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?

1 comment:

  1. Pascal, thanks for writing about how destructive "Constructivist" Math. I've seen how Constructivist math succeeds in CONSTRUCTING a lot of students who never master anything. I've seen the inefficient rabbit holes that students are lead down. It's no wonder that students struggle to pass the most basic exams. The NYS Regents exams for math have been thoroughly DUMMED down. The material is so basic compared to the material from 20 years ago. Plus the scale has become too generous: a few years ago, a student would need only about 50% correct to pass with a scaled score of 65. But on a recent Integrated Algebra 1 Regents exam, a student who got 46% correct was awarded with an average-sounding scaled score of 73. And of the 46% correct, many of the problems could be solved by using guess-n-check games (i.e., not algebraic solutions). Students have 3 hours to take Regents exams that they should be able to finish in 1 to 1½ hours so they have all the time in the world to "backsolve" from the answer choices. For the free response questions, it's rare to find one that is thought provoking or even remotely challenging.

    I hope that Bloomberg appoints a Chancellor who kills the Everyday Math curriculum and actually requires students to master the basics. If not, students will be further lead down the road to incompetence and won't be able to qualify for jobs that require basic (or not so basic) math skills.

    Best regards,

    Mark Kronenberg
    Founder, MATH 1-2-3®
    Specialists in Math Tutoring & Test Prep