Matrices Should Not Be Taught at the High School Level

On many topics in education I hedge based on context. With this statement, I feel firm.

Only a surface understanding is provided. Being able to add and multiply matrices is only a minute portion of their meaning. It’s like looking at Calculus and deciding to teach only how to find the derivative of a parabola. Some classes may get to solving a system of equations as a matrix, which then solidifies the concept in the student’s mind in a row-context. However…

…the understanding provided is misleading. To work with matrices in a genuine context (i.e. a linear algebra class, not a side topic in Algebra II) requires an understanding of vectors and summation notation leading to concepts of linear independence and dependence and beyond. Students stuck in a rut of thinking of matrices as a clever way to arrange numbers will get lost and confused; I know this because I was one of those students. I had to “de-program” what I was taught in high school before I could grasp what was going on. Some students will continue to think of matrices as only systems of equations long into their education, leading to…

…a removal of geometric intuition. A true understanding of the determinant requires connecting it with the volume of a parallelpiped, but a student who continues on the row-thinking path and avoids column-thinking (vectors) will quite often get there. Determinants remain an meaningless value. Further developments (vector spaces, orthonormality, etc.) become even more so. Finally…

…all the material in high school gets retaught anyway. Not only is it retaught as a refresher, but the material needs to be recontextualized so it meshes with how linear algebra really works.

Posting schedule

The little pumpkin is at 3 months, so it’s been a touch busy around here.

I still have some posts I want to work through, and I figure posting a schedule that people can thwap me about if I fall off will help.

June 23 (Tue): Matrices Should Not Be Taught at the High School Level
June 26 (Fri): What’s the Oldest Mathematical Artifact? (II)
June 30 (Tue): 10 Ways to Contend With Wolfram Alpha in Education
July 3 (Fri): What’s the Oldest Mathematical Artifact? (III)
July 6 (Tue): Is It Ever Mathematically Correct to Consider Multiplication Equivalent to Repeated Addition?
July 10: (Fri): What’s the Oldest Mathematical Artifact? (IV)

Sometime later in July: Is It Ever Educationally Correct to Teach Multiplication as Repeated Addition?
Sometime: The Mathematical Model for Optimal Memorization

Why You Want to Host the Carnival of Mathematics

The 53rd Carnival of Mathematics was hosted at The Math Less Traveled, and in theory Carnival #54 should occur this Friday.

However, it is quite possible this date will be skipped: as some have noted lately, the schedule has been highly irregular of late.

This is because of a lack of volunteer hosts. So this post is to convince you: you want to host the Carnival of Mathematics.

1. It’s easy. If you don’t want a lot of obligation, you can just collect the posts together sent to your email without a lot of fuss and hit send. Some have gone to lengths with their theming (or the traditional number trivia) but you can do as much or as little as you want.

2. You get to interact with members of the math blogging community. Some will send their submissions by email, with nice friendly notes. You get a warm fuzzy feeling.

3. You’re helping the community as a whole. You want more people interested in reading and writing about mathematics, right?

4. Your blog traffic will shoot up like a rocket. You may be familiar with the WordPress fastest growing blogs list. I might bring your attention to #17 on May 9, which happened to be the same day I posted the Carnival of Mathematics #52.

Interested? Email Alon Levy at alon_levy1 (at symbol) yahoo.com.

Teaching to the Limits of Wolfram Alpha

Maria H. Andersen has posted about a new article about Wolfram Alpha’s ramifications for education in The Chronicle of Higher Education. It’s fairly general about creating a new “math war”, but I’m starting to worry about the specifics: just how do we teach with this thing?

With graphing calculators I tried to convince students they needed to understand their graphs rather than have the calculators do all the work by giving cases where the calculator falls down, hard. (On a Texas Instruments machine, try graphing a logarithm or something with a lot of asymptotes.)

Here are three cases Wolfram Alpha can’t handle (as of now).

1. The alternate form of the quadratic formula

When I recently wrote about the alternate form of the quadratic formula I found Wolfram Alpha was able to derive the standard formula from scratch. I was then curious if it could get the alternate version, but (as far as I know) there is no way, even though the solution simply requires rationalizing the numerator. For example,

solve a=(-b+sqrt(b^2-4*x*c))/(2x) for x

gives Wolfram Alpha a blank.

Now, one might claim this instance isn’t useful, but for some computer calculations the alternate form ends up being more accurate. Thus somewhat ironically a calculation used to aid designing a technology algorithm has to be calculated by hand.

2. Graphing 3-D graphs that are not functions of z

A Twitter conversation came up recently about how to graph equations with x, y, and z that aren’t functions of z, that is like

x^2+y^2+z^2=1

Now, this particular special case of a sphere works, but even simply switching to

x^3+y^3+z^3=1

causes graphing to fail. If the z is isolated, however, one obtains the proper graph:

graph (1-x^3-y^3)^(1/3)

This is one case where Wolfram Alpha can isolate for z in the original case, but in general this case requires a conceptual leap to get from the start to the finish to enough of an extent that it would be hard to accomplish without knowing what’s going on with hand manipulation.

3. Assorted solve for x problems that Alpha can’t show the steps for

I’ve run across these more or less randomly, and while I think some cases will eventually be patched up, not all of them will.

solve 5=3/(2+x) for x

The issue

I’m especially worried about problems like case #1, where algebraic manipulation may be required no matter what system is in use. If we take a standpoint of letting CAS take over in general, how will students handle such cases? From my experience with Wolfram Alpha and other CAS systems, manipulation by hand is still sometimes required to get the computer to handle things correctly. However, this manipulation can’t be taught piecemeal, because each case is unique.

This creates a huge dilemma of allowing a CAS system yet convincing students that they need to learn minor calculations. A stockpile of problems like case #3 might be helpful, but to get to such examples one needs to practice simpler problems like 3x+5=2x-1 (which they can just have Wolfram Alpha do for them).

What’s the Oldest Mathematical Artifact? (I)

Candidate #1. The Lebombo Bone

A small piece of the fibula of a baboon, marked with 29 clearly defined notches, may rank as the oldest mathematical artefact known. Discovered in the early seventies during an excavation of the Border Cave in the Lebombo Mountains between South Africa and Swaziland, the bone has been dated to approximately 35,000 B.C. In a description of the bone, Peter Beaumont, an archaeologist who has done extensive work on Border Cave, has noted that the 7.7 cm long bone resembles calendar sticks still in use today by Bushmen clans in Namibia.
– from The oldest mathematical artefact by Bogoshi, Naidoo, and Webb

The above information has been repeated more or less verbatim across various sources which want to mark the beginning of math history.

But how accurate is it?

29 clearly defined notches

Count on the above picture: you will likely get a count of 29 or 30. The extra possible mark is on the left-hand side of the bone; it looks like it is possible 3 tally marks in quick succession. However, the middle “tally mark” is just a blemish on the image; taking a different image of the same stick and inverting the colors makes the picture clearer:

So: there are 29 notches, although the leftmost one is truncated enough one gets the impression the bone is broken. So does the count stop at 29 or does it continue?

There is some mathematical evidence that perhaps the leftmost notch is indeed the last: it is placed at roughly a 25 degree angle; the second steepest (tally mark #8) is only a 12 degree angle. I interpret this as the ancient tally-cutter “dragging” the diagonal mark, making it more likely to be a starting or ending mark.

29 also appears elsewhere in ancient counting, including a supposed lunar calendar in the Lascaux caves (painted 15,000 BC):

the bone has been dated to approximately 35,000 B.C

The latest data has put some the bones of the Border Cave at an earlier date than originally calculated in the 70s; however, the Lebombo bone has not received similar treatment.

resembles calendar sticks still in use today by Bushmen clans in Namibia.

Search for “calendar stick” and “Namibia” and the references are not to actual modern calendar sticks, but to the Lebombo bone. I was not able to find a single reference to modern calendar sticks — or an calendar sticks at all — in Namibia other than through mentions of this bone. Given how much Peter Beaumont knows about African archaeology, I’m going to presume this is a gap in the research literature, but it’s definitely a bizarre one.

The oldest mathematical artefact

This claim for “oldest” was made in 1987, but there are now other contenders for the title, including some discoveries made only announced this year; they will be the subject of future posts.

A simple application problem involving a rational expression

Dan Meyer wants to know what to do with rational expressions. Here is an example:

Of course, the students have a ready answer to this.

On conceptions of the variable

The Mathematics Education Research Blog has recently noted that some articles have been made freely available until July 31st.

I’d like to plug this article in particular entitled From arithmetical thought to algebraic thought: The role of the “variable”.

Many contemporary mathematics educators still maintain that the conceptions of the unknown and the variable as a thing that varies are the same, mainly because they represent unknown numbers and—on the level of symbols—they can be manipulated in the same way. However, the historical-epistemological analysis makes it possible to show, in a decisive way, that they belong to two different conceptual formations (see Radford 1996), unified under the name “variable”.

The authors do experiments complete with flow charts of student thought patterns:

With abbreviations meaning for example:

AL4.3: He/she adds a datum, but he/she considers that the bets are equal.
AL5: He/she translates the problem into a first degree equation with two unknowns.
AL6: He/she explicitly considers the bounds of the problem.

(See the appendix to the article for a full list.)

What would be involved in a multi-class blog?

Jackie’s ruminations about a classroom blog led me to thinking about what might be involved in a multi-classroom blog.

I’m not meaning my own classes all contributing together, or even disparate students able to submit to a collective blog, but a blog related to a single math subject where 4 of 5 or more classrooms across the world contribute and comment.

Potential good points:

* Getting to learn techniques that are taught in one country but not another.
* The possibility of establish global pen pals (or even study pals).
* Raising student global consciousness.
* The usual good points about blogs — getting students to write about mathematics.

Potential bad points:

* Lack of enthusiasm causing dead air — look at my own experience with students guest blogging as a reference. (Yes, I had rather a lot more than 3 students.)
* Mismatched curriculum; even taking two classes using the same textbook issues might arise from one class being on chapter 3 and another on chapter 5.
* Time zone issues.

What do you all think? Having some global class matchmaking site is one of my “dream tech” concepts. I should be able to pick whatever lesson I’m at — say, absolute value — and immediately bring up a list of classes all working on the same topic, and be able to set up a Skype connection on the fly or direct student emails to a communal help forum or –

Just, it’s silly. When I was looking for a laptop I logged on to a MUD I have frequented for 16 years, and an Austrian friend of mine was online and recommended a brand which my wife and I purchased the next day and are now happy with. Where’s the classroom equivalent? When will TIMSS results separating by country be blurred into meaninglessness?

Carroll Diagrams

From the department of teaching methods from outside the United States I’ve never heard of before, Carroll Diagrams:

[From this applet. Tip of the hat to Warrington ICT.]

I’ve used something like this before in geometry for sorting triangle types, but I never knew there was a name for it, nor did I think about sorting other math concepts in this way.

I started to brainstorm ways the diagram would work in higher math levels:

I know I’ve done sorting with graphic organizers before, but I don’t recall seeing this particular technique outside of geometry. Perhaps I’m just looking at the wrong textbooks.

The Alternate Form of the Quadratic Formula

It’s almost sacred in some classrooms, given , then:

It’s proven generally by completing the square; even Wolfram Alpha can do it.

But did you know there was an alternate form?

CHALLENGE: Prove it.