## Archive for the ‘Teaching’ Category

### It’s a Threeven Day!

March 3, 2011

Happy 3/3 everyone!

I just graded a bunch of proofs that √3 is irrational.  The proofs had a lot of holes in them.  This didn’t surprise me too much, in large part because the students weren’t math majors; rather, it was for a liberal arts math class taken largely as a gen ed requirement, and the whole proof by contradiction thing is really pretty scary and abstract for most people the first time around under the best of circumstances.

But actually, even when I’ve assigned this to math majors, they struggle.  They can have the proof that √2 is irrational right in front of them, be instructed that instead of even numbers they want to look at multiples of 3, and despite my Find and Replace instructions, they still don’t understand what to do.  The most common mistake is to replace “even” with “odd”.

In some ways this doesn’t surprise me, but in some ways it does.  Why is it such a conceptual leap to go from 2 to 3?  It’s a HUGE leap for many people.  And so I was pondering this while grading, and Batman suggested it might be because we have a special word for “divisible by 2″ but don’t for “divisible by 3″.  So you get, what, 10 years of reinforcement that there is just this one special way to divide the integers, and it doesn’t generalize.

What we need is a new word for these numbers.

And fortunately we have one:  threeven.  So 0, ±3, ±6, ±9, …  are all threeven, and the rest are…umm, not.  (Maybe we need two new words).  This word isn’t mine or even Batman’s; it actually was suggested by one of his students in response to this exact same problem.

As a bonus, it generalizes:  there’s fourven, fiven, sixen, seven-en (sev-en? )…as far as you want.     Which, admittedly, might not be very far but it still makes for a smoother sounding proof.

Happy threeven day!

### A cool sequence problem…

June 15, 2010

Our oldest son (nearly 10) posed the following challenge:

What comes next in this list?

1, 1, 1, 2, 2, 3, 2, 4, 3, …

Answer and rationale (his and mine) after the jump…

February 8, 2010

Here’s what happens when you didn’t get any all of your grading done this weekend (but the Saints won!!!!!!! Yay New Orleans!), and then instead of catching up before class your department gathers informally and starts to talk about Friday’s wacky trillion point grading scheme, and if there’s a class that lends itself to that, and someone mentions that Physics would really be the ideal course for this, because they have to deal with measures of scale so often.

Well, there’s only one place for this conversation to go:  the other extreme.  What if you had a class that, instead of having 500 points for the semester, or even 1,000,000,000,000, had only 1 point.  Total.  Exams could be worth 2/10 of a point, each homework assignment might only be worth 1/250 of a point, etc.  If someone had an unexcused absence, you’d knock 1/500 off a grade.   It might make it easier to defend points deducted, too, because you could say, “Look, I only took off 5 thousandths of a point for that mistake, so it’s hardly worth arguing about.”

Plus, this could be a learning experience because you could use Official Prefixes:  “This exam is worth 2 decipoints (not to be confused with decapoints)”.   The trouble is, most science students are pretty familiar with deci, centi, and milli so if you really want to make it…memorable….having 1 point for the whole semester is still too much.

The problem is, in looking online, we don’t really seem to have enough prefixes.  The prefixes start off simply enough (deci, centi, and milli being 1/10, 1/100, and 1/1000 respectively) but then decrease by factors of 1000:  the next smallest amount would be a micropoint, which is actually a thousandth of a millipoint, or 1/1,000,000 of an actual point.     Then there are nanopoints (1/1000 of a micropoint) and picopoints (1/1000 of a nanopoint) and while those words are fun to say, we’d need something closer in scale to a micropoint to be able to distinguish amounts, or else it becomes essentially a 1000 point grading scheme with a twist:  “This exam is worth 200 yoctopoints out of a total of 1 zeptopoint for the semester.”

I guess maybe this isn’t so practical after all.  And that’s truly a shame, because who wouldn’t want to write a grading scheme that used yoctopoints?

What has 8 legs and no room for anything else?  A yoctopus!

### Carnivals and Clowns. Clowns who shouldn’t be allowed to grade.

February 5, 2010

The Carnival of Mathematics #62 is up today at The Endeavor, and I can tell you right now that I’m totally jealous of the giant Dorito Sierpinski.    Now I’m looking forward to seeing Nerd High!

Speaking of Carnivals, though wonder of wonder we’re posting about #62 on the day it appears, there was also a Math  Teachers at Play #22 up at math hombre [hey, author John works with a friend of mine!  Yup, the math world is getting smaller by the second.]

So that’s the carnival news.  And clowns, you ask?  Well, I’m thinking that the clowns are the faculty of  my department, for  coming up with a grading scheme that’s so absurd I’m really tempted to use it in one of my classes next year.

Here’s the idea:  Suppose you teach a course and you want to have 3 exams each worth 20% of your grade, homework worth 10%, and a final worth 30%.  One way to do this is to set the midterms at 100 points each, the final at 150 points, and homework scaled to 50 points for a total of 500 points in the class.

So far so good, right?  The problem with this is that if you offer extra credit you have to be careful not to give too much — you wouldn’t award 10 points for being the first to speak in class, right?  (OK, you might, but that would be pretty generous.)  So if you want to be able to offer smaller amounts but not have them sound small, you need to have a larger total number of points.

How large?  How about 1 trillion points!  That ties in nicely to the scale of the national debt, which you can tie together with mathematical literacy and/or an interdisciplinary math/political science activity.   Tests are now worth 200 billion points.  The final is 300 billion.  And now, if a student gives a good answer in class you can off the cuff award them one million points of extra credit!  The student feels good — who doesn’t like to receive a cool million points in extra credit? — and you don’t even have to bother remembering to enter it in your gradebook.  On the other hand, if there are little errors on an exam that you want to point out but don’t necessarily want to penalize (forgetting to write parenthesis, for example, so that 2·(3x+5) is written as 2 · 3x+5 ) you could take off 50 million points.  That’s enough to get anyone’s attention.

I think in some of our classes this would be intimidating, so it’s probably not the best scheme in general.  But in other classes, especially the upper level ones, I think our majors would see this as amusing and, perhaps, a help in internalizing the scale of some of these numbers.

Like I said, tempting.

### A=B implies that 1=1, therefore…

February 2, 2010

I’ve ranted in the past about the fallacy of trying to prove an identity by starting with the equation itself, then manipulating both sides of the equation until you arrive at a valid identity.

While grading some homework this term (involving proofs of trig identities), I found the need to raise the subject again in class.  But my stock example, proving that -3 = 3 by squaring both sides, seemed too transparent.  I wanted something where the fallacy was solely due to proving that False implies True.

I ended up with the following example, which I like a lot, but which I’m certain has been rediscovered by others over the ages.  Still, it’s a good illustration of why we can’t prove identities in this way.

Claim: $\sin x = \cos x$, for all $x$.

Proof: Assume that $\sin x = \cos x$.  If we square both sides, this implies that $\sin^2 x = \cos^2 x$.  Furthermore, since equality is reflexive symmetric, it follows that $\cos^2 x = \sin^2 x$.

Finally, adding these two equations gives us $\sin^2 x + \cos^2 x = \cos^2 x + \sin^2 x$, which reduces to the equation 1=1.  QED.

The careful reader will note that squaring both sides is irrelevant, as is the Pythagorean identity for sine and cosine.  In essence we have a general proof that $A = B$ for any expressions $A$ and $B$:  If $A = B$, then by reflexivity symmetry we know that $B=A$, and thus $A + B = B + A$.  But since addition is commutative, this reduces to the identity $A+B = A+B$.

I prefer the slight obfuscation of the $\sin^2 x + \cos^2 x$ proof over the distilled simplicity of “reflexive symmetric plus commutative”.

### The Playground/Math Association

April 23, 2009

Russ  Lopez and his two buddies are the Defenders of the Playground.  I picture them with capes and swords, but actually they’re profs (Lopez from Boston University and the others from Tufts) who just studied the association between elementary school playgrounds and test scores.  According to BU Today today:

When Lopez studied the 2003 results of the fourth-grade English language MCAS (Massachusetts Comprehensive Assessment System), standardized tests that almost all public school students must take, he saw no discernible differences between children at the 70 schools with new playgrounds and children at schools with old playgrounds.

But when he looked at math scores, he saw a very different picture. In schools where fourth graders had new playgrounds, 25 percent more kids passed the math MCAS. And that remained true after he and his team controlled for factors such as demographics and the number of students receiving free or reduced-price lunches.

Of course, as the article goes on to explain, that doesn’t mean that building more playgrounds will automatically raise test scores — there could be other factors in play (so to speak).  But, especially in comparison to the English tests results, it’s certainly an interesting finding and I look forward to reading the follow-up.

The playground photo was taken by drk_faerie.

### Thinking mathematically

March 4, 2009

I spent an hour last night in the company of 8-year-old boys.  (My oldest son had a Cub Scout meeting.)  The discussion topic:  safety, including fire safety, and what to do in case of an emergency.

One of the boys in the group got in a “mood”, and every question that was posed to him, he’d twist around into a more extreme predicament:

Scout:  What if your FACE is on FIRE?!?!?

or

Leader:  What if you and your friend are walking across a frozen pond, the ice breaks, and your friend falls in?  What should you do first?

Scout:  What if there’s a giant glacier, and they fall in, but they’re too deep down to reach?  And you’re in the middle of no where, and you can’t go get help?

or

Scout: What if part of the roof falls down, and you’re stuck underneath, and you’re UNCONSCIOUS!  Then what do you do???

My reaction was “Hey, he’s thinking like a mathematician!”  He knows the stock answer that is expected, and he’s asking what happens if we change the hypotheses, considering a related problem where the conclusion doesn’t  follow.   HE’S DOING MATH!!!!!!!!!!!!!!!

And then I realized that no, he isn’t, he’s just being an eight year old at a cub scout meeting.

I’d  love if my students responded to my questions with phrases like “But what if we use fractions instead?”, or “But what if the coefficients are matricies? Can we still complete the square?”  etc….

So the challenge, as we prep for our classes:  find a way to ask questions with obvious answers, that will get students motivated to say “yeah yeah, but what about THIS situation?”, and aim to “lead them” (pushing rope comes to mind) toward the actual course content we want to explore.

If anyone has insights in designing lessons that exploit this inate human cussedness, I’d love to learn more.

### Is a Square a Rectangle?: Welsh edition

February 6, 2009

I’ve posted before about how even though in the US we define squares to be equilateral rectangles, there are many forces (often in the form of picture books on shapes) that treat squares and rectangles as distinct beings, and so it is really no surprise that many students reach college a little uncertain.  The mathematical definition doesn’t match the cultural one.

Anyway, this week in Geometry I was going over Euclid’s definitions, and I pointed out that he wrote explicitly that rectangles (oblongs) weren’t allowed to have four equal sides, which is different than the definitions we use today.

Then one of my students spoke up.  Katie is a senior math major getting certified to teach elementary & middle school, and this past fall she did part of her student teaching in Wales  (courtesy of a study abroad program here at the college).  She was in a 5th grade class, and according to their formal curriculum squares are not rectangles.  Indeed, Katie said that the definitions they used were pretty much the same as what appears in the translation of Euclid (rectangles have four right angles but can’t be equilateral; rhombi have  four equal sides but can’t have right angles; parallelograms have parallel sides, but can’t be equilateral or have right angles).

Similarly, when she taught about diamonds, she couldn’t call them squares even if they had four equal sides and four right angles.  She had to prove the same rules twice (say, that the diagonals of a square are equal, and that the diagonals of a equiangular diamond are equal) and when she drew them, she had to add a line to show whether the figure she was referring to was a square or a diamond.

That explains this Failblog post from last October:

(I’m not sure if this picture was from Wales, but she did see Shreddies in the store there.  She really liked the frosted kind.)

So now I’m wondering:  how are geometric figures (squares versus rectangles and the like) defined in other countries?

The image above is from Oliver Byrne’s way-cool color edition of Euclid’s Elements.

### Silly applications of factoring quadratic polynomials

January 13, 2009

One of the nicer talks that I attended at the Joint Mathematics Meetings in DC was given by Jeff Suzuki, on “A History of College Algebra in the United States During the Nineteenth Century“.   Suzuki’s talk focused on equation-solving, and he noted that factoring hadn’t been a significant equation-solving tool throughout that era.  Equations were solved by other means (e.g., for quadratics:  completing the square, or the quadratic formula;  for higher order equations, roots might be found/estimated by bisection or other iterative approximation schemes).

This reminded me of one of my pet peeves from teaching algebra.  Modern elementary algebra texts teach four ways of solving quadratic equations:  graphing (and finding x-intercepts by inspection), factoring, completing the square, and the quadratic formula.  Most current books aimed at the “college algebra” market emphasize factoring.  (One of my daily reads, jd2718, has an interesting take on the role of factoring in HS algebra.)

What irked me when last I taught college algebra were the “applications” of quadratic equations.  (Too) Many of the applications amounted to numerical exercises:  “Find two numbers whose product is 45 and whose difference is 4″ was a typical example.

The intent is that a student will introduce one or two variables, representing the two numbers as either x and x+4 or as x and y, and eventually arrive at the equation $x^2+4x=45$, either directly or via a system of equations.

From here, we solve the related equation $x^2 + 4x - 45 = 0$, by finding two numbers whose product is -45, and which add to +4:  that is, two positive integers whose product is 45 and whose difference is 4.

The usual technique for achieving this is to list all integer products which equal 45, and find the factorization by inspection.  But of course this amounts to a direct solution of the original word problem, without recourse to any algebra whatsoever.

Granted, there are some advantages to this algebraic method, which become clear when you consider situations with non-integer solutions.  For example, if we want to find two numbers whose sum is 6 and product is 10, trying a brute force attack is unlikely to work.  Setting up the problem algebraically, and reducing it to solving $x^2 -6x + 10 = 0$, allows us to apply another paradigm — completing the square — to eventually find the two numbers $3+i$ and $3 - i$.

I suppose that there just might be some slight pedagogical value in having students see that solving their quadratic equation is equivalent — literally and explicitly — to solving the original word problem.  But this amount of circularity always struck me as a bit daft, and I feared the day when an eye-rolling student in the back of the class was going to point out that the Emperor was underdressed.

### Recovering from the recession

December 11, 2008

I have a Modest Proposal for how to get the US (and World) economy out of the current recession.

It came to me last Friday, on our last day of classes, as I was walking across campus to Calculus II.  I wanted to talk about cool applications of the course content, as a way of summing up (so to speak) the semester: a course on techniques of integration, applications of integration, and infinite series.  I decided to go with something simple:  applications of geometric series.  And one of my favorites is computing the total amount of economic activity that ensues from a single injection of economic stimulus from the government.

Here’s the (fairly standard) story:  Suppose that each individual saves 10% of each dollar they receive, and they spend (recirculate) the other 90%.  Then for each $1000 (say) of government stimulus,$100 gets saved, and $900 is spent, becoming additional income for other individuals in the economy. But now consider that$900 on the rebound.  10% of it ($90) goes into savings, and the other 90% ($810) gets spent again, this time by its second owners.  And now of the $810, 10% is saved, 90% spent. And so it goes… This story leads to the following calculation. A stimulus of$1000 will lead to a total amount of economic activity equal to

$1000 + 1000(.9) + 1000(.9)^2 + 1000(.9)^3 + \cdots$

an infinite geometric sum, where each summand is 90% of the previous term.  Now a geometric series $a + ar + ar^2 + ar^3 + \cdots$ with common ratio $r$ converges to a finite sum provide $|r|<1$, and in that case the limiting value is $\frac{a}{1-r}$.   In the case of the $1000 above, the total amount of economic activity is $\frac{1000}{1 - .9}$, or$10,000.  [So the $700 billion stimulus package, under these assumptions, could lead to$7 trillion in economic activity, or roughly half of the US Gross National Product.]

The brilliant thought I had whilst crossing campus:  what if we drop the savings rate from 10% to a smaller number?  If instead of recirculating 90% of our income, each of us went out and spent 95%?  or 99%, or even… more???  This model predicts that the total amount of economic activity from a given stimulus is the amount of that stimulus divided by 1 – (the proportion recirculated).

So, what would happen if no one put any money into savings, and ALL of our income went directly into consumer spending?  In that case, we have a common ratio of $r=1$ in the geometric series, which now diverges, and the model predicts that any amount of government stimulus leads to an infinite amount of economic activity!

Woo hoo!  Let’s send out 5 cents in economic stimulus, and watch the American electorate spend us out of recession!!!

[inhale, exhale]

Ok, so that’s obviously wrong, which means the original discussion (standard fodder for all the calculus and precalculus texts I’ve seen in recent years) is also flawed.  And seeing what’s wrong is easier now in this extreme case.  Suppose the government gives Joe the Drummer a $10 stimulus check. Joe goes out and buys new drumsticks; the music store spends all of Joe’s$10 on rent; the landlord spends all of the $10 on utilities; the utility company etc…. What is missing in the series is the issue of time. Joe might take a day or two between getting his government rebate check and actually spending it. The music store owner won’t spend the$10 until the utilities are due on the 15th of the month; and so it goes.   $10 in stimulus in theory leads to an unending succession of financial activity, but it cannot do so in a finite period of time. What can happen in a finite period of time is a finite number of transactions. If we assume that over the course of the month (say), each dollar spent in stimulus changes hands a total of ten times, we get a truncated geometric series (again, with a$1000 initial stimulus, and 90% recirculation rate):

$1000 + 1000(.9) + 1000(.9)^2 + \cdots +1000(.9)^9 = \frac{1000( 1 - (.9)^{10})}{1-.9} \simeq \ 6513$

### Toilet Paper Math

May 8, 2008

Really, is there a more appropriate follow-up to yesterday’s featured theorem?

Last night young Quentin, age 4½, went to get some toilet paper to clean toothpaste out of the sink after brushing his teeth (because — get this — he likes to clean up after himself. I can hardly believe it.). As he pulled off a strip of TP, he suddenly held it against himself and got all excited: “This is as big as my belly!” I pointed out that his belly was three squares big, and asked how long his arm was. He measured, and exclaimed, “My arm is three squares long!” When he tried to measure his leg, it fell short so I suggested he might need one more square. He immediately went to the roll, counted off a strip four squares long, and held it against his leg. Yup, four squares worked.

The sink stayed dirty for a while after that while he went around measuring his hand (one square), our arms, etc. The nice thing about toilet paper is that he could take strips of various sizes and just pick the one that seemed best. His measurements weren’t exact (I’m not going to hire him to build a bookcase, for example) but he did seem to have the basic idea of measurement and that’s a topic that several K-6 teachers I’ve talked to say is the one that students need the most help with after number sense. (Speaking of which, Denise on Let’s Play Math had a great post Tuesday about helping kids learn number sense.) And I think non-standard measurement is one of the NYS math standards. [Quick check -- yup, it's 1.M.2, 1.M.11, 2.M.1, 2.M.10, and 3.M.10. I spent a while last year putting all the NYS math standards into Excel worksheets for easy searching and posted them here if anyone would find that useful.]

Thinking about blogging this, I googled “Toilet Paper Math” and found some other interesting ways to use toilet paper to do math. You can determine the least expensive choice of TP at the grocery store. You can fold it in half twelve times. You can find the thickness of a sheet of TP (although it seems like density — aka fluffiness — might make that inexact). You can calculate how much text you can print on a roll of toilet paper.

And finally, you can read about how Sir Roger Penrose sued the Kimberly Clark Corporation back in 1997 because one of the designs that was printed on Kleenex quilted toilet paper looked like Penrose tiles (see Wolfram’s Mathworld or this more detailed summary from Professor Richard H. Stern’s Computer Law Class at George Washington University.)

Update 5/10: I think these are the kinds of pictures that mbork was referring to below. In the first two examples the triangle has a right angle, but in the third the angle a bit larger.

### The Geometry Van

May 7, 2008

I taught Geometry this spring, and we spent about half the semester working through Euclid (Book I and a smattering of some others). We proved SAS (Side Angle Side), ASA, AAS, but not ASS (Angle Side Side). Because there is no ASS in Geometry.

Here <A=<A’, AB=A’B’, and BC=B’C’ but the two triangles are not congruent. All we get are bad jokes.

Except that’s not quite true! If <A=<A’, AB=A’B’, BC=B’C’, and BC≥AB, then the two triangles are congruent! In class, I referred to this theorem as ASS.

(The Hypotenuse-Leg Theorem is a special case of this, since the hypotenuses, as the sides across from the corresponding right angles, are certainly longer than the corresponding legs of the triangle).

We didn’t refer to this theorem very often, but it is, well, memorable. And so Adele, one of our majors, was thrilled when she noticed ASS written on a van this past weekend! She snuck a photo of it:

According to Adele and everyone who was with her, this is exactly how I would write the shorthand. We have no idea why it’s on this van. I like to think that this is a Secret Geometry Van, coming out to help students everywhere by providing them with extra theorems.