At the June 3 meeting of the Elementary Math Program Review Committee, a number of the teachers spoke in favor of the current program, Investigations, and against putting a different program in the hands of teachers in a pilot test of a different curriculum. Other teachers at the meeting remained silent, however, and one teacher spoke up to offer her mixed feelings about Investigations, prefacing her comments with, “I may be hanging myself here, but …”

At Board meetings last spring, curriculum support staff tried to explain why there were so many petition signers with negative impressions of Investigations by saying that many parents were confused and pressured into adding their names and soon after regretted having done so.  It is true that one petition signer did ask me to remove her name from the petition, but this was a SCASD teacher who had written effusively about her dissatisfaction with Investigations and made her request to be removed fewer than 24 hours later.

At the June 3 meeting the possibility of a teacher and/or parent survey was raised, and this is an excellent idea.  It is absolutely critical that this survey be conducted by an independent organization that can guarantee the anonymity of the survey respondents.  At Penn State, companies like this one are used to survey faculty and staff on workplace climate and the performance of department heads and deans.   A survey with names attached, or conducted by the district, makes sense only if the SCASD leadership is simply seeking justification for use of the current program.

At tonight’s meeting of the Elementary Math Program Review Committee, it was decided by Superintendent Richard Mextorf that the Committee would evaluate prospective math programs without fall 2010 pilot testing of those programs in SCASD classrooms. Several committee members had continued to express their concern that identification of programs to pilot test could not be properly accomplished in time for implementation of a pilot this fall.  Some of those same members suggested that the review process be paused or terminated while the 2nd edition of Investigations is fully implemented, given a chance to work, and its outcomes assessed.  It was also suggested that implementing a pilot test would be impossible without the current curriculum support staff, some of whom are now scheduled to be reassigned to fill classroom teacher vacancies as part of cost-cutting measures in the District.

Superintendent Mextorf asked for a show of hands for how many felt that the timeline was too aggressive, and a clear majority indicated that they did.  Following this, the Superintendent announced his decision that the review process would go forward, with Investigations being compared next spring to other candidate programs selected by the Committee.  He added that: (1) it was possible that the candidate programs might be evaluated in a mid-year pilot that would begin in January 2011 if the logistics could be worked out; (2) it would be important to evaluate how well the candidate programs and Investigations match up with the forthcoming Common Core Standards, which are certain to be adopted in Pennsylvania; and (3) the scores from PSSA tests taken by SCASD students in spring 2010 ought to be considered as part of  the program review.

Martin Gardner passed away last week at the age of 95. He was a pioneer of “recreational mathematics”, wrote a column called “Mathematical Games” in Scientific American for many years, and published many books filled with mathematical puzzles.

Here’s a puzzle from one of Gardner’s books that was posted on the New York Times site on the occasion of his 95th birthday:

Two missiles speed directly toward each other, one at 9,000 miles per hour and the other at 21,000 miles per hour. They start 1,317 miles apart. Without using pencil and paper, calculate how far apart they are one minute before they collide.

(Feel free to post your solution as a comment)

Now, missiles don’t speed directly toward each other at 21,000 mph in the real world; they follow parabolic paths (if air resistance is neglected) and move at half that speed at most.  Even if missiles did move in straight lines at realistic speeds, who cares about missile collisions aside from a few scientists and engineers who work on missile defense systems?  There is no doubt that this is a contrived problem, and many would question whether students can learn math by solving such problems.  This is a criticism often repeated by proponents of “reform” math programs like Jo Boaler or the person who recently sent an anonymous message to the SCASD Elementary Math Program Review Committee: “[A] good curriculum will focus on opportunities for application to real-world problems, because this is what makes learning “stick”.  A good curriculum, or implementation should minimize the use  of “contrived” problems (Two trains leave New York and Chicago at the same time…).”

I disagree with the implication that relevance to the student’s life equates to the potential for learning.  I think that such contrived problems – if they are thoughtfully devised – can foster good mathematical thinking by encouraging the solver to think in unfamiliar ways.  The problem above is a good example. Most people don’t associate math with creativity, but they are wrong about that, and abstract problems can be useful in promoting creative math.  As Douglas Hofstadter put it:

Martin’s columns and writings radiate a profound exuberance in the constant novelty of human thought. What comes through, even if it’s never explicitly expressed, is a kind of informal version of Gödel’s theorem for human thinking—a sense that creative minds will always one-up the pedestrian expectations generated by unimaginative, logic-bound thinking. There is an exultation in the breakout from expected patterns, the violation of seemingly ironclad laws, the making of wildly unexpected connections, the revelation that two seemingly identical properties are really quite different, and the counterexamples that make it all blindingly clear (at least for a moment—then you forget how it worked!)…. If nothing else, reading Martin Gardner should convince you that the human mind’s pathways of finding truths are as diverse and unpredictable as the pathways of evolution itself.

From Friday’s CDT:

About 50 percent of Pennsylvania’s math standards for third, fifth and eighth grades don’t align with the proposed national core standards, according to a study conducted by University of Pittsburgh professor Suzanne Lane.

The state Board of Education presented the information, and other study results, Thursday morning at a lightly attended forum at Mount Nittany Middle School, one of three taking place across the state.

The board of education will vote on adopting the national standards July 1.

Some of the variance between PA and Common Core is due to standards in one being absent from the other, but most is due to requirements appearing in different grades.  For example, compare the Grade 3 PA standard of:

2.2.3.B. Solve single- and double-digit addition and subtraction problems with regrouping in vertical form.

with the Grade 2 Common Core standard standard of:

2-NBT.13. Compute sums of two three-digit numbers, and compute sums of three or four two-digit numbers, using the standard algorithm; compute differences of two three-digit numbers using the standard algorithm.

If there are many instances like this of the PA standards lacking in rigor compared to Common Core, SCASD would do well to consider math programs that exceed the current PA standards rather than meet them, since it appears that adoption of Common Core in PA is inevitable if PA wants to compete with other states for additional federal funding.

UPDATE: This post referred to the draft version of the Common Core Standards.  The final version released on June 2 requires that the standard algorithm for addition (i.e., carrying) is taught, but by the end of fourth grade.

In October 2009 Gabriella and Paul Rosenbaum Foundation released a report entitled The Effect of Singapore Mathematics on Student Proficiency in a Massachusetts School District:  a Longitudinal Statistical Examination. It shows that the longer students had Singapore Mathematics as their curriculum the better they performed on Massachusetts’s tests.

North Middlesex Regional School District (NMRSD) is a rural school district near the border between Massachusetts and New Hampshire, serving the towns of Ashby, Pepperell and Townsend.  In response to poor student performance on state mathematics assessments, this district introduced a number of their teachers to the Singapore mathematics (SM) syllabus during a 2000 summer institute for pilot implementation that fall.

For effective implementation, new K-8 school curricula are most easily begun with K-1 or K-2, with another grade added each successive year.  However, worried about their entering high school students’ inadequate math knowledge, NMRSD chose to address these concerns with an SM pilot in their “feeder” middle schools (grades 5 to 8). The pilot program was quickly extended across classrooms and grades.  Kindergarten was added in the 2002-03 year and, as can be seen in Table 1 on page 10, every classroom in grades 1-6 was using the SM curriculum by 2005-06.  From this point on, Singapore math was established as the District’s official curriculum.

A summary of the report may be found here and the full report is here.

The Rye Neck Union Free School District in Westchester County, NY, just north of NYC,  has an total enrollment of 1,500, compared to 7,200 for SCASD.  In Fall 2009 this district, which had been using Growing With Mathematics, pilot tested three different curricula: Primary Mathematics (also known as Singapore Math),  enVision Math, and Math in Focus (marketed as “Singapore Math for U.S. classrooms”).

Math in Focus was recently recommended for adoption.   A description of the pilot can be found in this newsletter.

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.