Thursday, August 27, 2009

Multicultural math, anyone?

From I discussion I joined at

SJ quotes the following in support of there being substantial cultural effect on math learning:
There are those who suggest that mathematics is 'culture free' and that it does not matter who is 'doing mathematics' the tasks remain the same. But these are people who do not understand the nature of culture and it's profound impact on cognition.(Ladson-Billings, 1997, p. 700)

KB's response:

Ladson-Billings is not a cognitive scientist; I strongly suspect that cognitive scientists involved in empirical research on math acquisition, e.g., Stanislas Dehaene, would disagree with her general thesis about the "profound impact" of culture on cognition.


Did you read the study that was above quote? The quote was from a paper by Dr. Steven Guberman - University of Colorado at Boulder. Other research includes Supportive Environments for Cognitive Development. - The above research "Cultural Aspects of Young Children's Mathematics Knowledge" was used by NCTM.....all of this doesn't matter huh?


As I read it, the above paper mostly discusses cultural variations in pre-K mathematics exposure. These don't constitute a "profound impact" of culture on cognition, but rather the not-so-surprising fact that pre-K math preparation affects k+ math preparedness.


As far as left-brainers, my specific concern with claims of "profound" cultural impact is that they tend to further justify those practices that are particularly problematic for left-brain students: more group work; more hands-on activities; less conceptually challenging math...

... All because students from "other cultures" supposedly are culturally programmed not to be able to handle the kind of traditional math that's still taught in Europe--and the Indian subcontinent and East Asia, among other places around the world.

Monday, August 24, 2009

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

1. From the 2nd grade Investigations (TERC) Stickers, "Number Strings and Story Problems" unit, (assigned in late January):

Assessment: Number Strings

Use combinations that you know to solve these problems.
Show your work.

6 + 3 + 4 + 6 =

7 + 5 + 9 + 3 + 5 =

9 + 6 + 7 + 1 =

6 + 8 + 6 + 7 =

2. From the beginning of the 2nd grade Singapore Math workbook, volume 2 (of 2), Primary Mathematics 2B (Standards Edition), p. 17:

1. Add.

(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

Which problem set gives students more freedom and promotes more discovery learning?

Saturday, August 22, 2009

Does working in groups strengthen academic skills?

In my conversation with GC at (see last post), he offered the following reasons for requiring students to work in groups, even if they ask for more challenging work than what their classmates get:


The reasons are simple. I need that student to participate in cooperative learning and group activities for the sake of the other students (whom they can help in ways that I can't) and for their own sake (because being a peer tutor has been show to produce a more secure set of skills in a student).

In response to this, I wrote:


There are no randomized studies showing that social skills improve when students are forced to work in groups; if there were, then presumably my generation (which didn't do much work group at all), and students in non-Anglophone countries, have weaker social skills than younger Americans do.

GC's reply:

I think there's considerable support for the idea that participating in cooperative learning activities results in a stronger set of academic skills for the students. I'll quote Johnson, Johnson, and Stanne (2000):

Cooperative learning has been around a long time (Johnson, 1970; Johnson & Johnson, 1989, 1999). It will probably never go away due to its rich history of theory, research, and actual use in the classroom. Markedly different theoretical perspectives (social interdependence, cognitive-developmental, and behavioral learning) provide a clear rationale as to why cooperative efforts are essential for maximizing learning and ensuring healthy cognitive and social development as well as many other important instructional outcomes. Hundreds of research studies demonstrate that cooperative efforts result in higher individual achievement than do competitive or individualistic efforts. Educators use cooperative learning throughout North America, Europe, and many other parts of the world. This combination of theory, research, and practice makes cooperative learning one of the most distinguished of all instructional practices.

I then queried my friends at kitchentablemath about this study, who shared the following thoughts about whether working in groups strengthens academic skills:

From Catherine Johnson:

Teaching LD (Information and Resources for Teaching Students with Learning Disabilities) puts Cooperative Learning under "Use Caution" (i.e.: "practices for which the research evidence is incomplete, mixed, or negative).

Catherine also quotes the following from Learning LD, which suggests that only certain carefully orchestrate group work sessions are effective:
Whereas CL strategies typically involve two or more students working together to accomplish an assigned task, it is not synonymous with “group work.” Johnson and Johnson (1994) identified five elements critical to maintaining structure and student involvement in CL: (1) positive interdependence, which means students realize that group performance depends on the contributions of each member; (2) face-to-face promotive interaction, wherein students encourage and facilitate each other’s efforts to achieve; (3) individual accountability; (4) the use of interpersonal skills; and (5) group processing, which refers to groups’ reflections on how well they are functioning. Researchers emphasize that teaching students such interpersonal behaviors and monitoring their use are critical to the success of CL (e.g., Goor & Schwenn, 1993; Johnson & Johnson, 1992).
Catherine also quotes from John Hattie's Visible Learning: A Synthesis of Over 800 Meta-Analyses Relating to Achievement:
All of the many meta-analyses by the Johnsons and their colleagues show high effect sizes, whereas the others hover around the small to medium effects.


Johnson and Johnson (1987) argued also that cooperation was most effective among adults...


Cooperative learning is more effective in reading (Hall, 1988, d=0.44) than in mathematics (d=0.01), and Johnson et al. (1981) found that for rote decoding and correcting tasks, cooperation does not seem to be superior.

Anonymous adds the following key observation about the Johnson & Johnson study that GC cites in support of cooperative learning:

What's missing from the meta-analysis from U Minn is context. There's no sense of what types of learning tasks are being used to compare cooperative vs. individual learning. My experience as a parent and teacher is that cooperative learning is s technique that can be useful for specific types of learning, but that it is inappropriate for the majority of learning objectives.

Putting all this together, it seems that cooperative learning must be planned effectively, and used judiciously, if it is to have a positive effect on academic skills.

Also, might there be certain subtypes of children, e.g. left-brainers, who tend to learn better on their own, regardless of how effective the cooperative learning environment?

Thursday, August 20, 2009

Veteran teacher defends group work against working at your own rate

From a discussion I had with a teacher at

Why not simply allow each student to work through a math curriculum at his/her own rate? As a student, I attended a classes like this (in regular public schools, with large class sizes) and they were highly successful classes, precisely because they engaged each student at his or her level.

The idea of providing a curriculum that challenges and/or meets the needs of all students sounds to me like differentiated instruction (DI), which has become a major focus of professional development and curriculum design in the last few years. Carol Tomlinson is one of the more popular authors on the subject. I've been sent to three or four workshops on it in the past three or four years - and the general ed teachers I work with have been required to attend with me.

DI comes in flavors. There's DI intended to cater to different learning styles to ensure that kinesthetic learners don't have to rely on auditory or visual processes alone. There's DI intended to address the level at which students get challenged. The math lesson may be on probability; the curriculum we use will provide the teacher with a variety of tasks (of varying difficulty) that can be used with their students.

Both the math and the reading curriculum my district uses in the elementary grades build DI into each lesson.

Rate is a different issue. The adoption of a spiral concept in curriculum design makes *rate* seem like a problematic concept. Instruction is cyclical: we may spend a couple of weeks working on learning and using central tendencies in statistics with the fourth or fifth graders, then the curriculum moves to addition of improper fractions for a week, then it spends some time reinforcing student knowledge of geometric shapes and their properties, and a few weeks later it cycles back around to central tendencies.

Chances are good that the second and third graders are working on exactly the same concepts (though with simpler problems) if their teachers are on track with the pacing guides.

If a student came to me at any grade level (I've taught K-12) and said that they understood chapter 11 and had finished the work in it, and they asked me if they could go on to chapter 12 WITHOUT the rest of the class, I'd say no. [Emphasis mine] The reasons are simple. I need that student to participate in cooperative learning and group activities for the sake of the other students (whom they can help in ways that I can't) and for their own sake (because being a peer tutor has been show to produce a more secure set of skills in a student).

It's not that there is material to cover and the student could finish early, it's that there are standards (content standards) to meet. I'm happy if the kid can excede the standards for central tendencies; I'm not happy for the kid to move on to geometry while the rest of the class is still in statistics. In addition to math skills, I have to think about the student's social skills - their ability to work with with others.

DI can and should lead to engaging each student at their level. But it doesn't (and shouldn't) lead to anyone finishing the year's math curriculum sometime in March...

I know a lot of talented math buffs (the subjects of my forthcoming book) who are extremely frustrated by the practices you describe (which are unique to the U.S., and certain schools in Canada, Britain and Australia). They are severely under-challenged, especially with today's Reform Math (where the actual math is much, much easier than it used to be), hate working in groups, and resent being asked to teach math to other students. We are at risk of marginalizing (and under-preparing with respect to students from other countries) the next generation of potential mathematicians.

To this particular concern, GC did not reply.

How concerned are teachers about kids who say they are under-challenged and hate working in groups?

Tuesday, August 18, 2009

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

1. A late-October assignment from 2nd grade Investigations (TERC) "Counting, Coins and Combinations," Session 4.4:

Challenging Story Problems

Solve the problem. Show your work.
Write an equation.

Franco and Sally have 28 cherries and 13 grapes.
How many pieces of fruit do they have?

The teacher had 12 new pencils.
She bought 11 more pencils.
How many pencils does she have now?

Sally has 15 pennies.
Jake has 26 pennies.
How many pennies do they have in all?

2. From about 1/7 of the way through the 2nd grade Singapore Math Primary Mathematics 2B (Standards Edition), "Addition and Subtraction," p. 56:

Mrs. Kennedy made 95 apple tarts, 98 pineapple tarts and 57 orange tarts.
How many tarts did she make altogether?

She made _____ tarts altogether.

This chart shows the number of books in a library.

English books________________ 408
Science books________________ 274
History books ________________ 224

What is the toal number of books in the library?

The total number of books is _________.

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 cost of the camera is $_________.
The total cost of the camera and the watch is $ _________.

3. Extra Credit:

Why does the Investigations problem set, but not the Singapore problem set, declare itself "challenging"?

Which is more oppressive to students: making them fill in the blanks, or making them show their work?

Sunday, August 16, 2009

The lofty heights of the health-care debate: where are the cool-headed left-brainers?

So, on the one hand, we have Sarah Palin and others on the right somehow reading evil "death panels" into the Obama administration's proposed health care reform.

On the other hand, we have people on the left boycotting Whole Foods because its CEO wrote a Wall Street Journal Op-Ed piece opposing the proposed health care reform, not because of "death panels", but (in part) by arguing that Americans need to take more personal responsibility for their health.

And we have Princeton professor Melissa Harris-Lacewell reading racial overtones in such opposition by arguing on NPR's All Things Considered that "language of personal responsibility is often a code language used against poor and minority communities."

It seems that, at least when it comes to health care reform, one cannot open one's mouth without facing boycotts or accusations of racism or death paneling.

I can't think of a time when I've been more dismayed by public discourse. Please, let's turn down the thermostat and turn on the lights. Left-brainers, please speak up!

Friday, August 14, 2009

No Child's Critical Thinking Left Behind

The week's Education Week reports that U.S. Secretary of Education Arne Duncan plans to set aside $350 million of the $4.35 billion in discretionary aid in the Race to the Top Fund to improve student assessments:

Testing experts say that money could serve as a down payment for scaling up tests that would better measure students’ critical-thinking skills and improve teacher and student engagement in the assessment process.
Many education experts would like replace the multiple-choice tests that dominate today's No Child Left Behind Testing. Paraphrasing Randy Bennett, a scholar at the Educational Testing Service, Education Week notes:
Such tests... are not ideal for identifying whether students can take multiple pieces of domain-specific knowledge and analyze, integrate, and apply them in unfamiliar contexts..
Researchers familiar with international benchmarking argue that those critical-thinking skills are precisely the type that will be in demand as the global economy becomes increasingly knowledge-oriented.
Education Weekly cites two examples of such tests. First, there's the College and Work Readiness Assessment, a computer-based test used by private high schools:
A typical ... question might present examinees with a dossier of materials relating to a child who had a roller-skating accident at school. The materials could include newspaper articles, technical reports about the skates, data about competitors’ products, sales figures, medical reports, and the number of documented accidents. Then, the student would be asked to analyze those materials and write a memo about whether the skates are truly dangerous, and to justify his or her conclusions drawing from the information.
The second example is recently piloted subset of the 2009 National Assessment of Educational Progress in science, which used "interactive computer tasks" to prompt students:
to engage in the entire process of scientific inquiry, in which they must participate in a simulated experiment, record data, and defend or critique a hypothesis.
While such tests have typically been costly, because they must be scored by humans, Education Week cites experts as saying that advancements in technology could help score these tests:
The high costs of scoring such a complicated assessment with an almost unlimited number of answers... could be mitigated by advancements in natural-language-processing software­—essentially programming that proponents claim can judge written essays as accurately as human readers and reduce, though not eliminate, the need for costly human evaluation.
Even with what is still pie-in-the-sky technology (I've worked in Natural Language Processing!), the proposed new measurements sound dangerously subjective to me, and also highly language-intensive in ways that will disfavor bright, analytically-minded kids with language delays.

Also, wouldn't it be cheaper just to make the multiple choice questions (which, in many states, are notoriously simple) more challenging? Well-crafted multiple choice question can indeed measure higher-level thinking skills, as they do on standardized aptitude tests like the SATs.

Wednesday, August 12, 2009

Education Myths... & Politics

Here's my review of Jaye Greene's Education Myths:

There's just one key myth that this book doesn't shatter

And that one key myth is that the critics of the Powers that Be in education are, to quote one of the reviewers below, "right-wing" propagandists. Indeed, given the unfortunate political polarization of education policy in America, perhaps this book's greatest liability is the endorsement from Jeb Bush that appears on its cover.

For the most part, Jay Greene backs his claims up with references to specific studies, and one indication that he isn't distorting the data is that his critics haven't found fault with his data. The one exception I found was in his discussion of the Special Education Myth, in which he simply asserts that "any growth in neurological disorders caused by increased numbers of low-birth-weight babies has been more than offset by improvements in the prevention of such disorders in other areas, such as improved prenatal medicine, safe child car seats, and reductions in exposure to lead paint."

Greene does argue, convincingly, that the growth in special ed numbers is largely due to financially-motivated re-classifications. And, if students are generally less teachable than they used to be, it may be more because of teaching failures in the lower grades. But can we be sure that incoming kindergartners aren't less (or more) teachable than they used to be? It would be interesting to survey veteran kindergarten teachers--ones who've remained in the same schools for 20-30 years.

It would also be nice if Greene had included some of the myths that inform current teaching practices and curriculum choices--though these could fill a whole nother book.

These concerns aside, this is a hugely important book that convincingly debunks most of our most debilitating myths--and the left-wing and (yes!) right-wing assumptions that sustain them.

Monday, August 10, 2009

Math problems of the week: 1900's algebra vs. Interactive Math Program

1. The last four problems of the Quadratic Equations chapter of New School Algebra (published in 1898), p. 292:

23. A boat's crew row 4 miles down a river and back again in 1 hour and 30 minutes. Their rate in still water is 2 miles an hour faster than twice the rate of the current. Find the rate of the crew and the rate of the current.

24. A number is formed by two digits. The units' digit is 2 more than the square of half the tens' digit, and if 18 is added to the number, the order of the digits will be reversed. Find the number.

25. A circular grass plot is surrounded by a path of a uniform width of 3 feet. The area of the path is 7/9 the area of the plot. Find the radius of the plot.

26. If a carriage wheel 11 feet round to 1/4 of a second less to revolve, the rate of the carriage would be five miles more per hour. At what rate is the carriage traveling?

2. The last homework assignment in the World of Functions chapter of Interactive Mathematics Program, Integrated High School Mathematics, Year 4 (published in 2000), p. 344-345:

Now that The World of Functions is completed, it is time to put together your portfolio for the unit. Compiling this portfolio has three parts.
*Writing a cover letter summarizing the unit
*Choosing papers to include from your work in this unit
*Discussing your personal mathematical growth in the unit.

Cover Letter for The World of Functions:

For your cover letter for The World of Functions, focus on these two key ideas.
*The distinguishing characteristics of each of the family of functions you worked with, considered in terms of real-world situations graphs, tables, and symbolic representations.
*Methods of combining functions or transforming functions to create new ones.

For each of the important aspects of the unit, choose an activity that illustrates that idea to include in your portfolio.

Selecting Papers from The World of Functions:
Your portfolio from The World of Functions should contain these items.

*Brake!, "Brake!" Revisited, and Better Breaking
Include a statement of how your understanding of the "braking" situation developed over the course of the unit.

*Linear Tables, Quadratic Tables, and Exponential Tables
Discuss how your understanding of the relationship between tables and algebraic representations grew through these three activities.

*Activities discussed in your cover letter

*Homework 29: Beginning Portfolio Selection
Include the activities from the unit that you selected in Homework 29: Beginning Portfolio Selection along with your written work about those activities

*A Problem of the Week
Select one of the three POWs you completed during this unit (One Mile at a Time, A Spin on Transitivity, or It's Off to College We Go).

Personal Growth
Your cover letter for The World of Functions describes how the unit develops. As part of your portfolio, write about your own personal development during this unit. You may want to specifically address this issue.

How do you feel you have developed during this unit in terms of your ability to explore problems and prove conjectures in mathematics?
You should include here any other thoughts you might like to share with a reader of your portfolio.

3. OILF's Extra Credit

Write a letter discussing our country's personal growth in algebra teaching and learning over the past 100 years.

Saturday, August 8, 2009

Summer Math Projects: grade 5

Part I
You've dodged rhinos in JUMANJI, driven down the roads of LIFE, and experienced the ups and downs of CHUTES and LADDERS. Now, take a turn at your own game.
Using what you know and like about your favorite board games, create a math game that uses all the multiplication and division facts up through the twelve times table.

Here are some things to consider as you design your game:
What is the objective of the game?
How would players advance? Would they solve word problems, roll, or spin?
Would players need cards, spinners, number cubes, pencil and paper?

As you prepare your game "Package" be sure to include:
* A title for your game
* A colorful and creative game board
* All the materials needed to play: spinners, number cubes, cards, playing pieces, etc.
* A clearly written set of instructions

Once you have created your game, practice playing it with your family and friends so that you know what works and that players enjoy it.

Part 2
All incoming fifth graders will be expected to show mastery of multiplication facts through the twelves times table. One way to help learn the facts to the level of instant recall is through the use of "flash cards". Students should create their own personalized set using index cards and practice them often to increase their speed and accuracy.


When I was in school, teachers did not ask students to learn their multiplication tables over summer vacation. Rather, this one one of the things they drilled us on in the classroom (in fourth grade).

Nor did my schools require me to do "creative" projects over the summer, or stipulate that creativity be visual. Rather, the free time I had once school, homework, and multiplication mastery were over allowed me to do my own creative projects--where the best kind of creativity, the "personalized" kind, is possible.

Thursday, August 6, 2009

Summer Math Projects: 4th grade

Imagine that you are planning a Back to School picnic. Think about the types of food, drinks and supplies that you would like to have to accommodate the 25 students You only have $100.00 to spend Make a list of what you want, choose a supermarket that isn't too far, get a flyer and search for the best deals! Remember you want to get the most for your money! The goal is to get as close to $100.00 as possible without going over. Make a chart, using a poster board, with pictures from the ads of the items you will purchase. Make sure you invlude the cost of each item, how many you will purchase and the final cost for that item. At the bottom of your poster board, give the total amount you spent.
OILF's counter-assignment for teachers: estimate the amount of time parents of left-brain children will have to spend facilitating their children through this assignment.

Tuesday, August 4, 2009

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

1. The final word problem in the 3rd grade Fair Shares Activity Booklet (TERC Investigations):

In our classroom, we have 2 bookcases. Each bookcase has 4 shelves. On each shelf, there are 10 books. How many books do we have in our classroom?

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

2. The final multiplication word problem in the 3rd grade Multiplication and Division chapter of the Primary Mathematics 3A workbook:

A bicycle costs $385.A motorcycle costs 5 times as much as the bicycle.

(a) Estimate the cost of the motorcycle.

(b) Find the exact cost of the motorcycle.

3. Extra Credit

1. Investigations problems systematically require students to explain how they got their answers; Singapore problems do not. Yet Investigations, and other Reform Math programs like it, are the product of a society that prides itself on freedom and privacy. Why the uniquely American obsession with what's going on in students heads when they solve problems?

2. Have American teachers confused "explaining your answer" with "showing your work"?

3. Is it possible that sufficiently challenging problems--ones that students cannot easily do in their heads--will induce students to show their work automatically?

Sunday, August 2, 2009

Dropping the SATs: the hidden agenda and the underachiever

In an earlier post, I wrote about how eliminating the SAT/ACT requirements for college admissions disadvantages left-brainers, and about the lofty reasons cited by college admissions officials for doing so:

(1) "test scores appear to calcify differences based on class, race/ethnicity and parental educational attainment."
(3) "the contrast between opportunities and fancy suburbs and some of the high schools that aren’t so fancy"
(3) "academic research that suggests that test preparation and coaching results in an increase of 20 to 30 points on the SAT"

An article in this past week's New York Education Life suggests that colleges' actual reasons for dropping standardized tests may not be nearly so lofty:

(1) Doing so raises the number of applicants, allowing schools to claim to be a more selective than previously (even if the overall caliber of your applicants declines).
(2) Doing so raises your average SAT scores (because the students who report their SATs will tend to have higher scores).

Of course, as more and more schools succumb to such temptations, these effects will lessen. Meanwhile, the underachieving, high-testing left-brainer will face fewer and fewer college options.

...Until some colleges wise up and realize that a whole sector of talented students (budding but under-challenged mathematicians, engineers, programmers, and linguists, among others) is out there waiting, desperately, to be tapped.