SCASD Superintendent Richard Mextorf has already indicated that it is critical that SCASD’s elementary math program meet the new standards, whether it is the Investigations program currently in use or any new program considered for adoption in the District.

UPDATE: Virginia has said no to the national standards.  GA and DE are expected to adopt them, joining PA, NJ, WY, HI, MD, and OK.

This guy seems to me to be an excellent teacher who has found a great way to balance these approaches:

Dan Meyer is a young high school math teacher in Santa Cruz, CA, who has found creative ways to interest students in math through the use of “real life” examples and who strongly believes that highly structured problems in math textbooks are unhelpful.

In the above video, he sounds like a teacher who would prefer a strict constructivist approach like that employed in Investigations, but that turns out not to be the case.  On his blog, Mr. Meyer makes it clear that the benefits of strict constructivism are outweighed by the inefficiency that comes with it.  He prefers to follow up his examples with direct instruction:

I have to take great pains to point out to skeptical teachers that, even though you’re investing extra time on these kind of investigations, that investment into rigorous mathematical thought processes pays off huge dividends in direct instruction where you find students less impatient, more tenacious, and quicker to learn new skills by evaluating them against their existing intuition.

In a perfect world with unlimited time, he might like to stick with constructivism, but in this world Mr. Meyer believes that his use of direct instruction along with examples that appeal to students’ intuition results in a time savings of 75%:

I have determined my constructivism multiplier to be four, which is to say it takes me four times longer to bring a student to conceptual understanding through conversation and questioning in a social situation the student helped create than it does to get up in front of the class and simply give it to them straight, no chaser, through direct instruction and a handout of questions I wrote.

What I find maddening about conversations with committed constructivists (cf. the conversation here) is the reflexive assumption that educators choose direct instruction because they’re either power-drunk or self-obsessed or because they lack faith, courage, or high expectations. I can’t, personally, wave so dismissively at the massive institutional impediments to student-constructed learning.

What’s the Problem with TERC’s “Investigations”?

Parents who have complaints about “Investigations” are not arguing for rote memorization of math facts or “mindless” drilling.  No one has suggested that considering the concepts behind math operations is a bad idea.  Students and parents have been frustrated by “Investigations” overemphasis on written explanations of thinking, lack of the practice that that is necessary for computational efficiency, excessive use of calculators, failure to introduce the standard U.S. algorithm, simplistic homework, and poor preparation for pre-algebra, algebra, and higher math classes.  For further background on how programs like “Investigations” came to be, read this historical account by Cal State-Northridge mathematician David Klein.  For detailed explanations and critiques of TERC’s “Investigations” follow the links in the sidebar at right, particularly Bill Quirk’s site and the video “Math Education: An Inconvenient Truth”.

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 less 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.

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.

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.

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.

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.

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