Alternatives to stacking, II

I've received, both here and at kitchentablemath, a number of interesting non-stacking strategies for computing 825 - 267 that I had never thought of before. This made me wonder about just how far one can get without stacking. What happens when we add a few more digits?

For example, how about--using numbers randomly generated by my software program--885.66 minus 746.85?

My new question: Is there a way to subtract 746.85 from 885.66 that

1. Isn't massively facilitated by stacking one number on top of the other;
2. Only uses methods that grade school children can discover on their own;
and
3. Doesn't place excessive demands on short-term memory?

If so, please share it here!

Math problems of the week: 3rd grade Investigations (TERC) vs. French math

1. From 3rd grade TERC Investigations, "Things that Come in Groups," p. 89:

There are 26 crackers in a box. Each cracker can be split into 2 equal pieces. How many pieces will there be?

Show how you solved this problem. You can use numbers, words, or pictures.

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2. From Professeur Phifix, a web resource for French curriculum materials, "Multiplication and Division" problems for CE2 (3rd grade), translated from the French:

August has invited 4 friends over for his birthday. His mother bought him 3 cookie boxes each containing 25 cookies. After playing soccer, the children come to eat and celebrate August's 10th birthday.

Calculate the number of cookies that each child will eat.

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3. Extra Credit:

Using words, numbers, or pictures, explain why the Investigations problem, but not the French problem, stipulates that students must explain their answers.

Continental Math League Update:

Enthusiasm from students and parents; skepticism from teachers.

Specifically, about "stacking."  (Today's word for how we used to add, subtract, and multiply numbers by placing one number on top of the other.)

Kids love it.  And not just the ones on our team.  As Nancy Bea Miller writes:

When I showed one of my sons how I had learned addition, i.e. the "stacking" method, he was very impressed. "Wow, that's so cool! That works great! I wonder if my math teacher knows about this?" was his innocent comment.

Yes, she does, and she doesn't like it.  At least if she resembles the teacher who approached me after math practice yesterday and recounted the dismay she felt when she caught one of her students stacking numbers, thus abandoning the more "meaningful" and "faster" way he used to solve problems.

My co-coach and I tried to explain that the Continental Math League numbers are big enough, and random enough, that Reform Math's methods aren't faster and more meaningful, but inefficient and confusing. It's one thing to add 48 and 39 by reasoning that:

48 is 2 less than 50, and 39 is 1 less than 40, so add 40 and 50 and get 90 and then count backwards by 3 and get 87."

But take one of the problems we did at Continental Math League yesterday: 825 - 267. Restricting myself to the kinds of calculation that these second and third graders are able/expected to do in their heads, here's the most efficient non-stacking method I can come up with:  

The closest friendly number to 825 is 800, and the closest friendly number to 267 is 250.  825 is 25 more than 800.  250 is 10 more than 260, and another 7 gets you 267.  10 plus 7 is 17.  So 267 is 17 more than 250.  So subtract 250 from 800. Well, 800 minus 200 is 600, minus 50 more is 550.  Then subtract 17 from 25 by counting up from 17.  Seventeen plus 3 more is 20 plus 5 more is 25.  3 plus 5 equals 8.  Add 8 to 550* to get 558."

*By this point in the problem, how many people remember what they should be doing with this 8?

Anyone with a more efficient non-stacking method for subtracting 267 from 825 (no calculators allowed!) is invited to share it here.

Internet problems

Hope to be back tomorrow!

Math problems of the week: 5th grade Mathland vs. Singapore Math

1. From Mathland Daily Tune-Ups Grade 5, second-to-last problem set:

*There are 4 cups in 1 quart. How may cups are there in 6 quarts?
*There are 2 pints in 1 quart. How many pints are there in 4 quarts?
*There are 4 quarts in 1 gallon. How many quarts are there in 4 gallons?
*If there are 4 quarts in a gallon, and you have only 2 quarts, how many gallons is it?

2. From Grade 5 Singapore Math, Primary Mathematics 5B, end of "Measures and Volume" unit:

A rectangular container measuring 12 in. by 10 in. by 11 in. is completely filled with water. After 240 in.³ of water are taken out from the tank, what is the height of the water level in the container?

[picture of container with its measurements shown]

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Extra Credit:

1. Estimate the percentage of Singaporean 1st graders who can do the Mathland set.

2. Estimate the percentage of American 5th grade teachers who can do the Singapore problem.

An alarming report card on Investigations Math

Pat's teacher is 27 years older than Pat. Pat is 9. How old is Pat's teacher?
Pat's teacher is _______ years old.

This was one of the easiest problems on the assessment I gave the 2nd and 3rd graders who were trying out for the new after-school math team.

Answers among these mathematically inclined students varied alarmingly widely. The most common wrong answer was 27, but there were others.

Many students who got this problem wrong were able to do harder, multi-digit calculations later on in the assessment--harder calculations than what our school's Investigations curriculum offers them. From this, I can't help inferring that they're somehow finding extra-curricular opportunities to calculate.

And whatever, or whoever, the outside influence might be seems to be doing a better job than Investigations does.

For word problems--not hard calculations--are the pride and joy of the Investigations curriculum. 

And even if some of the wrong answers resulted from sloppy reading or poor reading comprehension, Investigations, which cares so much about language arts, should surely have addressed this by now.

Instead: after two or three years of this curriculum even many of the more mathematically inclined of the 2nd and 3rd graders of the highly educated parents that predominate at our school are unable to do a simple word problem involving 27, 9, and a comparison of two different ages.

Stacking, regrouping, and corrupting the children

At this fall's first "parent education workshop," our school's math consultant told us that we shouldn't show our children how to add numbers by "stacking" them one on top of another. Children don't understand how stacking and regrouping work, and would be better off devising their own solutions.

The many skeptics in this record-sized crowd were given no opportunity to ask questions. A few new converts, though, had chances to confess to being "stackers," and their humble admissions suggested hope that the sins of the fathers won't be visited upon their children.

Not if I can help it. This coming week, during our second math team practice, my co-coach and I will be showing the most mathematically inclined and advanced quartile of the second and third grade classes how to stack and regroup. Not only that, but we'll be picking this forbidden mathematical fruit from a particularly controversial Tree of Knowledge.

Our kids seem eager to be corrupted. The first problem set we sent home, plucked straight off this tree, was--if parent reports are accurate--devoured with reckless glee, as these 8 and 9-year-olds threw off the mathematical abstinence the higher powers in education have foisted on them.

Math problems of the week: grade 3 Trailblazers vs. Singapore Math

1. From 3rd grade Math Trailblazers, end of the fractions unit:

Look for fractions at home and in your neighborhood.  You might look in the newspaper (especially in the ads) or in magazines, in cookbooks, in the mail, or on signs.  Try to find at least six fractions.

Write about each fraction you find.  Tell what the whole is, and try to draw a picture that shows the whole and the fractions.

2. From the 3rd grade Singapore Primary Mathematics (3B) end of fractions unit:

Polly has 12 ribbons.  1/3 of the ribbons are red.  The rest are yellow.

(a) What fraction of her ribbons are yellow?

(b) How many ribbons are yellow?

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Extra credit:

What fraction of each assignment involves math?

Everything but the math curriculum

In the last week, I've seen two articles expressing concern about the nation's school children:  one about boys, and the other about girls. Both leave out one of the biggest underlying factors, namely, the grade school math curriculum.

First there's an article in Teacher Magazine about Peg Tyre's new book, The Trouble with Boys, which, quoting Teacher Magazine, "details the problems boys are facing in school and argues for a new, boy-focused “gender revolution.” As far as the curriculum's role in all this goes, Tyre faults it for being too narrow and test-focused.

Then there's an article in the New York Times about a forthcoming article in the Notices of the American Mathematical Society about how few American-born girls are competing in top math competitions like the Math Olympiad and the Putnam Math Competition. This article makes no mention of our country's new Reform Math, instead blaming the numbers on anti-nerd biases that especially ostracize female math students.

Once again, people are happy to blame everything but the math curriculum. But the boys I know who are languishing in school are doing so because of Reform Math's dumbed down math and emphasis on language arts.  

This may also explain the disproportionately small numbers of American girls--which is accompanied, though outnumbered, by a disproportionately small number of American boys--performing well in the top math competitions. Perhaps, outside of school, gender stereotypes still have people identifying and encouraging boy math buffs more than their female counterparts.  

And, as America's math classrooms dumb themselves down under the Reform Revolution, what happens outside of school--including ongoing sexist assumptions about math ability--wields an ever greater influence over who does well in math.

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

1. From one month into the 2nd grade Investigations curriculum, Coins, Coupons, and Combinations, "What Will You Need" worksheet:

1. Complete each sentence.

1. If I have 4, I will need ____ to make 10.

2. If I have 6, I will need ____ to make 10.

3. If I have 3, I will need ____ to make 10.

4. If I have 7, I will need ____ to make 10.

5. If I have 2, I will need ____ to make 10.

6. If I have 8, I will need ____ to make 10.

7. If I have 1, I will need ____ to make 10.

8. If I have 9, I will need ____ to make 10.

9. If I have 5, I will need ____ to make 10.

10. If I have 0, I will need ____ to make 10.

2. From one month into the 2nd grade Singapore Math curriculum: Primary Mathematics 2a, p. 40:

1. Add:

   648         436    700

+ 201 + 231 + 135

   540    625    213

+ 249 + 173 + 153

   107    445    657

+ 381 + 124 + 330

3. Extra Credit:

A generation from now, which country will be most able to extricate itself from a financial crisis?

Latin redux: A response to Constructivist language teaching?

Today's New York Times reports on the rising numbers of students enrolling in Latin classes.  

The resurgence of a language once rejected as outdated and irrelevant is reflected across the country as Latin is embraced by a new generation of students like Xavier who seek to increase SAT scores or stand out from their friends, or simply harbor a fascination for the ancient language after reading Harry Potter’s Latin-based chanting spells.

Coming at a time when increasing numbers of French, Spanish, and German classes are adapting Constructivist teaching methods (implicit learning through group conversations, skits, and interdisciplinary projects), the surge of Latin enrollment may have yet another partial cause.

Latin, after all, doesn't lend itself as naturally as living languages do to group conversations about high school social dynamics, or student-made travel guides and restaurant menus, or the marginalization of grammatical structure. And, while there's perhaps nothing preventing a Latin teacher from having students "decorate a tissue box with Latin vocabulary words," such assignments may not appeal to the kind of person who chooses to teach Latin in the first place.

For students and teachers who care about foreign language grammar, and who prefer explicit instruction to skits and crafts, Latin is the perfect antidote to current practices.

So you want to do a science experiment? Knock yourself out!

But wouldn't you rather write a personal reflection about science instead?

Every week J. has to do 15 points worth of science homework.  A "response sheet" gets you 5 points (this week's:  a "note to Josh" about your ideas about stream tables); a personal reflection about a science article earns you 7 points.  

But if you really want to do science, you can do the science experiment.  This week's involves the following:

Set up a scene with about 2 inches (5 cm) of sand or soil in the box.

Build and/or add one type of barrier, in the middle of the box.

Turn on the fan and let it run for five to ten minutes, depending upon the action you observe.

Photograph the changes in your soil or sand. Be sure to notice what happens both in front of and behind the barrier.

Repeat with several different barriers in the wind path. 

Building barriers; acquiring sand and soil; pouring it into boxes; blowing it; taking pictures; repeat?  Hmm... maybe we should do the personal reflection instead.

Except that we saw how J's teachers feel about his personal reflections....

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

1. From the end of the first unit of 6th grade Connected Mathematic Prime Time

A group of students designs card displays for football games. They use 100 square cards for each display. Each card contains part of a picture of a message. At the game, 100 volunteers hold up the cards to form a complete picture. The students have found that the pictures are most effective if the volunteers sit in a rectangular arrangement. What seating arrangements are possible? Which would you choose? Why?


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2. From the end of the first unit of 6th grade Singapore Math Primary Mathematics 6A

Simplify the following expressions:

12 + 8h – 6h =

9a + 1 -3a =

7 + 4k – 2 – 2k =

15x + 8 – 10x -3 =

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Extra Credit

Estimate the ratio of linguistic complexity to mathematical complexity in each problem.

To quote Connected Math, which would you choose, and why?

Autistic personal connections don't pass muster

Besides the narcissism of the "All About Me" unit, we have the narcissism of the "personal connections" paragraph that students are supposed to write after each night's reading.

Again, only neurotypical connections count.  My son's "personal connections" (about Harry Potter, Chamber of Secrets) earned him a D:

I never drink polyjuice potion before. I never do magic before. I liked that chapter because Harry and Ron realized that Draco Malfoy is not the heir of Slytherin.

His teacher's comment:  "Maybe J should find a book that he can make connections with to make the reflection section more meaningful."

Lurking within this neurotypical self-love, it seems to me, is a kind of subtle hatred of autism.