Saturday, January 31, 2009

Why teach long division?

Seriously, why teach it?

This is a genuine question, not a rhetorical one, and one that I've been wondering about lately.

Thursday, January 29, 2009

Math problems of the week: 4th grade Mathland vs. Singapore Math

1. From the final word problems in MathLand's Skill Power for grade 4 (pp. 218-219):

The answer is 85 cents. Write at least 6 different equations that have this answer.


Gina scored 12 points in last night's basketball game. She scored 1/6 of her team's points. How many total points did the team score?

A. 24 points. C. 72 points
B. 36 points. D. 84 points

2. From the final word problems in Singapore Math's 4th grade Primary Mathematics Workbook, 4B:

Karen earned $840 a month. She spent 2/5 of it on food. How much did she spend on food?


Alan used 3/4 of his money to buy a watch which cost $45. How much money did he have left?

3. Extra Credit:

The answer is MathLand. Write at least 6 different questions that have this answer.

Monday, January 26, 2009

Parenting in the 21st century

Current trends in popular psychology have psychologists telling parents that we're working too hard at the expense of quality time with our kids.

But current trends in education are detracting from this quality time as never before.

How often did our own parents:

1. face pressure to volunteer in elementary school classrooms, where the potential chaos of today's group-centered discovery learning and record numbers of field trips means that students require more adult supervision than ever before?
2. have to teach skills that schools once taught, like handwriting, phonics, and the standard algorithms of arithmetic?
3. have to drive kids around to libraries, and art supply stores, and other kids' houses for group-centered, arts & crafts-intensive research projects?
4. have to help our kids, as young as 6, through complicated, multi-step homework directions, interactive math games, and tasks like finding household objects that come in groups of 7 or 11?

My mother, like many of her generation, didn't work. She volunteered at school once or twice, but teachers generally turned down her offers, making it clear that they didn't need parental help. She relied on schools to teach us academics, and, once we started getting homework in 4th grade, could confidently expect that we'd be able to do most of it on our own.

She spent much of her time volunteering for civic activities, and, when we were at home, talking and playing games with us.

Saturday, January 24, 2009

Interactive science in college classrooms revisited

One of the concerns I expressed about MIT's new interactive science classes, namely about students being forced to work in groups, is one of many reasons why, as it turns out, MIT students hate these classes.

In her post on kitchentablemath, Allison cites the MIT newspaper:

Most students do not bother to hide their dislike for TEAL. Their list of grievances is long and oft-repeated: the physical set-up of small tables makes it difficult to see the lecturer, the numerous homework assignments are tedious, the in-class problems are gone over too quickly, the students strong in physics end up doing all the work, and so on.
Allison also cites one of the MIT students who commented on the New York Times story:
This is all geared towards collaborative learning, which is nice in theory, but what happened in my experience is that the people at the table who knew what they're doing would work through the problem, and I would be left in the dark in terms of where this equation came from and what that one means. The idea was to learn from eachother, except that I feel that we do plenty of this while working on p(roblem)-sets. Personally, I'd like classtime to be geared more towards learning from the teacher.
According to additional comments at the New York Times, class time is 5 hours a week, and attendance is mandatory.

One result: grades, just like in too many grade schools, favor the social students over those whose distaste for working in groups deters them from coming to class.

Thursday, January 22, 2009

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

1. From "Stickers, Number Strings, and Story Problems," tonight's Investigations homework:

Can you make ...?

Use three different numbers in each problem.
Use each number below at least once.

7, 8, 9, 1, 4
5, 3, 6, 2

1. Can you make 15 with 3 numbers?

_____ + _____ + _____ = 15

2. Can you make 16 with 3 numbers?

_____ + _____ + _____ = 16

3. Can you make 17 with 3 numbers?

_____ + _____ + _____ = 17

4. Can you make 18 with 3 numbers?

_____ + _____ + _____ = 18

2. From the middle of the 2nd grade Singapore Math curriculum, Primary Mathematics 2B, p. 17


(a) 183 + 99 =
(b) 246 + 98 =
(c) 199 + 99 =
(d) 206 + 98 =
(e) 99 + 556 =
(f) 98 + 235 =
(g) 99 + 408 =
(h) 98 + 399 =

3. Extra Credit:

1. For each problem set, relate the language arts challenge (understanding the directions) to the mathematical challenge.

2. Which problem set teaches a more useful mathematical skill? Compare the real-life and mathematical utility of:

a. Thinking of ways to "make" a two-digit number with three one-digit numbers, using each number at least once.
b. Adding different numbers that approach 100 to various two and three digit numbers and seeing the effects on place value.

Monday, January 19, 2009

Autism Diaries VI: a Macintosh by any other name

J. has discovered he can disgust his younger sister with the messy way he eats his favorite fruit, Macintosh apples: "Macintosh apples can 'ew' you!" So now he plays it up even more, opening his mouth even wider while eating; leaving bits up half-chewed Macintosh on the stairs; printing out images of cut-up Macintosh apples (the fruits of Google Images) to display around the house.

Things have gotten so out of hand that we finally told him that we won't be buying any more Macintosh apples for him until he cuts it out.

"Can we buy Empire apples, instead?" he inquired yesterday, while gearing up for the weekly trip to the grocery store with Daddy.

"OK," we agreed. Fruits are J's only vegetables, and we don't want a punishment to limit his nutritional intake.

Later, he and Daddy returned with a large bag of Empire apples. At least, that's what we thought they were.

Eventually, unable to hold back the satisfaction of reveling in his mischief with others, J explains that, while Daddy wasn't looking (and, apparently, while none of the store clerks were looking either), he'd switched all the stickers on the Empire and Macintosh apples on the top-most layer of their adjacent store bins.

Saturday, January 17, 2009

Interactive science in college classrooms

Reading an article in this past week's New York Times on how the MIT physics department has been:

...pioneering teaching methods drawn from research showing that most students learn fundamental concepts more successfully, and are better able to apply them, through interactive, collaborative, student-centered learning...
and has accordingly replaced the large lectures of its introductory physics classes with:
...smaller classes that emphasize hands-on, interactive, collaborative learning...
my first reaction was alarm and dismay.

All too often I've seen how, in education, "research shows" not what science shows, but what educators want it to show. And all too often I've seen how "hands-on, interactive, collaborative learning" has watered down the math and science curriculum in grade schools, favoring the gregarious over the shy or otherwise unsocial. So now, whenever I hear these buzzwords, I flinch.

But how alarming is this trend, which, according to the Times, extends beyond MIT to include Rensselaer Polytechnic Institute, North Carolina State University, the University of Maryland, the University of Colorado at Boulder and Harvard? The Times, of course, is not alarmed, but breathlessly enthusiastic--in the way of all recent articles about this kind of education reform:
Instead of blackboards, the walls are covered with white boards and huge display screens. Circulating with a team of teaching assistants, the professor makes brief presentations of general principles and engages the students as they work out related concepts in small groups.
Teachers and students conduct experiments together. The room buzzes. Conferring with tablemates, calling out questions and jumping up to write formulas on the white boards are all encouraged.
But, as I read, it occurs to me that college courses differ from grade school courses in ways that allow the best of both worlds: hands-on, experiential learning in the classroom, combined with abstract, linear learning (of the sort that many left-brainers prefer) outside the classroom, as students work through their textbooks. What's special about college classes, after all, is that, at 3 hours a week, the time students spend in class comprises a much smaller fraction of the overall time they spend engaging with a subject it does in grade school and high school. And, too often, the college lecture, however well-delivered, simply follows in lockstep with the textbook.

So, so as long as:

1. these modified courses cover the same amount of material as before, with the same overall rigor, and
2. students who work better on their own aren't forced to work in groups

these changes may actually be beneficial.

Unless they end up validating, and further entrenching, what our grade schools have done, in the name of "hands-on, interactive, collaborative learning," to math and science teaching--thus limiting the ability of American-educated freshmen to handle introductory college physics in general, whatever its incarnation.

Thursday, January 15, 2009

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

1. From the last three problems of the final multiplication/division unit in the 2nd volume of the 2nd grade Math Trailblazers workbook, Student Guide Book 2, pp. 340-341.

Each baby bird needs 5 raisins. Mother bird has 16 raisins. Draw the number of baby birds she can feed.

[large space for drawing]

How many baby birds ate raisins? _____


Each baby bird needs 8 seeds. Mother bird has 16 seeds. Draw the number of baby birds she can feed.

[large space for drawing]

How many baby birds ate seeds? _____


What did you see on your way home from school today? Use some of these things to write a multiplication or a division problem for the rest of your class to solve. Draw a picture to show what you are multiplying or dividing.

[even larger space for drawing]


2. From the last three problems of the final multiplication/division unit in the 2nd volume of the 2nd grade Singapore Math workbook, Primary Mathematics 2B, p. 56.

Peter has 9 balls. He wants to put the balls into 2 equal groups.

(a) How many balls are there in each group?

(b) Are there any balls left over?


Mary has 26 beads and some pieces of string. She puts 6 beads on each string.

How many beads are left over?


Natalie has a ribbon 37 in. long. She cuts it into pieces 4 in. long each.

How many inches or ribbon are left over?

3. Extra Credit

1. How many baby birds does the Trailblazers student need to draw?

2. Predict which students will earn higher grades on the Trailblazers homework: artists or math buffs.

3 (a). If you were a mother bird with a large brood, would you only feed some of your babies so that those babies could get the full quantity they "need," or would you spread the food around evenly and fly off to find more?

3 (b). Does a Trailblazers student who answers 16 and 24, respectively, for the first two Trailblazers problems receive full credit if s/he explains her answer and draws nice birds?

Tuesday, January 13, 2009

The 1940's: a heyday for algebra instruction?

Today's educators cast long ago math classes as mindless drill sessions, unenlightened by the "higher level thinking" that, so they claim, predominates, as never before, in today's "best practices."

But I've scoured today's algebra offerings and come up with little beyond the "cookbook style" memorize & plug-in evinced, for example, by the University of Chicago Math Project's approach to the Quadratic Formula.

Then I dredged up my mother's ancient algebra text, published in 1943. Check out some of these word problems from the introductory chapter, and compare them--and the actual higher level thinking they require--to what you find in today's algebra texts:

Express in symbols:
1. Two consecutive numbers.
2. Two consecutive even numbers.
3. Two consecutive odd numbers.
7. Henry's age three years ago if he will by y years old in two years.
20. The time it will take an automobile to go d miles at the rate of r miles per hour.

Sunday, January 11, 2009

Social skills and politics

A friend recently finished 10 years serving as a representative in the Connecticut legislature, and shared with me her informal legislative memoirs. Looking back, she realizes that she would have made much more of an impact if she'd spent more time schmoozing.

My friend would be the first to admit that she's not particularly social or charismatic (or able to remember names and faces), and when she was first elected, and then repeatedly re-elected, what impressed me was how her intelligence, hard work, and conscientiousness kept trumping what many might read as aloofness.

But where more social savvy would have paid off was in the legislature itself, where a bill, no matter how sensible, practical, and important, would never see the light of day, or would be amended and gutted beyond recognition, or would come up for a vote only when time was about to run out--unless you knew when, and how, and to whom to cozy up.

And could live with yourself afterwards.

Thursday, January 8, 2009

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

1. From the very end of the 3rd grade Everyday Math curriculum, Student Math Journal, volume 2, p. 316:

2,384 + 1 = ________
2,384 + 10 = ________
2,384 + 100 = ________
2,384 + 1,000 = ________
2,384 + 10,000 = ________

2. From the very beginning of the 3rd grade Singapore Math curriculum, Primary Mathematics 3A, p. 15:

Fill in the blanks:
5409 + ________ = 5419
5409 + ________ = 6409
5409 + ________ = 5410
5409 + ________ = 5509
5409 + ________ = 7008
5409 + ________ = 7098
5409 + ________ = 7998
5409 + ________ = 6999

Tuesday, January 6, 2009

Earning high grades in writing: "descriptive words"

If there's a subject in which I believe my 2nd grade daughter to be way above grade level, it's in writing. I believe this because she spends hours and hours of her free time inventing and writing down story after story after story, producing such gems as:
In the tunnle glowing lights shimered on the ceiling. It was blue. The wind was cool. It was like a cristal cave.
At school, however, where she's required to stick to nonfiction accounts of her personal life, she tends to produce stuff more like the following:
I went to my grandparents house. We drove there. It took a very long time to get there. I broget my doll there. When we got there I looked around the house for some paper.
A generation ago, when students were graded only on penmanship, spelling, grammar, and punctuation, this would have been fine. But today's "proficient" and "advanced" writing standards require students to "use descriptive words" that "paint pictures in people's heads."

Thus, lack of inspiration at school means writing skills assessed as "basic."

Sunday, January 4, 2009

Left-brained literary analysis

At the close of '08, as my daughter and I finished reading Charlotte's Web and Stuart Little, it struck me that part of what makes these books work so well is how certain key scenes, at once, are inherently intriguing, and set the stage for later scenes that would otherwise seem contrived, thus fulfilling Chekhov's maxim that:

If in the first act you have hung a pistol on the wall, then in the following one it should be fired. Otherwise don't put it there.
Consider, for example, the chapter in which Stuart Little's family searches for him after he gets rolled up in the living room blinds, which highlights the compulsiveness of older brother George:
George meantime went down cellar and hunted around to see if he could find the other entrance to the mousehole. He moved a great many trunks, suitcases, flower pots, baskets, boxes, and broken chairs from one end of the cellar to the other in order to get at the section of wall which he thought was likeliest, but found no hole. He did, however, come across an old discarded rowing machine of Mr. Little's, and becoming interested in this, carried it upstairs with some difficulty and spent the rest of the morning in rowing.
Entertaining in its own right, this scene also makes plausible George's subsequent rush to lower the blinds when the family starts worrying that Stuart is gone forever:
"George!" shouted Mr. Little in an exasperate tone, "if you don't stop acting in an idiotic fashion, I will have to punish you. We are having enough trouble today without having to cope with your foolishness."

But George had already run into the living room and had begun to darken it, to show his respect for the dead. He pulled a cord and out dropped Stuart onto the window sill.
Or consider how, in Charlotte's Web, the author gets the reader, along with Wilbur, to appreciate the miracle of web building prior to the first supernatural miracle, in a chapter in which Wilbur, provoked by Templeton, the rat, ties a rope to his tail and attempts to spin a web.

But how often do literature classes reverse-engineer a work of literature to see how the parts thus fit together to make an effective whole?

Even back when the close literary analysis held sway, in its preference for such rarefied techniques as imagery, metaphor, symbolism, and theme, it tended to shy away from the more mechanical, "workmanlike" aspects of writing.

Today's classes, which some secondary schools now call "Literacy" rather than "English," have taken us even further afield, with their blinkered, narcissistic focus on "personal connections," identity politics, and life lessons.

Friday, January 2, 2009

Math problems of the week: 3rd grade Pennsylvania Standards Test vs. Singapore Assessment

A. From the beginning of a sample placement test for the Pennsylvania Academic Standards, given at the end of 3rd grade (16 questions in all):

1. Which number has the greatest value?
200 1,001 30 1,100

A. 30
B. 200
C. 1,001
D. 1,100

2. Olivia looks at 4 different measurements. These are the measurements: 1 inch, 1 foot, 1 yard, 1 mile. Which measurement is the least?

A. 1 inch
B. 1 foot
C. 1 yard
D. 1 mile

B. From the beginning of the placement test for Singapore Primary Mathematics 3B, an exit exam for 3rd grade (18 questions in all):

1. Use mental calculation.

(a) 34 + 28 =
(b) 62 - 37 =
(c) 600 x 5 =
(d) 3600 ÷ 9 =

2. Fill in the blanks.

(a) 5 m - 3 m 45 cm = ______ m _______ cm
(b) 3408 m = ______ km ______ m
(c) 3 kg 250 g - 8 kg 600 g = ______ kg ______ g
(d) 3 L 6 ml = ______ ml

With standards like Pennsylvania's, why should our schools bother teaching anything but Reform Math?

Unless, of course, they care as much about their students as they do about their test scores.