At tonight’s School Board meeting, Pam Francis, Marybeth Irvin, and Deirdre Bauer presented an update on the Elementary Math Program pilot that is currently underway throughout the District. A report was presented to the Board detailing pilot activities to date and then the Board members asked questions.  The plan appears to be that in mid-March District administrators will assess the data generated by the pilot, including teacher and principal feedback, assessments of materials, reports from site visits to other districts, and parent feedback.  In April, this assessment would be completed and a recommendation would be presented to the Board.

Penni Fishbaine asked how parent feedback would be weighted in the final assessment, and the response was that while parent feedback is important, there is no specific weighting of the various elements planned.

Dorothea Stahl asked if consideration was being given to how a new program would be phased in beginning next year.  It was stated in response that such considerations were important but that such planning would have to wait until a recommendation is approved.

Another Board update is scheduled for March 14.

The creators of strictly constructivist math programs like “Investigations” believe that too much attention is given to “rote memorization” of math facts and algorithms, when students should be spending their time reasoning and solving problems, helping each other with the teacher serving as a “facilitator”.  As TERC’s Susan Jo Russell put it,

The focus in the elementary classroom is shifting towards an emphasis on mathematical reasoning and problem-solving in a true sense — thinking mathematically in order to solve a problem that you do not know how to solve. In this view, what makes a problem a problem is that it is problematic for the person engaging in trying to solve it. Further, the [NCTM] Standards and other current reform documents (e.g., National Research Council, 1989, 1993) emphasize that in order to solve problems, students must learn to describe, compare, and discuss their approaches to problems. Alternative strategies are valued, and multiple strategies — rather than a single, sanctioned approach — are encouraged. In order to learn, students must learn from each other, as well as from the teacher’s questions. They must communicate about their mathematics.

This view is enormously appealing – it would be wonderful if we could skip the learning of basic knowledge and proceed directly to exciting, “real-world” problem solving – but research in cognitive psychology does not bear out these hypotheses.  Writing in the AFT’s American Educator, University of Virginia cognitive psychologist Daniel Willingham explains:

“Rote knowledge” has become a bogeyman of education, and with good reason. We rightly want students to understand; we seek to train creative problem solvers, not parrots. Insofar as we can prevent students from absorbing knowledge in a rote form, we should do so. I will address what we know about this problem, and how to avert it, in a future column.

But a more benign cousin to rote knowledge is what I would call “inflexible” knowledge. On the surface it may appear rote, but it’s not. And, it’s absolutely vital to students’ education: Inflexible knowledge seems to be the unavoidable foundation of expertise, including that part of expertise that enables individuals to solve novel problems by applying existing knowledge to new situations—sometimes known popularly as “problem-solving” skills.

So how can teachers help their students use inflexible knowledge to form the basis for solving novel problems?  Willingham has several suggestions, including the use of examples (which is minimized in Investigations) and not being afraid of teaching facts:

Knowing more facts makes many cognitive functions (e.g., comprehension, problem solving) operate more efficiently. If we minimize the learning of facts out of fear that they will be absorbed as rote knowledge, we are truly throwing the baby out with the bath water.

The Center for American Progress has created an online tool for displaying something it calls “return on educational investment” for every school district in the U.S.  The methodology is explained in detail on the website, but essentially they are plotting student achievement as measured by standardized test scores against per pupil spending.  To the extent that test scores indicate the “return”, these graphs show which district get more while spending less, etc.

So, how does SCASD do?  We are firmly in the first quadrant among Pennsylvania districts – higher than average spending with higher than average achievement.  Our district tends to benchmark its performance against that of 13 other districts whose size and demographics are similar to SCASD’s.  Here’s that list:

Abington SD
Bensalem Township SD
Cumberland Valley SD
East Penn SD
Hempfield SD
Lower Merion SD
Norristown Area SD
North Allegheny SD
Pennridge SD
Seneca Valley SD
Souderton Area SD
Spring-Ford Area SD
West Shore SD

The District has moved ahead with implementing the pilot testing plan that was approved by the Board of Directors by an 8-1 vote on November 8.  The lone vote against the pilot plan came from Penni Fishbaine, who strongly supports testing new programs but also strongly objects to one of the candidate programs – enVision Math – being pilot tested only in conjunction with the current core program, Investigations II.  Some Board members who voted for the plan also expressed reservations about this aspect of the pilot plan:

“I don’t see that as a head-to-head comparison,” board member Chris Small said of the plan, later adding, “I’m not inclined to support this recommendation, where enVision is paired up with Investigations.”

Details of the pilot, including a list of participating teachers, can be found on the SCASD K-6 math web page.  Several means for assessing the programs are planned, including teacher and parent feedback, a review of the materials themselves separate from classroom piloting and conducted by teachers, and student testing.  A letter sent last week to parents of students in pilot classrooms described the testing this way:

Student achievement data will be derived from a locally developed assessment before and after each pilot unit is taught. These tests will resemble those that students typically take as part of their regular math instruction.

The actual piloting is slated to begin in January, with ongoing collection of feedback and assessment data followed by a recommendation to the Board in May 2011.  If the District can keep to this schedule, it should be possible to have a new core math program in place (if that is what is ultimately decided) for the beginning of the 2011-2012 school year.

Lisa Kosanovic is a math teacher (trained as an engineer) and parent in Holyoke, MA, who has written a good opinion piece in her local newspaper.  Kosanovic is no fan of Investigations:

I dislike Investigations both as a teacher and as a parent: My students come to me severely deficient in basic skills (Holyoke teachers were prohibited from supplementing the curriculum with traditional materials), and my younger son is unchallenged.

She also not enamored of the politics of the “math wars”:

The new mantra is that students must have both [basic skills and problem-solving], but my sense is that while everyone is saying this, few people really mean it. Proponents of the traditional approach, who claim to value problem-solving, praise the private “academic enrichment” programs that teach what I call “math-in-a-box.” In these programs, kids do hundreds of problems that become increasingly difficult, but there is no sense of what the math means, or how to use it. These programs turn kids into little calculators; I can’t imagine why a child would have any interest in math after doing 300 division problems.

On the other side, I hear people who claim to care about computational skills argue that we should put calculators into kids’ hands to avoid wasting precious class time on rote skills as basic as addition and subtraction, all the way up to using calculators to take derivatives and integrals in calculus. These people don’t seem to realize — perhaps because no previous generation was instructed in math this way — that all those rote skills impart a valuable sense of numbers that students need to solve problems.

I’m not sure what the private programs are that Kosanovic does not like – maybe she is referring to programs like Kumon.

My experience (here in SCASD, at least) doesn’t correspond to Kosanovic’s observation that many critics of strict constructivist programs like “Investigations” want to replace such programs with drilling and rote memorization.  I’ve talked to many parents here in SCASD who want a math change and I can’t recall a single one who didn’t take Kosanovic’s position, that a good math education should give attention both to conceptual understanding and to developing skills.  We’ve been saying the same thing on this web site since it was begun over a year ago.

The Common Core Standards for Mathematics were finalized in September 2010 and have been adopted by the state of Pennsylvania. The grade 5 Number and Operations in Base Ten (5NBT) standards list: Perform operations with multi-digit whole numbers and with decimals to hundredths: 5. Fluently multiply multi-digit whole numbers using the standard algorithm.

Keiser (2010) helpfully reports on outcomes for middle school students who were instructed using Investigations in Number, Data and Space during their elementary school math classes. Her study reports on sixth- to eighth-grade students assessed with the following problem:

“Problem 2: Consider the problem 26 X 45. First estimate what you think the product would be. Then solve for the exact answer using two different methods” (p. 221).

Her findings are disturbing:
2 digit multiplication for pqme

Sadly, only 62.2% of the approaches applied by middle school students trained with Investigations correctly calculated the correct answer to this basic multiplication problem. No errors were made using the partial product method, but it was used in only 7.3% of the students’ attempts. The standard algorithm was attempted by only 12.6% of cases and 57.8% of those answers were incorrect. Similarly, incorrect answers were obtained in 31.0% and 38.5% of students’ application of the array and cluster methods, respectively.

Overall, only 54.7% of the middle school students correctly answered the question. That is absolutely pathetic. What this tells me is that the highly touted flexibility and multiple approaches to problem solving that Investigations stresses have failed to produce students who can solve simple math problems.

Moreover, if only 1 in 10 even attempted to use the standard algorithm, it is would seem nearly impossible that grade 5 students instructed via Investigations will be able to meet the simple standard above: Fluently multiply multi-digit whole numbers using the standard algorithm.

Fall conferences are a great time to catch up with your child’s teacher(s) and learn more about their academic progress and what’s happening in the classroom. While I am delighted with my child’s teacher and the work they are doing together, I cannot escape that nagging notion that the funkiness of Investigations and its assessments aren’t giving us an accurate reflection of kids’ skills. Our latest personal example is a grade 1 assessment in which kids were asked to give all the possible combinations that make up 10. My student’s answer was:

9+1=10 1+9=10 8+2=10
5+5=10 7+3=10 3+7=10
2+8=10 4+6=10

OK, so a couple numbers were written backwards and 0+10=10 and 6+4=10 were missed. What’s odd to me is that this approach isn’t considered as ‘advanced’ because no pictures were drawn or words written. The funny thing is that complete and accurate number sentences/equations were used — which is certainly more advanced than drawing circles around pictures.

A similar proficiency issue is mirrored in a recent article in NCTM’s electronic newsletter (“Shifting our computational focus”, v 16 no 4, August 2010). Math educator Jane Keiser details middle school teachers’ concerns with students skills after years of TERC Investigations instruction (but strangely she misses the real concerns evident).

“We heard teachers complain that they have noticed a change in their current students’ abilities. For example, one teacher mentioned that students in the sixth grade did not seem to know what the term long division meant. When she would probe her students’ thinking, she realized that students could perform division but that they used other strategies to accomplish the task. Teachers noticed unfamiliar computational strategies for other operations as well,
and it often took students longer to perform a computation. ” (p. 2)

Given these and other concerns, the middle school teachers did mini-assessments with their sixth through eighth graders.

“Our first surprise was the amount of time it took our students to perform one simple computation. Some teachers reported that the following problem took the entire forty-two-minute class period for students to complete and share.”

Problem 1: Consider the problem 13 × 7. Create a real-life situation when you would use multiplication to solve this problem. Solve the problem, and be sure to show your work.

A whole period? This should take no more than 10 minutes to do and share. Of one teacher’s 90 students, here’s what they did to solve the problem:

Investigations clearly teaches and encourages use of arrays and clusters rather than the traditional algorithm. Sadly, approximately 1 in 5 was still using repeated addition in middle school. That’s a bit frightening, and if a whole period is required to answer a simple multiplication problem, proficient and efficient problem solving is not a high priority.

The more things change, the more they stay the same. Here’s a headline from the Marco Eagle in Marco Island, FL, which adopted Investigations 2nd edition two months ago:

“Creative solutions: Math just got a little easier at Marco Island’s TBE”

… and here’s a headline from the Buffalo News about the reactions of parents and mathematicians in Buffalo and Amherst, N.Y., which adopted Investigations 2 two years ago:

“Math forum to focus on curriculum”

The SCASD Elementary Math Program Review Committee concluded its work last night at a meeting that began with the last publisher presentation, by Math Expressions reps, and ended with a long discussion of which programs should go forward in a pilot-testing process that the District is currently working to define.

The Superintendent asked that the Committee’s recommendations be a true consensus, which is to say accepted by 100% of the committee members.  Remarkably, this was achieved last night (well, almost – one person abstained, neither rejecting nor approving the recommendation).  The Committee’s recommendation was for the District to evaluate the following programs:

1. Investigations (2nd ed.): This is the District’s current program and the Committee affirmed that it was our understanding from the start that any piloted programs would be weighed against the current program.

2. enVision Math: The Committee chose this program for piloting with the strong recommendation that the District consider supplementing enVision with Investigations materials or combining the programs in some way.  Such supplementation is facilitated by the fact that both are published by Pearson, who have created a “Joint-Usage Master Plan” so that Districts can do this easily.

3. Math Expressions: This program was specifically requested by inclusion by several SCASD teachers on the Committee.  Like enVision, it has a textbook, but it is probably not right to call it (or enVision any of the programs considered) a “traditional” program.  The development of all of these programs (I think) was funded by NSF and they all are authored by people who were instrumental in creating the NCTM Standards and Focal Points that have formed the basis of “reform” math programs that emphasize conceptual understanding.  Expressions and enVision differ from Investigations in that they seem to rely more on direct instruction in their materials and have some of the trappings that many will associate with a more “traditional” program, such as textbooks with worked examples.

The math publisher presentations scheduled for Wednesday evening have been pushed back to accommodate those who also want to attend the homecoming parade.  The first presentation on Wednesday will be the rep from McGraw Hill presenting Everyday Math at 7:30 pm.   The second presentation will be the rep from Houghton Mifflin presenting Math Expressions at 8:45 pm.

The location for the presentations is unchanged: the cafeteria at Mount Nittany Middle School, 656 Brandywine Drive in State College.

UPDATE: It now appears that the Math Expressions presentation for tonight has been canceled and possibly will be rescheduled.  EDM is still on for 7:30 pm.

UPDATE 2: The Math Expressions presentation has been rescheduled for Wednesday at 6:30 pm.