Greetings, mathaholics and mathphobics,
At PLML, we're very lucky to have assistance from Nan Ding, a Graduate student at Harvard whose expertise is - surprise! - math. She is sharing her insights as we develop our understanding and strategy around math, and helping us develop a system that we can use to identify critical math skills for teaching and learning.
I asked her to break down two perspectives on what makes math hard to learn - an American perspective, and a Chinese perspective. Here's her response, which I think you'll find fascinating:
Specifically for Chinese students, they may have high proficiency in solving math problems but they have low understanding about underlying mechanisms.
Chinese educators always hold the golden credo that practice makes perfect. Usually when a math teacher teaches something new, she/he will go quickly through the concept-analyzing and directly immerse students with a good number of exercises.
So the normal case would be like when a Chinese student is good at solving problems, he/she maybe don’t even know what they are operating on. Compared to a new set of teaching perspective-Teaching for understanding- proposed by Wiske (2005), such a way of teaching math could be named as “teaching for operating”.
As for US students, they may spend a lot of time working on real objects, which is super beneficial for knowledge understanding.
However, later they may feel reluctant to move one step further to the symbolic level. In other words, they may become too comfortable with real objects and feel the symbolic representations tedious and hard to understand.
Consequently, when they are still struggling with carrying and borrowing in addition and subtraction, their Eastern peers could flexibly operate multi-digit multiplication. A great metaphor to describe learners in two cultures might be like two types of workers on the assembly line. One type of workers (Eastern learners) know a lot about how to make the whole process move along but have no idea what they are producing. While the other type of workers could work on specific part but have loose control on the next step and what the whole process would look like.
So, educators, I bring it to you: what can we learn from these perspectives to improve our math instruction? What additional questions do you have for Nan?
Make it a teachable day,
--Dave
Comments
Remember TIMS
You may recall the TIMS study (Third International Math Study?) from the 90s. In that study researchers looked closely at videos of about 100 middle school math teachers each in Germany, the US and Japan and concluded that US teachers almost universally taught math as procedures, with little or no focus on underlying concepts, while their German and especially Japanese counterparts focused more on understanding concepts, and in Japan there was strong encouragement for students to explore a problem and come up with their own solution and then for the teacher to lead students through discussion to the best/most efficient procedure. It is laid out in the book, "The Teacher Gap".
d'oh -- that we need both.
d'oh -- that we need both.
Now, what hasn't been mentioned is that it's a whole lot easier to honestly assess mastery of the procedures. Therefore, the Chinese student who passes the test will know what the teacher thinks s/he knows.
In America, teachers often *teach* the concept ... er, that is, they present it. The student, however, often works really hard to figure out the procedure given the limited exposure to *that* and the limited practice. The student doesn't get the concept, either, but memorizes the answer expected (then forgets, since it's not connected to anything).
The Chinese student remembers those procedures and later on, at least has some raw material for figuring out the concepts (tho' it is also a challenge to crack the "what's the procedure?" mindset).
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