Saturday, April 19, 2008

The Relationship of Perimeter to Area

In answer to the Perimeter-Area Quiz from last week:

The area of a rectangle is always maximized, relative to perimeter, when the rectangle is a square.

To the left is an example where the perimeter of the rectangles remains a constant length of 24 units. The area increases as the sides of the rectangle get closer in length to each other.

To the right is an example where the area of the rectangles remains constant at 16 square units. The long thin rectangle has a perimeter of 34 units, the next rectangle has a smaller perimeter of 20 units, but the square's perimeter is just 16 units.

Both examples show that the maximum area of a rectangle, relative to its perimeter, occurs when the rectangle is a square.

Wednesday, April 16, 2008

The Case of the Unknown Perimeter

The Program for International Student Assessment report ranks American 15-year-olds 24th out of 29 in math among developed countries (see Education Next article).

Let’s see what that looks like in the American classroom. Recently, my Geometry students had a question in their textbook:

What is the perimeter of a regular hexagon with a side of 4 units. There was a picture, like the one on the left.

Some of my sophomore students had trouble with this problem because they didn’t know what perimeter meant. This is troubling because the concept of calculating perimeter, and the use of the word perimeter, are introduced in California in the 3rd grade math curriculum and are then reinforced in the 4th, 5th, 6th, 7th, and 8th grade curricula. They have worked on the concept of perimeter for seven years before entering my classroom.

When new math terms are introduce in classes, I talk about the roots of the words so that the students can connect mathematical terms with English terms. In this case, peri- means around and –meter means measure. Therefore it is curious that the students didn’t understand the question.

As I dug deeper, it became obvious that the students didn’t know what it meant to “measure around” something. I wondered if these particular students had ever held a tape measure in their hands let alone actually measured a room to calculate its area.

California started eliminating or changing vocational courses in schools years ago. Sewing, wood shop, cooking, drafting, and art, have been replaced by other courses. In the public school in Northern California, where I last taught, the cooking room was turned into a Cisco lab, the sewing room was turned into a computer lab, the drafting room was turned into administrative offices (during the school remodel), and the photography lab was moved to an area that floods in the winter rendering it useless. In any number of these “hands-on” courses, mathematical concepts, that would otherwise be abstract such as perimeter and fractions, have context, making them cognitively more accessible to the student. I’m not going to argue against computer classes but I believe that the gap, for some students, is in having a hands-on experience with math.

In the recent National Math Panel Report it is suggested that there are too many topics in early grades, that there should be fewer topics covered more thoroughly. I agree with the Panel, depth has suffered and useless topics are included in the math curriculum (7th grade students in California learn about Stem and Leaf Plots, a topic that I, with a degree in mathematics, had never heard of until I taught 7th grade math). We skim over topics without giving student the depth of understanding necessary to retain them. The result is 10th grade Geometry students who don't understand the concept of perimeter. Perhaps the Powers-That-Be should also be looking at reintroducing some of the enrichment programs that, at one time, were options in education in the United States.

Monday, April 14, 2008

The Mad Rantings of a Math Mother

Captive Audience

I don’t have a clue about what other mothers talk to their kids about when driving in the car but when my children were little, I routinely took the opportunity to quiz them about math facts. It seemed logical. All the moms were reading to their kids and playing Mozart to them, (even from outside of the womb… I live in Marin County, California), so why should mathematics be left out?

Now my kids are grown, and I’ve spent the last 15 years teaching math so I have an inkling that those were odd thoughts (versus even thoughts), and that the vast majority of mothers, fathers, parental units, guardians, nannies, au pairs, grandparents, and whoever else takes care of children, do not drive around in the car asking their charges, “What two numbers multiply to twenty and sum to nine?” let alone when the kids are in kindergarten and 3rd grade.

The “What Two Numbers” game helps build a foundation for when kids encounter the ever-so-challenging concept of Factoring Quadratics.

The “captive audience math challenge” is an effective way to develop math sense, just as reading to a child from an early age is an effective way to develop literacy.

Perimeter and Area of a Rectangle

Liping Ma, in her treatise Knowing and Teaching Elementary Mathematics, poses the following scenario: A girl told her teacher that she had discovered that when the perimeter of a rectangle is increased, the area also increases.

Is this correct? See side bar to vote.

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