Much can be said in defense of practice exercises, but when all is said and done,
very few students develop deep understanding from routine practice. For example,
compare these two approaches to the area of a trapezoid.

### Approach 1

### The teacher says: ”The area of a trapezoid is given by the formula h(b1+b2)/2, where h is the height, b1 and b2 are the bases. Here is a worksheet where you can practice this.”

The worksheet includes 20 examples, each with different numbers for the bases and the height. The students practice in silence. Many students like this, because they know exactly what to do. Other students don’t like it, because they find it boring. All of them know they will soon have to calculate some trapezoid areas on a quiz, and they hope that this practice will help them remember the formula. Some will make an effort to memorize the formula in preparation for the quiz. Most will have forgotten the formula a week, a month, or a year later. This is because they will not use the formula again, unless they take calculus many years hence. In any case, whether they remember it or not, doing the exercises does not help them understand the formula.

### Approach 2

Students are given a sheet of paper with a few copies of a certain trapezoid on it,
with all the measurements indicated, including the bases, the legs, and the height.
They are not given a formula. They are asked to find the trapezoid’s area. They
are allowed to use scissors to cut out the trapezoids, and if they want to, cut one
of them into smaller pieces that can be rearranged. Rearranging would allow them
to use formulas they already know, such as the one for the area of a rectangle, a
parallelogram, or a triangle. Students will almost certainly come up with different
strategies.

(In fact, see how many strategies you can find.)

Students who find one quickly can be encouraged to look for more. Some students may not like the activity, because they are not told exactly what to do. The teacher can offer hints to them, or encourage them to get help from neighbors.

Once some strategies have been found, the teacher can lead a discussion where students demonstrate their approaches. All strategies will reveal that the lengths of the legs do not contribute to the final answer. In fact all strategies will yield the same answer for the area. A general formula can be the final punch line: applying any of the strategies to a generic trapezoid always yields the same formula.

(Scissors are not absolutely necessary. For example, the activity can be carried out on paper, without any cutting. Whether that is preferable will depend on the specifics of a given class.)

###
Conclusion

Even if the first approach includes a brilliant teacher explanation of the formula, I claim that the second approach is preferable. Many students who cannot remember
the formula at some point in the future will be able to use one of the strategies that
came up in the course of the exploration, either to find a particular trapezoid’s area,
or to reconstruct the formula. This approach also carries the message that formulas
can make sense, that there are many ways to solve a given problem, and that not
everything needs to be memorized. A perhaps unexpected bonus is that the different
solutions to this essentially geometric problem yield different interpretations of the
formula, and some apparently different but actually equivalent formulas. Discussing
this can help improve symbol sense. Finally, if the teacher has an excellent explanation of the formula that was not
found by the students, nothing prevents him or her from sharing it.

(In fact, see how many strategies you can find.)

Students who find one quickly can be encouraged to look for more. Some students may not like the activity, because they are not told exactly what to do. The teacher can offer hints to them, or encourage them to get help from neighbors.

Once some strategies have been found, the teacher can lead a discussion where students demonstrate their approaches. All strategies will reveal that the lengths of the legs do not contribute to the final answer. In fact all strategies will yield the same answer for the area. A general formula can be the final punch line: applying any of the strategies to a generic trapezoid always yields the same formula.

(Scissors are not absolutely necessary. For example, the activity can be carried out on paper, without any cutting. Whether that is preferable will depend on the specifics of a given class.)

###
Conclusion

Even if the first approach includes a brilliant teacher explanation of the formula, I claim that the second approach is preferable. Many students who cannot remember
the formula at some point in the future will be able to use one of the strategies that
came up in the course of the exploration, either to find a particular trapezoid’s area,
or to reconstruct the formula. This approach also carries the message that formulas
can make sense, that there are many ways to solve a given problem, and that not
everything needs to be memorized. A perhaps unexpected bonus is that the different
solutions to this essentially geometric problem yield different interpretations of the
formula, and some apparently different but actually equivalent formulas. Discussing
this can help improve symbol sense. Finally, if the teacher has an excellent explanation of the formula that was not
found by the students, nothing prevents him or her from sharing it. *Starting with the hands-on activity does not prevent the teacher from offering an explanation*, but it does mean that more students will understand the explanation.
This is one of those eerie moments when someone's blogging about the lesson I taught just hours ago.

ReplyDeleteApproach 3:

Provide students an example and explanation of a midline cut, and give them time to practice making those midline cuts on various shapes. ("Draw a bunch of shapes with H Picciotto's Shape Tracer tool, and then draw the midline cuts.")

Then, I gave an assortment of triangles, parallelograms and trapezoids and asked students to find their area. I said that they should solve them however they'd like, but if they were stuck my advice today would be to draw a midline.

This is like Approach 2, but I think with more support. I'm not asking students to discover the midline cut. (Experience teaching this course shows me that kids often think to chop off shapes they recognize like triangles and rectangles, but the midline cut is harder to see.)

There's room between "tell kids a formula" and "let kids figure out whatever works." We can tell kids strategies that, while still being specific, are more generally useful (and simpler to remember) than an area formula.

I have no problem at all with Approach 3. It is not even ruled out by Approach 2. If you think dividing along the midline is important, you can always suggest that later. But really, this is a quibble. Your approach is based on your experience with students at your school. You trust the students to engage with the problem in their own way. You provide some leadership as the teacher, which fulfills your responsibility without smothering them. But in case it wasn't clear, I was not suggesting that it's right to stop at "whatever works". The advantage of students using familiar shapes is that, well, they're familiar. The whole point I was trying to make was that giving students a chance to think about the problem does not prevent the teacher from offering their own strategy. In fact it enhances that.

ReplyDeleteFor sure!

DeleteApproaches 2 and 3 also open up opportunities for kid watching. During a similar lesson last year I found two students who could not rotate or flip to put the original shape back together. I also had more than one student who looked at their rearranged shapes and sai they were bigger now. They thought that the actual area had changed. What are students missing through lack of hands on and talk? A lot more than they ever can from a formulaic teacher driven lesson I think.

ReplyDeleteYeah, when we do all the talking, we can fall into the delusion that the students are all listening, and that they all understand what we're saying. Seeing them work certainly disproves that! And it's so much more useful to know what they don't know early on, rather than wait for a test.

Delete