Thursday, October 29, 2009

Math problem of the week: 7th Grade 1920's math vs. Connected Math

1. From the last graphing problems in Hamilton's Essentials of Arithmetic: Higher Grades, Chapter II, "graphs" section (intended for "year 7"), p. 131:

Trace a curve to illustrate the following changes in temperature for a certain day: 7 A.M., 400; 9 A.M., 420; 11 A.M., 460; 1 P.M., 500; 3 P.M., 530; 5 P.M., 490; 7 P.M., 460

2. From the last graphing problems in 7th grade Connected Mathematics Variables and Patterns: Introducing Algebra, Investigation 5, "Using a Graphing Calculator," p. 68:

Write a letter to a friend explaining how to use a graphing calculator to make graphs and tables. Use a specific example to illustrate your explanation.

Tuesday, October 27, 2009

Sunday, October 25, 2009

"Left Brain Child:" Two Book Talks in Western New England

I'll be giving two talks about my book this week:

On Tuesday, 10/27, at 7:00 PM in Pittsfield, Mass (Chapters Bookstore).

On Wednesday, 10/28 at 7:00 PM in Manchester, Vermont (Northshire Books).

Please spread the word to anyone you know in the area who works in education, or who has a shy and/or unsocial and/or socially awkward and/or analytical and/or mathematically inclined child. I'm hoping for lively discussions.

Friday, October 23, 2009

Math problems of the week: 3rd grade Investigations vs. Singapore Math

Two ways to learn about the number system by "making numbers":

1. From the beginning of 3rd grade Investigations (TERC) Landmarks in the Hundreds booklet, p. 11:

Ways to Make ___

Choose a number to write in the blank above.
Find equal groupings that make your number.
Record your results in the chart.

Note: In class, we use cubes to make groupings. At home you might use beans, popcorn, pennies paper clips, pebbles, coins, or dots drawn on paper.

The number I am making is ____

Number of cubes in each group: ____
Picture of how you made your number with these groups

Skip count by the number of cubes in each group.

Number of cubes in each group: ____
Picture of how you made your number with these groups

Skip count by the number of cubes in each group.

Number of cubes in each group: ____
Picture of how you made your number with these groups

Skip count by the number of cubes in each group.

Numbers I tried that didn't work


2. From the beginning of 3rd grade Singapore Math, Primary Mathematics 3A, Standards Edition, "Numbers 1 to 10,000" unit, p. 13:

Use these cars to make six different 3-digit numbers.

[3] [9] [2]

The three digit numbers are:

Use these cards to make different 3-digit numbers.

[7] [2] [8]

The greatest number is __________.
The smallest number is __________.


What is the greatest 4-digit number that you can make using all the digits 1, 0, 3, 8?



What is the smallest 4-digit number that you can make using all the digits 7, 5, 2, 6?



3. Extra Credit:  

Compare the ratio of time expenditure to learning in each problem set.

Wednesday, October 21, 2009

Caging birds: the worst of both worlds

Beth’s comment on Monday’s post really captures the sad irony of today’s schools:

The tragic reality is that our schools don't do anything well. The fact that the humanities are suffering doesn't mean that mathandscience is taught well. The fact that the schools don't have the rigorous content and skills of the traditional approach doesn't mean that they encourage true creativity like the progressive approach.
They just don't do anything right. They've somehow managed to weld together the authoritarian, anti-creative, carrots-and-sticks approach of the worst kind of traditionalism, with the hollowed-out content and mushy thinking of the worst kind of progressivism.
Beth’s words returned to me last night, when I was marveling at some wildly creative white-on-black drawings that my daughter had recently produced at home: scenes teeming with ghosts, birds in flight, and worried children.“Do you ever get to draw this kind of thing at school?” I asked her, suddenly wondering whether the realism requirements that have persisted since kindergarten extend beyond writing (“realistic fiction”) to art.

“During indoor recess we can draw whatever we want to.”

“What about during regular class time?”

“Then we can’t draw imaginary stuff.”

“So during class time, ever since kindergarten, you haven’t been allowed to draw or write imaginary stuff?”

Apparently at the very end of 2nd grade they were allowed to write (and illustrate) a work of “imaginative fiction,” but my daughter ran out of time before she finished her story.

Further constricting her, most of the realistic fiction she’s tasked with producing is supposed to be about her personal life--a requirement she increasingly resents.

“Why do they want to spy on us?”

“They” may know something about her personal life, but her creative potential remains a well-kept secret.

Monday, October 19, 2009

Myths about left-brain schooling, II: media complicity

Mark Slouka's recent Harper's article about the supposed dominance of math and science in our schools prompted me to write the following letter to the editor:

If Mark Slouka [“Dehumanized”] were to visit an American grade school classroom, he would see that math and science do not rule the school. While classes called “math” and “science” still exist, they contain far less actual math and science than they did a generation ago. Indeed, Slouka’s observations about today’s reading assignments apply as well to assignments in math and science: intended, in Slouka’s words, to “provide students with mirrors of their own experience,” they have students connecting math and science to their personal lives rather than doing challenging problems. This worries many mathematicians, scientists, and parents, not because they want children, in Slouka’s words, to be “hired by Bill Gates,” but because we’re raising a generation of innumerate, scientifically illiterate citizens and turning off our brightest young lights in math and science.
But Harper's declined to publish any letters challenging the article's key assumption that math and science control our schools. Of the three letters they did publish, only one mentioned math education. Its author, a longtime "teacher of mathematics," writes:
I... choose to teach mathematics with reading assignments, art projects, oral presentations, even poetry--all to encourage critical thinking in my students, and to cultivate questioning minds.
Harper's doesn't seem to recognize that, thanks in large part to the many teachers like this one, math and science don't rule our schools.

Friday, October 16, 2009

Math problems of the week: 1930's Algebra vs. Interactive Math Project

1. The final problem in the pure algebra portion of A Second Course in Algebra (first published in 1937), from the "Binomial Theorem" chapter (which is followed by a chapter on logarithms and another on trigonometry), p. 335:

A man can travel from town A to town B by plane in 2 hours and 10 minutes, or by car in 6 hours and 30 minutes. Bad weather forces him to land when he is 65 miles from B and he completes the trip by car. If he traveled the same length of time in the car as in the plane, how many miles is it from A to B?

2. From the final homework assignment in the final algebra chapter of Interactive Mathematics Project Integrated High School Mathematics Year 4, "The World of Functions," p. 345:

Personal Growth

As part of your portfolio, write about your personal development during this unit. You may want to specifically address this issue.

How do you feel you have developed during this unit in terms of your ability to explore problems and prove conjectures in mathematics?
You should include here any thoughts you might like to share with a reader of your portfolio.

[The "conjectures" in question involve comparisons among the graphs of certain minimally modified functions, e.g., f(x), f(x) + b, and f(x+b)]

3. Extra Credit:

If your daughter is interested in a career in mathematics or science, would you worry more about the male perspective in the first problem, or the ratio of effort to learning in the second problem?

Wednesday, October 14, 2009

Right-Brain Children in a Left-Brain World, or Left-Brain Children in a Right-Brain World?

Checking out my book's Amazon page, as I can't help doing from time to time, I've learned that Raising a Left-Brain Child in a Right-Brain World is "frequently bought together" with Right-Brained Children in a Left-Brained World.

From such a sales paradox, two obvious questions emerge:

1. Why?
2. Is the world in question Right-Brained or Left-Brained?

Re question 1, one possibility is that people are simply satisfying their curiosity--e.g., about how these two books could co-exist. Or about how things could have changed so dramatically in the dozen years between the publication of Right-Brained Children in a Left-Brained World and that of Raising a Left-Brain Child in a Right-Brain World. Or, if things haven't changed, about whether it's right-brained or left-brained children who are more at sea in today's world.

Another possibility is that the two books appeal to overlapping groups of readers. To explore this, I acquired a copy of Right-Brained Children and read it last night. My conclusion: yes, there is indeed some overlap--in fact, quite a bit.

Right-Brained Children is about children with ADD. Author Jeffrey Reed, who has been working with such children since well before the term "ADD" became a household label, has long considered his clients as quintessentially right-brained.

But Reed's definition of "right-brained" not only differs mine, but overlaps somewhat with my "left-brained." Traits that he calls "right-brain" and I call "left-brain" include:

>being good at puzzles and mazes
>shying away from hugs
>being better at thinking of ideas if working alone rather than in a group
>being a late bloomer
>at the extreme, being on the autistic spectrum

I don't disagree that these traits are associated with ADD. Some researchers, indeed, have hypothesized that there's an overlap between autistic spectrum disorders and ADD. Nor do I believe that all children on the autistic spectrum are what I call "left-brain."

However, while Reed is basing his terms on what little has been scientifically concluded about brain hemispheres and personality traits (not much!), I'm basing my terms exclusively on the everyday vernacular, which commonly associates "left-brain" with puzzle skill, introversion, and working best independently rather than in groups.

One problem with Reed's dichotomy is that it raises more questions than it answers about the autism-ADD connection. He claims that the typical child with ADD is:

...extremely sensitive to your moods and expressions, reading your body language tone of voice, and look in your eyes far better than do most people. He can tell the moment you walk in the door if you had a good or a bad day at the ofice. If you're happy, he'll pick up on your giddiness; if you're on edge, he's apt to act out and show anger as well.
But for children with autism, such empathy is an area of weakness--indeed, it is a core deficit of autism. Nor do the ADD children I know strike me as more empathetic than their peers. Is it really the case that most of them are unusually right-brained in their ability to empathize with others?

As to the question of whether the world is right or left-brained, stay tuned for an upcoming post.

Monday, October 12, 2009

Right-brained science and disembodied facts

An article in last week's Education Week enthusiastically reports on an interactive science program in which scientists conducting research in the Phoenix Islands share their blogs entries, and correspond by email, with students in a marine biology class at a New Hampshire high school. The blog entries, says Education Week, provide:

--"first-person descriptions of topics they cover, such as coral-reef ecology and damage caused to them by pollution"

--"underwater photos and descriptions of abundant aquatic life—gray reef sharks, moray eels, giant clams, bohar snapper, and barracudas."

--"descriptions of scientific processes, like making observations and collecting data"

--"musings on life at sea: how to avoid the bends while diving, how to guard against infection, what the scientists are eating, and the researchers’ offhand reflections—on a rare species of bird or fish, or a glimpse of the Southern Cross in the night sky."

What could possibly be wrong with presenting such compelling material in such an interactive, real-time fashion? The problem is that this is a huge gamut of topics--from coral reef ecology to the bends--that emerged haphazardly in whatever order they happen to come up in blog entries and email mesages. Your typical high school student, meanwhile, is unlikely to have sufficient background knowledge to organize them systematically in long term memory.

Nor do the classroom follow-up activities appear to deepen students' systematic understanding. According to Education Week, their teacher, Ms. Mueller-Northcott, used the various blog entries to:

--"begin discussions and to prompt students to record journal observations about the scientists’ expedition. "

--"pose experimental-design questions to the teenagers: How could you study humans’ impact on coral reefs? Where would you do your research? What data and equipment would you need?

Journaling about a scientific expedition doesn't do much to deepen one's knowledge or conceptual understanding; as for experiemental design, this is something for experts, not novices, and involves questions far more subtle (and analytical) than desired location and equipment.

However, as Education Week reports, this particular venture is "just one of many aimed at connecting students through technology with scientists doing research in the field, an increasingly common practice in schools." The goals? To:

--"mak[e] scientific studies and careers more attractive to young people"

--"quash the stereotype of the scientist conducting obscure research in dreary isolation."

In other words:
The value for students does not come from scientists’ answering factual questions—that can be covered in class—but rather from the excitement of seeing a scientist at work: struggling, making breakthroughs, documenting joys and frustrations.

It's unfortunate that what the article brushes off in an easy aside--that facts "can be covered in class"--is happening less and less in today's classrooms.

But the real agenda of all of these interactive ventures isn't to teach scientific knowledge in any systematic way, but to promote science as anything but systematic (and analytical, and left-brained). As NASA engineer Heather Paul, a leading advocate for such programs puts it: “We need to work hard to dispel the myths.. [that] we’re brainiacs who sit in the lab all day... [In science] you have to be passionate about what you want to do. ... It’s not just a job, it’s a way of life.”

Like so many other well-intentioned but misguided right-brain fads in education, if only mere passion were all it took to make progress...

Saturday, October 10, 2009

Tae Kwon Do and the linear learning style

Of all recent situations I've been in, the one that reminds me most of what a linear, one-thing-at-a-time kind of person I am is when all three black belts, all zero red belts, all zero blue belts, and the one other green belt are all absent and I'm called upon to teach Tae Kwon Do class. This has happened twice so far, and will happen again soon.

As anyone familiar with martial arts classes knows, teachers typically take the class at the same time that they teach it. So here I am, simultaneously trying to kick and punch out the best kicks and punches I can model, count out each kick or punch from 1 to 10 in Korean, and scan the row of students facing me to make sure everyone's more or less "on target." I can practically feel what I imagine to be the unsually narrow bandwidth of my brain's intake circuitry straining to the breaking point, ready to short out at the slightest additional distraction.

In his "Why Students Don't Like School," Dan Willingham argues convincingly that there is no such thing as an auditory vs. visual learning style. But what about linear vs. holistic learners? I'm convinced I not only perform better, but also learn better, when things are presented to me one at a time, and I've heard many other self-identified "left-brainers" say the same thing.

But I'm still waiting to hear whether any empirical research backs this up. Or is it possibly the case that everyone learns better when things come one at a time?

Of course, when teaching (or taking) a martial arts class, one thing (or one muscle) at a time isn't really practicable.

Thursday, October 8, 2009

Math problems of the week: Systems of Equations in CPM vs. 1900's math

1. The only systems of equations that students are required to solve algebraically in the CPM (College Preparatory Mathematics) Algebra Connections "Systems of Equations" chapter (published in 2006):

y = 1160 + 22x
y = 1900 - 15x


y = 6 + 1.5x
y = 2x


y = 2x -3
y = -x + 3


y = 2x -3
y = 4x + 1


y = 2x - 5
y = -4x - 2


y = -x + 8
y = x -2


y = -3x
y = -4x + 2


y = 2x - 3
y = 2x + 1


y = -4x -3
y = -4x + 1


2. A subset of the over one hundred systems of equations in the Wentworth's New School Algebra "Simple Systems of Equations" chapter (published in 1898):

5x + 2y = 39
2x - y = 3


x/3 + y/2 = 4/3
x/2 + y/3 = 7/6


x + y - 8 = 0
y + z - 28 = 0
y + z - 14 = 0


6x - 2y + 5z = 53
5x + 3y + 7 = 33
x + y + z = 5


2x + 3y + 1 = 31
x - y + 3z = 13
10y + 5x - 2z = 48


1/x + 2/y - 3/z = 1
5/x + 4/y + 6/z = 24
7/x - 8/y + 9/z = 14


2/x - 3/y + 4/z = 2.9
5/x - 6/y - 7/x = -10.4
9/y + 10/z - 8/x = 14.9

3. Extra Credit:

(a) Discuss why CPM, but not New School Algebra, has to stipulate that the simultaneous equations be solved algebraically (rather than graphically or by "guess and check").

(b) Discuss the arithmetic and algebraic skills required by each problem set.

(c) Relate your answer in (b) to the final assignment in CPM's "Simultaneous Equations" chapter, the TEAM BRAINSTORM:

With your team, brainstorm a list for the following topics. Be as detailed as you can. How long can you make your list? Challenge yourselves. Be prepared to share you team's ideas with the class.

Topics: What have you studied in this chapter? What ideas and words were important in what you learned? Remember to be as detailed as you can.

Wednesday, October 7, 2009

How to ration high grades, part IV (addendum to Part III below)

3. Make sure that even the homework directions are clear only to someone who was paying attention in class.

Tuesday, October 6, 2009

How to ration high grades, part III

In addition to the strategies enumerated in Part I and Part II, there's also the strategy of making successful completion of homework dependent on paying attention in class:

1. Expect students as young as 8 to follow oral directions about which worksheets to rip out and put in their bags for homework. Anyone who fails to pay proper attention can then be given an incomplete.

2. Rather than basing homework questions on the material in a textbook, article, or information sheet that goes home in the student's backpack, base these questions on material that was only addressed during class time (e.g., "What did we learn in class today about rocks?"). That way, anyone who failed to pay enough attention during class will under-perform on the homework.

By favoring those who pay attention in class, you can keep high grades from certain students who might otherwise have earned them, specifically:

1. The bright kid who is bored in class and tends to space out.
2. The dreamy, developmentally skewed math/science/computer buff whose analytical skills far exceed his or her organizational skills and ability to pay attention.

Saturday, October 3, 2009

Actual scientific uncertainty opposed to the post modern notion that science (and math) is fraught with uncertainty:

Michael Brooks' 13 Things That Don't Make Sense: The Most Baffling Scientific Mysteries of Our Time

Just finished reading this fascinating, wonderfully researched book.

Yes, there are scientific mysteries--among them, some true bafflers. But that doesn't mean that there isn't a scientific explanation out there somewhere.

Another source of scientific uncertainty: fringe positions have sometimes proved correct. Much as we non-scientists would like to defer to the expert majority, this majority doesn't always get it right. As in all fields, there are egos, fads, and bandwagon effects.

But what's truly special about science as a discipline is that, eventually, errors are revealed and something closer to the truth emerges. Wouldn't it be nice if all human endeavors were like this?

Thursday, October 1, 2009

Math problems of the week: 2nd grade Investigations vs. Singapore Math

I. The entirety of the 2nd grade Investigations "Assessment: How Many More?" (session 2.6, "How Many Tens? How Many Ones?"), administered at the end of May:

1. Jake collects wizard stickers.
He has 46 wizard stickers.

How many more does he need to have 60?

a. Write an equation that represents the situation.

b. Solve the problem and show your work.

2. Sally has 76 marbles.

How many more marbles does Sally need to have 100 marbles?

a. Write an equation that represents the situation.

b. Solve the problem and show your work.

II. Two of the 15 problems in the second-to-last Review in 2nd grade Singapore Math, Primary Mathematics 2B (Standards Edition), p. 162:

9. There are 120 boys at a concert.
There are 85 more girls than boys.
19 girls and 16 girls wear glasses.
(a) How many girls are there at the concert?

(b) How many children are there altogether?

(c) How many children do not wear glasses?

13. Kelly has $6.80.
She wants to buy a photo album that costs $8.50.
How much more money does she need?

III. Extra Credit:

Compare the level of higher-level thinking involved in writing an equation that "represents" the situations described in the first problem set with the higher-level thinking involved in doing the multi-step problem and the three-digit problem in the second problem set.

Why does the first problem set, but not the second, explicitly require students to show their work?