## Transforming Mathematics Education: BAMAS, X

This past Saturday marked the tenth meeting of the Bay Area Mathematical Artists Seminars.  You might recall (see the post about Bay Area Mathematical Artists Seminars, VI) that at a recent meeting, we had a very stimulating dinner conversation about the future of mathematics education, with Scott Kim helping to guide the conversation.

Everyone was so engaged, it was unfortunate that the conversation had to come to an end.  So I invited Scott to lead a more formal discussion at a later meeting of the BAMAS.  We met at BAMAS member Stacy Speyer’s place — thanks for hosting, Stacy!

The discussion was quite animated.  Scott prepared a handout based on a lengthy blog post he wrote about various issues revolving around mathematics and mathematics education.  He graciously gave me permission to reblog his ideas.  The post is rather lengthy, so I’ll share it in installments.  You can go to Scott’s blog yourself if you can’t wait to read more.  So without further ado, I’ll let guest blogger Scott Kim take the wheel.  His original post was dated July 6, 2014.

# Navigating Math Education

Imagine that you are a sailor on a leaky boat that is on fire, sailing in the wrong direction, with a quarreling crew. Which problem would you fix first?

Well, that depends. If the leak is slow and the fire is raging, then you would put out the fire first. If the leak is gushing and the fire is small and contained, you would fix the leak first. It makes sense to fix the most urgent problem first.

What you would NOT do is fix one problem and declare victory. If your goal is to get to your destination safely, then you must fix ALL the problems, no matter how difficult. Anything less will not get you where you want to go.

Such is the situation with math education. The problems are so difficult and so numerous that it is tempting to fix one problem, and give up on the rest. And certainly we have to prioritize if we are to make progress. But if we are to get the ship of math education back on course, then we, collectively, must fix ALL the problems of math education. Nothing less will get us where we want to go.

Fixing all of math education may sound impossible or impractical. And indeed it is a formidable challenge. Well-meaning entrepreneurs who have launched successful businesses frequently grind to a halt when they try to start their own innovative schools. Resistance comes from all sides — standardized testing, textbook publishers, parents, administrators, government officials, and the students themselves trying to get into college.

But change is in the wind. America is losing its competitive edge, colleges are becoming impractically expensive, and the internet makes us dream of free education right now for everyone. I say we face the problem with eyes wide open, assess the full range of challenges we face, and look for the smartest moves that get us where we want to go.

With that in mind, here is my survey of the problems plaguing math education, and steps we can take to fix them. I’ve grouped the challenges into four levels that range from the tactical to the strategic: Mechanics, Meaning, Math, and Society.

## Level 1. Faulty MECHANICS (fire)

### The most obvious and urgent problem is that the mechanics of math are taught as a series of blink and you’ll miss it lessons, with little opportunity to catch up.

This one-size-fits-all conveyor belt approach to education guarantees that virtually everyone gradually accumulates holes in their knowledge — what Khan Academy founder Sal Khan calls Swiss cheese knowledge. And little holes in math knowledge cause big problems later on — problems in calculus are often caused by problems in algebra, which in turn are caused by even earlier problems with concepts like fractions and place value.

Here are three ways to fight the fire of poor pacing.

1a. Self-paced learning. The Khan Academy addresses the urgent problem of pacing by providing short video lectures that cover all of K-12 math. While the lectures themselves are fairly traditional, the online delivery mechanism allows students to work at their own pace — to view lectures when and where they want, and to pause and rewatch sections as much as they need. All lectures are freely available at all times, so kids can review earlier concepts, or zoom ahead to more advanced concepts. Short online quizzes make sure that kids understand what they are watching. And with an online dashboard that shows exactly how far each child has progressed, teachers can assign lectures as homework, and use class time to tutor kids one on one on exactly what they need.

Solution: the “flipped classroom.”

1b. Visual learning. I love the Khan Academy. My son hated it, because he, like many students, is a visual learner, and Sal Kahn’s lecture stick largely to traditional symbolic math notation. He would have done better with a visual experiential curriculum. Some kids are primarily audio or kinesthetic learners, some learn best socially. The bottom line is that different kids learn in different ways, and no one way is right for everyone. Education needs to address all learners, not just kids who learn in words.

Solution: teach every lesson three different ways.

1c. Testing for understanding. Nothing can change in education unless testing changes. Traditional standardized tests born of the No Child Left Behind era use multiple choice tests that assess only rote memorization of routine math facts and procedures. The new Common Core State Standards for mathematics, now entering schools across the nation, replaces standardized multiple choice tests with richer tests that include essay questions graded by human beings — a better way to assess mathematical understanding.

Solution: better assessment.

If we douse the fire of poor pacing in math education, we will increase test scores and student confidence. But there is more to mathematics than teaching the mechanics well.

I hope your interested is piqued!  Scott will continue next week….

## Circle Geometry

Today, I thought I’d share a little more about things learned along the way with my curriculum consulting.  As I mentioned before, I’m creating a series of online lectures for the Geometry unit.  This past week, the section I was working on (and will still be working on into next week) is Circle Geometry.

As I also remarked earlier, I’m using the University of Chicago School Mathematics Project’s textbook on Geometry as a reference.  In this text are many theorems about the measure of the angle between two intersecting lines in terms of the measures of the intercepted arcs.

This image is certainly familiar.

The question I had to consider was how to organize all these results in a coherent 5–7-minute lecture.  It turns out that there was too much for just one lecture, so I did spread it out into two.  But I still needed a flow.

Although the results were not new to me, I had never taught this topic before.  My main experience teaching geometry at the high school level was designing and teaching a course on spherical trigonometry as it applies to studying polyhedra.  So this gave me an opportunity to stand back and just think about putting it all together.

I was happy with what I came up with — an approach which could be classified under “combinatorial geometry.”  I decided to pose the following question:

In other words, if you have two intersecting lines, and you draw a circle so that it intersects both lines, what configurations are possible?

Looked at in this way, there are just two considerations:  whether the intersection of the lines is outside, on, or inside the circle, and whether the lines are secant or tangent lines.

It’s not difficult to make the enumeration, so I’ll just give it briefly here.  There is only one configuration if the intersection of the lines lies inside the circle, since both lines must be secant lines.

When the intersection of the lines is on the circle, one of the lines may be tangent, although both cannot be since there is a unique tangent at any given point on a circle.  And when the intersection of the lines is outside the circle, zero, one, or two of the lines may be tangent to the circle.

This enumeration allows for a systematic approach.  If you’ve ever worked through find the angle measures, you know that starting with the arrangement in the upper right corner is the way to begin.  I won’t go through all the details, but I will just indicate that the following figure is all you need:

This simple case is analyzed by considering $\angle QOR$ as an angle exterior to $\Delta POQ.$  The analysis of all the other cases builds from this.

I decided to include a discussion not found in the UCSMP text — continuity.  Of course this is not a topic which can be rigorously discussed at say, the 10th-grade level.  But why not give students an intuition of the idea?

This series illustrates the case that the intersection of the lines is outside the circle, and one of the lines is tangent.  We look at this as the limiting case of a series of pairs of secant lines.

This argument depends upon the fact that the measurement of all arcs and angles varies continuously as $S$ moves around the circle.  While, as mentioned, this cannot be addressed rigorously, it is a very intuitive argument.  Moreover, there are many different software packages you could use to make an animation of this process, and display all the arc and angle measurements as $S$ moves around the circle.

There is no reason not to introduce this argument.  In my pair of lectures, I used more traditional geometrical arguments as well.  It doesn’t hurt students to be exposed to a wide range of proof ideas.

I summarize all of these results in the following graphic.

The measure of the angle indicated with the red dot is half the measure of the intercepted arc, or the sum/difference of the measures of the intercepts arcs, shown in red and blue.  An arc in blue indicates its measure is to be subtracted rather than added.  I was very happy with this graphic.  I think that if a student followed the lecture, they could state every result just by looking at it.

This also proved to be a great segue into looking at the power of a point.  I thought I’d begin with the figure in the upper left, proving the usual theorem using similar triangles.

And now for another continuity argument!

This is a nice way to see that the power of any point on the circle is 0.  It is also a nice contrast to the theorems about the angle between the intersecting lines:  when $PT$ and $RT$ eventually reach 0, you’re not able to conclude anything about a relationship between $QT$ and $ST.$

This means that there is no theorem relating lengths of segments for the two cases when the intersection of the lines lies on the circle.   I use the following graphic to indicate this, with two cases grayed out when the power of the point offers no conclusion.

Now all of these results are the usual ones found in high school geometry textbooks; nothing new here.  But for me, just having to step back and think about how to put them all together was a fun challenge.

Again, I am surprised at how much I’m learning even though I’m just putting together a few slides on elementary geometry.  The process of writing these lectures is an engaging one, and I hope the students who will eventually watch them will benefit from a perspective not found in more traditional textbooks.

## What Is…A Polygon?

A haven’t made a post in quite some time in my “What is a Geometry?” thread.  In working on my online lectures in the section on Polygons, I of course needed to define just what a polygon is.  This turned out to be a little more challenging than I had imagined.  I thought that the issues that arose would make this discussion an interesting continuation of the “What is a Geometry?” thread.

In general, I think that the Wikipedia does a good job with mathematics — but specifically, the definition of a polygon leaves quite a bit to be desired.  I’ll reproduce it here for you:

In elementary geometry, a polygon is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain, or circuit.  These segments are called its edges or sides, and the points where two edges meet are the polygon’s vertices (singular: vertex) or corners.

OK, maybe not so bad of a start.  There are lots of examples given which fit this definition, but many which do not.  For example, this definition allows consecutive segments to lie on the same line, which is typically disallowed in most other definitions of polygons.

So maybe a clause may be added to the definition which does not allow this.  But then we encounter a polygon like this (I’m using screenshots from my lecture as illustrations):

My definition begins with a list of vertices — but the problem is still the same.  The vertex labeled “4” is on the edge joining the vertices labeled “1” and “2.”  Again, this is usually avoided.

And what about the following figure?

With the Wikipedia definition, a vertex can be an endpoint of more than two edges of a polygon.  Again, problematic.  There would be no way to distinguish this figure from a single polygon and two different triangles sharing a vertex.

Moreover, there is no condition saying that the straight line segments need to be distinct.  So the same segment might occur multiple times as an edge of a polygon.

None of these behaviors is illustrated anywhere on the Wikipedia page.  I’ve done some Wikipedia editing a while back, and would be interested in working on this page when I have more time to devote to such things.

So what is the fix?  I’m using the Geometry text of the University of Chicago School Mathematics Project as a reference, which is one of the most rigorous geometry texts around.  Here is their definition:

They remark that this is the definition used in 23 out of the 45 geometry text they surveyed.  And in fact, it is the rewrite of a definition in previous editions:  “A polygon is the union of segments in the same plane such that each segment intersects exactly two others, one at each of its endpoints.”  This definition was problematic, though, since by this definition, the following is actually a polygon!

Now this revised definition solves all of the problems above — but I couldn’t use it.  Why not?

One of the sections I’ll be writing lectures for is three-dimensional geometry — and (of course) I’ll be saying a lot about polyhedra in this section.  There are Platonic and Archimedean solids, as well as the Kepler-Poinsot polyhedra, like the small stellated dodecahedron shown below.

The faces of the small stellated dodecahedron are pentagrams, five meeting at each vertex.

But the UCSMP definition does not allow edges to cross.  Each edge meets exactly two others, at each of its endpoints.  So that means that an edge cannot cross another in its interior.

Now I just can’t talk about polyhedra without talking about nonconvex examples.  Sure, it is possible to talk about pentagrams with edges crossing as decagons without crossing edges.

But this would be the height of absurdity.  Besides the fact that none of the dozens of books and articles I’ve read on polyhedra in the past few decades ever do such a thing — and I’m sure none ever will.

So I had to go it alone.  I’ll share with you my definition — but I can’t say it’s the best.  The difficulty lies with being mathematically precise while still making the definition accessible to high school students.  Here it is:

A polygon is determined by a list of its vertices. Edges of the polygon connect adjacent vertices in the list, and there is also an edge connecting the last vertex in the list to the first one. All vertices in the list must be different. Finally, no three consecutive vertices of the polygon can lie on the same line, and no vertex can lie in the interior of another edge.

I don’t think this is too bad.  But there is still a subtle glitch, which I haven’t worked out yet, and which doesn’t necessarily need to be worked out at this level.  When I talk about triangles, for example, I allow cases where the sides have lengths 3, 4, and 7, for example.  But I qualify such a triangle by calling it a degenerate triangle.

Since a triangle is a polygon, a degenerate triangle should be a degenerate polygon, right?  The problem is that calling something a “degenerate polygon” gives the impression that it is actually some type of polygon.  But a degenerate triangle, by my definition, is not a polygon.  So when I use the term degenerate polygon, I’m not actually talking about a polygon….

So I’ll let you think this over.  I just wanted to share how surprised I was at how subtle the definition of something so “simple” could be.  An ordinary polygon.

If you find this sort of question intriguing, you might go online and research all the various definitions of polyhedron.  Convex polyhedra are easy to define (as are convex polygons), but when you get into the different types of behavior possible in the nonconvex cases, well, it becomes problematic.  In fact, no one, as far as I know, has ever come up with a satisfactory definition for “polyhedron.”  Might even do a blog post on that some day….