Monday, November 16, 2009

Everyday Math author defends his program against Katharine Beals

In today's Philadelphia Inquirer Letters to the Editor, excerpted here:

Katharine Beals' article on the use of "reform math" with students with autism contains many misperceptions about Everyday Mathematics that, as the program's coauthor, I want to clarify ("The 'reform math' problem," last Monday).

Everyday Mathematics was designed for general education students, but it has been effective in special education, including with students with autism.

Beals' claim that students spend large chunks of time working in unsupervised groups is untrue. A teacher supervises student group work at all times. While some assignments are "open-ended and language-intensive," many are not. A balanced curriculum needs simple exercises to build basic skills, as well as more difficult problems.

Beals writes that students "lose points for failing to cooperate in groups, explain their answers, and comprehend language-intensive problems." While decisions about how to grade students are made at the local level, many people believe it's reasonable to require students to work cooperatively, explain their work, and understand word problems.

Everyday Mathematics is not just a "sequence of themes," but a carefully organized sequence of lessons resulting in mastery of a specific set of goals. Its approach is well supported by research, the authors' experience, and decades of classroom experience.

Naturally, accommodations for teaching children with autism must be made, and that's what professionals always do. As with any tool, Everyday Mathematics must be used with professional judgment.

Andy Isaacs

Chicago

Saturday, November 14, 2009

Dysgraphia, dysteachia, dystopia

For some of the prototypically left-brain children I write about in my book, penmanship problems are common. They are worsened by the dearth of penmanship instruction in today's schools. One can ask the same thing about dysgraphia as more and more people are asking about dyslexia: how much of this is merely dysteachia?

Just as dyslexia (or dys-phonics-teachia) ultimately impedes higher-level reading comprehension, so does dysgraphia (or dys-penmanship-teachia) ultimately impede higher level writing. In struggling hands, ideas quickly bottleneck, choking off fluency.

Precisely this kind of writer's block is plaguing a gifted third grader I know. So his mom had him evaluated by an occupational therapist, who confirmed "dysgraphia." Mom brought this diagnosis to the school and asked penmanship tutoring. The answer? "No."

As it turns out, our school district (5th largest in the country) is not obliged to provide support for penmanship instruction... because penmanship isn't part of its official curriculum.

This, despite the fact that, in their many hand-written projects, students are routinely marked off for deficient "neatness."

Thursday, November 12, 2009

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

I. A sampling of problems from the 2nd grade Everyday Math Student Math Journal, Volume I, "Addition and Subtraction Facts," pp. 20-50.

Use > , <, or =.

6 + 7 ___ 15 - 4
5 + 8 ___ 8 + 5
18 - 9 ___ 5 + 4

-------------

Today is ________________
(month) (day) (year)

The date 1 week from today will be ____________

-------------

Use a number grid.

How many spaces from: 17 to 26? 49 to 28?

-------------

Which is heavier: 1 ounce or 1 pound?

-------------

Write an addition story.

-------------

Play Broken Calculator.
Show 17. Broken key is 2.
Show three ways.

-------------

Draw a rectangle around the digit in the tens place

349
406

-------------

Follow the rule. Fill in the missing numbers.

Rule: + 6

___in___out___
___2____8____
___3 ____9____
___5 _________
___9__________

-------------

Subtract. Use the -9 and -8 shortcuts.

13-9 = ___
14 - 8 = ___
...

II. A sampling of problems from the 2nd grade Singapore Math Primary Mathematics Workbook, Volume I, "Addition and Subtraction," pp. 31-68.

-------------

Compare two sets.
[squared-off picture of 11 flowers, labeled "A," next to squared off picture of 6 flowers, labeled "B"]

11 - 6 = ___
Set A has ___ more flowers than Set B.

-------------

Add

3 + 4 =
30 + 40 =
300 + 400 =

-------------

Subtract
7 - 3 = ___
70 - 30 = ___
700 - 300 = ___

-------------

Subtract.

689
- 32

786
- 73

-------------

Add
7 + 6 =
27 + 6 =
527 + 6 =

-------------

Add

264
+ 36

486
+ 54
...

-------------

A watch costs $167.
A camera costs $48 more than the watch.
What is the cost of the camera?
What is the total cost of the camera and the watch?

The total cost of the camera is $ ____
The total cost of the camera and the watch is $ _____

-------------

Add or subtract.

251
-170

358
+416

-------------

David collected 930 stamps.
He had 845 stamps left over after giving some stamps to his friends.
How many stamps did he give to his friends?


III. Extra Credit:

Discuss the phrase "a mile wide and an inch deep."

Everyday Math tells people to "trust the spiral." Do you?

Tuesday, November 10, 2009

Op-Ed in yesterday's Philadelphia Inquirer

Here.

Be sure to check out the comments that appear below the article.

For all the talking points that Reform Math proponents deploy in response to the general criticisms, I haven't yet seen any talking points that respond to concerns about children on the autistic spectrum. Has anyone else?

Since it's well-documented--and generally agreed--that AS children require structure, direct instruction, and discrete tasks, and that many of them have the potential to excel in math, and since the education establishment's purported missions include (1) mainstreaming and (2) catering to different learning needs, I believe this is a fruitful message to keep plugging.

Monday, November 9, 2009

Gifted Exchange

Laura Vankerkam of Gifted Exchange has honored me with an interview here.

She's also got a great piece on the BASIS Schools in Arizona. As Vanderkam notes, "the schools explicitly model their curricula on the best practices exhibited in other countries that routinely trounce the US in international comparisons."

Sunday, November 8, 2009

Mitosis, rote memorization, and the unchanging traditions of grade school biology

Before mitosis begins, the chromosomes and other cell materials are copied. [are copied? Who or what does the copying???] The pairs of centrioles, which are two cylindrical structures, are also copied. [Besides being cylindrical, what is a centriole, and what is its significance for mitosis???] Each chromosome now consists of two chromatids. [Remind us what a chromatid is!!!]

Mitosis Phase 1
Mitosis begins. The nuclear membrane brakes apart. [Why?] Chromosomes condense into rodlike structures. [Why is the new, rodlike structure important and significant?] The two pairs of centrioles move to opposite sides of the cell. [Significance?] Fibers form between the two pairs of centrioles and attach to the centromeres. [Remind us what a centromere is and why it is significant!]

Mitosis Phase 2
The chromosomes line up along the equator of the cell. [How??? and Why???]

Mitosis Phase 3
The chromatids separate [How?] and are pulled to opposite sides of the cell by the fibers attached to the centrioles. [This crucial event should be the centerpiece of the whole discussion of mitosis].

Mitosis Phase 4
The nuclear membrane forms around the two sets of chromosomes, and they unwind. The fibers disappear. Mitosis is complete.
From Cells, Heredity, and Classification (Holt, Rinehart and Winston), with my queries in brackets.

With all the questions it begs and explanations it lacks, this is little more than a list of terms and series of steps to memorize, with no obvious general concepts to guide or interest you. This approach seems to have a long history. It includes my own biology book of a generation ago, which is why I never pursued biology after 9th grade.

But now that my autistic son is studying it in middle school, I need to understand it better.

Only after multiple readings of the passage above did I sort of figure out what the underlying concepts were. (Perhaps if I were a more visual thinker, it wouldn't have taken me so long.)

Assuming that I'm more or less on target, it strikes that a more engaging introduction to mitosis might go somewhere along these lines (ideally generated by some sort of Socratic dialog, with accompanying illustrations):

We already know that cells consist of crucial elements, for example, the mitochondria and the chromosomes. We also know that for organisms to grow, their cells must divide. But is cell division as simple as a cell dividing itself into two? Bear in mind that each "half" of the cell must have all the crucial elements. This means that each element must be copied, and each half must end up with one copy of the element.

Making sure that each "cell half" has exactly one copy of a given element is particularly complicated when it comes to the chromosomes. Is it enough for each chromosome to make a copy of itself? Imagine what would happen if the chromosome copies simply swam around in the cytoplasm while the cell divides. Then what's to stop one half from ending up with two copies of chromosome 1 and no copies of chromosome 2, or vice versa? We already know how each chromosome contains different sets of crucial instructions for the cell, so the results of this kind of lopsided split would be disastrous.

So how can a cell make sure that exactly one copy of each of its dozen or more chromosomes ends up in each "cell half" before the division? Since the cell has no "brain" or other centralized information processor, as soon as the chromosome copy separates from its original, there's no way for the cell to "know" which copy goes with which original, and therefore no way to guarantee that each cell half gets exactly the right number of copies.

Well, suppose each chromosome copy remains attached to the original up until right before the cell divides. This preserves the information about which copy matches up with which original. Then suppose the chromosome pairs (original plus copy) all line up in such away that a simultaneous, symmetrical force emanating from each cell half can pull them apart, so that each original copy ends up in one half while its copy ends up in the other half.

Let's picture how this could happen. Imagine if the chromosome pairs line up along the equator of the cell, with one pair member on each side of the equator. Now imagine tentacles reaching out from the middle of the edge of each cell half and pulling at each chromosome pair from each side. If these tentacles are equally strong, and strong enough to separate the chromosome pairs, the result is just what we want: exactly one copy of each chromosome pulled into each cell half.

Friday, November 6, 2009

Math problem of the week: 5th Grade Trailblazers vs. Singapore Math

I. The first place value/multiplication problems in Math Trailblazers Student Guide 5, pp. 48-49:

Reach for the Stars

Mr Moreno's class is about to begin a unit on the solar system. Irma, Alexis, and Nila thought it would be fun to decorate the classroom. Mr. Moreno allowed them to stay after school to work on this project.

[Illustration of three girls in front of a blackboard and the following cartoon-bubble dialog]

Irma: Let's make a banner of stars.

Alexis: Great idea. We can make a banner with 2 rows of 30 stars.

Nila: Then how many stars do we need to cut out?

Irma: Since 2 × 3 is 6, I know 2 × 30 is 60.

1. A. Explain in your own words how Irma solved 2 × 30 = 60.
B. How would you solve 2 × 30 = 60? Explain your method to a friend.

[Illustration of the three girls in front of a blackboard that now has two long rows of stars on it, and the following cartoon-bubble dialog]

Nila: How about putting stars on the ceiling? Maybe we could get a parent to help us put them up?

Irma: First we need to know how many stars we need. Let's put a star on each ceiling tile. I counted 20 tiles wide and 30 tiles across. How many tiles is that?

Alexis: Looks like we have to multiply by numbers ending with zeros again!

2. Irma learned to look for patterns when multiplying numbers that end in zeros. Find the following products. Use a calculator if needed. Describe the patterns you see.

A. 2 × 3 =
B. 2 × 30 =
C. 20 × 3 =
D. 20 × 30 =
E. 20 × 300 =
F. 200 × 30 =
G. 200 × 300 =

II. The first place value/multiplication problems in Singapore Math Primary Mathematics 5B, p. 16:

Multiply.

(a) 254 × 10 =

(b) 692 × 100 =

(c) 93 × 40 =

(d) 57 × 1000 =

(e) 43 × 600 =

(f) 392 × 800 =

(g) 728 × 5000 =

(h) 8056 × 3000 =

III. Extra Credit:

1. Which activity leads to deeper mathematical understanding: calculator-facilitated pattern recognition, or pen and paper calculation? Which of these is a more important 21st century skill?

2. Estimate the reading comprehension skills necessary to identify the numerical typo in the second Trailblazers problem.

3. Enumerate the math skills necessary to do the Trailblazers vs. the Singapore problem sets.

4. Which problem set is more accessible to:
-Children with autism and/or language delays
-Children learning English as a second language
-Children with attentional delays/disorders