A math teacher in Alabama took the initiative to develop a “real world” math example in his classroom and was placed on leave and may be fired as a result:

A Jefferson County teacher picked the wrong example when he used as­sassinating President Bar­ack Obama as a way to teach angles to his geome­try students.

It is possible, of course, to use math from the “real world” (as opposed to abstract examples) in math classes without getting a visit from the Secret Service.  Use of such problems is a defining feature of strict constructivist programs like “Investigations” and “Everyday Mathematics” – they are supposed to make math more relevant to kids’ lives and thus increase their interest in math so that they will further develop their math skills.

It’s an idea that seems reasonable, but many mathematicians would argue that there is substantial value to learning math abstractly and that the assumption that concrete examples are more effective is wrong and harmful.  This view is supported by recent research from the Center for Cognitive Science at Ohio State that was published in the journal Science.  From a New York Times article on the research:

[M]any educators in recent years have incorporated more and more examples from the real world to teach abstract concepts. The idea is that making math more relevant makes it easier to learn.

That idea may be wrong, if researchers at Ohio State University are correct. An experiment by the researchers suggests that it might be better to let the apples, oranges and locomotives stay in the real world and, in the classroom, to focus on abstract equations, in this case 40 (t + 1) = 400 – 50t, where t is the travel time in hours of the second train. (The answer is below.)

“The motivation behind this research was to examine a very widespread belief about the teaching of mathematics, namely that teaching students multiple concrete examples will benefit learning,” said Jennifer A. Kaminski, a research scientist at the Center for Cognitive Science at Ohio State. “It was really just that, a belief.”

Dr. Kaminski and her colleagues Vladimir M. Sloutsky and Andrew F. Heckler did something relatively rare in education research: they performed a randomized, controlled experiment.

…

The students who learned the math abstractly did well with figuring out the rules of the game. Those who had learned through examples using measuring cups or tennis balls performed little better than might be expected if they were simply guessing. Students who were presented the abstract symbols after the concrete examples did better than those who learned only through cups or balls, but not as well as those who learned only the abstract symbols.

The problem with the real-world examples, Dr. Kaminski said, was that they obscured the underlying math, and students were not able to transfer their knowledge to new problems.

“They tend to remember the superficial, the two trains passing in the night,” Dr. Kaminski said. “It’s really a problem of our attention getting pulled to superficial information.”

The researchers said they had experimental evidence showing a similar effect with 11-year-old children. The findings run counter to what Dr. Kaminski said was a “pervasive assumption” among math educators that concrete examples help more children better understand math.

Education announces roundtables on Common Core Standards

The State Board of Education is announcing a series of roundtable sessions to discuss the national Common Core Standards.  In Pennsylvania’s Race to the Top application, the State Board outlined its commitment  to a transparent and public process around the adoption of Common Core academic standards in English language arts and mathematics.

With Common Core on the board’s agenda for its June 30-July 1 meeting, the board will hold a series of roundtables across the state – both to present preliminary results of a study comparing Common Core with Pennsylvania’s standards framework and to gather feedback from stakeholders.  The study is being conducted by professor Suzanne Lane of the University of Pittsburgh.  Dates and locations for the roundtables are below; each forum will begin at 10 a.m. and continue until all stakeholders are heard.

• Friday, May 21, University of Pittsburgh, Posvar Hall
• Thursday, May 27, Mt. Nittany Middle School, State College
• Wednesday, June 9, Philadelphia (location TBD)
  

Anyone who has questions or is interested in participating should contact the State Board office at (717) 787-3787.

Two memories of this math controversy in SCASD never fail to make me chuckle when I think back on them.  The first was a comment posted on the CDT site after the Ed Mahon wrote his first article on the subject, entitled “The Great Math Debate”.  The commenter wrote, “If you say ‘math debate’ over and over again, it sounds kind of funny.”

The second one happened at a math information session for parents last spring.  The district curriculum staff were telling the parents how much fun Investigations was for their children, and a parent raised his hand and said, “You know, it’s okay if my kids don’t have so much fun if they learn some more math.  They have plenty of fun at home.”

I thought of this last one frequently as I read “What’s Math Got to Do With It?” by Jo Boaler (Penguin, 2008).  Boaler, a professor of math education, has a lot to say that is relevant to our math discussions in SCASD, and the District curriculum staff recently recommended her book to the Board of Directors, and it was also anonymously recommended to the Elementary Math Program Review Committee.

In general, Boaler places much more emphasis on how much fun math students have than how much math they might or might not be learning.  Children in constructivist math classrooms, she reports, are “smiling and laughing” and “jump around” excitedly in sun-drenched classrooms.  They sigh, “I love this class.”  The teachers are “the happiest they had ever been.”

When students and teachers are denied constructivist math by “extreme traditionalist” parents, however, the party is over.  Teachers who are forced to use “traditional” or “passive” approaches are “demoralized and defeated” and their “cold, disinterested, and traumatized” students sit in rows and “work in silence.”

Parents looking for evidence of the success of strict constructivist approaches won’t find much here.  Boaler recommends TERC Investigations in an appendix and lists the research studies supporting the curriculum currently in use in SCASD:

Flowers, J.M. (1998). A study of proportional reasoning as it relates to the development of multiplication concepts. Unpublished doctoral dissertation, University of Michigan, Ann Arbor.

Goodrow, A. M. (1998). Children’s construction of number sense in traditional, constructivist, and mixed classrooms. Unpublished doctoral dissertation, Tufts University, Medford, MA.

Mokros, J., Berle-Carman, M., Rubin, A., & Wright, T. (1994). Full year pilot grades 3 and 4: Investigations in number, data, and space. Cambridge, MA: TERC.

That’s it – two unpublished dissertations and a report generated by the author of the curriculum, and not one peer-reviewed study.

Boaler describes her own research at length in “What’s Math Got To Do With It”, including a study demonstrating the benefits of constructivist math at “Railside School”.  She declines to provide the real name of the school, making it difficult to evaluate her claims or conclusions.  Fortunately a group of mathematicians have conducted an in-depth examination of Boaler’s research and found that “Prof. Boaler’s claims are grossly exaggerated and do not translate into success for her treatment students.”  Their report can be found here.

One final point:  Does anyone ever get good at anything without some hard work and, yes, even some unpleasantness?  No one is suggesting that any kids ought to hate math, but it is unrealistic to expect that our kids will develop true mastery while experiencing only fun along the way.

The Westwood Regional School District in northern New Jersey has an total enrollment of 2,600, compared to 7,200 for SCASD.  In Fall 2008 this district, which had been using the 2nd edition of Everyday Math, pilot tested two different curricula: Math Connects and the 3rd edition of Everyday Math.

A description of the process followed in Westwood can be found here, and a PowerPoint presentation may be downloaded from this page.

This suburb of Seattle has an total enrollment of 17,200, compared to 7,200 for SCASD.  In Fall 2007 this district, which had been using Investigations, pilot tested two different curricula: Math Expressions and Everyday Math.

A description of the pilot test can be found here, and the district’s web page covering the process, including a video and the final report, can be found here.

Yesterday we saw our 5,000th visit to this site since we started last September.  Interest in the site seems to be increasing in recent months – here is a plot of the number of unique visitors to PQME.org each month from September through April:

Stein, Kinder, Zapp & Feuerborn’s (2010) recent chapter on Promoting Positive Math Outcomes in NASP’s Interventions for Achievement and Behavior Problems in a Three-Tier Model including RTI provides a problem solving approach to educational challenges.  While the authors’ development of the Mathematics Problem Solving Inventory is still ongoing, their chapter details some very helpful questions we should be asking about potential alternative curricula under consideration in SCASD.

Regarding the evaluation of curriculum and instruction needs, specifically with regard to materials, textbooks, and organization: Are materials and instruction structured sufficiently to meet the needs of Tier 1  students?  They note that “improving mathematics performance requires attention to both content coverage and content mastery” (p. 534).

Their recommendations for mathematics curriculum evaluation (p. 538):

General program design

1. Do the lessons include objectives with measurable student behaviors?
2. Are newly taught strategies integrated with those previously taught?
3. Is there a balance between computation instruction and problem-solving instruction?
4. Is the program organized using a spiral or strand design?

Instructional Strategies

1. Are strategies explicitly taught in the program?
2. Are the strategies appropriately generalizable – neither too narrow nor too broad?
3. Are critical component skills taught prior to the strategy?
4. Are there adequate examples provided for instruction?
5. Are discrimination examples included?

Assessment

1. Does the program include a placement test with options for various starting points?
2. Do in-program assessments include recommendations for accelerations or remediation?
3. Are the in-program assessments carefully aligned with instruction?

Notably, constructivist, “reform” curricula typically are spiral in design and are common in the U.S.  Alas, spirally designed programs often lack adequate initial instruction and review to promote student mastery of skills.

According to the National Math Advisory Panel (2008), “A focused, coherent progression of mathematics learning, with an emphasis on proficiency with key topics, should become the norm in elementary and middle school mathematics curricula. Any approach that continually revisits topics year after year without closure is to be avoided.” (p. 22).

Hey, hey.

On Tuesday evening at Gray’s Woods Elementary School, the Elementary Math Program Review Committee held its first meeting, which began with introductions by facilitator Dr. Mark Dietz and Superintendent Richard Mextorf.  There was an introduction activity, followed by brainstorming in small groups on what would constitute an ideal math program.  A master list of about 25 ideal math program elements was compiled for the entire group. At the conclusion of the meeting, it was announced that volunteers would form a seven-member “steering committee” that would create a rubric based on the master list.  The rubric would then be used to evaluate math programs for potential pilot testing in fall 2010. As yet, the steering committee membership has not been announced. The next meeting of the entire committee is Tuesday, May 18.

While research on specific math textbooks may not be readily available, a tremendous amount of research clearly shows what works well (and what doesn’t) regarding mathematics instruction and math achievement. I finally got the book recommended in an earlier post, Hattie’s (2009): Visible learning: A synthesis of over 800 meta-analyses relating to achievement. It is excellent, and clearly details what works in education.

By way of introduction, a meta-analysis take groups of studies that have been done on a particular topic, innovation, or approach and combines their findings into a summary measure called “effect size”. An effect size of d = 1.0 indicates an increase of one standard deviation on the outcome (e.g., achievement), or advancing kids’ achievement by two to three years, or improving the rate of learning by 50%. An effect size of 1.0 means that, on average, the performance of students receiving that treatment would exceed 84% of students not receiving that treatment.

What does Hattie report for mathematics programs? Combining 13 meta-analyses covering 677 studies and 8565 people (2370 effects), the effect for mathematics programs is 0.45, a medium effect size that means they DO have an effect. Specifically, the highest effects were found when teachers provide feedback data or recommendations to students (d = .71) and peer-assisted learning (.62), explicit teacher-led instruction (.65), direct instruction (.65), and concrete feedback to parents (.43). Hattie states that “modern” mathematics that stress real-world problems and a high level of use of manipulatives (e.g., constructivist approaches) has an effect size of .24.  Effects are higher for teaching concepts (.36) and computation (.31) but not application (.06).

According to Hattie, “Overall, the presence of feedback, direct instruction, strategy-based methods, high levels of challenge and mastery has much effect on the learning of mathematics. That is, directive teaching makes the difference when teaching mathematics” (p. 147)

In a subsequent post, I’ll pull together Hattie’s findings regarding effect sizes for teachers as ‘activators’ and teachers as ‘facilitators’.