By Henry Borenson and Larry W. Barber
Note: The research presented herein, although it is restricted to one specific program, is sufficient to show that the use of math manipulatives, under the conditions specified herein, has the potential to be of tremendous value in the learning process.
In the August 2005 issue of the Scientific American, in her article “Mindful of Symbols,” Judy Deloach makes the observation, regarding the use of manipulatives in mathematics, that “If children do not understand the relation between the objects and what they represent, the use of manipulatives could be counterproductive.”
In the December 2005 issue, in response to a letter to the editor, she says further, “For many math educators, it is an article of faith that manipulatives help children learn fundamental mathematical concepts. Unfortunately, empirical research does not support that widespread belief. What research there is indicates that the current us of manipulatives is at best questionable and at worst counterproductive.”
It is certainly true that if children do not understand the relationship between the objects and what they represent that the use of the manipulatives would not be accomplishing their intended purpose. What happens, however, when the children do understand that relationship, and when the instructional system involving the use of the manipulatives follows a well thought-out sequential development of concepts? Since manipulatives are objects that can be seen, touched and moved, do they have the potential to clarify mathematical concepts beyond what can be done simply by using chalk on the blackboard?
If Ms. DeLoach were to look today at the research that has been done with a manipulatives-based program known as Hands-On Equations, she would see that there is solid evidence of the value of using manipulatives to clarify and indeed, to enable students to learn algebraic concepts. Not only would she see strong pre- to post-test results, but she would see that the use of manipulatives can enable children to learn algebraic concepts years earlier than they are normally presented using the traditional abstract methods of instruction.
The results obtained from our research confirm the statement made by Barbel Inhelder of Geneva, a student of Piaget, in 1959, “Advanced notions of mathematics are perfectly accessible to chidren of 7 to 10 years of age, provided they are divorced from their mathematical expression and studied through materials that the child can handle himself.”
As an example from Lesson #3 of Hands-On Equations, the students use pawns and numbered cubes, and a flat laminated balance, to set up and solve algebraic linear equations. For example, the equation 4x + 2= 3x +9 would be set up as shown at the left.
The students would then proceed to physically remove three pawns from both sides of the scale to balance the equation, and end up with the simplified setup, showing a blue pawn and a 2 cube on the left and a 9 cube on the right. From here, the students can easily see that the value of the blue pawn, or x, must be 7. Hence, the students kinesthetically carry out the subtraction property of equality, and thereby gain a solid understanding of this important algebraic concept.
The above equation, and others such as 3x = x + 12 and 2(2x +1) = 2x + 6, were among the questions administered to 418 4^{th}, 6^{th} and 8^{th} grade students in the United States to compare their results on the pre-test (prior to having the first six lessons of Hands-On Equations) and on a post-test following the first six lessons. A subsequent post-test was given after Lesson #7, in which the students did not use the manipulatives, but rather used the pictorial notation involving only paper and pencil.
All of the students involved in the study participated under the same experimental and testing conditions. The teachers of these students had all attended a full-day Making Algebra Child’s Play workshop conducted by a certified Borenson and Associates instructor. The teachers then implemented the program as instructed, following the specified sequence of concept development that went along with the use of the manipulatives. Each of the students used the game pieces and flat balance scale at their desks during the instructional segments, and in the post-test following Lesson #6. The pre-tests, post-tests, and time alloted for those post-tests were the same for all the groups.
N= number of students |
Pre-test |
Post-test after Lesson #6 |
Post-test after Lesson #7 |
Grade 4, n=123 |
30% |
84% |
88% |
Grade 6, n=190 |
48.2% |
92% |
93% |
Grade 8, n=105 |
64.8% |
87.7% |
88.8% |
From,“A Comparison of Algebra Achievement by 4^{th}, 6^{th} and 8^{th} Graders,” by Henry Borenson and Larry W. Barber
Note: The pre-test questions consisted of 1) 2x = 8, 2) x + 3 = 8, 3) 2x +1 = 13, 4) 3x = x + 12, 5) 4x + 3 = 3x + 6, and 6) 2(2x + 1) = 2x + 6. Similar questions were used for the post-tests.
From the table above we note that 1) the 4^{th} and the 6^{th} graders achieved at the same* level as the 8^{th} graders on a post-test following the first six lessons, with all groups scoring in the 84% to 92% range, and with all groups having a significant pre- to post-test gain, 2) the students were able to transfer their hands-on learning to a pictorial solution using only paper and pencil (post-test Lesson #7) and maintain their Lesson #6 post-test results and 3) all three groups did comparably* well on the post-test following Lesson #7, with all groups scoring between 88% and 93%.
* The means exhibited were so close that we did not even bother to do a t-test for significance.
The result obtained, showing that the 4^{th} graders, exposed to the same testing conditions as the 6^{th} and 8^{th} graders, did as well as the 6^{th} and 8^{th} graders leads to the conclusion that Hands-On Equations removes an age difference of up to 4 years as far as teaching these particular algebraic concepts is concerned. Hence, an important educational policy question arises from this study: Is there any need to wait until the 6^{th} or the 8^{th} grade to introduce these algebraic concepts, when 4^{th} graders can do as well?
It is clear from the above research results that the use of manipulatives, when students understand the relationship between the manipulatives and what they repressent and when they are used as an integral component of a well-structured sequence of concept development, has the potential to increase student achievement in mathematics. In particular, the Hands-On Equations instructional program, can remove an age difference of 4 up to years in the instruction of the algebraic concepts mentioned in this study.
Henry Borenson, Ed.D.