Friday, November 23, 2007

A Comparison of Algebra Achievement by 4th, 6th and 8th Graders

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 Conclusion section of the research report is noted below.
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



Sunday, October 21, 2007

JULIE BRENNAN REVIEW OF HANDS-ON EQUATIONS

Julie Brennan of LivingMath.net has written a review (click to see full review) about her experience in using Hands-On Equations with her children at home. In her review, she says:

"I cannot recommend this program enough, I am so impressed, and I have not raved about a math program I can think of without qualification. We are very eclectic in how we approach math (as this site can attest to), so math programs usually require a lot of tweaking to fit our family. This program, however, only required pacing adjustments, It’s easy to adjust it to your family’s pace and needs.



Delene, 9, and Kira, 7, use
"legal moves" to solve equations.







Because it is taught like a game, my 7 y/old could use it after she viewed the video clip of another (looks like 7 y/old) child doing the problem. And how can she resist game pieces and dice :o). Typical of this child who loves to draw, she went from the game pieces to the white board and demonstrated her “problem” there in just the same way a later lesson is supposed to teach her how to do this.


My 9 year old is totally ready for this and worked quickly and effortlessly through the first 3 lessons which got her to solving problems like this: 2x + 3 + 3x = x + 11 and x + 2 + 2x = x + 10 Within one month both she and her sister have easily moved into Level 2 lessons involving the white (negative x) pawn and learning the rules of simplifying by building zero sets. The kids solve the equations using the “legal moves” they have learned. In Level 2, you learn to do this by removing equal numbers of blue pawns representing x AND equal constant values from each side of the balance beam, so a problem like this one: 3x + 7 = 4x is with a quick swipe of 3 pawns from each side reduced to a 7 constant on one side and one pawn / x on the other. Obviously x = 7. She checks it by skip counting by 7s: If each x/pawn is 7, then 7-14-21-28 is one side, and 7-14-21-28 is on the other. 28 equals 28, it’s in balance.

"My biggest surprise was the reaction of my older boys to this. My 11-1/2 y/old has been “almost ready” for algebra for about a year. He used Singapore half the year and ALEKS math the 2nd half along with a lot of living math activities. He tried some of the algebra problems that ALEKS was giving him but just didn’t understand them, and I did tell him that some teaching at this point might be a remedy worth considering, but I wasn’t going to seriously pursue that until
he seemed ready.

"When he saw me using the Hands-On manipulatives with my daughter, he eagerly asked to join in. I had a 2nd set of game pieces and said fine, but after a while I realized I would need to work with him separately because he zoomed through the material in the first 4 lessons immediately once he figured out the first 2 sets of legal moves (the 2nd being the ability to subtract a constant “weight” from each side as long as they are the same amounts).

"What a reaction, my goodness. He was really excited because he saw how easy algebra was once you could mentally manipulate equations the way he was doing them with the game pieces. His leap from the concrete to abstract was almost immediate, but he had needed this concrete demonstration for the abstract to stick. In 2 weeks we went through two-thirds of the 26 lessons, which introduced the concept of a negative x and several other legal moves to form “zero sets” - a negative x plus a positive x is a zero set. The third section, Level 3, took longer as he needed to slow down working with both negative x pieces and negative constants. I thought he understood negative number reasonably well, but this level has made him much more comfortable working with negative numbers. Instead of memorizing rules applied to abstract numbers, he can *see* why these work. For example, to subtract a negative, it “turns into” a positive because the only way to subtract something that isn’t there is to add a zero set - a positive and negative of the same number. When you take away the negative, you are left with the positive number only. And yes, young children can learn this with the concrete method.

"The other really important aspect of this is that it teaches kids to think of a negative sign as an attribute of the number itself, vs. the subtraction operation elementary math usually teaches them. This is a critical factor in algebra success, as it allows one to move the negative number around within equations without making mistakes."


DJ, age 11 1/2, zoomed through
the first four lessons


In 2 weeks, DJ was 2/3
through the program

--------------------------------------------------------------------------------------------
Julie Brennan recommends the Home Packet . This includes an individual set of Hands-On Equations for use with one student at a time, the DVD (or VHS) Instructional Manual, and the Hands-On Equations Verbal Problems Book. This packet is offered at the special price of $125 for home schoolers and is available through Borenson and Associates, Inc.










Sunday, April 22, 2007

Algebra for Grade School Students

Click here to hear Dr. Borenson's 5-minute presentation to the National Math Panel in Chicago on April 20, 2007.


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.

Thank you.

Henry Borenson, Ed.D.


Appendix A

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

and

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


X1

X2

D

D2

Sum

122

740

618

622

Mean

.162

.984



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.

Appendix B

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