Concepts BEFORE Procedures

You Do The Math: Explaining Basic Concepts Behind Math Problems Improves Children's Learning

ScienceDaily (2009-04-12) -- Students benefit more from being taught the concepts behind math problems rather than the exact procedures to solve the problems. The findings offer teachers new insights on how best to shape math instruction to have the greatest impact on student learning.

2nd Math Task Force Meeting

The 2nd NJDOE Math Task Force meeting has concluded. The morning was spent discussing common ground on calculators and alternative algorithms. After much discussion, a few sentences were agreed upon with respect to calculators. The agreement was:
Students should be able to calculate (+, - , x, ÷ ) without calculators
Students should develop proficiency with mental mathematics
Judicious use of calculators
Students should select appropriate tools

Alternative algorithm discussion continued without a clear common ground about the use of "the" standard algorithm.

In the PM, three groups were formed which each group editing and analyzing one standard (number, geometry, and algebra only). These groups will continue their work at the next (and final) meeting on April 20th.

What happens after April 20th is anyone's guess.
One DOE senior staffer even said and I quote: "We don't have a plan."
Another DOE senior staffer said: "A writing committee might been formed to write another set of standards."

Anti-Reformists & Common Ground: Not Likely

The purpose of the NJDOE task force is to reach common ground on the ongoing math discussion in NJ.  I am beginning to believe, however, that reaching common ground is not really the goal of the anti-reformists in NJ.  To date, I have resisted direct confrontation with these anti-reformists.  However, they have now decided to call my University (several times) and now have made open public records act (OPRA) to find out information about my grants.  Although I have nothing to hide and am very proud of my grants, why would the anti-reformists make such requests?  (To learn more about my grants, see newspaper articles in the Courier and the Chronicle.)   One can only imagine that the anti-reformists want to use such information to disqualify me from the NJDOE task force (even though DOE and the Commissioner is well aware of my successful grant activity.)  Seems that they really don't want to reach consensus, they want to remove anyone who disagrees with them.  There is further evidence of this as below appears an email that I received from one of the anti-reformists.  This email was received when the Concerned Mathematics Educators of NJ was initial formed.  The rhetoric is both disturbing and distressing and show the lengths that the anti-reformists will resort to.  Common ground is certainly not their goal.

I have reviewed the information on your web site and would like to comment.  Given the name of your organization, it is not surprising that the vast majority of those endorsing the positions on your web site are not individuals who are practicing mathematicians, engineers or scientists.  This is a serious flaw.
As educators, your role should be one of teaching, and not driving, the curriculum.  The curriculum goals should be driven by the expected outcomes, which in this case are success in the fields dependent upon proficiency in mathematics.   I find it abhorrent that mathematics educators are hoping to continue to dictate the content of math standards that will drive curricula throughout the state of NJ.  It is becoming increasingly evident that there is a clear divide between those who teach mathematics (or teach mathematics teachers), and those who actually use mathematics in their daily life as scientists, engineers, or in other professions. 

The failures in this educator-driven approach to curriculum are reflected in the continued decline of the United States dominant position in science, engineering and mathematics. Since the NCTM recommendations of ~20 years ago were developed into various curricula, the level of mathematics competency within the US has been abysmal.  Now, with the State of NJ poised to draft new core curriculum content standards for math that eschew many of the components of the NCTM recommendations, your group hopes to continue to hold NJ's students back by holding fast to the status quo.

Your listing of endorsers includes some individuals who have financially benefited from the standards development process.  These individuals have engaged in activities such as:  received grants, acted as consultants for school districts and/or the State of NJ to draft or select standards and/or curricula, or provided training to teachers and administrators regarding the selected standards or curricula.  In other industries, a situation where those who help to make policy then also receive financial benefit as a result of those policies could be considered a conflict of interest.  This is truly distressing. 

I urge the members of your organization to actually LISTEN to the parents, the taxpayers, the mathematicians, the scientists, the engineers, the IT professionals, the economists and others in NJ who are becoming increasingly concerned with the standards that your organization wishes to perpetuate.  I urge the members of your organization to review the studies showing a disconnect between what math educators believe to be important and what is actually important.  It's time for your organization and its membership to realize that the standards that you wish so dearly to hold onto are resulting in NJ's students being poorly-equipped for college or life.  

Please, please, please--for the sake of our children, STOP the pursuit of sub-standard mathematics standards and curricula and partner with parents, mathematicians, scientists and engineers to provide a solid mathematics education for NJ's students.

What is TRULY distressing is the rhetoric and tactics used by the anti-reformists. Common ground - very unlikely.

Constructivism vs Rote Learning in Science

A conversation with Dr. Eric Mazur (the Balkanski Professor of Physics and Applied Physics at Harvard University and an internationally recognized scientist and researcher) entitled Using the ‘Beauties of Physics’ to Conquer Science Illiteracy by Claudia Dreifus appeared in the NY Times in July 2007.

Dr. Mazur discusses rote memorization versus constructivism in his University classroom.

Dr. Mazur states:

It’s important to mentally engage students in what you’re teaching, he explains. We’re way too focused on facts and rote memorization and not on learning the process of doing science. From what I’ve seen, students in science classrooms throughout the country depend on the rote memorization of facts. I want to change this. The students who score high do so because they’ve learned how to regurgitate information on tests. On the whole, they haven’t understood the basic concepts behind the facts, which means they can’t apply them in the laboratory. Or in life.
Today, by having the students work out the physics problems with each other, the learning gets done. I’ve moved from being “the sage on the stage” to “the guide on the side.”

National Math Panel Part2

The National Math Panel report continues to get huge "play" by the anti-reformists.  However the report is far from accurate.  Another critique of the report can be found by Jere Confrey (pictured), Alan P. Maloney, and Kenny H. Nguyen entitled Breaching the Conditions for Success for a National Advisory Panel. Confrey, Malonet and Nguyen identify six conditions for success for the work of high-level national panels and identify breaches in these conditions in the recent Foundations for Success: The Final Report of the National Mathematics Advisory Panel (2008). They question the trustworthiness, validity, and intellectual integrity of its findings and advice to the nation because of (a) an inappropriate composition of Panel expertise and biased selection of literature (Condition 1); (b) failure to appropriately and consistently apply methodological standards (Condition 2); and (c) inconsistencies between the task group and subcommittee reports and the final report (Condition 6). In asking what difference these breaches make, the authors recount recent events suggesting that these breaches have already contributed to degrading the national discussion of curriculum standards for K–12 mathematics education.

First, they review the composition of the panel members as follows:

The responsibility for the composition of the NMAP was unusual, considering the charge. The chair of the Panel was a chemist and distinguished emeritus university president. The other Panel members included six psychologists, four mathematics educators (one of these added during the final year), four mathematicians, one special educator, one middle school mathematics teacher, one policy researcher, and one reading researcher (U.S. Department of Education, 2008). This means that, of 19 members, only 5 (the mathematics educators and the mathematics teacher) regularly had sustained interactions with mathematics instruction at the K–12 level; fewer than half the Panel members had documented academic preparation in mathematics.

And what about the research cited by the report:

Despite the NMAP report’s focus on the learning of mathematics, of the total of 466 journal articles referenced in the task group’s report, mathematics education articles constituted only slightly more than 10%, whereas psychological accounts (i.e., those published in journals on psychology, child development, and educational psychology) constituted approximately 70% of the references. Of the 10 journals with 10 citations or more each, 9 (90%) were psychology-oriented.3 Only one leading research journal in mathematics education had more than 10 citations.

And the recommendations & conclusions:

Perhaps even more fundamental, however, are the ways in which the breaches of the conditions for success may have distorted the particular recommendations in the report. It will take much more time and a variety of critical reviews to discern the extent of the damage in this regard. Because the U.S. Department of Education was responsible for the composition and the oversight of the NMAP, it should be held responsible for breaching the conditions for success of the Panel and, in effect, undermining the potential benefits to the country of this endeavor. The result was a loss of opportunity for improving mathematics education in our nation.

Fordham Institute Nonsense

Many anti-reformists often refer to a report entitled The State of State Standards 2006 by Fordham Institute as evidence of the failure of NJ Mathematics Standards. of course these same anti-reformists always ignore the tremendous performance of NJ students on the NAEP (but that will be a future post). The Fordham report has been reviewed by The Education Policy Research Unit at the Arizona State University. The full review can be found here. An excerpt is below.

The reviewed report assigns grades to the content standards of 49 states and the District of Columbia, on an A-F scale, and uses those grades as a basis for criticizing schools for lack of progress in improving standards. This review found no evidence supporting the validity of the grades and also found no evidence of a relationship to student academic performance, contrary to the report’s conclusions. The report’s claims in support of its grading practice were selectively data-mined and were seriously lacking in methodological rigor. Policymakers and educators would be ill-advised to base any decisions about policy or practice on the grades assigned by this report.

In summary, these three analyses were selectively mined from data gathered by Fordham – data which themselves are flawed and for which there is no evidence of validity. No rationale for Fordham’s unorthodox ad hoc analyses is provided, and those analyses are sorely lacking in methodological rigor. Indeed, the post-hoc massaging of the data reaches the point of absurdity, as the authors search for some approach to the data that might lend support to Fordham’s conclusion that content standards of the kind it rates highly do result, in fact, in improved student performance

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Response to the National Math Panel

As the NJDOE math task force gets ready to meet for a second time next Tuesday, some members believe that NJ must follow the National Math Panel (NMP) report and recommendation. However, the NMP report was not exactly a balanced report by any means. Jo Boaler wrote a wonderful review of the report. Dr. Boaler is the Marie Curie Professor of Mathematics Education at the University of Sussex in England. Her article (When Politics Took the Place of Inquiry: A Response to the National Mathematics Advisory Panel’s Review of Instructional Practices) uncovers some of the politics behind the report. Boaler writes:

One of the most critical areas the panel chose to consider was the impact of different teaching approaches on student learning. In framing its review, the task group posed this question: “How effective is teacher-directed instruction in mathematics in comparison to student-centered approaches?” (Gersten et al., 2008, p. 12). The task group’s assertion that student-centered teaching involves teachers handing over the teaching of mathematics to the students is remarkable. Indeed the task group itself acknowledged that such definitions are “extreme” (Gersten et al., 2008, p. 30). The task group admits that it found no studies in which students were doing the teaching of mathematics, and it quotes from the National Research Council’s report Adding It Up(Kilpatrick et al., 2001), summarizing the different and complex forms of teaching in which student-centered teachers engage. The idea that showing and explaining would be absent from student-centered teachers’ pedagogy suggests that the task group members either held deep misunderstandings about student-centered teaching or that certain members of the group were engaged in a more politicized exercise. Even the strongest advocates of different teaching approaches would probably find the task group’s trivialized definitions of teaching to be inadequate, making any subsequent reviews of research redundant. The task group’s advocacy for an “extreme” and unrealistic definition of student-centered teaching was blamed on conversations held with teachers. This is particularly ironic given the Panel’s admonition to the readers of its report to recognize only experimental evidence.

Boaler continue her review of the report and its recommendation by stating:

More seriously perhaps, the National Mathematics Advisory Panel’s report presents a case of a government controlling not only the membership of a panel chosen to review research—a panel dominated by educational conservatives rather than mathematics researchers—but the forms of knowledge admissible in the public domain. In its adherence to government directions (Gersten et al., 2008, p. 213), resulting in the disregard of the field of mathematics education research, the Panel’s report communicates the view that the government, rather than academic researchers, should decide on the forms of knowledge that are legitimate in our pursuit of understandings about ways to help children learn (U.S. Department of Education et al., 2003). When governments step in to control research and knowledge production, limiting the methods used by researchers and the forms of knowledge acceptable, to the extent that a whole field of research is invalidated, then it is time to acknowledge that America’s celebrated freedom—of thought and inquiry—has been dealt a very serious blow.

If you want a copy of the full article, just send me an email and I will send you the file.

Balanced?

The Star Ledger ran an article on Sunday entitled: A new approach to an old math problem (The state looks to balance traditional and reform methods to improve learning). Once again the Commissioner was quoted as stating:

"I think there was a movement away from children mastering the basics of math -- the basics of division, multiplication, the basics around percentages. We began to rely on calculators," Davy said. "We need to wind up with a balance."

With all due respect, Mrs. Davy, movement away from mastering the basics? Where? I have visited hundreds and hundreds of classrooms across NJ and there is no elementary school teacher who has moved away from kids learning their basic facts. No where, no place. Such rhetoric is very disappointing and it shows a complete lack of reality of what is happening in the field. She also stated we began to rely on calculators? Who is the we? Do you blame the teachers of NJ? Do you blame their students? Calculator usage has always been balanced in the NJ standards. Always. It always seems to come back to calculators. That somehow these $1 tools cause students to not master their basic facts. It is such a sad and tired argument. Cathy Liebars, in contrast to the Commissioner, states:

"The traditional way of teaching math is to drill them until they memorize. That produced a lot of adults who hate math or don't understand math," said Cathy Liebars, a mathematician at the College of New Jersey and one of the founders of the Concerned Math Educators. "Kids don't just have to memorize things. They can learn basic facts while they are solving problems." And calculators help students "when the focus is not on doing the computing but on how to figure out how to solve the problem," Liebars said. "Teachers help them understand when it is appropriate."

I only wish the Commissioner was so eloquent.

Algebra for All?

It is very easy for politicians and various parental groups to push for more algebra (algebra for all in 8th grade and algebra II for all in high school) and the rhetoric sounds so nice. But, the reality is much different. A March 2009 study in Education Week highlights this. The article: Algebra-for-All Policy Found to Raise Rates Of Failure in Chicago states:

The Chicago school district was at the forefront of that movement in 1997 when it instituted a mandate for 9th grade algebra as part of an overall effort to ensure that its high school students would be “college ready” upon graduation.
The policy change may have yielded unintended effects, according to researchers from the Consortium on Chicago School Research, based at the University of Chicago. While algebra enrollment increased across the district, the percentages of students failing math in 9th grade also rose after the new policy took effect.
By the same token, the researchers say, the change did not seem to lead to any significant test-score gains for students in math or in sizeable increases in the percentages of students who went on to take higher-level math courses later on in high school.
“This policy that Chicago tried in 1997 seems to be sweeping the country now and not a lot of thought is being given to how it really affects schools,” Elaine M. Allensworth, the lead researcher on the study, said in an interview.

The simplistic rhetoric of algebra for all and high standards for all does not address the mathematical readiness of students and the often irrelevant traditional algebra curricula. We must stop the rhetoric and discuss the real issues such as a curriculum that addresses the needs of the 21st Century (contextual application not rote drill) with the full integration of technology.

The evils of Technology

As I work on the draft of a white paper on calculators for the NJDOE math standards task force, I am reminding that the discussion about calculators can be extended to any technology including the Internet. Wired Magazine had a wonderful article: The Critics Need a Reboot. The Internet Hasn't Led Us Into a New Dark Age.

When in doubt, blame the latest technology. Socrates thought the advent of writing would wreak havoc on the powers of the mind.

I think Socrates was wrong and those who blame calculators ...