Tuesday, January 29, 2008

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 6th 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?

Tuesday, January 22, 2008

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 4th 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

Thursday, January 03, 2008

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 4th, 6th and 8th 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 4th, 6th and 8th 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 4th and the 6th graders achieved at the same* level as the 8th 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 4th graders, exposed to the same testing conditions as the 6th and 8th graders, did as well as the 6th and 8th 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 6th or the 8th grade to introduce these algebraic concepts, when 4th 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