Sunday, March 30, 2008

A COMMENT ON THE NATIONAL MATH PANEL REPORT

The Final Report of the National Advisory Math Panel noted that many students "have difficulty grasping the syntax or structure of algebraic equations and do not understand the procedures for transforming equations or why transformations are done the way they are." It is indeed true that for many students algebra is a foreign language. Many students simply do not understand the meaning of the symbols used in algebra. Some students succeed by memorizing rules or procedures for solving equations.

All students, however, would benefit from instruction in algebra that made the concepts visual and hands-on. This is where Hands-On Equations comes in. A study recently completed, "The Effect of Hands-On Equations on the Learning of Algebra by 4th and 5th Graders of the Broward County Public Schools, shows that such instruction can be provided as early as the 4th and 5th grade. Of the 195 students from the regular classrooms which participated in this study, more than 80% of the students experienced success in solving equation such as 3x = x+12 and 4x+3=3x+9 after seven lessons of instruction. On a retention test administered three weeks later with no Hands-On Equations instruction in the interim, the students did equally well.

The students learning via Hands-On Equations develop an intuitive and indeed a kinesthetic sense of important algebraic principles, such as the subtraction property of equality, by physically removing three blue pawns, representing the x's, from both sides of the balance scale.

If students beginning an Algebra 1 course have not been fortunate enough to have had this hands-on experience earlier in their educational career, it is still important for the regular high school algebra teacher to provide this experience to the students. Even a few short lessons can demystify basic algebraic equations and how to solve them.

Ideally, though, it is best to provide this hands-on experience earlier on, say in grades 4 to 6. Indeed, the ability to solve such equations should be a prerequisite, in the view of this educator and publisher, for a student to enter an Algebra 1 or even a pre-algebra class. If the students have had Hands-On Equations they will have no trouble at all solving these types of equations with the game pieces, and then pictorially using only paper and pencil. (The retention test noted above was administered without the game pieces.)

The Task Group on Conceptual Knowledge and Skills noted, "Without any doubt, the foundational skill of algebra is fluency in the use of symbols." Students working with Hands-On Equations develop a high level of comfort in working with algebraic linear equations of increasing complexity with unknowns on both sides of the equation. If, in addition, the students develop strong computations skills, as advocated by the Panel, the success level of such students in Algebra 1 should be significantly higher than has been the case in the past. Borenson and Associates, Inc. hopes to conduct research in this area in the 2008-2009 academic year with algebra 1 students.

If your district has a large number of students failing algebra 1, and you would like to participate in a research study to determine if Hands-On Equations instruction can make a difference in student success when they repeat the course, please send a note of inquiry to info@borenson.com.

Friday, March 21, 2008

BROWARD COUNTY HANDS-ON EQUATIONS RESEARCH STUDY

The Effect of

Hands-On Equations®

on the Learning of Algebra by 4th and 5th Graders of the

Broward County Public Schools

by Henry Borenson and Larry W. Barber


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

(The full 30 page research report may be found here. If you wish a hard copy please send $10 to cover postage and handling to Borenson and Associates, Inc., PO Box 3328, Allentown, PA 18106).

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.

The authors wish to thank Miriam Sandbrand, Mathematics Curriculum Specialist, K-5, for her efforts in coordinating this study and to the teachers who participated in this study.


GENERAL SUMMARY

BROWARD COUNTY RESEARCH STUDY

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

TEST QUESTIONS FOR STUDY #131MA

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