Closing the Achievement Gap


Results of Instructional Alternatives in Teaching Multiplication

Suzanne E. Sax, Ph.D.

Study conducted at UC, Berkeley

A study was conducted at an inner-city elementary school in Oakland, CA.  48 third grade students were tested on a battery of 10 tests to ascertain their ability levels in two cognitive domains — conceptual and associative.  Students in each ability grouping were randomly assigned to one of two computer-delivered instructional treatments:  Program C used primarily a “conceptual” method and involved a limited amount of reading for concept mastery; Program A used primarily an “associative” technique with multiplication problems presented in factor groups (e.g., 0 x 0 = 0, 0 x 1 = 0, 0 x 2 = 0, etc.; 1 x 0 = 0; 1 x 1 = 1, 1 x 2 = 2, etc.; 2 x 0 = 0, 2 x 1 = 2, 2 x 2 = 4, etc.). Both instructional programs were effective in achieving the terminal objective — that students be able to find products up to 20 of two numbers.  However, Program C resulted in significantly better performance (On a 20-item test:  MeanC = 17.6; MeanA = 12.1) while Program A required half the time to complete (MeanC = 328 minutes; MeanA = 164 minutes).

Some of the most remarkable outcomes of the study were serendipitous and therefore not subjected to the rigors of statistical analysis.  Problems occurred relating to basic reading and simple numeration. 

Most of the students could not read with comprehension the words used to ask questions.  They appeared to be at the associative level in both reading and numeration.  They could sound out words and define them, but they weren’t “incorporating” them in context.  They could count, and recognize numbers, but didn’t understand relationships.

When reinforcements (e.g., “Good.” “Right.” “Correct.” “Wrong.”) appeared, they did not make the connection that those words related to their performance.  Thus, they did not know when they had erred or answered correctly.  Furthermore, they transposed words and/or skipped words, thus changing the meaning of a sentence.  Many students read word by word, thereby failing to understand the sense of a given question. 

When students were told their scores at the end of the subtests, they had no idea what the numbers meant.  For example, the program might state, “You scored 2 correct out of 6.”  If I asked, “How many did you get wrong?” they would answer either “2” or “6.”  Similar querying confirmed the lack of true grasp of numbers as quantitative entities.  Thus, as in reading, students appeared to be operating at an associative level rather than a conceptual one in their understanding of numbers.

Measured IQs ranged from the 60’s to 154.  Parenthetically, the student who scored 154 on the 10-test battery was not able to read with comprehension, and his teacher had no idea that he was “bright.”

Toward the end of the experiment, all students taking Program C were reading with comprehension.  In addition, the slowest students, who had severe comprehension difficulties with both arithmetic and reading at the beginning, often progressed quickly through the last two lessons, which were “difficult” in terms of content, more abstract, fading the use of graphic representations to support the arithmetic concepts. Furthermore, students were required to find a missing “factor” of an equation, which is a division concept and precursor to algebra.  Of special interest, it was those “slow” students who volunteered to come during their recess, lunch periods, and after school in order to finish the lessons in the limited (two-month) time allotted before the removal of the terminals.


While both methods resulted in learning, Program C, which taught the concepts of multiplication and required reading with comprehension, was deemed preferable, due to the benefits that could transfer to other learning. Program C is the Multiplication course on the EduThink.us website.

It is our hope that further research in math and reading, based upon the study results, will prove valuable in closing the achievement gap of at-risk chidren that currently exists.