Friday, February 27, 2009

math problems of the week: 2nd grade investigations vs. French Math

1. From the middle of the 2nd grade TERC Investigations curriculum:

2. From the beginning of the 2nd grade French Math curriculum (worksheet from the Professeur Phifix website):

Is everyone using the equivalent of Singapore Math but the U.S.?

Wednesday, February 25, 2009

Why teach the traditional long division algorithm?

There are two general long division algorithms that I'm aware of: the Partial Quotients algorithm as used in Everyday Math and other Reform Math programs, and the long division algorithm that many of us grew up with (including its European variations).

The partial sums algorithm has the advantage of being much more conceptually transparent to students.

But the long division algorithm has its own unique advantages. First, it is the most complex, multi-step algorithm that students encounter in all of arithmetic: divide, multiply, subtract, bring down; divide, multiply, subtract, bring down. As bky points out, unlike the partial sums algorithm, the steps are precise and predetermined, so that it's an algorithm in the strictest sense. All this, among other things, is good exposure for our future computer programmers--a concrete reference point, for example, for the principle of Recursion.

Also, because the traditional long division algorithm is not at all transparent, students, in order to remember what to do when, need to think hard about what they're doing and why. With enough practice, they start to grasp the underlying principles, including some subtle aspects of place value, and the inter-relationships between place value, division, multiplication, and subtraction.

Indeed, in the spirit of true Constructivism, a child who does the long division algorithm is experientially constructing his or her knowledge of these higher level concepts.

Why discourage this?

Third, the traditional long division algorithm lends itself more readily than the partial quotients algorithm (which tends to deal in integers) does to converting fractions into decimals. As David Klein and James Milgram argue in this paper, referred to me by Niels Henrik Abel, repeated execution of the long division algorithm helps you understand why all fractions convert to repeating decimals, and helps you see connections between the length of a given repeating decimal and the properties of the corresponding fraction. More generally, the authors suggest, performing the long division algorithm helps students build conceptual understanding of what a real number is.

Indeed, anyone who's considering ditching the traditional long division should read Klein and Milgram's paper first... and make sure they understand it.

Monday, February 23, 2009

Why teach long division: part II

In my previous post, I discussed one reason for teaching a general algorithm for long division: namely that reliance on calculators and estimation doesn't give you enough feedback about wrong answers.

Reason #2 for teaching a general algorithm for long division is that this lays a foundation for a key topic in algebra: polynomial division. Algebra teachers, including Mrs. H, report that in teaching how to divide polynomials, it's helpful to begin with concrete examples of the algorithm in question that involve numbers instead of variables.

Indeed, with complex problems in general, it's helpful to start students off with analogous problems involving small, simple numbers. Since many general algebraic procedures are analogous to general arithmetic ones, all the more reason to teach the standard algorithms to mastery--and to start doing so well in advance of algebra.

Next up: reasons for teaching the traditional long division algorithm in particular.

Saturday, February 21, 2009

Why teach long division: part I

Three weeks ago, I asked Why teach long division? So far, I've received 8 thought-provoking replies.

My thoughts are provoked into thinking that we should teach long division.

But first, there are really two questions here, and one has an easier answer than the other.

The one with the easier answer considers any general algorithm for multi-digit division to be an instance of "long division." Besides the long division algorithm that most of us learned as students, we now have the partial quotients method, popular, for example, with Everyday Math.

So the first question is, should we renounce (or minimize) general multidigit division algorithms and rely on a combination of calculators and estimation skills?

Here, the answer is a resounding "no."

Reason #1 pertains to the general problem with doing math primarily via calculators and estimation skills. This is the problem of "poverty of feedback." Briefly put, if your estimation skills tell you that the result on your calculator screen is wrong, there's no feedback about what you did wrong. You don't know if you accidentally pushed the wrong key, or whether you're accurately typing in the wrong sequence of keys, or whether you made a mistake in setting up the problem in the first place (e.g., mistranslated a word problem into the wrong equation). There's simply no paper trail to go back over.

If, on the other hand, you work out the problem by pen and paper, you can review your calculations and see if you made an arithmetic mistake or an algorithm error; or if, instead, you'd better go back further and see if you set up the problem wrong.

I'm a linguist by training, and I see this "poverty of feedback" problem, as well, in programs that purport to teach grammar, like Rosetta Stone and Laureate Learning. Here, as with calculators, the learner simply clicks (on the picture that goes with a sentence). Because the software code can't tell from a simple click what the basis for the person's mistake was (e.g., whether it stemmed from deficiencies in vocabulary, or deficiencies in grammatical parsing), all the program can do in response is to indicate that an answer is "right" or "wrong". As an alternative, I've developed a program called GrammarTrainer. Here, you construct a sentence rather than clicking on a picture, which enables the software code to analyze your answer and give you detailed linguistic feedback about what exactly you're doing wrong. Preliminary empirical results suggest that this feedback is crucial.

As with language, so, too, with math: simple clicks make meaningful feedback difficult; meaningful feedback is key.

Thursday, February 19, 2009

Math problem of the week: 2nd Grade Trailblazers vs. Singapore Math

1. The final assignment in 2nd grade Trailblazers Student Guide Book 2, Unit 11, Ways of Subtracting Larger Numbers, p. 299:

Difference War

Players: two or more

Materials: digit cards or regular playing cards, paper, pencil (If you use the regular playing cards, remove the tens and face cards and use the aces for the number one.)

Winner: Player with the most cards at the end of the game.

*Deal out four cards to each player.
*Each player makes a subtraction problem with the four cards.
*Each player solves his or her subtraction problem.
*The player whose answer is the smallest takes everyone's cards and puts them aside.
*Keep dealing four more cards to each payer and making problems.
*When all the cards are gone, the player who has collected the most cards wins.

Play several games. Then, write about what happened. Tell how you think you can win.

2. The final assignment in 2nd grade Singapore Math Primary Mathematics 2B, Unit 7, Addition and Subtraction, p. 25:

(a) 180 - 99 =
(b) 302 - 99 =
(c) 556 - 99 =
(d) 848 - 99 =
(e) 205 - 98 =
(f) 467 - 98 =
(g) 780 - 98 =
(h) 632 - 98 =

3. Extra Credit

Which activity is more mathematically engaging, and to whom?

Tuesday, February 17, 2009

Autism Diaries VII: When the home becomes the world, and the world, the home

I got my latest credit card statement the other day and I must say, I was shocked, shocked to discover a number of unauthorized charges for computer game downloads.

It seems J has memorized one of my credit card numbers, along with the security code.

I promptly closed out the account, informed J that he's taken care of his birthday presents, and subjected him to various consequences. I'll also call up the companies to see if they'll refund me, but if past experience is predictive, they'll laugh in my face. And I'll try to keep my wallet in a safer place from now on.

It's weird not to be able to let your guard down in your very own home.

Similarly weird is when members of your own extended family become like the world at large, giving you obvious or impractical autistic parenting advice that they think you haven't already thought of, or that they think is reasonable to ask of you.

"Why don't you teach him the Golden Rule," suggests a cousin. "Why don't you monitor his emails and Internet posts," suggests an uncle.

I've of course long grown tired of 24-7 monitoring, apologizing, etc.. I stay sane, rather, by thinking of autism not as a problem that is confined to specific families, but one that, like it or not, affects the entire world--up close and personal. Many people, sooner or later, will be touched by autistic person, whether or not they happen to live or work with him or her--whether at a restaurant, on a bus, or over the Internet.

Sunday, February 15, 2009

"Education is All In Your Mind"

Today's New York Times has published my letter in response to Richard Nisbett's January 8th New York Times Op-Ed piece.

Beyond the concerns I expressed there, I have a broader worry about the effects of Nisbett's Op-Ed on classroom education. Specifically, I worry that his theory--that fostering positive student attitudes is more effective than teacher training and curriculum content--will only further entrench current practices.

Already, too many educators think it's all about how students perceive themselves, how comfortable they feel in the classroom, and whether they relate what they're learning to their futures, and that actual academic content matters little. You can bet that, in these circles especially, Richard Nisbett's piece will get loads of coverage.

Refreshingly, today's New York Times has an Op-Ed piece by Nicholas Kristof that touts the importance of good teachers. Kristof cites a Department of Education study, published this month, that shows, in his words, "no correlation between teacher certification and teacher effectiveness."

But Kristof, like Nisbett, makes no mention of curriculum content, and, in fact, goes on to suggest that teachers' academic training matters little: "Particularly in lower grades, it also doesn’t seem to matter if a teacher has a graduate degree or went to a better college or had higher SATs."

When will a major newspaper run a piece that exposes the impoverished content of today's Reform Math programs, and raises the possibility that both what your teacher knows, and what you learn, makes a difference?

Saturday, February 14, 2009

Ideas on helping children with hard word problems

As I work with my daughter through Singapore Math word problems, I try to keep track of the more successful strategies I've used to help her when she's stumped. Of course, I'm not a certified teacher, so I wonder if any of these are ever covered in Education School math methods classes?

In case they aren't, I thought I'd share a couple here.

1. In some cases, a combination of large numbers and multiple steps may intimidate the child enough that he or she doesn't know where to start. If this is the case, have your child begin with a parallel problem that involves small, easy numbers.

2. Sometime so much is going on in a problem that the child doesn't know which operation to perform when. In this case, charts may be helpful.

Let's see how this works with a given exercise, namely, a word problem from the end of Primary Mathematics 3B:

Isabella cut a rope 414 m long into pieces. Each piece was 9 m long. She divided each piece into four equal parts. How many parts are there?
Using Strategy 1, we might have our child first do the following problem:
Isabella cut a rope 20 inches long into pieces. Each piece was 4 inches long. She divided each piece into two equal parts. How many parts are there?
Using Strategy 2, we first ask:

What are the units (denominations) in this problem?
Answer: a 414-meter rope; 9-meter rope pieces; "parts" of the 9-meter rope.

We then have our child make a chart, heading each column with one of the units:

414-meter rope | 9-meter rope pieces | parts of 9-meter rope pieces|

We then ask what quantities go in each column. We have our child fill in the known quantities, leaving the unknown quantities blank or "?":

414-meter rope | 9-meter rope pieces | parts of 9-meter rope pieces|

It should then be transparent that the first step is to fill in column 2, and that this involves dividing 9 into 414. We then get:

414-meter rope | 9-meter rope pieces | parts of 9-meter rope pieces|

Going from the middle column to the right-hand column is less transparent, so we reread the word problem to see how "parts" and "pieces" are related, and if necessary, ask hypothetical questions involving low numbers (if there were one "piece," how many "parts" would there be?) and record the answers in a second chart, until the pattern is established and we return to the unknown:

pieces | parts

We then ask which unit we know the value for, and have our child fill in this value in the appropriate column:

pieces | parts

It should then be transparent how to find the remaining unknown and solve the problem.

Thursday, February 12, 2009

Math problems of the week: 6th grade Connected Math vs. Singapore Math

1. The final word problems on percent in Connected Math's "More about Percents," Bits and Pieces II: Using Rational Numbers, p. 26:

In a survey of 100 dog owners about their pets' habits, 39% said that their dogs eat bugs. How many dog owners surveyed said this?

When 300 tarantula owners were surveyed, 26% said they let their spiders crawl on them. How many tarantula owners surveyed said this?

In a survey of 80 students, 40% said they had a savings account of their own. How many students surveyed said this?

During a survey of 80 students artists, about 6% said they had sold at least one of their works of art. About how many students surveyed said this?

2. The final word problems on percent in Singapore Math's "Percentage," Primary Mathematics 6A, p. 64:

A dress was sold for $42 after a 30% discounts. What was the usual price of the dress?

The price of a television set was increased by 10% to $2420. What was the price before the increase?

In a school choir, the number of boys was increased by 20% to 60 and the number of girls was increased by 20% to 60. Find the overall increase or decrease in membership of the choir.


3. Extra Credit:

Estimate the percentage increase in mathematical understanding effected by bug and spider humor vs. problems that require higher level mathematical thinking.

Tuesday, February 10, 2009

Heroes of children's lit: champions of right-brained learning

As I revisit children's literature with my daughter, I'm realizing that its most engaging heroes are poster children for all of today's right-brained trends in classroom education. From Huckleberry Finn to Pippi Longstocking, they shun school, mock traditional classroom authority figures, and brim with real-life wisdom and problem solving skills that they acquired experientially through hands-on, self-guided learning.

On the other hand, this learning takes place not in group activities in the classroom, but while sailing down the Mississippi or out on the high seas.

And Pippi doesn't know her "pluttification" tables, or write nearly as well as the award-winning author who narrates her life. Cf:

vs. the paragraph that immediately follows:
Tommy and Annika were so happy they began to skip and dance. They understood perfectly well what was printed on the card although the spelling was a little unusual. Pippi had had a great deal of trouble writing it. To be sure, she had not recognized the letter i in school the day she was there, but all the same she could write a little. When she was sailing on the ocean one of the sailors on her father's ship used to take her up on deck in the evening now and then and try to teach her to write. Unfortunately Pippi was not a very patient pupil. All of a sudden she would say, "No, Fridolf"--that was his name--"no, Fridolf, bother all this learning! I can't study any more now because I must climb the mast to see what kind of weather we're going to have tomorrow."
Of course, Huck Finn writes quite well, but I suspect he has a little help from someone who spent more time in school than he did.

Friday, February 6, 2009

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

1. The quiz given to Investigations 3rd graders at a model school on January 16, 2009:

Today is the 82nd day of school. How many more days until the 100th day of school? Explain your answer in numbers or words. Write an equation for this problem and put the answer in the answer box.

Number equation __________

Show and explain your solution:

Answer Box: __________


2. The first word problem on the Singapore Math 3B placement test, a test assessing mastery of the first half of the 3rd grade Singapore curriculum:

The difference between two numbers is 456. If the larger number is 854, what is the smaller number?


3. Extra Credit:

Discuss the correlation between being asked to show your work and having work to show.

Thursday, February 5, 2009

Against open-ended projects: arguments from psychology

The projects that dominate today's "project-based learning" pose problems for those I call "left-brainers:" linear, analytical thinkers who struggle with big-picture thinking. The more open-ended the projects, the more they have trouble even knowing where, and how, to begin.

Now research from psychology, as reported in a recent Economist article, suggests that open-endedness may pose problems for all students:

People act in a timely way when given concrete tasks but dawdle when they view them in abstract terms.
Researchers led by Sean McCrea, of the University of Konstanz in Germany, compared how speedily students completed such tasks as explaining how they might go about keeping a diary (concrete) vs. asking why someone might want to keep a diary (abstract). The results:
[A]lmost all the students who had been prompted to think in concrete terms completed their tasks by the deadline while up to 56% of students asked to think in abstract terms failed to respond at all.
Given this, today's teachers might consider how many of their students are actually completing such assignments as "invent a culture," "make up a game using everything you know about math," or, indeed, "write about why people keep diaries." Perhaps, often, it isn't the student, but a more future-thinking, grade-obsessed parent, who is doing most of the work.

Tuesday, February 3, 2009

School Science Fairs: Right-brained obstacles to left-brainer recognition

When my autistic son first started participating in the mandatory city-wide middle school school science fair, I figured this scientifically-minded kid would naturally distinguish himself--after all, he's been doing science experiments since he was about 2--and thus enhance his chances for admission to a decent magnet high school.

But two years have gone by, and he's won no prizes.  In fact, he hasn't even gotten past the first hurdle: being chosen as the one or two students to represent his class.

Perhaps some of his classmates are very strong competitors, I'd simply assumed.

Then, this morning, a friend of mine who happens to be a school science fair veteran explained to me how it works.

To make it past that first hurdle, it turns out, you have to be elected by the majority of your classmates.  And for this, you are evaluated not on the scientific merits of your experiment, but on the quality of your presentation.

Thus, graphic design and public speaking skills trump scientific talent, further reducing what few opportunities remain for left-brainers to distinguish themselves.