Hands-On Equations Research, Interim Report Nov. 19, 2007
A Comparison of Algebra Achievement by 4th, 6th and 8th Graders
By Henry Borenson and Larry W. Barber
The full report may be found on the research page of www.borenson.com)
CONCLUSIONS These three studies demonstrate that 1) Each of the combined group of 123 4th graders, 190 6th graders, and 105 8th grade students achieved a large and significant gain from the pre-test to the post-test following Lesson #6, and 2) This significant gain was maintained on the post-test following Lesson #7, where the students did not use the game pieces (rather, they used the pictorial notation learned in Lesson #7). These results demonstrate that students who learn the HOE methods of solving equations can be equally successful with or without the game pieces. In other words, the students are able to transfer their hands-on learning to the pictorial method presented in Lesson #7, which uses only paper and pencil, and be equally successful in solving the equations. Looking at the above combined group results again, in the chart below, we note the consistency in the scores on both post-tests for each of the three 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% |
We make the following observations: 1) Hands-On Equations seems to be grade-blind, i.e., students at either the 4th, 6th or 8th grade will do equally well with the program 2) We note the gradual increase in the pre-test scores going up from the 4th to the 6th to the 8th grade. It is reasonable to assume that this difference is due to the regular mathematical instructional content that the students had in the intervening two years (for the 6th graders) and four years (for the 8th graders) 3) We note that the post-test score following Lesson #7 for the 4th graders exceeded the pre-test scores for the 8th graders. It is reasonable to inquire whether the first seven lessons of HOE provides a higher level of competence on these particular algebraic concepts and skills than the regular math curriculum does in the intervening two years (for the 6th graders) or four years (for the 8th graders). In order to explore these questions further, we intend to carry out the above study with a larger group of 8th graders, and also to conduct the same study with 7th graders.
Several very important questions arise from the above research: Is it possible that 4th graders, exposed to seven lessons of HOE, can achieve at a higher level than 8th graders (who have not had HOE) on the basic algebraic concepts tested in this study? If this result is confirmed with larger numbers of students, is the critical factor that these concepts are not presented in the regular math curriculum? Or, is it that they are presented but the traditional methods of instruction do not compare in their effectiveness to the methods used in HOE?
Additionally, since the above study suggests that 4th graders do as well as 6th and 8th graders on these algebraic concepts (when presented via Hands-On Equations), it is clear that no purpose is served in holding off instruction on these concepts until the 6th or 8th grade. Hence, the concepts tested in this study, many of which have been traditionally taught at the 8th or 9th grade, can be presented to students as early as the 4th grade, via HOE, with an expectation for a high level of success.
TEST QUESTIONS FOR STUDY #59a, 102B and 105a
| Pre-Test Questions 1. 2x = 8 2. x + 3 = 8 3. 2x + 1 = 13 4. 3x = x + 12 5. 4x + 3 = 3x + 6 6. 2(2x + 1) = 2x + 6 Post -Test after Lesson #6 1. 2x = 10 2. x + 3 = 8 3. 2x + 2 = 10 4. 3x = x + 4 5. 4x + 3 = 3x + 9 6. 2(2x + 1) = 2x + 8 | Post-Test After Lesson #7 1. 2x = 6 2. x + 3 = 10 3. 2x + 1 = 7 4. 3x = x + 2 5. 4x + 3 = 3x + 7 6. 2(2x + 1) = 2x + 10 |
