A geometry lesson inspired by a silvering company – and a rant about computerized learning programs

Here is some information that teachers at quite a few different levels could use* for a really interesting geometry lesson involving reflections involving two or more mirrors, placed at various angles!

Certain specific angles have very special effects, including 90, 72, 60, 45 degrees … But WHY?

This could be done with actual mirrors and a protractor, or with geometry software like Geometer’s Sketchpad or Desmos. Students could also end up making their own kaleidoscopes – either with little bits of colored plastic at the end or else with some sort of a wide-angle lens. (You can find many easy directions online for doing just that; some kits are a lot more optically perfect than others, but I don’t think I’ve even seen a kaleidoscope that had its mirrors set at any angle other than 60 degrees!)

I am reproducing a couple of the images and text that Angel Gilding provides on their website (which they set up to sell silvering kits (about which I’ve posted before, and which I am going to attempt using pretty soon, on a large astronomical mirror I’ve been polishing for quite some time)).

At 72º you see 4 complete reflections.

When two mirrors are parallel to each other, the number of reflections is infinite. Placing one mirror at a slight angle causes the reflections to curve.




Rant, in the form of a long footnote:

* assuming that the teacher are still allowed to initiate and carry out interesting projects for their students to use, and aren’t forced to follow a scripted curriculum. It would be a lot better use of computers than forcing kids to painfully walk through (and cheat, and goof off a lot) when an entire class is forced to use one of those very expensive but basically worthless highly-centralized, district-purchased computer-managed-instruction apps. God, what a waste of time – from personal experience attempting to be a volunteer community math tutor at such a school, and also from my experience as a paid or volunteer tutor in helping many many students who have had to use such programs as homework. Also when I was required to use them in my own classes, over a decade ago, I and most of my colleagues found them a waste of time. (Not all – I got officially reprimanded for telling my department chair that ‘Renaissance Math’ was either a ‘pile of crap’ or a ‘pile of shit’ to my then-department head, in the hearing of one of the APs, on a teacher-only day.

Keep in mind: I’m no Luddite! I realized early on that in math, science, and art, computers would be very, very useful. I learned how to write programs in BASIC on one of the very first time-share networks, 45 years ago. For the first ten years that my school system there was almost no decent useful software for math teachers to use with their classes unless you had AppleII computers. We had Commodore-64’s which were totally incompatible and there were very few companies (Sunburst was one) putting out any decent software for the latter. So when I saw some great ideas that would be ideal for kids to use on computers to make thinking about numbers, graphs, and equations actually fun and mentally engaging, often I would have to write them my self during whatever free time I could catch, at nights and weekends. Of course, doing this while being a daddy to 2 kids, and still trying to teach JHS math to a full load of students (100 to 150 different kids a day at Francis Junior High School) and running a school math club and later coaching soccer. (I won’t say I was a perfect person or a perfect teacher. I believe I learned to give better math explanations than most, didn’t believe that you either have a ‘m,ath gene’ or you don’t, at times had some interesting projects, and at times was very patient and clear, but had a terrible temper and often not good at defusing things. Ask my kids or my former students!) Later on, I collaborated with some French math teachers and a computer programmer to try to make an app/program called Geometrix for American geometry classes that was supposed to help kids figure out how to make all sorts of geometric constructions and then develop a proof of some property of that situation. It was a failure. I was the one writing the American version, including constructions and tasks from the text I was currently using. There was no way I could anticipate what sorts of obstacles students would find when using this program, until I had actual guinea pig students to use them with. Turns out the final crunch of writing however many hundreds of exercises took place over the summer, and no students to try them on. Figuring out hints and clues would require watching a whole bunch of kids and seeing what they were getting right or wrong. In other words, a lot of people’s full time job for a long time, maybe paying the kids as well to try it out so as to get good feedback, and so on. Maybe it could work, but it would require a lot more investment of resources that the tiny French and American companies involved could afford. We would have really needed a team of people, not just me and a single checker.

I find that none of these computer-dominated online learning programs (much less the one I worked on) can take the place of a good teacher. Being in class, listening to and communicating logically or emotionally with a number of other students and a knowledgeable adult or two, is in itself an extremely important skill  to learn. It’s also the best way to absorb new material in a way that will make sense and be added to one’s store of knowledge. That sort of group interaction is simply IMPOSSIBLE in a class where everybody is completely atomized and is on their own electronic device, engaged or not.

Without a human being trying to make sense out of the material, what I found quite consistently, in all the computerized settings, that most students absorbed nothing at all or else the wrong lessons altogether (such as, ‘if you randomly try all the multiple choice answers, you’ll eventually pick the right one and you can move on to some other stupid screen’; it doesn’t matter that all your prior choices were wrong; sometimes you get lucky and pick the right one first or second! Whee! It’s like a slot machine at a casino!).

By contrast, I found that with programs/apps/languages like Logo, Darts, Green Globs, or Geometer’s Sketchpad, with teacher guidance, students actually got engaged in the process, had fun, and learned something.

I find the canned computer “explanations” are almost always ignored by the students, and are sometimes flat-out wrong. Other times, although they may be mathematically correct, they assume either way too much or way too little, or else are just plain confusing. I have yet to detect much of any learning going on because of those programs.


Bad ‘rigorous’ Common-Core-style geometry problem

Here is a problem put out by Houghton-Mifflin and labeled ‘test prep’ that I came across as a worksheet for a 9th grade Geometry student I’m currently helping here in Washington, DC.

How many things can you find wrong with this question?

goofy proof sss question

If you have a hard time reading the question, it asks, “Why is segment DE the perpendicular bisector of segment FC?” (FC wasn’t drawn by the makers of the worksheet; that was added by the student right before I took the photo with my phone.)

Here is what I find objectionable:

(1) Segment DE is in fact NOT the perpendicular bisector of segment FC. It might be perpendicular, if EB’A’D is a rectangle, which is nowhere given in the problem, but even if the skinny space between the two triangles is a rectangle, DE does not intersect segment FC’ at its midpoint.

(2) We should not be asking students to prove things that are not true.

(3) The first two paragraphs of verbiage* are unnecessary, but if we take them at face value, you could end up with either of these two diagrams instead:

another possible sss question

or else the following:

yet another poss sss q

In neither case is it true that DE is even perpendicular to undrawn segment CF, even though I definitely used ‘rigid motions’ to transform triangle ABC into triangle DEF.

(4) In geometry classes, an apostrophe after a capital letter means something: that a point is the result of some sort of transformation performed on the point that lacks the apostrophe. In this problem, points A, B, and C should not be written as A’, B’, and C’.

(5) Here is a much more straightforward problem: You are given the congruences shown in the diagram. Explain why (or “Prove that”) segment HJ is the perpendicular bisector of segment IK.

better sss question


Maybe this is what Sandra Stotsky was complaining about when she dissented from approving the Common Core math standards, objecting to making transformations (such as rigid motions or dilations) the core of geometry .

*For those of you unfamiliar with the term, a rigid motion means a rotation (turn), a reflection (flip), or a translation (slide) — they  normally change the location of a figure without changing its size, area, or angles. And, yeah, I taught geometry to 8th and 9th graders for many years, so I know a little bit of what I’m talking about.

More Problems With DCPS Curriculum and DC-CAS

Upon taking a closer look at the DCPS standards and the DC-CAS, I submit that they should probably both be ignored by any teacher who actually wants to do right by students. If you are doing a good job teaching the things that students should actually know, it won’t make much difference on their DC-CAS scores. Conversely, if you teach to the DC-CAS, you are short-changing your students.

Case in point: Standards in Geometry and Algebra 1 ostensibly covered on the 10th grade DC-CAS. Recall that all 10th graders at this point in DCPS have supposedly finished and passed Algebra 1, and are enrolled in at least Geometry by 10th grade.

I have prepared a little chart giving the standards (or learning objectives) for Geometry: the ones listed in the DCPS list of learning standards, and the number of questions that I found on the page of released DC-CAS questions that supposedly address that standard. There is almost no correlation at all. In fact, if you threw a dart at the topics and chose them randomly, you would do a better job than the test-writing company did.

Published in: on March 23, 2011 at 12:37 pm  Leave a Comment  
Tags: , , , , ,

More Weird DC-CAS Questions

I began looking at the released 10th grade math questions today, and as usual I found some weird ones.

Here is one, where the only difference between answer C and D is the color scheme (D fits the colors in the graph, C doesn’t). Both of them have the math correct. Is the color scheme all that significant? Is that what we are testing for now?

Here’s another one, which merely asks students to tell the difference between a mean, a median, and a mode. Wait a second – isn’t that one of the 6th, 7th, and 8th grade standards?

Here are the exact wordings for the various “standards” that involve mean, median and mode:

For 6th grade: “6.DASP.1. Describe and compare data sets using the concepts of median, mean, mode, maximum and minimum, and range.”

For 7th grade: “7.DASP.1. Find, describe, and interpret appropriate measures of central tendency (mean, median, and mode) and spread (range) that represent a set of data.”

For 8th grade: “8.DASP.1. Revisit measures of central tendency (mean, median, and mode) and spread (range) that represent a set of data and then observe the change in each when an “outlier” is adjoined to the data set or removed from it. Use these notions to compare different sets of data and explain how each can be useful in a different way to summarize socialphenomena such as price levels, clothing sizes, and athletic performances.”

And for Algebra 1: “AI.D.1. Select, create, and interpret an appropriate graphical representation (e.g., scatter plot, table, stem-and-leafplots, circle graph, line graph, and line plot) for a set of data, and use appropriate statistics (e.g., mean, median, range,and mode) to communicate information about the data. Use these notions to compare different sets of data.”

Why is DCPS testing such a low-level skill in Algebra 1? And why do we insist on loading the curriculum with the same eleventy-umpteen standards each year, only varying by an adjective or adverb or phrase or two? Is it because we assume that nothing at all gets learned in any year, so that teachers have to yet again re-teach EVERYTHING all over again, starting from nothing?

Published in: on March 23, 2011 at 12:28 pm  Comments (1)  
Tags: , , ,
%d bloggers like this: