• +1 or +3.  A bonus question appeared at the end of a test.  ”Write Free +1 and you’ll get a bonus point on the test, no questions asked.  Or write Go For +3 and if two-thirds of the class does it, all who do will get 3 bonus points.  Saying anything out loud voids the bonus for everyone.”  I was pretty surprised that in most classes, not enough students went for the +3.  One student wrote Free +3 and got nothing…
• Choose A or B.  In class, students were given this choice: “Write A and you’ll get a point for everyone in the class who writes A.  Write B and you’ll get twice as much as anyone who writes A.”  After a long debate, most students picked B.  Far fewer students picked B when the choices were revealed publicly instead of privately.  (Social commentary followed…)
• The probability you’ll get this question right.  Still one of my favorite bonus problems of all time, this came from my wife: “If train A leaves Denver at 50 miles per hour, and train B leaves Chicago at 60 miles per hour, what is the probability that you’ll get this question right?”  Students who wrote 1 or 100% were right, students who wrote 0 were wrong, and anything in between was resolved with a coin or random number generator.  (There’s a fun discussion over whether a student who writes 0 can be right by being wrong… but if they’re right then they’re wrong…)
• The dollar auction.  This one is evil.  See http://en.wikipedia.org/wiki/Dollar_auction for a full description.  In one class, a student successfully “escaped” by outbidding the other player significantly.   In another class, a third student jumped into the auction midstream allowing another to escape.  If I were still teaching, I would repeat the process using the bidding systems by Beezid and other “penny auction” sites (http://en.wikipedia.org/wiki/Penny_auction).  Hopefully by exploring these auction styles with students they will avoid being scammed for real money later in life.

What else?

Take a ruler or any stick.  Mark it off into three equal parts.  Then, perhaps in a different color, mark it off again into five equal parts.  This divides the ruler into how many total parts?

(The mathematician’s version: Consider a number line from 0 to 1.  Mark all multiples of 1/3 and 1/5.  How many intervals are produced?)

Start over.  On another ruler, mark it off into four equal parts, then again into ten equal parts.  How many total parts this time?

On another ruler, mark it off into twelfths and thirtieths.  (This is close to what happens with inches and centimeters, but not quite.)  How many parts now?

If you mark off the ruler into M equal parts, then again into N equal parts, how many parts are there in total?

Can you think of ways to extend either this problem or the rectangle-diagonal problem?

We’d love for readers to be able to explore these problems, so resist the urge to provide answers in the comments. Instead, we’d love helpful suggestions and ideas about different ways to think about them, successful or not. If you’d like to provide a full solution, do so with a pingback to your own blog!

Picture a pool (okay, a rectangle) made of white square blocks, surrounded by a border of black square blocks.  Is it possible to use exactly the same number of white and black square blocks?  If so, find all the ways.  If not, prove it can’t be done.

Bonus points for creating an interesting 30-second video to motivate this exact question and no others!

More bonus points: think of some other questions that extend this problem, and let everyone know.  There are lots of interesting ones!

(Thanks to Matt Chedister and the PROMYS for Teachers crew for bringing up this interesting problem.)

We’d love for readers to be able to explore these problems, so resist the urge to provide answers in the comments. Instead, we’d love helpful suggestions and ideas about different ways to think about them, successful or not. If you’d like to provide a full solution, do so with a pingback to your own blog!

http://www.nytimes.com/2011/08/25/opinion/how-to-fix-our-math-education.html

presents a plan to “fix math education.”

I’m disappointed.

The arguments are variations on themes that have surfaced (and been debunked) many times over the past century.

First of all, the authors set up a straw man in their characterization of Common Core: a codification of school mathematics as a highly abstract program, devoid of context, that introduces “the mysterious variable x, which many students struggle to understand.” In fact, a major difference between Common Core and the state standards that have plagued us for at least a generation is that Common Core strives for meaningful mathematics, where algebra is understood as a way of expressing generality and precision. The very first standard in the Common Core states that “Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution.” For one example, and there are many, the Common Core says, “Reading an expression with comprehension involves analysis of its underlying structure. This may suggest a different but equivalent way of writing the expression that exhibits some different aspect of its meaning. For example, p + 0.05p can be interpreted as the addition of a 5% tax to a price p. Rewriting p + 0.05p as 1.05p shows that adding a tax is the same as multiplying the price by a constant factor.” Understanding that different recipes for calculating the same thing can have different uses (in this case, 1.05p is much more efficient than p + 0.05p, something that’s important in computer calculations) shows the power of abstract reasoning in calculation systems.

This simple example also debunks another claim of the editorial: “There is a world of difference between teaching `pure’ math, with no context, and teaching relevant problems that will lead students to appreciate how a mathematical formula models and clarifies real-world situations.” Common Core is full of examples of ways to apply algebra and geometry to all kinds of situations.

What the authors really object to is the way Common Core organizes its standards around mathematical themes and conceptual categories. But this is precisely what makes the standards so compelling. Common Core describes an approach to precollege mathematics based on mathematical coherence, showing how a small number of general-purpose ideas can be used to build a textured and intricate edifice of results and methods that have utility all over mathematics, science, and everyday life. The editorial asks us to “Imagine replacing the sequence of algebra, geometry and calculus with a sequence of finance, data and basic engineering.” I just did the exercise and concluded that the result would be a program organized around a set of special-purpose techniques that would likely be out of date by the time students graduate college. What’s more, to get them to use the specialized tools, we’d have to include some shallow and formulaic versions of classical topics like geometric series, expected value, and recursive algorithms.

This stance about organizing programs around mathematical themes isn’t just based on the experience of my colleagues and me. Schmidt and others have analyzed curricula in “high performing” countries and they describe the curricula there as “organized around the ways that ideas are organized within the discipline.” That’s what one finds internationally.

The editorial ignores another distinguishing objective of Common Core: Equipping students with what the CCSS calls standards for mathematical practice, and what we at EDC call mathematical habits of mind. This gives them, as it has for centuries, the mental wherewithal to deal with problems that don’t yet exist.

When I started teaching high school, I thought that mathematics was an ever-growing body of knowledge. Algebra was about equations, geometry was about space, arithmetic was about numbers; every branch of mathematics was about some particular set of mathematical objects. Gradually, I came to realize that what my students (some of them, anyway) were really taking away from my classes was a style of work that manifested itself between the lines in our discussions about triangles and polynomials and sample spaces. I began to see my discipline not only as a collection of results and conjectures, but also as a collection of habits of mind.

This realization first became a conscious one for me when my family and I were building a house at the same time that I was researching a problem in number theory. Now, pounding nails seems nothing like proving theorems, but I began to notice a remarkable similarity between the two projects. It’s not that house-building requires applications of results from elementary mathematics (it does, by the way), but that the two projects required the same kinds of thinking. In both theorem-proving and house-building, you perform thought experiments to visualize things that don’t (yet) exist, to predict results of experiments that would be impossible to actually carry out, to tease out efficient algorithms from seemingly ad-hoc actions, to deal with complexity, and to find similarities among seemingly different phenomena.

This is not to say that other facets of mathematics should be neglected; questions of content, applications, cultural significance, and connections are all essential in the design of a mathematics program. But reorganizing a mathematics program around this or that set of topics or applications is just the wrong way to do it.

Then there’s the question of student engagement. I get nervous when grown-ups try to predict what will hook kids. The only general rule is that students, like all people, get great satisfaction from figuring things out. My colleague Paul Goldenberg talks about the popular puzzle books that one finds for sale at the checkouts of many supermarkets. Deborah Schifter, another colleague, likes to say that people take delight in their own mathematical thinking. At the high school level, I’ve seen over and over how students—all kinds of students—get hooked on something just because it engages their intellect.

In the field tests of early versions of our precalculus course, we held an advisory board meeting of high school juniors and seniors (many of them “very weak” in terms of traditional measures). This was at the end of the first term; up to that point, students had been experimenting with recursively defined functions, modeled in a CAS on their calculators, finding closed forms for such functions, proving that their closed forms and the recursive models were equal on the non-negative integers by mathematical induction, and then doing a bit with Lagrange interpolation. At the meeting, Wayne Harvey, another project member, asked the question, “How is this mathematics course different from others you have taken?” Four kids answered, almost in unison, “It’s more realistic.” That response was startling, even to us, because “realistic” is usually taken to imply everyday or other “real world” contexts, and the activities the kids were talking about were purely mathematical. But what the students meant was that it felt more like real work, more like the kind of thinking they must do when they are solving a real problem. What mattered was that they got a chance to exercise their own creativity. What mattered was how, not where, their mathematics was used.

This issue of viable and engaging contexts is complicated for a couple reasons.  Many of the students in my high school classes came from situations that many of us would find hard to imagine; the last thing they cared about was how to balance a checkbook or figure the balance on a savings account. But they loved solving problems. For another thing, reality is relative. The authors claim that “it is through real-life applications that mathematics emerged in the past, has flourished for centuries and connects to our culture now,” and I agree. But the best mathematicians and scientists I know, and the students in my classes who really got it (and these were not necessarily the “good students”)—see the power and satisfaction one can derive from doing mathematics—all see mathematics as part of their real world.

1. A bag contains four green chips and three white chips.  You win if you can pull all the green chips before pulling all the chips of a different color.  Find the probability of winning the game.
2. Now two red chips are added to the bag.  The rules are the same.  What is the probability of winning this game?
3. On “The Price Is Right”, the “3 Strikes” game is played with five white chips and one red chip.  White chips are not replaced, but the red chip is.  What is the probability of pulling all five white chips before pulling the red chip for the third time?  (In the real game, after pulling a white chip the player must make a decision that might return the white chip back to the bag, so the probability of winning the full game is significantly lower.)

We’d love for readers to be able to explore these problems, so resist the urge to provide answers in the comments. Instead, we’d love helpful suggestions and ideas about different ways to think about them, successful or not. If you’d like to provide a full solution, do so with a pingback to your own blog!

How many rhyme schemes are there for six-line poems?  ABACAB is one of them.  So are AAAAAA and ABCDEF.

(The fine print: if a new line doesn’t rhyme with any previous line, it must use the next available letter in the alphabet.  BAAB is wrong: it should have been ABBA. ABACAF is wrong too.  The first letter is always A.)

We’d love for readers to be able to explore this problem, so resist the urge to provide answers in the comments. Instead, we’d love helpful suggestions and ideas about different ways to approach the problem, successful or not. If you’d like to provide a full solution, do so with a pingback to your own blog!

PCMI 2011 Day 9 (July 15, 2011)

Five more problem sets to go!  Have a great weekend.

PCMI 2011 Day 8 (July 14, 2011)

PCMI 2011 Day 7 (July 13, 2011)

A brief description of the “salt activity”:

• Get a big salt dispenser, a box, a paper cup, and a cardboard cutout of a triangle.
• Put the triangle on the upside-down cup inside the box.  This steadies the triangle.  Make sure it is flat.
• POUR SALT everywhere until the triangle is completely covered with salt.  More.  Put as much salt as you possibly can on the triangle.
• Observe.
• Ideally, do it over again with more different triangles to help understand what is happening.

PCMI 2011 Day 6 (July 12, 2011)

If you’re not attending PCMI, “Ning” is a reference to a social networking service used by the conference.