We all sat through many, many hours of math classes, and we “did math” daily through many years of school. But have you ever stopped to ask yourself what it really means to “do math”?
In 1989 the Mathematical Sciences Education Board (MSEB) published a landmark book titled Everybody Counts, and on p. 31 there appears this wonderful quote: “Mathematics is the science of patten and order”.
Most of us understand that “doing science” is more that just getting quick answers. Rather, we associate with science many verbs such as these:
Of course, there are many more verbs that could be added to the list, but what is important in all of them is that they indicate a process of “figuring out” or “making sense” of the science. All of the verbs require active participation and deep thinking about the ideas. It is virtually impossible to be a passive observer and “do science”. Science is really about figuring things out, and it always begins with some problematic situation.
Mathematics should be this way also. Students should regularly be required to figure out the mathematics of a situation, to work to solve mathematical problems. And what is amazing about this process is that, in doing that hard work of problem solving, students come to build an understanding of the mathematical concepts. My personal term for this is “noodling” — I guess I derive that from the idea of “using your noodle.” 🙂
Traditionally in North America we show students how to do something (e.g., multiply decimals with tenths such as 3.5 x 2.3) and then have them practice a page of practice problems. But rarely do we them give questions that require them to show an understanding of the mathematics behind the procedure, to demonstrate with models or with diagrams what is going on in the multiplication. Just why does multiplying tenths times tenths give us an answer in hundredths? Sadly, many students (or adults!) can’t answer that — they just know they are supposed to “count the decimal places in the numbers being multiplied and count that many in the answer”.
Later, as students go on to use calculators more, few often stop to look at the answer displayed and ask themselves if the answer is reasonable. Clearly the answer to 3.5 x 2.3 should be between 6 (3 x 2 — both numbers rounded down) and 12 (4 x 3 — both numbers rounded up). But if a wrong number was accidentally entered or a decimal left out in putting in the data, few students ever notice.
A single mathematical problem for which students have to “noodle out” the answer is almost always a better plan than having students to a page of practice problems that has no conceptual meaning for them. We need to provide ways for students to build an understanding of mathematics.
Let’s give students the opportunity to really “do math” in a meaningful way!