The Effect of
Hands-On Equations®
on the Learning of Algebra by 4th and 5th Graders of the
Broward County Public Schools
Hands-On Equations Interim Report: March 17, 2008
A Study of the Strength of Acquisition of Algebraic Concepts by 4th and 5th Graders via Hands-On Equations and a Measure of the Retention of the Pictorial Notation
ABSTRACT
The Broward County Public Schools agreed to participate in a research study to determine the effectiveness of the Hands-On Equations® program in providing its students with a successful experience with algebra. The study sought to determine whether the 4th and 5th grade students of the district could learn to solve equations such as 3x = x + 12 and 4x + 3 = 3x + 6, equations normally presented in the 8th or 9th grade. If the students were successful with these concepts, they would have overcome at an early age one of the obstacles to the learning of algebra.
The teachers who participated in this study received a full day of training in the use of the program. The workshop they attended, the Making Algebra Child's Play® workshop, was conducted by a certified Borenson and Associates, Inc. instructor in the fall of 2007. Immediately after instruction, the teachers administered a pre-test to their students, and then proceeded to teach the first seven lessons (Level I) of Hands-On Equations. They also administered two post-tests and a three-week retention test.
This report presents the meta-analysis conducted on six 4th grade regular classes, three regular 5th grade classes and five gifted and talented 5th grade classes, a total of 14 classes involving 326 students. The Appendix includes the test results for other classes participating in the study. For various technical reasons explained in the report these additional classes could not be included in the meta-analyses.
Since the teachers and students participating in this study were representative of those in the district as a whole, the results shown herein are indicative of the results that would be expected if the Broward County Public Schools were to implement the program district-wide in the 4th and 5th grades.
GENERAL SUMMARY
A total of 326 students from 14 different classes were included in this study. The raw scores and percentage scores are shown below. We note that the average 4th graders saw their scores triple from the pre-test to each of the post-tests and to the retention test; the average 5th graders saw their scores more than double from the pre-test to these post-tests and to the retention test.
| | Pre-test | Post-test after Lesson #6 | Post-test after Lesson #7 | 3-Week Retention Test after Lesson#7 |
| Grade 4, n=111 Study #131MA Regular students | 26.8% (m=1.61) | 84.2% (m=5.05) t(P, P6)=20.50 | 84.2% (m=5.05) t(P, P7)=20.45 | 81% (m= 4.86) t(P, P7-R3)=19.49 |
| Grade 5, n=84 Study #138MA Regular students | 37.7% (m=2.26) | 88.3% (m=5.30) t(P, P6)= 19.62 | 88.5% (m=5.31) t(P, P7)=17.09 | 84.7% (m= 5.08) t(P, P7-R3)=14.71 |
| Grade 5, n=111 Study #139MA Gifted/Talented | 78% (m=4.68) | 95.3% (m=5.72) t(P, P6)=8.06 | 95.3% (m=5.72) t(P, P7)=8.14 | 94.2% (m= 5.65) t(P, P7-R3)=6.05 |
These three meta-analyses demonstrate that 1) Each of the combined group of 111 regular 4th graders, 84 regular 5th graders, and 111 gifted and talented 5th graders 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 confirm the results of previous studies conducted with 4th, 6th and 8th graders that students who learn the Hands-On Equations (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.
Additionally, the current study showed that after a three-week period of no HOE instruction, the students performed essentially the same as they did three weeks earlier on the Lesson #6 and Lesson #7 post-tests. Since the three-week retention test was conducted without the use of the game pieces, the current study demonstrates that 4th and 5th grade students are able to retain the methods they have learned in the program and are able to solve algebraic equations using the pictorial notation even after a period of three weeks without HOE instruction.
In summary, the results obtained in this study are consistent with previous studies which show that when teachers who have been trained in the Hands-On Equations program instruct their students in the use of the program, and go through the first seven lessons of the program as prescribed, the students learn the algebraic concepts presented, they do well on the posts-tests, and they remember what they learn, with or without the use of the game pieces.
Appendix 9
Item Analysis
Below, we show the percentage of students who obtained the item correct on the pre-test vs. the percentage of students who obtained the comparable item correct on the three-week retention test for each of the three meta-analyses.
Grade 4, n =111. Study #131MA Regular Students
Percentage of Students with Correct Item Response
| | Equation | Pre-test | Retention-test |
| Question #1 | 2x = 8 | 48% | 92% |
| Question #2 | x + 3 = 8 | 70% | 89% |
| Question #3 | 2x + 1 = 13 | 22% | 82% |
| Question #4 | 3x = x + 12 | 9% | 86% |
| Question #5 | 4x + 3 = 3x + 6 | 8% | 79% |
| Question #6 | 2(2x+1) = 2x +6 | 8% | 59% |
Grade 5, n =84. Study #138MA Regular Students
Percentage of Students with Correct Item Response
| | Equation | Pre-test | Retention-test |
| Question #1 | 2x = 8 | 68% | 94% |
| Question #2 | x + 3 = 8 | 87% | 93% |
| Question #3 | 2x + 1 = 13 | 42% | 83% |
| Question #4 | 3x = x + 12 | 15% | 89% |
| Question #5 | 4x + 3 = 3x + 6 | 10% | 87% |
| Question #6 | 2(2x+1) = 2x +6 | 5% | 74% |
Grade 5, n =111. Study #139MA Gifted/Talented Students
Percentage of Students with Correct Item Response
| | Equation | Pre-test | Retention-test |
| Question #1 | 2x = 8 | 97% | 99% |
| Question #2 | x + 3 = 8 | 95% | 96% |
| Question #3 | 2x + 1 = 13 | 87% | 99% |
| Question #4 | 3x = x + 12 | 71% | 91% |
| Question #5 | 4x + 3 = 3x + 6 | 71% | 95% |
| Question #6 | 2(2x+1) = 2x +6 | 46% | 85% |
Appendix 4
| 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 |
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