Thursday, March 10, 2016

Memorizing is Essential

The origin of reform math started in the 50s, probably earlier. The idea that memorization of math facts hinders enthusiasm for learning and stifles intellectual curiosity is not supported by evidence. Also, the idea that students learn math better through discovery activities by focusing on patterns is not accurate, either. 

These wrong ideas, which are progressive ideology, can also be traced back to the 1950s or much earlier, yet they persist today via reform math in state standards. In 1961, the Greater Cleveland Mathematics Program’s Teacher’s Guide (GCMP), “She [the teacher] knows that the young child is always at his best when he is in the process of discovery. She realizes that adhering rigidly to narrowly prescribed bounds and learning facts in neat categories may seem efficient, but this very efficiency may destroy enthusiasm for learning and stifle intellectual curiosity.” Really? 

The problem is that none of the reform math ideas above are supported by evidence. In fact, the memorization of single-digit math facts, which may not be much fun, is essential for using the standard algorithms and for enabling higher-level thinking (i.e., problem-solving in math). Memorization is critical and does not stifle intellectual curiosity or destroy enthusiasm. Also, discovery learning is among the least efficient methods to learn mathematics. These old anti-math ideas (memorization hinders and stifles; discovery learning is the best way) are prevalent today, and, like before, have not been supported by the cognitive science. (I guess drill-for-skill to automate combinations of core competencies on the beam or bars somehow destroys a gymnast's enthusiasm and stifles her intellectual curiosity.) 

FYI: In Singapore, first-grade students memorize the addition facts and use those facts for simple multiplication as repeated addition; e.g., 3 x 4 is 4 + 4 + 4 or 12

Here is a summary of "thinking steps from GCMP:
7 + 9 = 7 + 10 - 1
7 + 9 = 10 + 7 - 1
7 + 9 = 10 + 6
The class should know that adding 10 and subtracting 1 is the same as adding 9."

It is incorrect mathematics. As written, it doesn't follow the order of operations convention (left to right rule). By convention, 7 + 10 - 1 means to add 10 +  7 first, then subtract the 1 from 17, which is 16. Of course, one could rewrite it as 10 + (7 - 1), then it would be okay (Do Me First). Frankly, I would have memorized 7 + 9 = 16 (automation) from the start and skipped all this nonsense that clutters my working memory. Below is some more nonsense.
The calculation of single-digit number facts in working memory is inefficient and often overloads the working memory. Children should not calculate single-digit number facts in working memory; they should memorize them, so they stick in long-term memory for instant use in calculating and problem-solving.

Greater Cleveland Mathematics Program - GCMP
The assumption (rearrangement property) complicates 
simple arithmetic.
My Comment on GCMP: Calculating sums by expanding and rearranging is not necessary. Expanded notation (place value) might help the student understand the meaning of 435, which should have been taught in the 1st grade, but when you combine (add) 435 with 364, the standard algorithm is far superior because it sets up the correct structure from the start. Breaking down numbers and rearranging everything are not efficient methods for addition because they make simple arithmetic more complicated. In contrast, the structure of the standard algorithm automatically arranges ones added to ones, tens added to tens, and hundreds added to hundreds, etc. In short, why break down numbers and then rearrange them by place value as in GCMP?


The standard algorithm is a paradigm of efficiency and does not require breaking down or rearranging. Any method that makes simple arithmetic more complicated is inefficient and should be discarded.  

Indeed, GCMP insists that its way of addition has structural similarity to the standard algorithm; however, they are not that similar because the processes are different. In GCMP, the processes involve breaking down numbers then rearranging them, etc. In the standard algorithm, the only process is retrieving single-digit math facts from long-term memory instantly. Moreover, the GCMP complicates simple arithmetic and wastes valuable class time. Fast calculating skills are just as important as the major ideas and applications in mathematics. 

A good mathematics program requires three ingredients:  
(1) Skill in calculation (standard arithmetic),
(2) Good understanding of the ideas of mathematics, and
(3) Comfortable facility in using mathematics in many different situations (applications).
Trio: Skills:Ideas:Uses

My Teach Kids Algebra program was a supplemental program
 that focused mostly on the ideas of mathematics, which for grades 1 to 5 had included: fractions, variables, equations, functions, negative numbers, and graphs. Dr. Robert Davis said that understanding takes time and grows gradually. "The way we understand any particular thing [including mathematical ideas] will necessarily change over time. A 1st grader's understanding of place value takes a time to develop. It is different from a 5th grader's understanding. Unfortunately, the three ingredients are weak in many of today's modern schools. TKA required standard calculations and avoided nonstandard, alternative  algorithms from math reforms. Standard algorithms should be taught in the 1st gradeThe basic ideas presented in the 1st grade (Spring 2011) were equality, variable, variable substitution, equations, input-output model, figuring out function rules, building tables (x-y), plotting graphs in Q-I (linear), perimeter and area. 

Reference: Dr. Robert B. Davis, The Madison Project, 1957 
Grade 2

Second Grade TKA, (40 + students)
Lesson 7 (April 26, 2011)
Table Building #1, #2

Title-1 City School

I am thinking of a number. If I double it and add five, I get 11. What is the number? etc.

The table-building idea came from Getting Ready for Algebra (Alan Osborne, 1988), a middle school curriculum that was calculator based. I selected easier numbers so that 2nd graders could calculate them. Almost all the algebra ideas taught involved calculation skills, starting with true and false equations. Also, I did table building in 1st grade (#1). First graders also made x-y tables, figured out function rules, and plotted the x-y number pairs in Q-I. (Note. First graders also figured out perimeters of rectangles using the doubling idea: P = w + w + l + l). Calculators were not used in TKA. I blended calculation skills, with ideas, and uses.


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