BROWARD COUNTY HANDS-ON EQUATIONS RESEARCH STUDY

The Effect of

Hands-On Equations^®

on the Learning of Algebra by 4^th and 5^th 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 4^th and 5^th 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 4^th and 5^th 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 8^th or 9^th 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 4^th grade regular classes, three regular 5^th grade classes and five gifted and talented 5^th 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 4^th and 5^th 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 4^th graders saw their scores triple from the pre-test to each of the post-tests and to the retention test; the average 5^th 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 4^th graders, 84 regular 5^th graders, and 111 gifted and talented 5^th 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 4^th, 6^th and 8^th 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 4^th and 5^th 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

The Guardian Review (UK)

Catch them young

Fear + loathing = algebra. Unless you're one of the thousands of 9-year-old Americans to have discovered that algebra = fun

Polly Ghazi
Tuesday October 24, 2000
The Guardian

The equation on the blackboard reads 2 (3x + 1) = x + 22. "Who wants to explain how they solved this?" asks the teacher, Vicki Fisk. A forest of young hands shoots up. Jack, aged 10, is chosen. He jumps up and runs to what looks like a colourful toy set up on a desk in front of the blackboard. He rearranges the blue pawns (representing x) and red numbered cubes set out on a plastic balance beam to produce the right answer, amid nods from his classmates at Somerset Elementary in Maryland, US.

"It's easy and it's fun. I really enjoy maths lessons now," confides Richard Kingdom, nine, whose family moved from Wiltshire to America two years ago. "Last year I was trying to do algebra in my head and I found it very difficult. Now I can take the pieces away with my hands and make the two sides balance and I understand what it's all about."

Had he stayed in England, Richard would not have been taught such a complicated equation until he was at least 12 years old. But in the US, a new philosophy of demystifying algebra - a subject that traditionally terrifies pupils - by starting children younger is producing highly encouraging results. And for thousands of American teachers it is a new breed of educational toy that is making all the difference.

The balance beam system used by Vicki Fisk, for example, has been made a mandatory part of maths lessons for eight- to 11-year-olds in Maryland's Montgomery County, one of the US's largest school districts. Nationwide, tens of thousands of teachers have had training in using the system, know as Hands On Equations, which was developed by Dr Henry Borenson, a Pennsylvania maths teacher.

"I wanted to literally make algebra child's play," explains Borenson, who developed his system with the help of children with learning difficulties. "We have had a lot of feedback that using the equipment greatly boosts children's self-esteem. Teachers who struggled themselves with algebra have called it a revelation."

America's National Council for Teachers of Mathematics is spearheading the drive to begin teaching some high school maths concepts, algebra in particular, to children as young as six. According to the council's president, Lee Stiff, results over the past five years have been impressive, aided by educational toys such as Borenson's. "We have evidence from a number of states showing that nine-year-olds are doing better at basic algebra than older kids who come to it cold," he said.

The American approach was welcomed last week by British maths experts, many of whom believe algebra is introduced too late into UK schools. Roger Fentem, a maths educator who trains primary and secondary teachers at the College of St Mark and St John in Plymouth, described as "astonishing" the prowess shown by Vicki Fisk's class. "In Britain we would expect a bright 12-year-old or an average 14-year-old to solve that equation," he said.

Barry Lewis, director of UK Maths Year 2000, launched in January to "challenge the national fear of figures", agreed. "The leap from physical numbers into abstract qualities such as using letters in equations is the critical place where maths leaves many students behind," he said. "We support making algebra accessible and exciting at as young an age as possible. This kind of educational toy, which emphasises the concept of balance as the central principle of algebra, is spot on."

Such educational toys, however, are not generally used in British schools although many secondary-school textbooks and maths computer games use balance beam illustrations to teach algebra. What's more the national numeracy strategy, launched in September 1999 to raise basic maths standards, explicitly excludes algebra teaching in primary schools. As a result, children do not start learning even the most basic algebraic equations or formulae, such as 5 plus x = 8, therefore x = 3, until they reach 11 or 12.

"Really children should be learning such formulae as soon as teachers start asking them "five plus what is eight?" which is usually around seven or eight years old," said Fentem. "Part of the way ahead is to raise primary school teachers' knowledge of basic algebra and their confidence in their ability to link numeracy lessons with algebra. Based on the success of the numeracy strategy so far, I believe we should see great strides in algebra standards among British children over the next few years."

Making such strides is not just about academic success. American research suggests that pupils who drop out of algebra are less likely to achieve successful careers in well-paid fields such as computing and engineering. And a recent British study found that students achieving maths A-level went on to earn incomes roughly 20% higher than fellow students with only arts A-levels.

Meanwhile, a pioneering maths-teaching programme developed by Exeter University's School of Education will give some insight over the next few years into whether teaching algebra early could be as successful here as in the US. Forty primary schools around the country are using specialised lesson plans to introduce simple algebraic concepts to children as young as five. "I believe teaching algebra at 11 or 12 is far too late. Pupils find it very scary because they have no foundations in place to understand what the x and the brackets mean," says Professor David Burghes, who heads Exeter's School of Education and sat on the government's national numeracy task force.

"We find that five-year-olds have no problems understanding that 5 plus a square box equals 7 and then working out that the box equals 2. In fact they love it. But my views are not mainstream in Britain. I am out on a limb."

Back in Montgomery County, Vicki Fisk has no doubts about the pluses of teaching algebra at an age when some children are still learning to read. "The kids just love it - and they learn very fast," she says. "I have a friend who teaches seven- and eight-year-olds using Hands On Equations and they actually start crying when they have to miss a maths lesson!"

A class of nine- and 10-year-old American children individually solved these eight equations in just under half an hour. Can you do better? (Answers at bottom of page)

1 3x + 2x = 10

2 4x + 2x = x + x + 20

3 3x + 5 = x + 19

4 2(3x + 1) = x + 22

5 5x + 2 = 3x + 12

6 2x + 1 + x + 4 = x + 16 + x

7 5x - 3x + x + 8 = 2x + 1 + x + x

8 2(x + 4) = x + 10

Join in our online debate on school maths at 12.30pm. Should it really be compulsory up to GCSE level, or are there better and more useful ways of developing logic and reasoning skills? The authors of two recent books from the Institute of Education, Why Learn Maths? and The Maths We Need Now, will be live online to answer questions and fight their corners.

• Answers, x=...

1 2
2 5
3 7
4 4
5 5
6 11
7 7
8 2

The Old Schoolhouse Product Review

Algebra! That one word can strike fear into any child's heart, but it doesn't have to be that way. Algebra can be taught to a child as young as third grade, or an older student or parent that needs a hands-on, easy-to-understand method. Let me introduce you to Algebra: Hands-On Equations developed by Henry Borenson.

Each student kit includes a visual that is laminated to be durable. It illustrates the two sides of an algebra equation and its equality by using an image of a balance scale. The student manipulates the equation by using colored pawns and colored number cubes. He/she is able to physically show the problem with the manipulatives and find the answer by making both sides of the balance scale equal. Problems begin with simple concepts and end with such problems as 3x-2(-x+4)=x+(-32). The instruction manual is clear and easy to use. Videotapes are available that teach each lesson and would be great for independent work for the student. After the student uses the manipulatives to solve the problem, he/she then completes a worksheet for extra practice.

With this method anyone, including myself, can be taught algebra without the frustration. For the first time in my life, I actually understood how to do algebra and why it works. You and your child do not have to fear algebra. With Algebra: Hands-On Equations, the solution to algebra is in your hands! I give it an A+!

-- Product Review by: Debra Cogburn, Lead Product Reviewer, The Old Schoolhouse Magazine

Here's another Algebra: Hands-On Equations review!

The Algebra: Hands-On Equations Learning System is a visual and kinesthetic teaching system for introducing algebraic concepts to students in third through eighth grades. Supposedly, the patented teaching system developed by Dr. Henry Borenson, enables children, as early as third grade, to access algebraic concepts normally presented in the seventh through ninth grades. My first thought was let's round up the kids, put on the course Instructional VideoManual and see what happens.

That’s just what I did. I sat three children, ages five, nine and 11, in front of the television and fired it up. After an initial groan or two when it was discovered that they were about to watch something educational, the room became quiet. These children were glued to this video watching the demonstrator present problems and solutions.

My 11-year-old was in control of the remote so he stopped the video each time a new problem was given. He and his nine-year-old brother worked the problem independently, then compared answers, and then started the video rolling to see if their answer jived with the one presented. I was amazed. They loved the video, the manner of instruction, and solving these problems. We completed the first five of 26 lessons before taking a break. What fun. What an education in just a few short minutes.

When we turned the video off, my 81-year old mother said, “Is that all we get to see?” I had no idea she was also watching and working the problems. She held up her paper and announced that she hadn’t missed one yet. The only one who was totally unenthused about this product was my five-year-old, but then Dr. Borenson wasn’t gearing his system to five-year-olds. I considered our home test of this product a total success.

If you’re interested in teaching your children fundamental algebraic concepts, I wouldn’t hesitate to recommend Algebra: Hands-On Equations. Also, if you’re one of those people who is terrified of the thought of teaching algebra to your children, this is the product for you. Please consider Hands-On Equations when preparing to teach basic algebraic concepts.


-- Product Review by: Dr. Heather W. Allen, Senior Analyst, The Old Schoolhouse Magazine

MONTESSORI REVIEW

HANDS-ON EQUATIONS REVIEW

Heidi Anne Spietz
American Montessori Consulting

Dr. Henry Borenson has masterfully created Hands-On Equations®, a system for helping children understand algebraic linear equations. This visual and kinesthetic teaching system for introducing algebraic concepts to students in grades 3 to 9 is very consistent with Montessori principles.

By manipulating pawns and cubehs children learn how to set up equations on a replica of a balance scale. They achieve "balance" by performing "legal moves". The visual aspect of this exercise is splendid, for the child can actually see how the equation is solved. Children learn to solve simple equations and progress to learning advanced equations which involve positive and negative integers.

I first became aware of the Hands-On Equations program in 1995 at a NCME Montessori conference in Newport Beach, CA. I was impressed then, and continue to be impressed now with the recently released Hands-On Equations Instructional DVD Manual, which allows the learner to pause any lesson on the DVD at any point in time and practice the concepts presented. I spent ample time reviewing the DVD and the newly released Hands-On Equations Verbal Problems Book.

In concert with the Montessori approach, the Hands-On Equations program encourages students to solve new problems but also provides them with the opportunity to review and practice previous concepts as well. Detailed instructions for each lesson are available for the presenter.

The Hands-On Equations Learning System, along with the DVD manual and the Hands-On Equations Verbal Problems Book is a package which is perfect for use in many learning settings. This combination of items is called the Hands-On Equations Home Packet, and is available for $125. Teachers wishing to actually view the Hands-On Equations lessons before presenting them can spend time reviewing and applying the concepts presented. By so doing, the teacher will feel more confident when she provides assistance during the actual presentations and practice sessions.

Montessori knew that students would need to review concepts from time to time. Because the program is very suitable for individual viewing, the student can progress at a rate comfortable for him. The hands-on aspect of the exercises is sure to appeal to children who have participated in advanced Montessori math exercises. If a student progresses with the lessons but finds along the way that he needs to review a concept, he can simply program the DVD to replay the lessons.

Dr. Borenson's presentations are extremely clear. He has an engaging manner which will be a real relief to the student who has felt intimated by algebra in particular, as well as math in general. Gifted students will benefit by being able to set their own pace. All students will enjoy seeing their peers actually working with the Hands-On Equations materials.

For additional information about this fine, innovative learning program please visit www.borenson.com.

Heidi Anne Spietz
American Montessori Consulting
http://www.amonco.org

Inspirational Stories: Hands-On Equations

From Lisa Hoeflich, Depew, New York. Submitted January 28, 2008

I have two inspirational stories I’d like to share regarding my experience in using this program. First, I have been teaching for 15 years and have never before seen such impressive results at the completion of a unit. I am a fifth and 6^th grade enrichment teacher and I used the program with 161 students; including regular, gifted, and learning disabled.

The Light Goes On

I’d like to call this story “The light goes on” because it really did all suddenly click for a sixth grade girl I’ll call Sue. She is new to our district and is very shy. When I look back at her efforts, she was definitely a late bloomer. She remained dedicated to this program and struggled throughout the first six lessons. It took her twice as long to work on problems. She was getting very few correct. I kept working with her on the side. Modeling and redoing problems step by step. It wasn’t easy. She was getting down on herself, but she was determined to stick with it.

At the end of lesson #6, amazingly it all made sense; all at once. I saw the “lightbulb go on above her head”. She was so excited that she finally “got it”. I won’t forget the smile on her face. She and I were both so proud that she kept persevering. It was a long road. But somehow, she zipped through the problems on lesson #7 getting them all perfect!

My Three New Shining Stars

My room arrangement is five round tables and students sit together and work in groups. In one of my 10 classes, I have 3 inclusion boys sitting at one table. This makes it easier for the inclusion teacher to offer them assistance all at once. These sixth graders also enjoy sitting with their friends. I have to say that these three boys, I’ll refer to as Joe, John, and Jim, are rarely seen as leaders or positive role models. They struggle, find it difficult to remain on task, and at times, can be silly. They do not experience much success. When I introduced this new unit to the class, they really seemed to perk up and be interested in what I had to say. As we started through the lessons, THEY were the ones who were “getting it” and solving the problems accurately. They were quick. They would stand up and clap, smile, and dance after they checked their problems. They were the ones who listened to my directions, watched my modeling, and learned from my steps. Other kids who normally excelled, tried to do things their own way, take short cuts, and were making careless mistakes. Soon, my three new shining stars, Joe, John, and Jim, were my “teacher’s helpers” and checked other students’ problems for accuracy. I enjoyed watching them look over the gifted students’ papers. Quite a reversal of roles! In their reflection statements at the end of the quarter, they would write, “…I made so much progress in hands-on equations, …I felt like a leader… I was so fast… I felt good.” I wonder if they’ll have any more experiences like this?

HANDS-ON EQUATIONS WEBINARS:Spring 09

Take the workshop from the comfort of your school. All you need is a high speed internet connection and a computer headset, or microphone and speakers, and you are ready to go! (No video camera needed) Click here for a Join Meeting Test to verify that your operating system and computer is compatible with the meeting requirements.


Making Algebra Child's Play Introductory Webinar:
One Session Only
March 16 or March 23, 3:15 - 5:00 pm. Registration Fee: $75.

Day1: Making Algebra Child’s Play® (for teachers of grades 3 – 8)

This webinar consists of four sessions:

Thursday, March 12, 19, 26 and April 2 3:15 pm– 5:00 pm EST

Note: You may sign up for both the Day1 and Day2 webinars. Each Day1 online session will provide the background needed for the corresponding Day2 online session. Individuals signing up for Day2 only should have prior or concurrent experience with Hands-On Equations, or should have previously attended a Day1 workshop.

Day2: Hands-On Equations® Verbal Problems Workshop (for teachers of grades 6 -8)

This webinar consists of four sessions:

Tuesday, May 5, 12, 19 , 26 3:150 pm – 5:00 pm EST

Registration Information: The webinar consists of four sessions, each from 3:15 to 5:00 pme EST. The registration fee for each of these online workshops is $125 per person. The registration fee includes the Hands-On Equations Learning System (complete program for use with one student) for Day1 and the Hands-On Equations Verbal Problems Book and a set of student game pieces for Day2. The workshop materials will be mailed to you in advance. Registration deadline is two weeks prior to the first session. You may register via mail, fax, phone or online at www.borenson.com/workshops. Submit three paid registrations and receive the 4^th registration for free! Registration is limited to 20 participants per webinar.

Please register online at www.borenson.com.

Borenson and Associates, Inc., P.O. Box 3328, Allentown, PA 18106

(800) 993-6284 * Fax (610) 398-7863

www.borenson.com

Do Math Manipulatives Make a Difference?

By Henry Borenson and Larry W. Barber

Note: The research presented herein, although it is restricted to one specific program, is sufficient to show that the use of math manipulatives, under the conditions specified herein, has the potential to be of tremendous value in the learning process.

In the August 2005 issue of the Scientific American, in her article "Mindful of Symbols," Judy Deloach makes the observation, regarding the use of manipulatives in mathematics, that "If children do not understand the relation between the objects and what they represent, the use of manipulatives could be counterproductive."

In the December 2005 issue, in response to a letter to the editor, she says further, "For many math educators, it is an article of faith that manipulatives help children learn fundamental mathematical concepts. Unfortunately, empirical research does not support that widespread belief. What research there is indicates that the current us of manipulatives is at best questionable and at worst counterproductive."

It is certainly true that if children do not understand the relationship between the objects and what they represent that the use of the manipulatives would not be accomplishing their intended purpose. What happens, however, when the children do understand that relationship, and when the instructional system involving the use of the manipulatives follows a well thought-out sequential development of concepts? Since manipulatives are objects that can be seen, touched and moved, do they have the potential to clarify mathematical concepts beyond what can be done simply by using chalk on the blackboard?

If Ms. DeLoach were to look today at the research that has been done with a manipulatives-based program known as Hands-On Equations, she would see that there is solid evidence of the value of using manipulatives to clarify and indeed, to enable students to learn algebraic concepts. Not only would she see strong pre- to post-test results, but she would see that the use of manipulatives can enable children to learn algebraic concepts years earlier than they are normally presented using the traditional abstract methods of instruction.

The results obtained from our research confirm the statement made by Barbel Inhelder of Geneva, a student of Piaget, in 1959, "Advanced notions of mathematics are perfectly accessible to chidren of 7 to 10 years of age, provided they are divorced from their mathematical expression and studied through materials that the child can handle himself."

As an example from Lesson #3 of Hands-On Equations, the students use pawns and numbered cubes, and a flat laminated balance, to set up and solve algebraic linear equations. For example, the equation 4x + 2= 3x +9 would be set up as shown at the left.

The students would then proceed to physically remove three pawns from both sides of the scale to balance the equation, and end up with the simplified setup, showing a blue pawn and a 2 cube on the left and a 9 cube on the right. From here, the students can easily see that the value of the blue pawn, or x, must be 7. Hence, the students kinesthetically carry out the subtraction property of equality, and thereby gain a solid understanding of this important algebraic concept.

The above equation, and others such as 3x = x + 12 and 2(2x +1) = 2x + 6, were among the questions administered to 418 4^th, 6^th and 8^th grade students in the United States to compare their results on the pre-test (prior to having the first six lessons of Hands-On Equations) and on a post-test following the first six lessons. A subsequent post-test was given after Lesson #7, in which the students did not use the manipulatives, but rather used the pictorial notation involving only paper and pencil.

All of the students involved in the study participated under the same experimental and testing conditions. The teachers of these students had all attended a full-day Making Algebra Child's Play workshop conducted by a certified Borenson and Associates instructor. The teachers then implemented the program as instructed, following the specified sequence of concept development that went along with the use of the manipulatives. Each of the students used the game pieces and flat balance scale at their desks during the instructional segments, and in the post-test following Lesson #6. The pre-tests, post-tests, and time alloted for those post-tests were the same for all the 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%

From,"A Comparison of Algebra Achievement by 4^th, 6^th and 8^th Graders," by Henry Borenson and Larry W. Barber

Note: The pre-test questions consisted of 1) 2x = 8, 2) x + 3 = 8, 3) 2x +1 = 13, 4) 3x = x + 12, 5) 4x + 3 = 3x + 6, and 6) 2(2x + 1) = 2x + 6. Similar questions were used for the post-tests.

From the table above we note that 1) the 4^th and the 6^th graders achieved at the same* level as the 8^th graders on a post-test following the first six lessons, with all groups scoring in the 84% to 92% range, and with all groups having a significant pre- to post-test gain, 2) the students were able to transfer their hands-on learning to a pictorial solution using only paper and pencil (post-test Lesson #7) and maintain their Lesson #6 post-test results and 3) all three groups did comparably* well on the post-test following Lesson #7, with all groups scoring between 88% and 93%.

* The means exhibited were so close that we did not even bother to do a t-test for significance.

The result obtained, showing that the 4^th graders, exposed to the same testing conditions as the 6^th and 8^th graders, did as well as the 6^th and 8^th graders leads to the conclusion that Hands-On Equations removes an age difference of up to 4 years as far as teaching these particular algebraic concepts is concerned. Hence, an important educational policy question arises from this study: Is there any need to wait until the 6^th or the 8^th grade to introduce these algebraic concepts, when 4^th graders can do as well?

It is clear from the above research results that the use of manipulatives, when students understand the relationship between the manipulatives and what they repressent and when they are used as an integral component of a well-structured sequence of concept development, has the potential to increase student achievement in mathematics. In particular, the Hands-On Equations instructional program, can remove an age difference of 4 up to years in the instruction of the algebraic concepts mentioned in this study.

Henry Borenson, Ed.D. Allentown, PA, and Larry Barber, Ph.D. Bloomington, IN

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 4^th, 6^th and 8^th 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

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.