Presentation to the National Mathematics Advisory Panel
Chicago, IL, April 20, 2007
By Dr. Henry Borenson, President
Borenson and Associates, Inc.
Mr. Chairman and members of the panel: I thank you for this opportunity. My name is Dr. Henry Borenson, President of Borenson and Associates, Inc. Some twenty years ago, as a middle school math teacher, I was concerned with the difficulty students were having learning algebra abstractly. I determined to find a way to simplify the concepts, to make them concrete and visual, and to make them accessible to grade school students.
After two years of experimentation, working with children, including LD children, I developed a system known as Hands-On Equations. This is a system which uses games pieces, a flat laminated scale, and a specific progression of ideas to enable students as early as the third grade to physically represent and solve algebraic linear equations, the type of equations which until then were typically taught in the 8th or 9th grade.
Since 1995, Borenson and Associates has conducted more than 1500 Making Algebra Child’s Play workshops throughout the United States. In these workshops, teachers of grades 3 to 8 see learn how to introduce the concept of a variable, the concept of an equation, the subtraction and addition properties of equality, and other key algebraic concepts.
A key part of these workshops is a student demonstration with local 4th and 5th grade students. More than 1500 times since 1995, the teachers attending our seminar have seen how in three lessons 4th and 5th grade students, even so called “low ability” students, can learn to solve an algebraic linear equation such as 4x + 3 = 3x + 9.
In a study to determine teacher confidence level to teach algebraic linear equations to their lowest achieving students, Barber and Borenson (2006) discovered that only 16% of 751 teachers from grades 3 to 8 attending a Making Algebra Child’s Play workshop felt they would be successful using the traditional abstract teaching methods, while 98% expressed confidence of success if they were to use the Hands-On Equations materials. See Appendix A.
In an ongoing series of studies involving multiple student characteristics and multi-site replication, supervised by Dr. Larry W. Barber, formerly Director of Research for Phi Delta Kappa, we have found significant pre test to post test gains for 2nd grade gifted students, regular 6th grade students and 9th and 10th grade low achieving students.
Recently we completed a study involving four 5th grade inner-city classes comprising a total of 111 students. The pre to post test results showed a large and highly significant increase in scores. The combined mean increased, in percentage terms, from 44.8 % on the pre test to 85.3% on the post test. On a three week retention test, provided three weeks after the post test—with no Hands-On Equations instruction in the interim—the mean was 78.6%. When compared with the pre test score of 44.8%, this increase was found to be statistically significant with a t-value of 13.71. We are talking about 5th grade inner city students succeeding with important algebraic concepts. This study may be found in Appendix B.
We believe we have provided evidence that Hands-On Equations system of instruction significantly and positively impacts upon teacher self-confidence in their ability to introduce algebraic linear equations to their students, and evidence that the program makes a measurable difference in student learning. We believe it is possible and it is important for students to gain the perception that mathematics is a subject they can understand, and a subject at which they can excel. In Hands-On Equations the students need not memorize a set of procedures in order to obtain an answer. They can use their creativity to apply general algebraic principles in the manner that best suits them. We ask the Panel to consider recommending Hands-On Equations as a supplementary program that is effective in introducing grade school students to basic algebra.
Henry Borenson, Ed.D.
The Teacher Study Results*
First Problem: To obtain a measure of teacher confidence in teaching algebraic concepts to 80% or more of the students in their lowest achieving class using traditional instructional vs. HOE, and to compare the results.
We wished to identify whether or not teachers attending the Making Algebra Child’s Play seminar, the large majority of whom are elementary and middle school teachers, are confident that they can successfully teach algebraic concepts to at least 80% of the teacher’s lowest achieving class, via two different modes of instruction: the traditional mode of instruction and HOE, and then to compare these responses to see if there is a significant difference. The response to the traditional mode was obtained on the pre-test, prior to the beginning of the seminar treatment. The HOE response was obtained at the conclusion of the 6th lesson of the seminar, approximately 2 ½ hours into the full-day Making Algebra Child’s Play seminar.
To accomplish this objective, the teachers were asked to respond anonymously to the following question on their onsite questionnaire (see Appendix), prior to the beginning of the seminar:
Please indicate “Yes” or “No” to the following question.
“I am confident that using the traditional method of teaching algebra, I am able to teach 80% or more of the students in my lowest class how to understand and solve these two questions.
2x + x + x + 2 = 2x +10
2(x + 4) + x = x + 16
Following the conclusion of Lesson #6, they were asked to respond, also anonymously, to the following question on the same questionnaire:
Please indicate “YES” or “NO” to the following question:
“ I am confident that using the Hands-On Equations system of instruction, with each student having their own set of game pieces, that I would be able to teach 80% or more of the students in my lowest class how to understand and solve the two equations shown above.”
Both questions essentially asked the teacher, “Are you confident that you can teach these algebraic equations to 80% or more of the students in your lowest class using this specific method of instruction?” The teachers had to select either a “Yes” or a “No” response.
For this line of research we found 751 teachers who filled out both the pre-test and the post-test. All but 46 of these respondents were elementary and middle school teachers. In order to quantify the data we assigned a 1 to each “yes” response on the pre and post-test and assigned a 0 to each “no” response. The comparison was between the mean of the yeses on the pre-test and the mean of the yeses on the post-test. The pre-test mean was .162. The post-test mean was .984. We then calculated a t value (we used the t for paired observations) between the two means: the t value was 57.81. The post-test mean was over 6 times larger than the pre-test mean.
* This study is taken from the report titled, “Borenson Hands-On Equations Research Designs and Interim Results: December 2006. Effect of Making Algebra Child’s Play® Seminar on Teacher Self-Concept and Student Achievement” by Larry W. Barber and Henry Borenson. This study may be found in full at www.borenson.com
Teacher Study Data Results: Problem #1
N= 751, t = 57.81
The statistic used above was the difference between the Means for Paired Observation and Equated Groups (“t-test for paired observation,” Edwards 1963) (Barber et. al 1988). The formula used came from the Edwards book (p. 281) and was applied to test the difference between the group mean on the pre-training self-report and the group mean on the post-training self-report from the same teachers. For this first analysis we simply analyzed the data on all teachers who gave both a pre and post response.
Conclusion: We note that only 16% of teachers coming to the Making Algebra Child’s Play seminar expressed confidence that they would be able to teach 80% or more of the students in their lowest classes the solution to equations such as 2x + x + x + 2 = 2x +10 and 2(x + 4) + x = x + 16 using the traditional teaching methods. In light of the significant relationship which research shows to exist between teacher efficacy, i.e. teacher belief, and student achievement (see page one of this study), this result is important. If this result turns out to be representative of teachers nationwide, it would suggest that the use of the traditional methods of instruction is not likely to accomplish the goal of successfully teaching the above concepts to 80% or more of the students in our lowest achieving classes. On the other hand, by the end of the 6th lesson of the seminar, 98% of the participants at this seminar, the majority of who were elementary and middle school teachers, expressed confidence that, using the Hands-On Equations system of instruction, with each child having their own set of manipulatives, they would be able to teach these concepts to 80% or more of the students in their lowest class.
Hands-On Equations® Research
Interim Results, Study # 33b, Mar. 30, 2007.
The Effect of Hands-On Equations on the Learning of Algebra
By Title I Inner City Students in the 5th grade.
By Larry W. Barber and Henry Borenson