Tuesday, July 17, 2007

Developing Conceptual Reasoning in Geometry Via Standardized Test Questions

Ok, here's a fairly typical SAT-type of question that requires application of fundamental ideas in geometry. Some students 'see' a way to find the value of x in less than 15 seconds. These students have strong conceptual ability and are confident of their knowledge and reasoning. They are not thrown by a question that is somewhat different from the textbook problems they've seen.
Some of the issues as I see them are:
(1) How do we raise the knowledge/experiential base and confidence of those students who cannot seem to find a solution path and inevitably give up?
(2) How do we extend the thinking of those talented students who solve the problem in short order and just sit there complacently?

Perhaps, beyond these considerations and the instructional strategies employed is the bigger issue of PROVIDING FREQUENT CHALLENGES FOR OUR STUDENTS THAT GO BEYOND NORMAL TEXTBOOK FARE.
Where does one normally find these types of challenges in school curricula? Embedded in a natural way in the regular set of textbook exercises that can be routinely assigned? OR are they labeled as Standardized Test Practice at the end of a section or in a separate part of the chapter or text? OR are they found primarily in ancillary materials provided by the publisher? Can you guess where I think they should be and if they should be labeled? Should they be stand-alone multiple-choice questions or more open-ended with several parts that go beyond the 'answer?'

Before I suggest an activity based on this innocent-looking problem, I invite readers to consider a variety of methods of solution (so far I've observed about 6-7 'different' approaches) and how one might go beyond this standardized test question to deepen student reasoning. To remove the element of surprise and focus attention on process, the 'answer' is 90... (sorry!).

Have fun but if you 'see' the answer in 15 seconds or less, don't stop there!


mathmom said...

Ok, it took me a few more than 15 seconds, but the easiest way I see to solve this problem is to draw QR and then look at the sum of the interior angles of PQR and SQR. Let's call the two unlabeled angles in SQR y and z. The sum of the interior angles in PQR is 40+(20+y)+(30+z) = 90+y+z. Since that has to sum to 180, we know y+z=90. Then since x+y+z also has to sum to 180, x has to be 90 degrees.

Another way to solve it would be to look at sum of the interior angles of the quadrilateral PQSR. I'm not sure that the usual interior angles formula applies for non-convex polygons, but since this one can be easily broken into two triangles by drawing in segment PS we do know that the sum of the angles in that quadrilateral must be 360. We have 90 already, so the big angle QSR inside the figure must be 270 degrees, and x must be 360-270 = 90 degrees. (Is there a word like complementary and supplementary for angles that sum to 360?)

If you count drawing PS and then figuring out that angle PSR plus angle PSQ must sum to 270 as a different way than the above, then that's 3. But I don't have anything else.

You don't need my comments as to whether problems like this should be included in the regular curriculum; you can guess what my answer will be just as I can guess yours. ;-)

As to how to help students get better at this, it is hugely a matter of experience, IMO. But there are also "heuristics" you can suggest when looking at geometry problems. Look for triangles. Add lines to create triangles. Look for right angles. Look for special triangles (30/60/90, isosceles, equilateral, etc.). Making it second nature comes from experience, but I think students do need to have a list of strategies to draw upon.

I don't do much geometry with my elementary/middle school groups (and what we do tends to be perimeter/area/volume and not angles) but we do have a list of strategies (mostly for non-geometry problems) in our toolboxes to look at when no ideas leap to mind, such as: find a pattern, try a simpler problems, act it out, guess and check, make a picture, make an organized list, use logic, remember a similar problem. I will note that nowhere on my list is "use a memorized formula" though that sometimes is the right way to solve a problem. But this list is for when they don't already know or remember a formula for solving the problem.

Totally_clueless said...

(1) S+x = 360, and S+P+Q+R = 360. Thus, x=P+Q+R =90.
(2) You find S first and then find x
(3) More complicated methods with constructions etc, gradually increasing the amount of complexity.

I agree with Mathmom that experience is the way to develop intuition.

Extension: Give the figure, and the problem is to show that point S is not the center of a circle passing through P,Q and R


Totally_clueless said...

Further extension: With the figure given, the exercise is to determine whether PS is longer or shorter than the radius of the circle that passes through P, Q and R.


mathmom said...

The problem isn't just in math education: This NYT science article discusses a very similar problem in teaching physics:

From what I’ve seen, students in science classrooms throughout the country depend on the rote memorization of facts. I want to change this. The students who score high do so because they’ve learned how to regurgitate information on tests. On the whole, they haven’t understood the basic concepts behind the facts, which means they can’t apply them in the laboratory. Or in life.

On a physics exam, the student will see a diagram and they’ll classify it. Then, it’s simply a matter of putting the right numbers in the right slots and, sort of, turning a crank. But this is algebra. It is not physics. When you test the students later on the concept, they can’t explain what they’ve just done.

This saddens me. In my laboratory, we’ve made some important discoveries. Several were accidental — serendipitous. If we’d only functioned on the standard knowledge, we wouldn’t have recognized what was before us.

I can solve TC's exentsion using a theorem I only recently learned about the relationship between an inscribed angle and the central angle subtended by the same arc. With that plus some intuition, I can solve TC's second extension as well. I think those would be tough for students without much circle geometry experience, though

Jackie said...

My method was the same as the first method posted by totally clueless.

If there is time, I'm going to give it to my students tomorrow.

As for TC's extension and mathmom's second comment, this might make a great lead in to geometry of circles. On second thought, maybe I'll use this Monday as an intro problem into our circle unit!

Dave Marain said...

hey folks--
It's such a pleasure to sit back and watch outstanding educators take wing with these 'trivial' problems!

mathmom's and jackie's comments are deeply appreciated. tc's extensions to circles are 'awe-sum' (still can't get math puns out of my system).

As educators we celebrate all students' efforts and, of course, when we enable this kind of problem-solving we will forever be surprised by the ingenuity of some of our students. Teachers who routinely create such environments come to expect these flashes of insight but we never cease to be amazed.

On the other hand, who does not appreciate the beauty of tc's first solution! What makes that approach particularly elegant? How often does that type of thinking arise when solving diagram problems? Student NEED to see that method in order for their reasoning to devleop.

The problem is that students need to see a succession of these problems, incrementally developed, in order to help the majority of develop proficiency. Pls quote me the sourcebook for this! If it existed would instructors be more willing to devote more attention to rich problem-solving? OR would there simply not be enough time in the curriculum to include such 'extras'? As long as secondary math classrooms are content-driven and quantity is valued over quality, this will not change. There seems to be more receptivity to this approach at the middle school level.
However, you know I always swing back toward the middle:
There also needs to be a strong balance maintained between skills development (goal: automaticity with percents, decimals, fractions just as there should be for basic facts -- what a pipe dream, huh?) and deeper thinking at a more conceptual level. Great teachers have always managed to do both or at least they never give up trying!

Now I did observe a few more student approaches, one of which surprised me. I will share these later...

mathmom said...

The problem is that students need to see a succession of these problems, incrementally developed, in order to help the majority of develop proficiency. Pls quote me the sourcebook for this!

How about this:
Mathematical Circles (Russian Experience)

It is meant for extracurricular "Math Circles" for 12-14yos, yet I don't think you'll find it too elementary for many of your high school students. I think it's close to the kind of thing you are looking for.

If it existed would instructors be more willing to devote more attention to rich problem-solving? OR would there simply not be enough time in the curriculum to include such 'extras'?

I think "not enough time" is a common "excuse" for not doing this, even more so than "don't have the materials." Somehow we need to get a "can't afford not to" mentality going.

I read something a while back about how each stage of mathematical education is designed mainly to feed the next stage. There is certain content you will need to teach for your students to be prepared for the next level. BUT, those problem solving skills are arguably more important than any particular piece of content.

I think "the basics" are important. But the basics and problem solving are not separate. The problem solving applies the basics in novel ways. It can help show when the basics are forgotten, or not properly understood.

Considering that things take a while to sink in anyhow, how much less content do you really think the average teacher would cover if they gave up one lesson per week (or per 2 weeks?) to non-routine problems?

I know I didn't really learn that kind of problem solving until my first semester in college. I did math contests and I did fairly well on them, but I was shocked at how easy and downright "routine" the "grade 13" (this was in Ontario, Canada, when HS went through grade 13) math contest was after only 1 semester of college calculus, in which I learned "nothing" I hadn't already covered in high school calculus, or so I thought...

Mark and Shelli said...

This is interesting. I drew a line through the quadrilateral from P, creating two smaller triangles. The "larger" of which has angles of 20 and 40, so the new angle that I just created is 120 degrees, which means the new angle in teh smaller triangle is 60 degrees. Now, in the smaller triangle, I have angles of 30 and 60, meaning the angle next to x is 90. Since they form a linear pair, x=90.

I like it :) I'm sticking this one into my "triangle sum" pile.