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.”

And,

“… 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. 

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.”

And,

“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.