80 mph

My brother showed me this video --- can't remember if we've ever posted it here.

What I love about it is that the girlfriend actually comes up with a guesstimate that's pretty close to correct, which reminded me of something I once read about the difference between really good math students versus the "works hard" variety.

To wit: the really good students devise elegant and efficient solutions and proofs. The 'works hard' students go on wild goose chases.

Works-hard students do get to the answer or the proof eventually, but the process isn't pretty. (Speaking as a person who has spent a lot of time teaching herself math, I relate.)

The process may not be pretty, but it can be funny.



Another thing: this exchange is a brilliant example of "inflexible knowledge" in action. The young woman isn't transferring the meaning of "per hour" to a different phrasing of the same situation.

Speaking of inflexible knowledge, in Atlantic City this weekend we had a semi-galling episode of Failure to Transfer. My father-in-law has been deaf for years, and Ed and I -- and Andrew -- have had iPads for at least two. We use Andrew's iPad to communicate with him via typing (Andrew types, too). Yet it had never occurred to us that we could do the same for Ed's dad.

Two years to make the connection!

Immediately after I'd had the blinding revelation that iPads work for old people with hearing loss as well as young people with autism, we began plotting and scheming how to get an iPad for Ed's dad (would he use it?? Which one should we get?? The big one?? The little one that would take up less space on a dinner table?? Etc.)

It took me a good 5 minutes to figure out that Ed's dad does not need an iPad. He can talk.

We had a chuckle over that & then an hour or so later Ed raised the subject again.

We have spent a LOT of years living with people who can't talk.

"How I Used Math to Beat a Traffic Ticket"

At RealClearScience

Very fun.

Revolutionizing Math at the School of the Future

(Cross-posted at Out in Left Field)

A front page article in Monday's Local News section of the Philadelphia Inquirer profiles a math class at Philadelphia's Microsoft-funded High School of the Future, whose teacher, Thomas Gaffey, placed second in Microsoft's U.S. Innovative Education Forum and was a semi-finalist in its Worldwide Innovative Education Forum. In Gaffey's ninth-grade algebra class there are:

No textbooks, no paper, no chalk, no desks, and no assigned seats.

Instead, students use laptops while sitting in rolling chairs at trapezoidal tables spaced out in hexagonal classrooms.

Just how newsworthy this sounds to you depends on whether you think chair mobility and table shape have a big influence on learning, on whether you've been following current trends in education over the last 50 years, and on how unusual you think it is for a teacher to "encourage his students to find answers to their own questions" and engage with them in exchanges like these:

"Is this an obtuse triangle?" one student asks.

"Well, what can you tell me about an obtuse triangle?" Gaffey replies.

"One of the angles has to be more than 90 degrees," the student answers.

"Are any of the angles here like that?"

"Yeah. Oh, I get it now!"

As the Inquirer explains:

This snippet of student-driven discussion is a glimpse of the style and approach that have earned Gaffey national and international recognition.

Student-driven? Who's asking most of the questions? But I'm splitting hairs here. What I should be asking is: Why does this kind of exchange warrant international recognition?

To fair, it wasn't this, specifically, that earned Gaffey his honors. Rather:

Les Foltos, one of the judges who reviewed Gaffey's work, was impressed by his emphasis on "actively engaging students in solving real-world problems." As Gaffey puts it, "If we want to teach math to learners, we should teach math how it is actually used. It doesn't matter how much you know. It matters what you can do."

Ah yes, "real world problems." Again, only if you've been out of touch with the last half century of educational reform, and with today's Reform Math in particular, will this strike you as revolutionary. Here is Gaffy's version of real world math:

In his classroom on a recent Tuesday, Gaffey's challenge to his "learners" - as students in the Parkside public school are called - was to estimate Earth's land area.

To solve the problem, the class first covered basic concepts about area and polygons - shapes with three or more straight sides.

Gaffey then asked, "If a shape has four sides, is it always a polygon?"

Learners who answered yes (the wrong answer) were asked to redefine what a polygon is, while those who answered no were asked to draw a four-sided shape that was not a polygon on the class "smart board."

Gaffey drew a shape with three straight sides and one curved side.

"Is this a polygon?" he asked.

"No," the class responded.
...

The class drew lines through each of the continents, chopping them up into complex polygons, then simple polygons.

The final phase was to derive formulas for the areas of the simple polygons, and add up the areas.

This sort of problem is not particularly new, as a quick survey through now-standard textbooks like Everyday Math and the Interactive Math Program makes clear. And it's been around long enough to have garnered some serious criticism--specifically in what Barry Garelick calls its "just in time" approach to teaching.

Among other things, "just in time" often means serious delay. For example, one would hope that students would already know the formulas for the areas of simple polygons, and how to derive them, well before they hit 9th grade.

But because so many students are so far behind where they should be, there is one thing in which I and Gaffey are in whole-hearted agreement. In Gaffey's words, as cited by the Inquirer:

"Math education, more than any other subject, is in need of drastic reform."

From Russia with love: real wor[l]d problems

A friend of mine discovered this paper, Word Problems in Russia and America, by Andrei Toom. Quite a find. A long read, but worth it if you need any moral support in your personal fight against fuzzy math. Here is an excerpt:

The high-school part of [ed. American NCTM 1989, I think is being referred to]“standards” contains a list of topics to increase attention, where the first place is given to “the use of real-world problems to motivate and apply theory” (p. 126). What is a “real-world problem”?

Browsing through “standards”, I found quite a few statements about these mysterious critters. On p. 76 (middle-school part) it is said:

“The nonroutine problem situations envisioned in these standards are much broader in scope and substance than isolated puzzle problems. They are also very different from traditional word problems, which provide contexts for using particular formulas or algorithms but do not offer opportunities for true problem solving.”

What? What did they say about traditional word problems? What a nonsence! With their narrow experience the authors pretend to set standards! Are they aware of the rich resourses of excellent traditional word problems around the world? Let us read further:

“Real-world problems are not ready-made exercises with easily processed procedures and numbers. Situations that allow students to experience problems with “messy” numbers or too much or not enough informations or that have multiple solutions, each with different consequences, will better prepare them to solve problems they are likely to encounter in their daily lives”.

Pay attention that the author uses future tense. This means that he or she has never actually used such problems in teaching and never observed influence of this usage on his or her students’ daily lives. He or she has not even invented such problems because he or she does not present any of them. Nevertheless, he or she is quite sure that these hypothetized problems will benefit students. What a self-assurance!

After such a pompous promise it would be very appropriate to give several examples of these magic problems. Indeed, we find a problem on the same page, just below the quoted statement. Here it is:

Problem 48: Maria used her calculator to explore this problem: Select five digits to form a two-digit and a three-digit number so that their product is the largest possible. Then find the arrangement that gives the smallest product.

This is a good problem, although rather difficult for regular school because having guessed the answer, Maria needs to prove it. But the author never mentions the necessity of proof. What does the author expect of calculator’s usage here? It can help to do the multiplications, but it cannot help to prove. It seems that the author expects Maria to try several cases, to choose that one which provides the greatest product and to declare that it is the answer. But what if the right choice never happened to come to her mind? This is very bad pedagogics. Also let us notice that Maria is expected only to “explore” this problem rather than to solve it. According to my vision, exploration is the first stage towards a complete solution. Do the authors expect Maria ever to attain a complete solution? Do they want children to solve problems or just to tamper for a while?

But let us return to our main concern: so-called “real-world problems”. Notice that this problem has none of the qualities attributed to these mysterious critters on the same page: there is neither too much nor not enough information and there are no multiple solutions, each with different consequences.

One colleague noticed that the book still contains some problems described on page 76. Indeed, there are, but in another document. Here is one of them:

Problem 49: You have 10 items to purchase at a grocery store. Six people are waiting in the express lane (10 items or fewer). Lane 1 has one person waiting, and lane 3 has two people waiting. The other lanes are closed. What check-out line should you join?

I have never read any report about usage of this problem. Also I have never read any solution of this problem. Irresponsibility again!

What about problems with too much or not enough informations, they attract much attention in Europe lately, but European scolars want children to treat them critically and in many cases to refuse to solve them! Take for example that famous problem, after which Stella Baruk named her book [Baruk]. In the late seventies, the following problem was given to 97 second and third graders of primary school in France:

Problem 50: There are 26 sheep and 10 goats on a ship. How old is the captain? [Baruk], p. 25

76 children (out of 97) presented a numerical answer obtained by tampering with the given numbers. For instance, they might add the numbers and declare that the captain was 36 years old. Educators of several European countries (France, Germany, Switzerland, Poland) are very preoccupied by the fact that children “solve” unsolvable problems. The European educators would be very pleased if children refused to solve such problems with a comment like “It cannot be solved”. The European educators are quite right. But the same is true of what the “Standards” call “real-world problems”.

The most sound reaction to the problem 49 is “I don’t know”. But what a grade will an American student get after that?

Chocolate Pecan Pie & Fractions

I have been baking chocolate pecan pie almost every Christmas for over 20 years. Tonight my 5th grade daughter helped me with this recipe:

CHOCOLATE PECAN PIE
1 pie shell, unbaked
Filling:
1/4 cup (1/2 stick) unsalted butter
2 ounces unsweetened chocolate
3 large eggs
1 cup sugar
3/4 cup dark corn syrup or sugar cane syrup
1/2 teaspoon pure vanilla extract
3 tablespoons bourbon or rum (optional)
1/4 teaspoon salt
1 1/2 cups pecan halves
Preheat the oven to 350 degrees F.
To make the filling: melt the butter and chocolate in a small saucepan over medium-low heat, remove from heat and let cool. Beat the eggs in a large mixing bowl until frothy and then blend in the sugar. Stir in the syrup, vanilla, bourbon, salt, and the melted butter mixture until well blended.
Arrange the pecans on the bottom of the pie crust and carefully pour the egg mixture over them. Bake until the filling is set and slightly puffed, about 45-50 minutes. Test for doneness by sticking a thin knife in the center of the pie, if it comes out pretty clean, you're good to go. Transfer the pie to rack and cool completely before cutting.

We made two pies because we’re having 15 guests for dinner on Christmas Day. My daughter instantly converted all the fractions to the quantities needed for two pies. She was faster than I was. It may not seem like such a great achievement to some, but to me it was wonderful.

Thank you, Kumon!

Real world vs mathematics for math's sake

Arguments about how math should be taught frequently include the issue of "real world" math problems. I.e., students need relevance, otherwise they'll tune out. This attitude excludes a whole host of problems that one might find in a geometry book, let's say. So a problem that is real world and makes use of the Pythagorean Theorem is OK, but a proof of the Pythagorean Theorem is not. Well, no, people might object to my extension. So let's use another example. "For any point within an equilateral triangle, the sum of the perpendicular distances of the point to each respective side of the triangle is constant, and equal to the altitude of the equilateral triangle."

This wonderful theorem probably wouldn't qualify as "real world" and even though understanding the theorem and how it is proven opens up higher order thinking skills and mathematical reasoning, a math reformist would not teach it in its pure form. They would have to find some application, however contrived, to provide relevance.

Math for math's sake is out. All math must be relevant. In fact, when the Fairfax County Public School Board (in Virginia) was adopting math textbooks back in 2001, they argued about the criterion for "real world" applications. During the ensuing debates, two school board members found the following criterion too narrow: "Materials and concepts are related to real world situations". They argued for, and lost, the following (excerpted from the minutes of a School Board meeting found here):

"Mrs. Brickner moved, and Mrs. Thompson seconded, to amend the main motion to add the words, “mathematics and” to the eighth bullet under criteria #2, so that it would read, “Materials and concepts are related to mathematics and real-world situations.”

"Mrs. Brickner said that a mathematics textbook could not relate everything to a real-world situation; that there should be a balance between the presentation of math concepts and their relationship to the real world to help students understand the need for those concepts; that applications of mathematics should come from both within mathematics and from problems arising from daily life; that a strict application of the criteria, as originally written, would cause an evaluator to find a textbook less effective than they otherwise might on the basis that the text did not wholly focus on the real world; and that the objective was clearly to teach math concepts and skills.

"The motion to amend the main motion to add the words, “mathematics and” to the eighth bullet under Criterion #2, so that it would read, “Materials and concepts are related to mathematics and real-world situations” failed 4-7, with Mr. Braunlich, Mrs. Brickner, Mr. Reese, and Mrs. Thompson voting “aye”; with Mrs. Belter, Mrs. Castro, Mr. Frye, Mr. Gibson, Mrs. Heastie, Mrs. Kory, and Mrs. Strauss voting “nay”; and with Mrs. Wilson absent. "

So, the upshot is that math textbooks must base all examples and applications on real world situations; mathematics for mathematics sake does not count. That would make calculus textbooks rather challenging to write!