Monday, November 16, 2015

Lesson 5-6: The Parallel Consequences (Day 56)

Today's lesson focuses on proving the Parallel Consequences -- that is, statements of the form, "if two lines are cut by a transversal, then ..."

This lesson will be set up almost exactly like Dr. Franklin Mason's Lesson 4.4. We wish to prove the converses of the Parallel Tests. We do so by using my favorite trick for proving converses -- we use the forward theorems along with a uniqueness statement. The uniqueness statement we need is the Uniqueness of Parallels Theorem -- in other words, Playfair.

It's possible to prove all of the Parallel Consequences by using the respective test plus Playfair -- so we'd prove the Corresponding Angles (CA) Consequence using CA Test plus Playfair, the Alternate Interior Angles (AIA) Consequence using AIA Test plus Playfair, and then Same-Side Interior Angles Consequence using that test plus Playfair.

But Dr. M only proves one of the consequences using Playfair -- he then uses vertical angles and linear pairs to derive the other consequences, as is traditionally done. We've seen that students should definitely be familiar with using one of the consequences to prove the others. Dr. M uses the Alternate Interior Angles Consequence to prove the others, but as we've discussed before, I'm changing this to the Corresponding Angles Consequence instead.

I looked back at last year's lesson and compared it to what I'm doing this year. In my first lesson after posting the Fifth Postulate last year, I posted some properties of two types of quadrilaterals, isosceles trapezoids and parallelograms. This year, we've already proved the isosceles trapezoid properties.

As for the parallelogram properties, due to the way I've reorganized this course, Lessons 7-6 and 7-7 can now be posted basically intact. I'm saving 7-6 and 7-7 to be the last two lessons taught in the first semester, just before the final. I want the last lessons to be something that's important but tricky to remember, so that it will still be fresh in their minds for the big test. The parallelogram tests certainly qualify here, as do the concurrency theorems (some of which use parallelograms in their proofs). So today's lesson will focus on the Parallel Consequences themselves.

Here we go again -- this is now the third straight post in which I discuss traditionalists, as the fallout from the Atlantic article written by Drs. Katharine Beals and Barry Garelick continues. This article has caught the attention of many math teacher bloggers -- including the King of the Math Teacher Blogosphere, Dan Meyer:

This post has 91 comments -- the most drawn by a Meyer post since his three-act lesson on volume back in September, "Dandy Candies." (I posted Meyer's "Dandy Candies" here on my blog in March.)

Meyer writes:

Math answers aren’t math understanding any more than the destination of your car trip indicates the route you took. When five people arrive at the same destination, asking how each arrived tells you vastly more about the city, its traffic patterns, and the drivers, than just knowing they arrived.
Their other exemplar of understanding-without-explaining is strange also. Mathematicians advance the frontiers of their field exactly by explaining their answers – in colloquia, in proofs, in journals. Those proofs are some of the most rigorous and exacting explanations you’ll find in any field.
Those explanations aren’t formulaic, though. Mathematicians don’t restrict their explanations to fragile boxes, columns, and rubrics. Beals and Garelick have a valid point that teachers and schools often constrain the function (understanding) to form (boxes, columns, and rubrics). When students are forced to contort explanations to simple problems into complicated graphic organizers, like the one below from their article, we’ve lost our way.
[emphasis Meyer's]

Whenever I discuss another article or blog post, I like to discuss some of the comments. Meyer already has a "Featured Comments" section in his post, so I don't need to repeat those comments. I do point out that both authors of the original article posted comments on Meyer's post. Here are some excerpts of their comments:

  1. The American approach is to build conceptual understanding
    through time-consuming student-centered discovery of multiple solutions and explanations of relatively simple problems. An internationally more successful approach is to build conceptual understanding through teacher-directed instruction and individualized practice in challenging math problems.
    It would be interesting to see how American students would do on the Finnish and Chinese exams as compared with their Finnish and Chinese peers. [Full comment available at -- dw]
  2. on 13 Nov 2015 at 9:34 amBarry Garelick
    I did in fact major in math (though I am not a mathematician). But in elementary school I was not yet a math major and received mostly C’s in arithmetic and was poor in aspects of it. It was not as intuitive for me as you assert, but I was able to build on procedural knowledge to gain understanding as I progressed through school. I would not have been able to provide good explanations of how I solved problems in lower grades, though as I learned more in high school, I likely could. The accusation of selection bias is just that–an accusation. Our article included links to referenced peer-reviewed papers that show that procedural fluency and conceptual understanding work in tandem. We do not negate the importance of conceptual understanding in teaching math. The math I learned in elementary school included the conceptual underpinning. I was not taught in the “rote” fashion as math teaching of that era is often mischaracterized.
    I appreciate the interest in the article and the conversation on this blog. [Full post at]

I repeat Beals here: "An internationally more successful approach is to build conceptual understanding through teacher-directed instruction and individualized practice in challenging math problems." And I re-emphasize what I wrote last week -- it's more successfully only if the students actually do the challenging math problems. Even Garelick, who struggled in elementary school math, went on to do the work in high school. I'm talking about the students who simply won't do the work when they are in a teacher-directed classroom.

But, as Meyer points out, a third major traditionalist entered his comment thread. That traditionalist is Dr. Ze'ev Wurman. Indeed, Meyer devotes an entire post to an exchange between Dr. Wurman and math teacher Brett Gilland, who represents the progressive side of the debate:

Meyer says of the debate, "I'm sure that everyone walked away feeling like their side won, but one side is wrong about that." He never states though which side is the winning side. I can't be sure, but here's one thing I say about Meyer -- his famous "three-act lessons" hardly sound like lessons of which a traditionalist would approve.

As far as my own opinion is concerned, now I'm starting to lean towards the majority in the first thread who argue that the amount of work needed for that middle-school percent problem is ridiculous and excessive. I agree with the first commenter in that thread:

Great points. Understanding is the goal and simply swapping one rigid formulaic representation for another does not ensure understanding. Creative flexibility as we move toward the goal seems like good advice.

It goes along with what I wrote last week about making students draw lines down the equal signs in their equations in Algebra I classes -- I wouldn't force students to draw the lines or deduct points, but I'll try always to draw the lines myself, and suggest the same to students who regularly get wrong solutions to their equations.

To conclude this post, Gilland posted a few sample Algebra I, Geometry, and Algebra II questions, and Beals replied that she hopes to blog about some of those questions as part of her "Math Problems of the Week" series. If she writes about any of the Geometry problems Gilland posted, then of course I'll mention it here on my own blog as well.

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