Monday, May 16, 2011

The Case for Standard Algorithms

One of the most controversial tenets of constructivism is that teaching basic skills to mastery with the use of the standard algorithms prevents children from developing a conceptual understanding of the broader issues. Quoting Constance Kamii from her 1997 article in the Constructivist magazine (Kamii, 1997):

“the teaching of [standard] algorithms is harmful for two reasons: a) Algorithms force children to give up their own thinking, and b) they “unteach” place value and therefore hinder children’s development of number sense.”

One recognizes here the dilemma raised by the cognitive load theory: storing basic skills in long term memory vs. overloading the working memory with problem-solving search. Prof. Hung-Hsi Wu of the University of California at Berkeley summarizes this sequencing issue (Wu 1999):

“This bogus dichotomy would seem to arise from a common misconception of mathematics held by a segment of the public and the education community: that the demand for precision and fluency in the execution of basic skills in school mathematics runs counter to the acquisition of conceptual understanding …

The precision and fluency in the execution of the skills are the requisite vehicles to convey the conceptual understanding … It is the fluency in executing a basic skill that is essential for further progress in the course of one’s mathematics education. The automaticity in putting a skill to use frees up mental energy to focus on the more rigorous demands of a complicated problem.”

To address why the four standard algorithms of arithmetic (addition, subtraction, multiplication, and long division) should be preferred and taught in all programs and schools, a committee of the American Mathematical Society (AMS) wrote a report in 1998 protesting the lack of rigor in all NCTM-approved reform math programs. The following report excerpt is from a letter signed by over 200 eminent mathematicians/scientists and sent to then Secretary of Education Richard Riley (see Okun 2001 and Milgram 2000):

“We would like to emphasize that the standard algorithms of arithmetic are more than just "ways to get the answer"–that is, they have theoretical as well as practical significance. For one thing, all the algorithms of arithmetic are preparatory for algebra, since there are (again, not by accident, but by virtue of the construction of the decimal system) strong analogies between arithmetic of ordinary numbers and arithmetic of polynomials. The division algorithm is also significant for later understanding of real numbers.”

Moreover, Prof. H. Wu asks whether the two fundamental principles of any algorithm can be ensured in today’s classrooms. Clearly, the average class size makes constructivist methods impractical at best and dangerous at worst (Wu 1999):

“What is left unsaid is that when a child makes up an algorithm, the act raises two immediate concerns: One is whether the algorithm is correct, and the other is whether it is applicable under all circumstances. In short: correctness and generality. In a class of, say, 30 students, asking the teacher to carefully check 30 new algorithms periodically is a Herculean task. More likely than not, some incorrect algorithms would slip through, and these children would come out of this encounter with mathematics with no understanding at all.”

Finally, and irrespective of class-size and other constraints, Mr. Ocken says (Ocken 2001):

Throughout history, it has been a challenging task for adult mathematicians to devise reliable and efficient algorithms that apply to general classes of operands. Any claim that today’s K-8 students can do the same should be subjected to the most intense scrutiny.”

In other words, why reinvent the wheel?

The Case against Constructivism and for Direct Instruction

In the last post we saw that Constructivism revolved around three main ideas: 1) Let children ask their own questions and construct their own solutions; 2) Provide little if any guidance and do not correct them if/when wrong; and 3) Do not teach traditional algorithms as they automatically annihilate all independent thinking.

To present the case against constructivism, I will address the first two points in this post and address the third point on traditional algorithms in a separate post.

1)     Minimal Guidance vs. Direct Instruction

In 2006, a paper written by Kirschner, Sweller, and Clark (KSC 2006) generated a swell of controversy over these very theoretical and philosophical issues. The reader is urged to read the series of papers and commentaries located at the bottom of this link Direct Instruction versus Constructivism Controversy.

The constructivist theory rests on the belief that learning and human cognition is shaped mainly through personal discovery, independent problem-solving and free thinking. Science has shown this to be true in the acquisition of “biologically primary knowledge”. For example, we learn to speak, listen, recognize faces, and interact with others very early on and mostly through immersion and experimentation in various social environments… not through constant and explicit instructions from parents/guardians/etc.

While that might be true of primary knowledge, constructivists have taken it one step further and believe the same is true for “biologically secondary knowledge” which includes most everything else.

And that’s where the rubber meets the road…

A significant body of research, including numerous controlled empirical studies, has thoroughly discredited that constructivist belief. According to KSC (2007):  

“There is no theoretical reason to suppose or empirical evidence to support the notion that constructivist teaching procedures based on the manner in which humans acquire biologically primary information will be effective in acquiring the biologically secondary information required by the citizens of an intellectually advanced society. That information requires direct, explicit instruction.”


“… we have not evolved to effortlessly acquire the biologically secondary knowledge such as the operation of a base 10 number system or scientific theories that are characteristically taught in educational institutions. That information passes through working memory and so requires conscious effort. It must be explicitly taught; indeed we invented educational institutions in order to teach such knowledge, and the manner in which it is taught needs to take into account the characteristics of working memory, long-term memory and the relations between them.”

In other words, constructivists rely on a misconception of how differing instructional and pedagogical methods interact with our current knowledge of the “human cognitive architecture”, especially in the case of young students.

2)     Cognitive Load Theory

Over the past half century, research on “cognitive load theory” has shown that we learn and recall best when our long-term memory is amply supplied with clear instructions on how to solve problems rather than when we overload our working memory by attempting to solve problems creatively without having relevant information stored in our long-term memory.

As per KSC (2006):

“Our understanding of the role of long-term memory in human cognition has altered dramatically over the last few decades… long-term memory is now viewed as the central, dominant structure of human cognition.

Working memory is the cognitive structure in which conscious processing occurs. Working memory has two well-known characteristics: when processing novel information, it is very limited in duration and in capacity.

The limitations of working memory only apply to new, yet to be learned information that has not been stored in long-term memory. In contrast, when dealing with previously learned information stored in long-term memory, these limitations disappear.”

“Solving a problem requires problem-solving search and search must occur using our limited working memory. Problem-solving search is an inefficient way of altering long-term memory because its function is to find a problem solution, not alter long-term memory. Indeed, problem-solving search can function perfectly with no learning whatsoever (Sweller, 1988). Thus, problem-solving search overburdens limited working memory and requires working memory resources to be used for activities that are unrelated to learning. As a consequence, learners can engage in problem-solving activities for extended periods and learn almost nothing (Sweller et al., 1982).”

The fact that the constructivist pedagogical methods rely almost entirely on “working memory” at the expense of “long-term memory” are (or should be) their fatal flaw.

“These memory structures and their relations have direct implications for instructional design.

Recommendations advocating minimal guidance during instruction proceed as though working memory does not exist or, if it does exist, that it has no relevant limitations when dealing with novel information, the very information of interest to constructivist teaching procedures.

Any instructional recommendation that does not or cannot specify what has been changed in long-term memory, or that does not increase the efficiency with which relevant information is stored in or retrieved from long-term memory, is likely to be ineffective.”

And after sparring with reformists (Schmidt and Hmelo-Silver) on these issues, KSC (2007) reiterates the finding that:

“After a half-century of advocacy associated with instruction using minimal guidance, it appears that there is no body of research supporting the technique. In so far as there is any evidence from controlled studies, it almost uniformly supports direct, strong instructional guidance rather than constructivist-based minimal guidance.”

Finally, and this confirms a lot of anecdotal evidence heard and read about, constructivist reform programs leave behind a lot of children that may not have “constructed” the right solution in the first place and may also reinforce mistakes made at the outset.

Again from KSC 2006:

“… Not only is unguided instruction normally less effective; there is also evidence that it may have negative results when students acquire misconceptions or incomplete or disorganized knowledge.

Note that some parents ask whether the two instructional methods can be combined to extract and deliver the best of both. When I asked Prof. R. James Milgram, Mr. Ze’ev Wurman, and other math experts that question, they all replied in the negative mainly because of the inherent pedagogical contradictions between the two methods. The post on Curricular Spiraling shows that its difficult if not impossible to combine a pedagogical method based on mastery-first with one based on incremental-learning over years. 

Friday, May 13, 2011

The Constructivist School of Thought

Parents from my daughter’s school and our local park often ask me what are the philosophical and scientific underpinnings of the so-called “Constructivist” school of thought. It seems odd to them that these reformists could be so misguided in their beliefs and yet be so successful in the “math wars”. After all it has been almost three decades since their agenda has taken hold – certainly since the “NCTM standards” were nationally implemented in the early 1990s. Surely, these parents believe, the reformists must be onto something good and must have the best of intentions in mind. In other words, there must be significant research proving their agenda correct and the leaders of the education community must know better what is best for our children’s education than mere parents or teachers bent on old-school traditionalism…

I appreciate these questions because I battled with them myself (and still do on many issues). It does seem unbelievable that the reformists would be so misguided for so long and yet still be all around… but after having read many papers, reports, surveys, etc. not only am I more skeptical of the constructivist philosophy but I am now suspicious of and opposed to their opaque agenda. Their willful disregard for all expert opinions and empirical evidence militating against their programs is borderline criminal and definitely unethical in my book. While another post will address why I think the reformists have and still thrive despite their horrendous track record, this post presents the Constructivists’ main arguments and the next post will offer the case against.

This presentation relies on a paper by Constance Kamii published in The Constructivist in 1997 and titled “52 X 8: The Importance of Children's Initiative”. Please note there are many more primary sources interested readers should review. I selected this one because it neatly summarizes the main points but mainly because it was regularly distributed in the early 2000s by Manhattan District 2 Math Director Lucy West to parents at information meetings held by Ms. Karen Feuer, President of Community School Board 2, to educate school parents on the philosophy of TERC Investigations.

Today, and this is the interesting part, Ms. Feuer is principal of PS110, my daughter’s school, and Mrs. Lucy West has just been hired as our math consultant to supplement our Everyday Mathematics curriculum with another reform math program called Math in the City. How interesting can it get? The same cast of characters, in different roles 12 years later, and still promoting the same reform math programs… in fact, promoting new math reform programs to “supplement” older math reform programs… can you square that circle?

The reformist school of thought is generally called “Constructivist”… because it is largely based on Jean Piaget’s theory of constructivism which holds that children are better wired to arrive at or “construct” their own solutions to problems rather than being passive recipients of direct instructions from a teacher standing at the blackboard ruler in hand…

As Ms. Kamii says (pp. 7-13):

Piaget showed that children acquire logico-mathematical knowledge not by internalizing rules from the outside but by constructing relationships from within, in interaction with the environment.”


Children’s minds do not work in the fragmented manner by which textbook writers organize their texts. Children go much farther and more naturally, with greater joy, if they are encouraged to pose their own questions and answer them in their own way.”

Thus the question becomes how do teachers foster children’s inherent ability to construct their own solutions? Ms. Kamii lists the 3 main constructivist steps:

1)       “… by picking up on what they say... the questions they ask are more developmentally appropriate than those found in textbooks… because they come out of their level of thinking.”

2)       “… by refraining from teaching conventional algorithms and, instead, encouraging children to invent their own procedures for solving problems. Algorithms force children to give up their own thinking.

Here it’s important to note that the authors say “the only children who have not been crippled by conventional algorithms are the brightest, most advanced minority in each class, who could make sense of the algorithms.” This advanced minority, for the authors and the reformists in general, is male, white or Asian. As we will see in another post, this is important to understand because it explains why reformists benefit from such wide political support

3)       “… by refraining from saying that an answer is correct or incorrect and, instead, encouraging children to agree or disagree among themselves. When the teacher decrees that an answer is correct, all thinking and all initiative stop.

And there you have it… the three guiding principles of the constructivist school of thought that stand diametrically opposite of what most of us learned and how we were taught in school. Not that that is necessarily bad in and of itself… yet the fact that it seeks to destroy what has been tested and fine-tuned over centuries of teaching with good results seems counter-productive and irresponsible, especially in light of the disastrous math test results and international rankings this generation of students has achieved under this reform.

Clearly something is not working as planned: 30 years and bad results should provide ample evidence that its time to stop this experiment and change course… but it doesn’t. In fact, the reformists have more power and influence than ever. Why? That is the real question.

My next post will present the case against Constructivism.

Wednesday, April 20, 2011

Prof. R. James Milgram rejects the adoption of Core Standards in Texas … or any state

Prof. R. James Milgram testified last week in support of a bill that would prevent Texas from adopting the Common Core State Standards. Why?

First, although he was a member of the CCSS Validation Committee, Prof. Milgram says:

“There are a number of extremely serious failings in Core Standards that make it premature for any state with serious hopes for improving the quality of the mathematical education of their children to adopt them.”

Second, Texas is considered to have finalized one if not the top mathematics standards of all states in the country. Therefore adopting the Core Standards would be a downgrade.

So what about New York?

Texas House of Representatives

Committee on State Sovereignty

Meeting: 04/14/11
10:30a.m. - 7:02p.m.

Written Testimony of R James Milgram, Professor Emeritus, Stanford University, member of the Common Core Standards Validation Committee

I would like to testify in support of the bill Rep. Huberty filed, HB 2923, to prevent the so called Core Standards, and the related curricula and tests from being adopted in Texas.

My Qualifications. I was one of the national reviewers of both the first and second drafts of the new TX math standards. I was also one of the 25 members of the CCSSO/NGA Validation Committee, and the only content expert in mathematics.

The Validation Committee oversaw the development of the new National Core Standards, and as a result, I had considerable influence on the mathematics standards in the document. However, as is often the case, there was input from many other sources – including State Departments of Education – that had to be incorporated into the standards.

A number of these sources were mainly focused on things like making the standards as non-challenging as possible. Others were focused on making sure their favorite topics were present, and handled in the way they liked.

As a result, there are a number of extremely serious failings in Core Standards that make it premature for any state with serious hopes for improving the quality of the mathematical education of their children to adopt them. This remains true in spite of the fact that more than 35 states have already adopted them.

For example, by the end of fifth grade the material being covered in arithmetic and algebra in Core Standards is more than a year behind the early grade expectations in most high achieving countries. By the end of seventh grade Core Standards are roughly two years behind.
  • Typically, in those countries, much of the material in Algebra I and the first semester of Geometry is covered in grades 6, 7, or 8, and by the end of ninth grade, students will have finished all of our Algebra I, almost all of our Algebra II content, and our Geometry expectations, including proofs, all at a more sophisticated level than we expect.
  • Consequently, in many of the high achieving countries, students are either expected to complete a standard Calculus course, or are required to finish such a course to graduate from High School (and over 90% of the populations typically are high school graduates).
Besides the issue mentioned above, Core Standards in Mathematics have very low expectations. When we compare the expectations in Core Standards with international expectations at the high school level we find, besides the slow pacing, that Core Standards only cover Algebra I, much but not all of the expected contents of Geometry, and about half of the expectations in Algebra II. Also, there is no discussion at all of topics more advanced than these.

Problems with the actual mathematics in Core Math Standards As a result of all the political pressure to make Core Standards acceptable to the special interest groups involved, there are a number of extremely problematic mathematical decisions that were made in writing them. Chief among them are:

1. The Core Mathematics Standards are written to reflect very low expectations. More exactly, the explicitly stated objective is to prepare students not to have to take remedial mathematics courses at a typical community college. They do not even cover all the topics that are required for admission to any of the state universities around the country, except possibly those in Arizona, since the minimal expectations at these schools are three years of mathematics including at least two years of algebra and one of geometry.
  • Currently, about 40% of entering college freshmen have to take remedial mathematics.
  • For such students there is less than a 2% chance they will ever successfully take a college calculus course.
  • Calculus is required to major in essentially all of the most critical areas: engineering, economics, medicine, computer science, the sciences, to name just a few.
2. An extremely unusual approach to geometry from grade 7 on, focusing on rigid transformations.  It was argued by members of the writing committee that this approach is rigorous (true), and is, in fact, the most complete and accurate development of the foundations of geometry that is possible at the high school level (also probably true).  But
  • it focuses on sophisticated structures teachers have not studied or even seen before.
  • As a result, maybe one in several hundred teachers will be capable of teaching the new material as intended.
  • However, there is an easier thing that teachers can do – focus on student play with rigid transformations, and the typical curriculum that results would be a very superficial discussion of geometry, and one where there are no proofs at all.
Realistically, the most likely outcome of the Core Mathematics geometry standards is the complete suppression of the key topics in Euclidean geometry including proofs and deductive reasoning.

The new Texas Mathematics Standards

As I am sure you are aware, Texas has spent the past year constructing new draft mathematics standards, and I was one of the national reviewers of both the first and second drafts. The original draft did a better job of pacing than Core Standards, being about one year ahead of them by the end of eighth grade, so not nearly as far behind international expectations. Additionally, they contained a reasonable set of standards for a pre-calculus course, and overall a much more reasonable set of high school standards.

There were a large number of problems as well – normal for a first draft. However, the second draft had fixed almost all of these issues, and the majority of my comments on the second draft were to suggest fixes for imprecise language and some clarifications of what the differences are between the previous approaches to the lower grade material in this country and the approaches in the high achieving countries.

It is also worth noting that the new Texas lower grade standards are closer to international approaches to the subject than those of any other state.

I think it is safe to say that the new Texas Math Standards that are finally approved by the Texas Board of Education will be among the best, if not the best, in the country. (I cannot say this with complete certainty until I have seen the final draft. But since I am, again, one of the national reviewers, this should be very soon.)
So it seems to me that you have a clear choice between
  • Core Standards – in large measure a political document that, in spite of a number of real strengths, is written at a very low level and does not adequately reflect our current understanding of why the math programs in the high achieving countries give dramatically better results;
  • The new Texas Standards that show every indication of being among the best, if not the best, state standards in the country. They are written to prepare student to both enter the workforce after graduation, and to take calculus in college if not earlier. They also reflect very well, the approaches to mathematics education that underlie the results in the high achieving countries.
For me, at least, this would not be a difficult choice. So for these many reasons I strongly support HR 2923, and hope the distinguished members of this committee will support it as well.

Respectfully, R. James Milgram

Mr. David Steiner: Do constructivist/progressive education policies best serve our children?

Mr. Steiner resigned 3 weeks ago as NY School Commissioner along with Mrs. Cathie Black, the NYC School Chancellor. While Mr. Steiner's reasons for resigning, after only 2 years, are hotly debated in the education blogosphere, it is clear that he resented the education graduate schools' one-sidedness in favor of the constructivists/progressivists. Did that position presage his early exit?

Copied below is an article written by David M. Steiner in 2004, while at Boston University.

The New York Sun, May 27, 2004

David M. Steiner - Mr. Steiner is the chairman of the Department of Administration, Training, and Policy Studies at the School of Education, Boston University

Are you forever grateful to that teacher who stood out from the rest? Your gratitude is well founded: Research confirms what common sense tells us: that even a single talented teacher can make a profound difference.

Small classes, modern facilities, and equipment, and all the other things we look for in a school, have much less effect on student achievement than the quality of the teachers. Even poverty becomes less important when good teachers are placed in the classrooms of disadvantaged students.

What distinguishes a good teacher? Here, too, research confirms common sense. Teachers who are smart, highly literate, and know their subject well have the greatest effect on the achievement of their students.

Yet we all have known teachers who had all of these qualities, but were dull and uninspiring in the classroom. We know, too, that our public schools, especially those in the inner city, pose particular challenges to teachers.

Imagine yourself standing in front of a classroom: two-thirds of your students do not speak English as a first language; half come from homes with a single parent struggling to make ends meet; over the course of the year, there is a steady stream of students departing your class and joining it. You know your subject, but can you teach it?

This is where our nation's 1,400 schools of education enter the picture. Their role is to provide the link between knowing your subject and teaching it effectively. At the undergraduate and/or the graduate level, these schools offer a sequence of courses that has been approved by the state as the route to a teaching certificate. While there are other paths to teaching ("alternative certification"), the great majority of public school teachers are prepared for the classroom in a school or department of education.

What are students taught in these education programs? Surprisingly, almost nobody in the last 20 years has examined the coursework that education schools, as well as states, require as a preparation for teaching. Doing so is not easy: Some schools put their syllabi on the Web, some do not. Many have extremely complex programs - determining what students are required to take as part of their professional preparation often requires considerable detective work. Nevertheless, with the help of my research assistant, Susan Rozen, I decided to try.

We reviewed syllabi in 16 schools of education, 14 of which were ranked by U.S. News and World Report in the top 30 in the nation. We looked at the sequence of courses required in each school for the initial teaching license, only reporting the results when we were able to obtain the syllabi for all of these courses.

By analyzing the required readings, the assignments, and the instructors' stated intentions for their courses, we were able to offer a first portrait of what future teachers are studying in schools of education. Our work has been published in "A Qualified Teacher in Every Classroom?"(Harvard Education Press).

By noting what readings were commonly required and what were generally absent, and through an analysis of the requirements for the student-teaching experience, we raised questions about the rigor, the ideological balance, and the thoroughness of these programs.

A brief explanation: There is a deep division among those who engage in and write about teacher preparation. One school of thought, represented by such figures as Eric Donald Hirsch Jr. and Diane Ravitch, argues that teachers should focus on the basics.
Like piano teachers who stress the discipline of scales and finger technique before encouraging deeper interpretive performance of demanding music, Mr. Hirsch and Ms. Ravitch argue that the best education - especially for the least advantaged - requires direct teaching of the three R's and the other elements of cultural literacy (to borrow Mr. Hirsch's term).The attainment of such knowledge and skills should then be assessed through state tests.

By contrast, another school of thought stresses what is called "constructivism" and "progressivism." Broadly speaking, constructivism is the view (drawn from the work of John Dewey and Jean Piaget) that the teacher should not be a "sage on the stage" but a "guide on the side" encouraging children to discover and create according to their natural impulses. Progressivism is the idea that teachers should focus on the particular voices and experiences of repressed minorities, tailoring instruction accordingly.

In educational theory today, these two ideas are often fused into one view - constructivist-progressive - that is opposed to high-stakes testing and state-mandated, standardized school curricula.

Given the divide between "back to basics" and the "constructivist-progressive" models, one would expect education schools to expose students to both points of view. Our research (which covered 165 syllabi of required courses in the foundations of education, the teaching of reading, and teaching methodology) strongly suggested, however, that at many of our highest ranked schools of education, the constructivist-progressivist arguments are being taught to the almost complete exclusion of the other, direct instruction model.

We found that texts by Mr. Hirsch and Ms. Ravitch and other likeminded authors were required readings in only one or two compulsory courses in all of those we examined. Yet in the majority of programs that required any philosophy of education, education policy, or educational psychology, readings from John Dewey, Henry Giroux, or Howard Gardner were prominently featured.

We also found noted problems in the courses where students gain teaching experience. Only three out of 59 such courses we reviewed, in 11 different schools, used audio or video recordings of students' practice teaching. Moreover, schools of education generally use adjunct appointees, not regular faculty, to supervise and evaluate student-teachers.

Finally, we found very little evidence in any of the programs we reviewed that teachers were being prepared to teach in a high-stakes testing environment.

A first study is never definitive, and the very difficulty of getting the relevant data should provoke reasoned discussion about our research. Naively perhaps, we were not prepared for the outright denunciation of our work by education faculty. Some professors of education argued that course syllabi should not be taken seriously - to which our response is to wonder whether they say as much to their students when they hand out syllabi at the start of a semester. Others said our standards for reviewing required readings were personal and political. Our judgment that future teachers should be exposed to both sides of the major educational debate is, however, no more personal or political than the practice of restricting that exposure to only one viewpoint. Most strikingly, however, has been the reluctance - to date - of our critics to offer an affirmative defense of what they teach future teachers.

If the courses we analyzed are typical of those required at the great majority of schools of education, we confront a paradox. At the same time as states are putting in place high-stakes testing and accountability regimes for students and schools, they are authorizing teacher-preparation programs that teach distrust and even opposition to this same regime.

While it is an open question whether the preparation of teachers should be governed by prevailing national and state education policy, is it right that state mandated programs teach largely criticism of that policy? It is hard to see how such an incoherent approach best prepares teachers to help our children succeed in school.

Wednesday, April 6, 2011

Singapore Math: Great and Quick Results in California and Kentucky

It is interesting to see that more and more schools across the nation are tired of the limitations and awful test results of our Reform or New Math curriculum, including TERC and Everyday Math. The fact these school districts are being pro-active and successful at adopting alternative math programs is very encouraging. Hopefully their wisdom will spread across the nation.

Below are two articles on how school districts in California and Kentucky adopted Singapore Math with successful results. Please note the adoption dates: it doesn’t take years to get good results.

I want to thank Beth Schultz of the American Math Forum for bringing the Santa Catalina article to my attention and Cassandra Turner’s blog for doing same with Kentucky article.

A word of caution: Some math experts rate “Math in Focus” (the one adopted in Fayette, Kentucky) as much less rigorous than “Primary Mathematics” (the one adopted in California). Nevertheless, while I’m not an educator or math expert, I believe if a school had to choose between any “Reform Math” program (TERC, EM, etc.) and Math in Focus, the latter should be the choice each and every time.

For a fuller understanding of what the Singapore Math “mastery first” and “sequencing” program is all about, please take a look at this document.

From The Herald:

April 5, 2011

“It seems a small thing, concentrating on basics, making sure students master concepts before moving on, but proponents of Singapore Math say the system is dramatically improving math scores at Santa Catalina School.

“Davidson said a lack of emphasis on teaching basic skills in the lower grades is partly to blame for poor test results at schools nationwide. For example, students are expected to grasp algebra and geometry without having mastered fractions and ratios.

Singapore Math specialist Bill Davidson said the curriculum builds upon preceding levels of knowledge to achieve mastery.

"It's like rungs in a ladder," he said. "Before you move up to the higher rung, you have to master the rungs below. We go into more detail covering fewer topics."

Davidson called math a fundamental skill connecting to all other types of learning. "Math teaches you how to think," he said. "I want our students to learn to be great thinkers."

And in Fayette County, Kentucky, the school district adopted “Math in Focus”, with such good results that it is expected to be adopted by other school districts

Editorial, April 2, 2011

“But despite all the praise “new math” received from educators when it was first introduced, it has not helped improve the math scores of American students. Just the opposite, in fact. The math scores of American students continue to decline when compared to the scores of students in other nations at a rate that should alarm us.

Singapore math was introduced in the Lexington schools as a pilot program in 2009 and the math scores of at least three schools have increased significantly.  That led to the Lexington school board to approve an expansion of the pilot project.

“… expect Singapore math to spread across the state, and if it succeeds in raising the math scores of Kentucky kids, “new math” soon will be old ...  and forgotten.”

Saturday, April 2, 2011

Curricular Spiraling and Distributed Practice

While researching the scientific evidence supporting the Reform Math (Everyday Math) claims on curricular spiraling and discovery learning, Mr. Ze’ev Wurman was kind enough to select a few quotes from the most important publications and reports available and to highlight how Everyday Math misinterprets the research. I post his explanations, quotes, references, and Mr. Wurman’s bio (at bottom).

Mr. Ze’ev Wurman and I are members of the American Math Forum (AMF). We are thankful for the opportunity AMF has provided in making this Q & A session possible.

Curricular Spiraling and Distributed Practice:

How Everyday Mathematics Misinterprets the Research.

Reading Everyday Mathematics justification for distributed practice[1], which is also frequently called spaced learning, one reads:
From the beginning, accordingly, Everyday Mathematics was designed to take advantage of the spacing effect. An explicit attempt was made to ensure multiple exposures to important concepts and skills, spread over two or more years. As the First Grade Everyday Mathematics teacher’s manual states, “If we can, as a matter of principle and practice, avoid anxiety about children ‘getting’ something the first time around, then children will be more relaxed and pick up part or all of what they need. They may not initially remember it, but with appropriate reminders, they will very likely recall, recognize, and get a better grip on the skill or concept when it comes around again in a new format or application—as it will!”

Yet distributed practice in cognitive science literature is not about children "pick[ing] up part or all of what they need" over "two or more years." It is about distributed practice rather than distributed instruction, and the distribution, or spacing, should occur over period of weeks rather than years[2].
For example, this is what the National Advisory Mathematics Panel[3] wrote about distributed practice:
Distributed practice should naturally occur as students progress to more complex topics. However, if basic skills are not well learned and understood, the natural progression to complex topics is impeded. (p. 4-xxv)
At all ages, there are several ways to improve the functional capacity of working memory. The most central of these is the achievement of automaticity, that is, the fast, implicit, and automatic retrieval of a fact or a procedure from long-term memory. … For other types of information, including much of mathematics that is taught in school, automaticity is achieved only with specific types of experiences, including practice that is distributed across time (e.g., Cooper & Sweller, 1987). (p. 4-5)
The fast and efficient solving of arithmetic combinations and execution of procedures requires considerable practice that is distributed over time. (p. 4-39)
Research on learning in general … indicates a benefit for practice that is distributed across time, as contrasted with the same amount of practice massed in a single session. Pashler, Rohrer, Cepeda, and Carpenter (2007) provide a recent review, including discussion of one area of mathematics. Initial experimental studies with mathematics, specifically teaching probability, are consistent with the more general literature. ... The course work results suggest that the distributed review and integration—which was likely to have occurred more consistently for algebra than geometry—of the material across years contributes to the retention of the material throughout adulthood. (p. 4-87)
Retention of algebraic skills into adulthood requires repeated exposure that is distributed over time. This occurs as core procedures and concepts are encountered across grades. In much of mathematics, distributed practice should naturally occur as students progress to more complex topics. However, if basic skills are not well learned and understood, the natural progression to complex topics is impeded. This is because students will continue to make (and potentially practice) mistakes. As an example, procedures for transforming simple linear equations are embedded in more complex equations and thereby practiced as students solve them. The practice will not be effective, however, if students incorrectly transform basic equations, as they often do. (p. 4-89)
Here is what the IES Practice Guide has to say on the subject of spacing learning over time:
To help students remember key facts, concepts, and knowledge, we recommend that teachers arrange for students to be exposed to key course concepts on at least two occasions—separated by a period of several weeks to several months. Research has shown that delayed re-exposure to course material often markedly increases the amount of information that students remember. The delayed re-exposure to the material can be promoted through homework assignments, in-class reviews, quizzes (see Recommendation 5), or other instructional exercises. In certain classes, important content is automatically reviewed as the learner progresses through the standard curriculum (e.g., students use single-digit addition nearly every day in second grade math), and this recommendation may be unnecessary in courses where this is the case. This recommendation applies to those (very common) course situations in which important knowledge and skills are not automatically reviewed.[4]

What research indicates is that once a topic has been taught, a distributed exposure to it, over period of weeks and perhaps few months and in the form of practice or various quizzes and test, is conducive to fluency and long term retention. And that this is particularly important for topics that are not routinely practiced already in the process of subsequent learning. The research does not find that topics should not be taught to a full understanding and mastery when instruction occurs. Rather, it says that fluency and retention will be better achieved by distributed practice. Contrast this with the interpretation that Everyday Mathematics offers, where it instruct teachers not to expect mastery when taught and it re-teaches essentially the same content over period of 2-3 years. That is exactly what the National Advisory Mathematics Panel warned against: “Any approach that continually revisits topics year after year without closure is to be avoided[5].”

In summary, Everyday Mathematics repeatedly misinterprets the research about distributed practice and applies it in support of distributed instruction in their spiral curriculum, a practice that is not supported by research.

[4] Organizing Instruction and Study to Improve Student Learning: A Practice Guide. Institute for Education Sciences, 2007, page 5.
[5] First recommendation of The Final Report of the National Mathematics Advisory Panel, U.S. Department of Education, 2008.

Mr. Ze'ev Wurman worked over 30 years in the high tech industry, most recently as the Chief Software Architect with Monolithic 3D, a semiconductor start-up in the Silicon Valley. He has a long involvement with mathematics standards and assessment in California and served on the 1997 Mathematics Framework Committee and on the STAR Mathematics Assessment Review Panel since its inception in 1998. He was a member of the 2010 California Academic Content Standards Commission that evaluated the suitability of Common Core’s standards for California. He was a member of the Teaching Mathematics Advisory Panel to the California Commission on Teacher Credentialing. Between 2007 and 2009 Wurman served as a Senior Policy Adviser to the Assistant Secretary for Planning, Evaluation, and Policy Development in the U.S. Department of Education. Wurman has B.Sc. and M.Sc. degrees in Electrical Engineering from the Technion, Israel Institute of Technology.

Q & A with Jim Milgram

Prof. R. James Milgram is Professor of Mathematics Emeritus at Stanford University.  He was one of the authors of the previous California Standards, and the main reviewer of the NCTM Curriculum Focal Points.  Additionally, he served on the National Board for Education Science that oversees the IES – the research arm of the U.S. Department of Education, and he also served on the NASA Advisory Council, as well as the Validation Committee for the new Core Standards.

Prof. R. James Milgram and I are members of the American Math Forum (AMF). We are thankful for the opportunity AMF has provided in making this Q & A session possible.

1.       Do you prefer Traditional Math over Reform Math?

Anyone interested in majoring in a technical area should have a much more traditional program than the reform programs. Though there is nothing intrinsically wrong with some of the reform ideas, the implementations are worse than horrible at this time.

2.       Do you prefer Singapore Math over Everyday Math (EM)?

There is a pretty good program hidden inside EM.  But no more than 1 in 500 teachers are capable of locating and delivering it.  However, that one teacher would almost certainly be able to do better on her own.

Singapore math, on the other hand, is very solid mathematically and in terms of the problems students are given.  But there are some limitations.  It isn't quite as effective for ALL students as the pure Russian program.  Also, there are many elementary school teachers who don't know enough mathematics to deliver the Singapore program effectively, and need extensive professional development.  (Singapore adopted the Chinese curriculum in 1984, but this was the program the Chinese adopted from Russia in 1955. However, the Russian program requires teachers who are even more mathematically knowledgeable than does Singapore.)

3.       If you had to pinpoint two/three main deficiencies in EM and Singapore, what would they be?

There are no major deficiencies in the Singapore program, just a few points where it could be better than it is.  On the other hand, the recommended lessons in EM are mostly useless.

4.       Does it make sense to try to merge the two programs?

Not from my perspective.

5.       Do you think using EM is actually doing a disservice to our young generation and impairing our future economic competitiveness?

In general, yes.

6.       Do you think Parents/Educators/Mathematicians would be well advised to reject EM if they had the opportunity to do so?


7.       How can you explain why EM has spread rapidly like a prairie wildfire?

For many years I've felt that when school administrators get their "advanced" administration degrees from the ed-schools, a critical component should be SERIOUS courses in statistics and data analysis.  But it seems that there is virtually no chance this will ever happen.

Right now, EM is being used in the Palo Alto school district, and as expected, recent data seems to show that the most seriously affected students are those with the most limited financial resources.  The wealthier parents can send their kids to after-school tutoring programs that are result oriented, not dogma driven.  It is worth noting that, once it became clear that Palo Alto would choose EM, it seems that Kumon and the others opened branches in the Palo Alto area.

8.       As you know, I do not believe there is an organized “conspiracy” purposely thrusting a deficient program on our children… although I know there are some shenanigans going on for sure… Yet, I also find it hard to believe that a “bureaucratic bungle” would last so long and is actually growing. For example, our NYC gifted and talented “Anderson School” dove in the Koolaid pool in 2009… Although the top notch “Hunter College Elementary School” adopted Singapore Math in 2007 and other public schools (e.g. PS132) are also adopting Singapore Math versions.

All in all it seems to me that if the science behind EM is borderline fraudulent (meaning that EM has twisted the scientific evidence on “spaced” or “distributed” practice and instruction methods to fit its reform agenda) and if most mathematicians are opposed to EM and if most parents are frustrated with EM and tutoring their kids … then why is this program still around? I call it “institutional inertia” and “high barriers of entry”… but even that doesn’t satisfy me.

Do you have any ideas? Are we hostage to a “conspiracy”?

Actually, as mathematicians we should not make judgments one-way-or-another about teaching methods.  This is not our area of expertise.  We - especially our foreign born and trained colleagues - may think that the pedagogy is extremely strange, but we are clearly not qualified to discuss this issue as experts.

However, we don't have to discuss pedagogy in this case.  We can see that the mathematical content of the program is absolutely minimal if it is delivered as intended by the authors.  As a result, we do object as content experts, but our input is ignored by school administrators.  After all, we've never been in "the classroom."

So, one has to take a long term look at what happens.  Among the districts that started using EM 15 - 20 years back, none, to my knowledge, still use it.  And a number of districts, such as, e.g. Philadelphia, seem to quietly trying to get rid of it without losing too much face.

When we first started this fight, EM was just the best of a really horrible collection of such programs.  We managed to get rid of the others in California, and, as far as I can tell, when we said that EM was the best of these programs, though all were extremely problematic, this was interpreted by the ed-schools and school administrators as saying that "EM is the best."

9.       When you say "EM was just the best of a really horrible collection of such programs", I take it you include TERC and Mathland among them, right?

Yes, I include TERC and Mathland in that group.  But, to be fair, most of the traditional U.S. programs were not terribly good either.

10.   You say "Among the districts that started using EM 15 - 20 years back, none, to my knowledge, still use it." Are you +/- certain of that fact?  It appears that District 2 in NYC has been using TERC since late 90s. In other words, it seems Reform Math (TERC or EM) is spreading, not shrinking... Is there any hard data you know of that I can use/quote?

All I know is most of the original list of districts, and for this set, I'm virtually certain that none are using the program now.  But I don't have rigorous data.

11.  You say "we don't make judgments one-way-or-another about teaching methods" Yet by favoring traditional math, are you not making a judgment that "teaching one topic until mastery has been achieved" is better than say the spiraling method? Isn't the difference between Traditional and Reform also a difference in pedagogy?

There is one very annoying aspect of EM in that it virtually forces teachers to use specific pedagogical techniques.  I don't like any of this, but I refuse to say whether it's right or wrong.  However, what bothers me most of all - and I can be regarded as an expert here - is the low level of the math that the intended program presents.

12.  You say “this was interpreted by the Ed. schools and school administrators as saying that ‘EM is the best’.” Didn’t you and other advisors dispute the misinterpretation? How can they do this with impunity? That’s what I don’t understand… how can they steamroll over learned math experts and frustrated teachers and parents? That makes no sense to me. For a few years? Yes. For 20+ years? No way. How can that be? Is it really because some/most teachers and teacher instructors are math-phobic and thus … perpetuate a deficient program they can actually understand and teach?

Yes, a number of us stated our reservations about the program repeatedly and with explicit examples of the kinds of issues involved.  However, those statements were almost universally ignored in the ed.-schools and the country’s school districts.