?”
This student, who was in his mid to late thirties, responded
immediately, “If all my computers go down, I have a much bigger problem that
what two plus two is.” And for the career he was planning, he was completely correct.
As developmental math teachers we are dealing with students
from many different backgrounds, and many students who are returning to college
as adults. Because of this the standard answers often given by primary and
secondary teachers don’t work for many if not most of our students. This is
even truer now that all of us have a computer in our pocket.
Any type of appeal to authority; e.g. "Because the department says we will do it this way"; is only going to breed resentment in our students. What we need to do is get our students to be partners in the process: to buy into why we are not using calculators.
This idea, that teaching to adults takes their active involvement is not new. It goes back to at least the late 19th century, and it was popularized in education circles in the 1980's by Malcom Knowles and goes by the name andragogy (as opposed to pedagogy).
The basic idea, distilled down, seems fairly obvious. As adults we use what we already know when learning new things. We need to be motivated to learn. And we need to be able to see that what we are learning is going to help us achieve our goals.
So, how can we justify not using calculators in developmental math classes?
This has been a struggle for me over the years. Like a lot of teachers, I started teaching by doing what I had seen others do, and I used different versions of, "We it do it this way because," to poor effect. As the years went on, I began to think more about why the policy was in place. One thing that I would stay to my students was that they needed to have a firm grasp on the arithmetic so that if they ‘fumble fingered’ on the calculator they would be able to at least tell if the answer “didn’t look right.” But that still seemed to be a less than convincing reason.
It wasn’t until I heard an interview of Dr. Keith Devlin of Stanford University by Krista Tippet on her show On Being that the light bulb went on.
What I really want to do, within the confines of a developmental course, is to get my students to see the beauty of mathematics, to think mathematically. I want help them see at least a glimmer of the beauty I see in mathematics. Sadly, in the Algebra classroom it can be hard to get away from just teaching the rules and share that beauty with students.
So, the insistence on not using calculators, can we fit it in to helping students develop mathematical thinking? I thinking about this I have found a book that I need to add to my collection, Mathematical Thinking: How to Develop it in the Classroom (Isoda, M and Katagiri, 2012). In the sample chapter there are a couple examples of teaching mathematical thinking using arithmetic. Their examples used multi-digit multiplication, but I have used the same approach with single digit addition.
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9 + 1 = _____
9 + 3 = _____
__ + __ = ____
What pattern do you see?
Can you come up with another equation that fits this pattern?
Can you justify, that is show, that your equation is the
correct answer?
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Then you can move from there to larger problems. Again asking them to explain the pattern they see and to justify their work.
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11 + 19 = ___
12 + 19 = ___
43 + 29 = ___
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Now, the how of the style seems fairly straight forward, the issue then arises of how to implement that in the developmental classroom. I will discuss that in a later article. If you have any ideas, or examples from your own classroom I welcome them in the comments below.
