Friday, July 31, 2009

Summer reading project, 3rd grade

Have been procrastinating all month on this one, but soon we'll have to bit the bullet:

In How to be Cool in Third Grade, Robbie, the main character, discovered that the best way to be "cool" in third grade is not by wearing jeans or by avoiding his mom's kisses at the bus stop. Rather, he learned to be cool by letting his best qualities (being smart, nice, and funny) shine through. Create a poster that in some way shows your three best qualities. Be as creative and colorful as possible! On the poster, please also write a few sentences that describe how your three best qualities will be helpful in your third grade classroom this year.
What if a child's best qualities don't include postering and visual creativity?

In particular, how well do the project's self-esteem promoting strategies serve this child?

Wednesday, July 29, 2009

Programming, Grammar, and Left-Brained Obsessions

To left-brainers like myself, few things are more absorbing than grammar rules and computer programming.

Combine them into a grammar software program, and throw in a deadline, and the absorption becomes an all-encompassing obsessive compulsion that, every so often, led to a kind of left-brained Nirvana.

For the last two weeks, my blog-posting frequency (not to mention my sleeping, eating, and parenting hours) hit record lows as I re-vamped, re-structured, and re-programmed the GrammarTrainer.

Among the results are the following new demos, which I invite you (or your children) to enjoy at your leisure:

Level I
Level II
Level III
Level IV

Monday, July 27, 2009

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

From two of the most related problem sets on graphing:

1. From a problem set entitled "The End of a Function," from a subsection entitled "Going to the Limit," from a chapter entitled "The World of Functions," in Interactive Mathematics Program Integrated High School Mathematics, Year 4, p. 289

1. Look at examples of polynomials of different degrees. In each case, try to figure out what happens to the y-value as x increases in absolute value. (The result may depend on whether x is positive or negative.)

A graphing calculator can help you to some extent, but the graph that a calculator shows can only give you part of the picture. Thus, you should come up with algebraic or numerical explanations for your conclusions about the end behavior of your examples.

2. Next, look at functions from other families: exponential functions functions from the sine family, rational functions, and any other families you want to consider. Again, take notes on your results and any general conjectures that you make.

2. From the problem set entitled "Exercise 70A," from a subset entitled "The solution of Quadratic Equations by Means of Graphs," from a chapter entitled "Quadratic Equations; Radical Equation," in A Second Course in Algebra (first published in 1937), p. 253.

Solve the following equations graphically and find the values of x which determine maximum and minimum points on the curve.

1. x3 + 3x2 - 15 = 0
2. x3 - 7x2 + 16x -8 = 0
3. x3 - 12x -12 = 0
4. x3 + x2 - 1 = 0

3. Extra Credit

Compare the grades that the underachieving math buff is likely to receive on each assignment.

Wednesday, July 22, 2009

Autism Diaries XIII: A handbook of practical jokes

When autism reduces your exposure to pop culture, you have to reinvent some common pranks yourself. I'm no longer sure which of these are common, and which are original to J, but my list has grown long enough for a post:

1. Lean a glass against the inside of its cabinet door so that when someone opens the cabinet it comes crashing down.

2. Wherever two switches control one light, position each switch in a position halfway between "on" and "off" so that nothing happens when people flick the other switch.

3. Loosen everything on your mother's bike so that it just holds together until she takes it out the door and tries to ride it.

4. Loosen cabinet handles so they come off when people pull them.

5. Unscrew cabinet handles and reassemble them on the other side of the cabinet door.

6. Dig a narrow hole in the yard, insert the hose nozzle into it, and turn the water on full blast.

Then there are the more technologically complex pranks that probably aren't common:

7. Reprogram the remote that controls your bedroom fan and light to have the same code as the kitchen fan so that you can control the kitchen fan and light from your bedroom.

8. Program the phone to block calls from certain familiar individuals.

9. Program the phone alarm (who knew it had one?) to go off at 3:15 in the morning.

10. Hack, hack, hack.

The only pranks of J's I know for sure he didn't invent himself are those he got from Captain Underpants, his favorite being Ketchup packs under the toilet seat.

Monday, July 20, 2009

Math problem of the week: 3rd Grade Everyday Math vs. Singapore Math

***Various delays have shifted OILF's problem of the week to Monday!***

1. One of the first dollars and cents problem in 3rd grade Everyday Math, Student Math Journal Volume 2
(p. 244):

1. Enter $3.58 into your calculator. The display shows ________

2. Enter the following amounts into your calculator.
Record what the display shows.
Don't forget to clear between each entry.

Price Display
$2.75 _____
$1.69 _____
$12.32 _____

Make up prices that are more than $1.00
_____ _____
_____ _____
_____ _____

3. Enter 68 cents into your calculator. The display shows _____

4. Enter the following amounts into your calculator.
Record what you see in the display.

Price Display
$0.10 _____
$0.26 _____
$0.09 _____

Make up prices that are less than $1.00.
_____ _____
_____ _____
_____ _____

5. Use your calculator to add $1.55 and $0.25

What does the display show? ____

Explain what happened. ___________________________________
_____________________________________________________
_____________________________________________________

2. From the first dollars and cents problem set in 3rd grade Singapore Math, Primary Mathematics Workbook, 3B (p. 68):

Write each amount of money in figures:

(a) five dollars and sixty-five cents
(b) ten dollars and eight cents
...
(e) three hundred twenty dollars and four cents
(f) one thousand thirty dollars

Write each amount of money in figures:

(a) $0.80
(b) $1. 36
...
(g) $44.55
..
(i) $37.05

3. Extra Credit

a. Which involves more math: translation between numbers and calculator displays, or translation between numbers and words?

b. Does the age of calculators revise these priorities?

Wednesday, July 15, 2009

Self-determination in math: who benefits?

Today I learned from a friend that her son B didn't make it into the highest-level math class at his new high school, despite being by far the strongest math student in his middle school (as measured by grades and standardized test scores).

My friend faults B's middle school math teacher, who (to her credit) recognized that the Interactive Math Project curriculum was way too easy for B. Her solution? Let B do whatever he wanted to during math class. So B mostly goofed off or read novels.

There are a number of education experts out there (for examples on line, see Bruce Smith at education@change.org) who feel that kids learn best when determining their own course of study. In fact there's an entire education model, the Sudbury Schools, constructed around this premise.

Self-determination may work well for the most diligent, driven kids. But most young students, however good at math, require specific assignments and incentives.

In letting them study what they want to, how many B's are we letting through the cracks?

And, however much they currently enjoy their academic freedom, how will they feel once they face competition from those who had, at the very least, a challenging textbook to work through and strong classmates with whom to compete?

Here's an exchange I had with Bruce Smith about how well Sudbury Schools train children in math:

KB:
Do they learn how to derive the Quadratic Formula, graph rotated ellipses, and calculate the area under a curve? In other words, are they able to jump into college-level calculus upon graduation? Please tell more! I'd love to hear more about the math curriculum in particular.

BS:
Sudbury students can do absolutely as much math as they like. In fact, they can even go on to get a PhD from MIT and become chair of the Math Department at the University of Oregon (to cite the career of one graduate). Of course most won't go that far, but then, most don't need to.

As for "the math curriculum," there's no particular sequence any Sudbury student must follow. All areas of knowledge are open to study using whatever methods work best for each student, who may seek help and consult experts when they wish.

When the curriculum is life itself, and finding one's place in it, the Sudbury experience is that people will learn (a) all they need to be successful in life and (b) the depths and nuances of subjects for which they have a personal passion.

The fact that he doesn't answer my question (about content), but a different question (about preference), doesn't exactly disabuse me of my concerns.

Monday, July 13, 2009

Slower but more aware: deficits in gut instinct

For those of us with below-average right-brain instincts, there is a silver lining.

This was brought home to me, yet again, these past few weeks as I taught my "Autism, Language, and Reasoning" course and discussed Uta Frith's Autism: Explaining the Enigma, in which she discusses Mike Anderson's model of human intelligence.

Citing Frith:

In Anderson's model the cognitive system works like a small company with dedicated specialists and a responsible head office. When one of the specialists is ill, one of the company directors steps in but, having no special talent or training, will do the job less well.
The company director takes over, for example, if the public relations specialist is ill.

But who is this company director? I'm guessing it's our conscious selves, who must work out logically, via our basic information processing mechanisms, which social rules pertain in a given situation.

While these conscious selves do the job less well than the dedicated specialist, we get much better with practice, and also gain conscious access, in the process, to many intriguing rules of human behavior to which our more sociable counterparts, while slavishly following them, are completely oblivious.

Saturday, July 11, 2009

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

1. The last three word problems (of 10 in all) from Chapter 2 of A Second Course in Algebra, (originally published in 1937), p. 84-5

Two boys took a trip into the country on a bus which average 18 miles per hour. They walked home over the same route at the rate of 4 miles per hour. The entire trip took 3 hours and 40 minutes. How far did they go?

A boy can row 15 miles upstream and back in 8 hours. His rate upstream is 3/5 of his rate downstream. Find his rate downstream.

A train traveled 180 miles from P to Q. Forty-five miles from Q it met with an accident which delayed it 30 minutes. It then continued at half its usual speed and arrived at Q two hours late. What is its usual speed?

2. The only three word problems in the most closely matching problem set in Interactive Mathematics Program, Year 4 (Chapter 4 ), p. 287

An Average Drive

Amparo took a trip on a warm, sunny day to her favorite spot overlooking the ocean. That spot was 100 miles from home. Amparo knew that in order to be home on time, she needed to average 50 miles per hours for her overall trip.

1. As Amparo reached her destination, she realized that she had made good time while still staying within the legal speed limit. She figured out that she had averaged 60 miles per hour on her trip there. What should Amparo's average speed be on the return trip so that her average speed for the round trip will be 50 miles per hour.
(Warning: The answer is not 40 miles per hour!)

2. On another trip to the same spot, Amparo took a more leisurely pace. When she reached her destination, she realized she was late! She had averaged only 25 miles per hour getting there. What should Amparo's average speed be on the return trip so that her average speed for the round trip will be 50 miles per hour?
(Warning: The answer is not 75 miles per hour!)

3. Amparo makes this round trip frequently, and her goal is to average 50 miles per hour for the round trip. However, her average speed for the trip to the ocean varies.

Let x be her average speed (in miles per hour) on the way to the ocean. Find an expression in terms of x that tells Amparo what her average speed should be on the return trip for her average speed for the whole trip to be 50 miles per hour.

3. Extra Credit
After they successfully complete the second problem set, how likely do you think it is that Interactive Math Program students will be able to do the first problem set without hints?

Thursday, July 9, 2009

Autism Diaries XII: Pragmatics and Perspective Taking

Even if no other aspect of language is impaired in autism, there's pragmatics, or the ability to hold "normal" conversations. Knowing how much information to supply, for example, is inextricably caught up in perspective taking (a.k.a. Theory of Mind)--one of the core deficits of autism.

But J has grown so adept at figuring out how much information to supply that I'm wondering if the process has become nearly as automatic is it is for us "neurotypicals."

For example, in talking about someone's home with his younger sister (in the course of teaching her how not to get checkmated in four moves), he spontaneously (and consistently) referred the house not as H's house (as he does when discussing it with me), but as L's house. H is my adult friend; L is her son--a little boy with whom J's sister often has play dates. It seemed as if J had calculated that his sister would think of this house as L's house rather than as H's house.

Here's a more complicated example:

Ever since Jonah completed GrammarTrainer Level III, he's understood "cleft sentences," or sentences of the form "It was Daddy who did that." But GrammarTrainer is not PragmaticsTrainer (stay tuned!), and so he never learned when it might be appropriate to use a cleft structure. Therefore, since they're more complex than their unclefted counterparts, he hasn't bothered actually using them.

Until yesterday. After he slammed shut a door that his father had left open for both of us, and after I'd had to let myself in with my key, he said to me (a twinkle in his eye): "It was Daddy who closed the door."

His pragmatics were perfect. That someone had closed the door was obvious--indeed, it was surely the event his mother was brooding about as she irritably fumbled for her keys. Who had closed the door, on the other hand, wasn't so clear (well... unless you know J), and was perhaps the question his mother was asking herself. The cleft structure's function is to embed old information (that someone closed the door), and to highlight new information (that it was Daddy), and J somehow knew to do just that.

Along with many other people, I've often suspected that high functioning autistic individuals are able to function socially by working out logically what the rest of us do by gut instinct. But now I'm wondering whether, if you practice it repeatedly enough, some of this logic can become gut instinct--just like the algorithms of arithmetic do (at least for the lucky few who still have the opportunity to practice them sufficiently).

Certainly J, who has made practical jokes his life's mission, has more motivation than many people do to practice working out--over and over and over again--the complex logic of perspective taking.

It will serve him well.

Tuesday, July 7, 2009

Debunking Constructivism: What's bad for the goose is bad for the gander

Every once in a while, an empirical study comes along that suggests that one or another of those Constructivist practices that shortchange left-brainers (or so I claim) may, in fact, be bad for students in general.

Dan Willingham's book Why Don't Students Like School presents a whole bunch of these experimental results. Together, they challenge the notions that:

1. Students need to learn inquiry, argumentation, and higher-level thinking rather than lots of facts.
2. Integrating art into other subjects enhances learning. So does integrating computer technology.
3. Children learn best through self-guided discovery.
4. Drill is kill. Multiple strategies are better than a single strategy practiced multiple times.
5. Students learn best when constructing their own knowledge.
6. The best way to prepare students to become scientists and mathematicians is to teach them to solve problems the way scientists and mathematicians do.

The experiments that Willingham cites show that, in fact:

1. "Factual knowledge must precede skill"

"Data from the last thirty years lead to a conclusion that is not scientifically challengeable: thinking well requires knowing facts... The very processes that teachers care about the most--critical thinking processes such as reasoning and problem solving--are intimately intertwined with factual knowledge that is stored in long term memory..."

2. Teachers "must design lessons that will ensure that students are thinking about the meaning of the material."

Here, Willingham cites two examples where this failed to happen. In one case, his nephew was asked to draw pictures to represent the plot elements in the book he'd read, which "meant that my mephew thought very little about the relation between different plot elements and a great deal about how to draw a good castle."

In another case, groups of high school students were allowed to use computers for their research projects, and quickly got caught up in Power Point. "The problem," Willingham writes, "was that the students changed the assignment from 'learn about the Spanish Civil War' to 'learn esoteric features of PowerPoint'" like animations, videos, and unusual fonts.


3. "Use Discovery Learning with Care."


"If students are left to explore ideas on their own, they may well explore mental paths that are not profitable. If memory is the residue of thought, then students will remember incorrect 'discoveries' as much as they will remember the correct ones.
"

Willingham suggests that the ideal environment for discovery learning is when feedback is immediate, as when you are learning how to use a computer:

When you make a mistake, it is immediately obvious. The computer does something other than what you intended. This makes for a wonderful environment in which 'messing around' can pay off. (Other environments aren't like that. Imagine a student left to 'mess around' with frog dissection in a biology class.)
4. "It is virtually impossible to become proficient at a mental task without extended practice."

In particular, extended practice helps students extend and abstract concepts in ways that don't otherwise come easily. "Working lots of problems of a particular type makes it more likely that you will recognize the underlying structure of the problem, even if you haven't seen this particular version of the problem before."

5. "Students are ready to comprehend but not create knowledge."

"Experts create... Experts not only understand their field, they also add new knowledge to it... A more modest and realistic goal for students is knowledge comprehension."


6. "Cognition early in training is fundamentally different from cognition late in training."

Willingham observes that the best way to train a novice to eventually become an expert is knowledge comprehension and extended practice.


----
Now we just need the education experts to drop fact-free inquiry and argumentation and pay attention to Willingham's results.

Sunday, July 5, 2009

Summer math projects: 2009

Please share your summer math projects here!

Here's ours--for incoming 7th graders:

A Mathematical Scavenger Hunt at the Library

Go the the city library--either the main library or a branch library. Do the following activities and record all of this information in an attractive booklet or on a poster. Plan ahead since you may need more than one visit to do everything on the list

1. Draw a sketch of the front of the library on 8.5 X 11 paper. Show the windows and doors. Estimate the width and height of the building and show these dimensions on your sketch. Explain the strategy you used to make your estimate.

2. Go to a room in the library. Make a sketch of the floor plan of the room. Estimate the length and width of the room. What is your estimate of the area of the room? Explain the strategy you used to make your estimate

3. Find a section of the books that you like. Write down the types of books you chose. Place your forearm along the shelf and count how many books there are from the tip o your elbow to the tip of your fingers.

4. Estimate the number of books in this room. Explain what strategy you used to come up with your estimate.

5. Find a chart showing the Dewey decimal numbers for the categories of books in the library. Copy the information to the chart.

6. If you do not already have one, sign up for a library card.

7. Check out a non-fiction book that you would like to read. List its title, author, and Dewey decimal number.

Bring the project to school on the first day. Your teacher will use the data you have collected for class.

------

Just reading through this list exhausts me. From it emerges another, no less exhausting, list:

1. Multiple trips with my autistic son to the library--is there any other public place in the universe where it's more important that I somehow prevent him from being disruptive?

2. Yet more arts and crafts tasks required of someone with no motivation for such activities, and who must be prodded step-by-step through each and every one.

3. Yet more verbal explanations required of someone who not only struggles to verbalize his thinking, but does math in his head without verbalizing it.

4. Yet more watered down math for someone who's been measuring and estimating for years, and who will find nothing intriguing about Dewey decimals and books per forearm.

5. Best case scenario: five hours of my time, and J's, under maximum stress and tedium for both of us. How many other far more enjoyable ways might there be to spend this time--mathematically--with my math-loving boy?

-----

I'm not alone in these gripes; even parents of neurotypical children are moaning over this assignment. How I wish teachers could overhear our conversations.

And how I wish I could assign teachers the following summer project:

Consult with the parents of your students before assigning a summer project to us--for who, all too often, ends up doing most of the work? Then, in the words of the Scavenger Hunt project above, "use the data you have collected."

Friday, July 3, 2009

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

Preliminary division word problems:

1. 3rd Grade Investigations Landmarks in the Hundreds, p. 39
Share a Dollar

Solve the following problem about splitting up a dollar.

Ten third grade students were playing out at recess, and they found $1.00. No one claimed the $1.00, and their teacher told them that if they could figure out a fair way to split up the money, they could keep it. How did they split it up? How did you get your answer?

Now write your own problem about splitting up a dollar.

On the back of this sheet, solve your own problem. Be prepared to share your problem with some of the students in your class.

2. Primary Mathematics 3B, Multiplication and Division, p. 79-80

5 children share 40 cherries equally. How many cherries does each child get?

Mary saves $4 a week. How many weeks will she take to save $24?

Matthew earned $45 in 5 days. He earned the same amount each day. How much did he earn a day?

3. Extra Credit

Which problem set would your 3rd grader rather do?

Wednesday, July 1, 2009

Autism Diaries XI: furthering backfirement

Last night, for the first time in his life, J laughed himself to tears.

This milestone occurred after I got so fed up with J's attempts to raise the temperature in our kitchen that I decided to reveal to him how we'd been deceiving him, for years, about the kitchen ceiling fan.

The back story: years ago, tired of him constantly turning the fan on fast, we'd arranged for his aunt to send him an email message, ostensibly from the "fan company," warning owners that the fans would break down if turned on when the temperature was too low. To maintain proper fan health, the email went on, owners shouldn't turn on their fans until the surrounding temperature reached 78 degrees, and shouldn't turn them on fast below 85 degrees.

Ranking fan longevity over fan-on-fast, and not yet email-savvy enough to recognize email fraud (let alone perpetrate it himself), J obediently monitored the kitchen thermostat and kept the fan speed--or lack thereof--within proper parameters. And he continued to do so for years.

But that didn't stop him from using all means necessary to raise the kitchen temperature as high as possible. In summer, this has meant nightly struggles over the kitchen windows, which J is constantly conniving to open in the heat of the day and to close in the cool of the evening. Finally yesterday, amid this summer's first heat wave, I'd had enough.

He took the news well. In fact, already familiar with the term "backfire," he reveled in all the ways our attempts to save energy and keep the kitchen comfortable had done just this--to the point of laughing himself to tears. His only consternation came when I pointed out that he'd laughed himself to tears, whereupon I quickly explained that tears don't always mean unhappiness, but sometimes the exact opposite (for what better feeling in the universe is there than side-splitting mirth?).

To allow him to further savor the moment, I encouraged him to call up his impostering aunt and tell her what he'd learned. Only then did I hear the full litany of strategies he'd used to raise the kitchen temperature: not just opening and closing windows, making up excuses to cook things in the oven rather than the microwave, and opening the oven door whenever the oven was on and we weren't looking, but also turning up the thermostat, setting the dishwasher to "extra-heated dry," and turning on the exhaust fan to draw in heat.

In pretending to be the fan company his aunt, he informed her, had "furthered backfirement."

In this moment of parenting defeat, I, too revel. I revel not just in my son's mirth, but in three things that many autism experts say that children with autism can't do: understand, appreciate, and participate in pretense; understand metaphor (this was no literal backfiring); and manipulate language creatively, grammatically, and pragmatically appropriately--yielding a playful bureaucratese entirely of his own making.