Thursday, September 22, 2011

Math problems of the week: 1960's vs. 21st Century geometry

Second set of introductory problems:

I. From Weeks Adkins A Course in Geometry (1961), pp. 11-12 [click to enlarge]:

II. From Discovering Geometry: An Investigative Approach (2003), pp. 8-9 [click to enlarge]:

III. Extra Credit:

1. Which is more important in the 21st century: precise definitions or artistic designs?

2. Estimate the volume and mass of the Discovery and Akins texts. Hint: one of these texts has twice the volume and three times the mass of the other.


Deirdre Mundy said...

Hey Katherine- I just came across something that might appeal to your son--

Around here, we have a blacksmith's shop that starts offering classes at age 11. You might want to check around Philly for something similar-- I know until now he's mostly been all about electronics, but the combination of fire, hammers, metal and the chance to make really useful things might appeal to him.....

Mnemosyne's Notebook said...

Katharine - I must confess to having strong and conflicted feelings about these two examples. My personal preference is for the rigor of the first, yet having recently read a biography of the great physicist Paul Dirac, I am reminded of the power of drawing and visualization in mathematics and science (and art, obviously). I'm sure the second text is weightier, and may have much that is fluff. But I don't think the example shown completely fluffy. Developing the skills to imagine the symmetries of organic molecules is a pretty advanced cognitive process, one that needs to begin somewhere.

I'm teaching 8th grade math for the first time this year (after having taught pre-calculus and calculus to high schoolers) and I would like to introduce fractals into the coursework along with tilings and visual pattern recognition. I think the beauty and charm of the figures could help some students get a sense that visualizing physical problems is a completely appropriate way to problem solve, that equations are not the only way to think about such things (even if they are my preferred way). But, I am as yet unsure of how might I go about doing this in a way that fulfills the "rigor and relevance" requirements of my administration. I don't think that a bit of "math appreciation" is bad from time to time.

I was fairly skeptical about using algebra blocks to aid in teaching solving two step and multi-step equations, but in action they seemed like a good way to help students who could not quite make the jump to abstract manipulation of 'x.'

So, the drawing of the curvy figures that contain only straight lines, and the iterations and reflections of the Sierpinski triangle can spur the creativity and curiosity of students who don't have a taste for abstract symbol manipulation.

I'd like to sprinkle some more of that into my class.

On the other hand, with the block scheduling at my school, I count every minute of instruction and want it used to help my students master the subject matter of the standardized tests that will affect me, my students, and my school.

Anyway, thanks for the provocative post. You touched on something that I've been mulling over for a while

Katharine Beals said...

@Deirdre, Thanks for the suggestion. J was fascinated by an old iron working outfit he saw last spring. Noticing that all the tools were themselves made of iron, he asked how the first-ever iron tool was made.

Katharine Beals said...

Mnemosyne's Notebook, Thanks for your thoughts. I agree that it shouldn't be all or nothing. My sense though is that the logic is a big huge part of what's special and useful about learning geometry (including preparation for future math and science classes), and that doing this justice requires a lot of cumulative time and careful practice. So while I might sprinkle in some fractals and symmetry, I'd focus mostly on applied logic. Unfortunately, that logic--those Euclidean postulate-based proofs--seem to have completely vanished from today's geometry.

Mnemosyne's Notebook said...

Katharine - My own recollections of geometry are of enjoying both the logic of the proofs and the complex figures that can be constructed with simple implements. I agree that both are important for future work in math, engineering, and science. My daughter's 10th grade geometry class (two years ago) seemed to skimp on the drawing, while retaining some of the logic and proofs. A lot of time in geometry was spent reteaching the basics of linear equations and their graphs - stuff that was covered (and should have been mastered) in algebra the previous year. I think that reteaching things for the 10th grade NCLB tests contributes to the dumbing down and fluffiness - and I suspect it is one of the reasons textbook makers come up with their absurd books. There are other pressures, as you well know.

I recall looking for new calculus and pre-calculus textbooks when I was teaching at a small private school. All of the books were so busy in their layouts, with too many methods for solutions. The college counselor at the school remarked that all of the students wanted these overdone, confusing layouts - it was what they liked looking at. They may have liked it (I decided to keep our 10 year old textbooks), but in my observations, none of the students had the power of concentration to pull anything useful out of such visually busy books. I showed one of the texts to the mother of a student with ADHD and she told me that he would have found it completely unreadable. So, this crud has crept even into textbooks for the advanced students. A mile wide and an inch deep. With flashy billboards along both shorelines.

As someone who actually enjoys math, my preference is for the rigor of the proofs that you describe, but in my 22 year career as an engineer, before becoming a teacher, I would say that the knowledge of geometric construction and visualization was often of more value in engineering problem solving than the "hey, this is an example of Side-Angle-Side, I know what to do now." Though, that was also valuable.

I agree with you on the value of rigor, I just wanted to point out the value of skillful visualization - the second textbook brought that out to me. Perhaps the best solution is to kill all of the spiralling/review, teach the previous subjects to mastery, and devote the whole year to real geometry - proofs and constructions both. Here's hoping...