The kinesthetic component of Hands-On Equations is one of the facets of the program which produces such strong student gains. Hence, the use of the Smartboard should not replace the physical elements of the program. Students should still use the game pieces at their desks and the teacher should still use the Teacher's Demonstration Scale at the front of the classroom.
Click to on the arrow to play.
The Smartboard application is simply another means to illustrate the teaching problems of each lesson. For example, the teacher can illustrate the simultaneous removal of three pawns from each side of the scale.
Each of the examples that are presented in each lesson of the Hands-On Equations red, blue and green booklets is presented on its own slide. In addition, there is a blank template slide if the teacher wishes to provide additional examples on the Smartboard. The Table of Contents enables the teacher to go directly to the desired lesson.
A number of the lessons have teaching points to be presented to the students as a summary of the lesson. These are available via a pull-out tab such as that shown below.
As noted, the Hands-On Equations program is fully effective without the Smartboard. Indeed, the Smartboard application can be considered a luxury item. Hands-On Equations for the Smartboad is expected to be available for download purchase.
Cost: for one teacher $125. License for five teachers: $500
Borenson and Associates, Inc. 800-993-6294 info@borenson.com
Students in grades 4 to 8 who have had the first seven lessons of Hands-On Equations score above 80% on a post-test consisting of the above questions. Wouldn’t you like to know if the students in your Algebra 1 classes -- near the end of the academic year -- can do at least as well on these basic equations? If so, we invite you to participate in this preliminary study.
We can provide you with the instruction for administering the survey, the Equation Survey, Teacher Questionnaire and Summary Form. You may ask the teachers to return the Summary Form and Questionnaire to you, or they may submit them directly to us at the address noted on the document. We will analyze the results and submit a report back to each teacher.
Why would you wish to conduct this study?
The ability to work with simple algebraic linear equations is crucial to success in Algebra I. Knuth, Alibali, et al (2008) quote Carpenter, Franke, and Levi (2003) saying that “a limited conception of what the equal sign means is one of the major stumbling blocks in learning algebra. Virtually all manipulations on equations require understanding that the equal sign represents a relation.” If your Algebra I students cannot solve the vary basic equations noted above, is it any wonder that they are having difficulty with the subject?
We invite you to participate in this preliminary study to give us initial feedback on how Algebra I students compare to younger students who have had Level I of Hands-On Equations. Your assistance is appreciated.
Sincerely,
Henry Borenson, Ed.D.
References
Carpenter, Thomas, Megan Franke, and Linda Levi. Thinking Mathematically: Integrating Arithmetic and Algebra in the Elementary School. Portsmouth, NH: Heinemann 2003
Knuth, Eric J, Martha W. Alibali, Shanta Hattikudur, Nicole M. McNeil, and Ana C. Stephens. “The Importance of Equal Sign Understanding in the Middle Grades.” Mathematics Teaching in the Middle School. Vol. 13, No. 9, May 2008:514-519
HOW TO PARTICIPATE:
Administer the survey to one or more of your Algebra 1 classes. Complete the teacher questionnaire and summary form (no student names please), submit it to us, and as a token of our appreciation we will provide you with a complimentary copy of the Hands-On Equations Verbal Problems Book containing more than 350 verbal problems, including consecutive number, age and distance problems. If you wish to participate, mail your inquire to info@borenson.com and we will send the survey documents to you. Thank you.
Hands-On Equations Instructor Kendra Jensen just completed conducting a Day1 and Day2 workshop in Hot Springs, AR.
The comments below, submitted by the instructor, are typical of the high value that workshop participants place on this training.
Hot Springs, AR
Day 1
Tuesday, April 21, 2009
Thank you for sending me to Arkansas! I learned about it's geography by driving into Hot Springs on a winding, mountainous highway for the last 2 hours of the drive! Beautiful scenery, challenging drive. All was fine.
Participants:
Our participants readily volunteered. Their teaching experience ranged from 10 years to 28 years and they were just as eager to learn as an educator with less experience. Great group.
From Bryant, Carol taught 7th grade; Heather 8th; Jaime was a special ed teacher who teaches with them. Elisabeth was also a special ed teacher.
Our two participants not from Bryant were Shannon who teaches K-6 gifted and Kittena who is a K-12 Math Coordinator.
Day 1 teacher quotes and conversations:
Carol wanted to know early on if there were worksheets and teacher manuals. Heather was excited about the smartboard application.
"This is so cool ... When I saw the demonstration at my school, it was too fast."
Lesson 7: "I've always done the numbers first. I guess you get stuck in your ways. I like that HOE teaches the removal of pawns first." Heather repeated this when asked what the highlight of the morning session was for her.
Shannon knows of a school who uses HOE and has Level l taught in 3rd grade, Level ll in 4th and Level lll in 5th grade. They like it.
Shannon especially liked the fact that we're reaching so many learning styles with this approach!
Kittena loved both days! "I have a monthly math focus for various grades. I demonstrate or model teach on that focus topic. I'll use HOE for this! I'm going to buy a kit for each school (or teacher?)." Kittena took such detailed notes because she plans to present this to the many teachers she helps. I told her to enquire about overheads from our office.
Day 2
Wednesday, April 22, 2009
Additional Participants:
For Day 2, Michelle joined us and teaches 6,7, and 8th grades.
Corrie and Laurie were 3rd and 4th grade teachers from the same school. They said they registered for the half day. They were a little concerned about the level of math necessary but were encouraged to find their 3rd and 4th graders could do many of our verbal problems.
Day 2 teacher quotes and conversations:
The Arkansas Benchmark test was mentioned several times. "Oh, if our kids had HOE before that test, they would have aced it!"
Kittena, the Math Coordinator, expressed a concern about open response questions on the Benchmark. There are two people grading a student's answer to the open response. Each grader has 4 points to give for a total of 8 points. "How would individual graders view the pictoral solution. Is there enough there to earn a full 4 points?" Heather, an 8th grade teacher responded enthusiastically, "The student has to solve and then verify the answer. For word problems, there's also the full sentence answer to show understanding. I definitely think there's enough here to earn each grader's 4 points."
When asked what they liked most about this approach, the participants replied:
"I like setting game pieces on the scale phrase by phrase through the word problem." It makes sense.
"This makes fractions more meaninful."
"I love the fact there's so much going on here. There are so many topics buried in this program!"
The highlight of Day 2 for me personally was bringing volunteers up to demonstrate the travelling scooters and trains. These participants loved it and fully participated. I heard several "This is good!" "They'll understand this." - type comments.
Observations:
I used the Effective Learning transparency #13 from Day 1 to help in the review for Day 2.
These participants noticed two minor discrepancies on the transparencies. On transparency # 32, the verbal problem is actually number 6 not 5 on VPB page 42. and on transparency #37, the verbal problem is number 10 not 9 on VPB page 57.
The Velda Rose Hotel was a grand hotel in its day. My room was nice. The meeting room showed signs of age ... with worn trim and scratched doors, even a broken window down the hall from it. The hotel smelled of smoke when you walked in the front door. I asked Kittena Bell, a participant who stayed overnight there, what she thought just in case I was being too picky ... maybe it was a local attraction. She told me she actually called a friend to come stay with her because she wouldn't stay in such a place by herself. She also said "it wasn't a positive draw for such an awesome workshop."
This price includes the $34.95 Hands-On Equations Learning System
Here is your opportunity to learn about this powerful teaching technology which is enabling grade school students to solve algebraic equations normally not presented before the 8th or the 9th grade.
Experience this whole-brain instructional approach which uses a visual means to demystify equations and which uses gestures to solve the equations. With this system, your students in grades 3 - 8 will learn to solve (and understand) equations such as 4x + 3 = 3x + 9 and 2(2x + 3) = x + 10. The webinar will also provide an introduction to using the game pieces to solve word problems.
To take advantage of the special price, please enter code "0325Web31" when you register online.
For additional information about Hands-On Equations, please go to www.borenson.com.
You may also call Borenson and Associates, Inc. at 800-993-6284.
GSAS 530_ 01:Hands-On Equations as a Model of Instruction
1 Credit
One or two graduate professional development credits provided by Regent University are available at an additional cost of $185.00 per credit/hour. Attend one of the Day1: Making Algebra Child's Play Workshop (for teachers of grades 3 to 9), or the Day2: Hands-On Equations Verbal Problems Workshop (for teachers of 6 to 9) held throughout the United States, and complete additional course work to obtain this credit. Below is the course description for Day 1. The course description for Day2 is yet to be posted. (See end of this entry for registration procedure.)
Regent University School of Education
Professional Development Course
GSAS 530¬_ 01: Hands-On Equations as a Model Instruction 1 Credit
Professor: Jenny Sue Flannagan
Ed.D. Adult Learning & Staff Development, Regent University
Ed.S. Administration and Supervision, University of Virginia
M.Ed. Curriculum and Instruction, University of Virginia
B.S. Biology/Earth Science/Space Science, Longwood University
E-mail: Jennfla@regent.edu
Course Description: This course will focus on the Hands-On Equations (HOE) method of teaching algebra. Educators participating in this course will understand how research should inform teaching practices in mathematics and will reflect on the use of this method of instruction to enable children to demystify the learning of algebra.
Purpose - Educator as a Professional: A professional teacher can defend his or her practices based on the "state -of-the-art." The best learning is "learning by doing." That is, students learn best when they construct meaning for themselves out of the course content. The central aim of this course is to provide the impetus for educators to implement and reflect on how Hands-On Equation is an appropriate model of instruction for mathematics.
Course Objectives: 1. Demonstrate knowledge of how children develop and learn by providing opportunities that support the use of visual and kinesthetic approaches to the teaching of mathematics
2. Demonstrate knowledge of an ability to implement the Hands-On Equations model of instruction, and to conduct and evaluable pre- and post-tests
3. Demonstrate the ability to assess student self-perception prior to and after instruction
4. Demonstrate the ability to monitor self-perception regarding a mode of instruction
Course Requirements/Evaluation Procedures: 1. Prior to taking the Hands-On Equations workshop, indicate your feelings about mathematics and algebra. For example, “I like mathematics because…” or “I like teaching math because...” or “My feelings about mathematics are…”. Please include a response to “When I think of algebra and teaching algebra I feel…”
2. Participate in the full-day Making Algebra Child’s Play Workshop.
3. Administer the pre-test for Level I to at least one class
4. Administer an assessment tool to give you feedback on the student self-perception of their ability to do mathematics, for example, students could respond to, “I like mathematics because….” Or “I do not like mathematics because..” or similarly, “I think I am good at math because…” or “I do not think I am good at math because. Another example, “When I hear the word ‘algebra’ I think (I feel)……
5. Teach the first six lessons of Hands-On Equations
6. Administer the post-test after Lesson #6*
7. Teach Lesson #7 of HOE
8. Administer the post-test after Lesson #7*
9. Administer the post-assessment tool of #4 above.
10. Self-administer the post-assessment instrument mentioned in #1 above
Note: the teacher may administer either the Post-test after Lesson #6, the post-test after Lesson #7, or both. If you wish the statistical analysis to be conducted by Borenson and Associates, Inc., simply send them the summary forms at P.O. Box 3328, Allentown, PA 18106.
a. Participate in all on-line discussion forums and complete online activities in a timely fashion. There will be two discussions during the course.
b. Write a two to three page reflection paper that articulates the following:
• What gains did you see in the students’ pre/post test?
• What changes did you see with regard to students’ beliefs about their ability to do algebra and mathematics?
• What were your initial feelings/beliefs about teaching algebra and how did these feelings/beliefs change over the course of using the Hands-On Equations method of instruction?
• If you feel this method of instruction was successful with your students, what particular elements of the instructional system do you give the most credit?
Grading Scale (Point Breakdown): 1. Online Participation: 25 points
2. Administration and posting of pre/post test classroom data: 20 points
3. Administration and posting of student survey data: 20 points
4. Completion of written report regarding use of instructional
model: 25 points
5. Completing all assignments on time: 10 points
Total possible points = 100 points
Grading Information: It is important that all submissions be presented by the deadline. Late submissions will not receive full credit.
If you are interested in this opportunity, you will need to complete a registration form and send it via fax (if paying by credit card) or postal mail (if paying with a check). Please e-mail Molly Waters at mollree@regent.edu to request a form or visit http://www.regent.edu/acad/schedu/profdev/ to download a form. You will then send the completed form to Molly Waters at Regent University:
Molly Waters
Regent University
1000 Regent University Drive
Suite 266, Administration Building
Virginia Beach, VA 23464
Or fax it to Fax is 757- 352-4147
To receive the one credit from Regent University you must: 1. Complete the full-day Making Algebra Child’s Play Workshop.
2. Complete the Regent University registration form (send it along with payment to Molly Waters at Regent University.
3. Once you are registered you will receive an email with directions on how to log into the course.
4. Complete all “Course work” as described in the course syllabus. You can access the syllabus at www.regent.edu/mathscience. To receive a passing grade in this course and earn the credits from Regent University, you must complete either the one or two days Hands-On Equations workshops and complete all assignments in the online course.
Contact Molly Waters, Professional Development Programs Coordinator, at Regent University for additional questions regarding this opportunity.
This publication of blackline masters contains more than 200 Level I verbal problems (taken from the Hands-On Equations Verbal Problems Book), one per page to make it easy fo the teacher to use with an image projector or to copy and distribute to the class. An example is shown below:
VPB Pg. 10/Ex. 28; Solution Pg. 20/Ex. 28
28. Bill had two water jugs that each held 7 gallons. Tom had one jug that held 2 gallons and three others that held equal amounts. If all the jugs together held 28 gallons, how much did each of the three equal jugs hold?
VPB Pg. 25/Ex. 5; Solution Pg. 30/Ex. 5
5. Jamie guessed a number. Rob guessed a number that was double the sum of 3 and the number guessed by Jamie. If the sum of the numbers guessed by Jamie and Rob is 18, what are the numbers?
by Douglas K. Brumbaugh, Enrique Ortiz, Regina Harwood Gresham - Education - 2006 - 328 pages
Hands-On Equations Hands-On Equations is another manipulative that helps develop algebraic concepts. These materials may be used to solve equations like the ...
by Mary E O'Shea, Gratz College, RTC Division - 2003
A research project submitted to the faculty of Gratz College in partial fulfillment of the requirements for the degree of Master of Arts in Education, 2003.
... balance beam with top like objects to represent variables and dice to represent numbers. Your math lab can order the Hands On Equations by going to www. ...
Hands-On Equations Hands-On Equations is an introduction to algebra, using manipu- latives; and by the time kids work their way through the first few ...
by Julia M. Jacobsen, Jan Fay Kress, Jane C Belcher - Education - 2007 - 157 pages
... more inquiry Already noted this on 2 other forms . . . math manipulations in general and hands on equations Workshops are wonderful Electricity Inquiry ...
by Margaret Wayne Gosfield, California Association for the Gifted - Education - 2008 - 307 pages
...Hands-on Equations, Mortenson Math, Math-U-See, and more. Maybe If We Ignore It, They'll Outgrow It! Preferred learning styles don't affect just ...
by Phyllis Jean Perry - Education - 1997 - 127 pages
Activity suggestions for groups of four promote cooperative, logical thinking. Uses maps, science, and mapping skills. ** Hands-on Equations. Allentown, Pa. ...
by Joan Franklin Smutny, Sarah Von Fremd - Education - 2008 - 344 pages
(The kit Hands- On-Equations uses an excellent approach for teaching this topic. ) Creative Constructions Student-designed structures made from stacks of ...
by Luis Ortiz-Franco, Norma G. Hernandez, Yolanda De la Cruz - Mathematics - 1999 - 168 pages
... Who Had Seen and Used Various Types of Manipulatives Type of Manipulative Percent Algebra tiles 26.8 Fraction bar 51.9 Hands-On Equations® 41.4 Geometry ...
... DC 20004 William R. Fry, Executive Director Oldest and largest organization of Americans for legal reform Hands-On Equations National Media Coordinator ...
by United States Congress. House. Committee on Education and the Workforce. Subcommittee on Early Childhood, Youth, and Families - Education - 2000 - 78 pages
... development and training in the areas of Hands on Equations and Everyday Counts Calendar Math, as well as training in a new physical science curriculum. ...
by United States Congress. House. Committee on Education and the Workforce. Subcommittee on Early Childhood, Youth, and Families, United States, United States Congress. House. Committee on Education and the Workforce. Subcommittee on Postsecondary Education, Training, and Life-long Learning - Mathematics - 2000 - 265 pages
... full implementation of Hands-On Equations and the Wil- liam A Mary Reading/ Language Arts program; some exciting elementary school curriculum initiatives ...
... (Hands On Equations: This is a simple, inexpensive, yet useful manipulative program to help children "get the idea" behind math calculations and ...
The examples below are taken from the Hands-On Equations Verbal Problems Book. This book contains more than 300 verbal problems including number, consecutive number, age and distance problems for all three Levels of Hands-On Equations. A sampling of the types of problems presented in the book is shown below. Within each section of the book the problems are graduated in increasing order of difficulty. This makes the book a valuable resource for teachers in grades 4 to 6, as well as for teachers of pre-algebra and Algebra I students. (The number in parenthesis indicates where the problem can be found in the verbal problems book.)
Level I
1. Kathy's plant grew the same amount in January and February. In March, it grew 3 inches. If the plant grew a total of 13 inches during these three months, how much did it grow during each of the other months? (Page 8/7)
2. Heather can buy 4 pizzas for the same price as 2 pizzas and 8 one-dollar drinks. How much does each pizza cost? (Page 9/16)
3. Celeste is 12 years older than Rosa. In four years, she will be twice as old as Rosa will be then. How old is each now? (Page 58/17)
4. Charlene has a container 1/2 filled with pennies. She realizes that if she adds 12 pennies to the container, it will then be 2/3 filled. How many pennies does the container hold? (Page 77/18)
5. The average speed of an express train is 14 miles per hour more than 1/3 the speed of a freight train. In two hours the express train travels the same distance as the freight train in three hours. Find the average speed of each train. (Page 102/18)
NEW: HANDS-ON EQUATIONS VERBAL PROBLEMS BLACKLINE MASTERS FOR LEVEL I More than 200 Level I verbal problems taken from the verbal problems book, one per page (VB-BM1.......................................................................................................$25)
Level II
1. The sum of two numbers is 10. Twice the first, increased by the second number, is 10. Find the numbers. (Page 27/18)
2. Jim has two lists of three consecutive even numbers. The sum of the first number on each list is 10. If twice the second number on the first list has the same value as the first number on the second list, what are the two set of consecutive even numbers? (Page 43/18)
3. If Jim's age is added to Sandra's age, the sum is 18. If twice Jim's age is subtracted from Sandra's age, the difference is 3. How old is each? (Page 68/22)
4. Charlotte has a total of 18 coins consisting of dimes and nickels. If the number of nickels is 12 more than the number of dimes, how many of each coin does she have? (Page 92/28)
5. Bobby can paddle a canoe at 3 miles per hour. For 1 hour, Bobby paddles with the current and travels 4 miles further then when paddling back against the current for one hour. What is the canoe's speed when it travels with the current? (Page 104/29)
Level III
1. When a number decreased by 4 is doubled, the result is the same as the number increased by 6. Find the number. (Page 28/25)
2. Charlotte has two lists of consecutive odd numbers. The sum of the first number on each list is 10. When the 4th number on the 1st list is doubled and then subtracted from the first number on the second list, the result is the same as the second number from the firs list, decreased by 14. Find the two sets of numbers. (Page 44/28)
3. Ten years ago, Marlene was 6 years older than 1/3rd of her present age. How old is she now. (Page 60/30)
4. Two-thirds of a collection of 90 coins consists of nickels. Of the remaining coins, the number of dimes is 10 more than 1/3rd the number of quarters. How many of each type of coin is in the collection? (Page 78/24)
5. A private plane flying for two hours meets a headwind that reduces its speed by 20 miles per hour. If it took the plane a total of 5 hours to travel 440 miles, find the speed of the plane prior to meeting the headwind. (Page 106/40)
Hands-On Equations to be implemented in 1,000 classrooms in Africa and Asia
Contact: Henry Borenson, henry@borenson.com, 800-993-6284, Or Patti Davis, patriciadavis@opportunityeducation.org, 402-660-2174
October 28, 2008--Borenson and Associates, Inc., an Allentown, PA firm known for its innovative algebra teaching system that enhances learning performance in U.S. schools, has been selected to provide its program to more than 1,000 classrooms in Asia and Africa.
Under the agreement, Hands-On Equations®, which is the name of the proprietary algebra program developed by Henry Borenson, will be implemented in 5th, 6th and 7th grade classrooms in Uganda, Tanzania, South Africa, Ghana, Sierra Leone, India, Sri Lanka and Nepal by Opportunity Education, a charitable foundation that provides educational tools for children in developing countries around the world.
“We are delighted to be chosen to provide our program to these students in developing nations,” said Borenson. “I’m confident the teachers in these classrooms will be very satisfied with the success they will see in their students.”
Borenson’s algebra teaching system uses manipulatives – similar to game pieces – to help students visualize an equation. Students use the manipulatives to make chess-like moves to balance an equation. “The program helps demystify the language of algebra and helps take an abstract concept and make it an intuitive process,” he said.
The Hands-On Equations® program has been introduced to over 25,000 teachers in the U.S. and is used in schools across the nation.
Opportunity Education chose to distribute Hands-On Equations® because the organization’s director of education, Patti Davis, had used the program as a classroom elementary school teacher, and had seen the improvement in her students’ abilities to learn algebraic principles and retain the skill set.
“I recommended Hands-On Equations® because I had had personal success using it in my classroom,” said Davis. “We chose the program because it is the easiest to understand, has the most applicability in classrooms in other parts of the world, and produces results.”
Providing the learning program to schools in developing nations supports the organization’s overall mission to supply curriculum and materials to classrooms in order to give children in developing countries the advantages of a U.S. education. “Providing the program and the materials the teacher needs to use to teach the program is imperative to help these countries meet their educational standards,” says Davis.
Opportunity Education was founded by Joe Ricketts, founder of TD Ameritrade. It is a charitable foundation that provides educational tools for children in developing countries around the world. For more information, go to www.opportunityeducation.org.
Borenson and Associates was founded by Henry Borenson in 1986. Borenson had taught honors mathematics classes at the Stuyvesant High School in New York City. For more information, go to www.borenson.com.
Complete instructions for administering pre- and post-tests for Level I are now available online. These instructions include a Summary Form for the teacher to record student scores and compare class results on each test, as well as class performance on each question.
On the pre-test, especially with students in grades 3 - 5, the scores are expected to be very low since these concepts and examples are not normally presented at these grades. The teacher should not be too surprised by low pre-test scores. If the teacher has gone through the training and implements the program as designed, he/she will see significant gains on the Lesson #6 post-test. This improvement is expected to be maintained as the students move away from using the game pieces to the pictorial notation in Lesson #7.
Pre-Test- 15 minutes
Post-Test After Lesson #6
(Students may use the game pieces on this post-test- time limit: 15 minutes)
Post-Test after Lesson #7 (Students do not use the game pieces-- just the pictorial notation) Time Limit: 15 minutes
Public Hands-On Equations workshops are held in most states. On-site staff development is also available. The coaching mode of instruction is used to assure teachers leave the workshop ready to implement the program in their classrooms!
Learn about this powerful program for introducing students in grades 3 -8 to significant algebraic concepts in a fun, game-like manner. See the remarkable results students achieve.
Bring a nationally certified Hands-On Equationsinstructor to your district to provide first-class Staff Development on-site!
(See District Leaders Response to Our Workshops Below)
MAKING ALGEBRA CHILD’S PLAY®(Grades 3 – 8)HANDS-ON EQUATIONS® VERBAL PROBLEMS WORKSHOP(Grades 6 – 8; ½ day workshop grades 4 – 6)
FREE ONSITE STAFF DEVELOPMENT! Purchase thirty class sets of Hands-On Equations for Teacher and Thirty (30) Students and receivethe training absolutely free!
Call 800-993-6284 to schedule your onsite workshop!
What District Leaders Say About Our Workshops
"Our teachers loved the workshop, Making Algebra Child’s Play!Students and teachers caught on to the math concepts very quickly and the use of the manipulatives to clarify operations was great!"
Dr. Robin Gillespie, Supervisor of Gifted Education Jefferson County Schools (Grades 3-5) Birmingham, AL August 25, 2008 24 Participants 9.96 rating
"I thoroughly enjoyed the workshop!I believe it is a great enhancer for all students, but especially for visual and tactile learners.It really helped me to see how algebraic problems help to visually teach students." Shaundala Summers, Campus Magnet Coordinator TASD7 (Grades 4&5) Texarkana, AR August 11, 2008 11 Participants 10 rating
"We enjoyed the seminar and found it to be beneficial to our math program.The manipulatives and basic principles help to visually teach students." Anita Corum, Elementary Curriculum Director Princeton Christian School (Grades 3 - 8) Homestead, FL August 6, 2008 16 Participants 9.14 Rating
"I coordinated 32 workshops this summer and Tina Weiner’s workshop was among the best.All the participants were very happy with the results." Paul Farrer, Academic Outreach Coordinator Institute for Advanced Learning Research (Gr. 3-12) Danville, VA July 29, 2008 15 Participants 9.53 rating
Borenson and Associates, Inc., P.O. Box 3328, Allentown, PA 18106
Level III Distance Problem, Page 107/44 and Solution Page 133/44
This problem has been modified as shown below.
This may be considered an honors problem
for students in grades 6 - 9.
The solution remains the same as that shown in the book.
44.A motorboat, after traveling for one hour in tranquil waters, begins to go downstream with the current, thereby increasing its speed by 2 miles less than one-third of its original speed. After going downstream for one hour to reach its destination, and after unloading one of the passengers, it reversed direction and went upstream against the same current for one hour to arrive at a refueling station. At that point the motorboat still needed to travel an additional 36 miles to reach its starting point. Find the speed of the motorboat in tranquil waters, and the distance it traveled to arrive at its destination. [III, 25]
Word problems, also known as verbal problems, provide the students with the opportunity to think through a situation in depth. Short cuts such as "clue words" will not serve the students well, as every so called "clue word" is often not a clue at all. For example, teaching that the word "is" means "equal," presents a difficulty with a problem such as, "Four times a number is increased by 2...." Certainly no equal sign is involved here.
Hence the use of clue words, although designed to be helpful to the students and to serve as a crutch, is actually a disservice to the student since it does not convey to the student that there are no short-cut to doing verbal problems, rather the student must think!
There are methods, however, that the student can use to help break down the problem and represent the various elements. The following example shows how this can be done with one such problem using Hands-On Equations. This problem is taken from the Hands-On Equations Verbal Problems Book. Using Hands-On Equations, this problem is accessible to students as early as the 4th grade.
Theresa could purchase four small gifts and a $3 doll for the same price as three of the same small gifts and one $5 doll. What was the price of each of the small gifts?
Solution:
We let the blue pawn represent the price of each of the small gifts. The price of four of the small gifts would therefore be represented by 4 blue pawns. The $3 doll would be represented by a red 3 cube. And likewise for the other side.
The setup for the problem therefore looks as follows:
From here, we can use legal moves (remove three blue pawns from each side) to simplify the setup.
From this simplified setup we can see that the blue pawn is worth 2. Hence, The cost of each small gift is $2.
Check: $11=$11
ATTENTION TEACHER OR HOME SCHOOLER:
If you would like to provide these types of problems to your upper elementary and middle school students, you may wish to obtain the HANDS-ON EQUATIONS VERBAL PROBLEMS BOOK, which has more than 250 number, coin, age and distance problems, as well as general story problems, for all three levels of Hands-On Equations, along with solutions!
Additionally, if you are a teacher in grades 3 to 8 you may wish to attend a Making Algebra Child's Play workshop this season, In this workshop, you will learn how to use the Hands-On Equations program to solve equations, and also how to apply the concepts to verbal problems.
If you have already attended a Making Algebra Child's Play workshop, or are already using Hands-On Equations in your classroom or in your math program, and you are teaching in grades 6 and up, we encourage you to consider attending the Day2 Hands-On Equations Verbal Problems Workshop. In this workshop, you will review the key ideas of Hands-On Equations and you will also see how to apply these ideas to solve a wide variety of consecutive integer, age, coin and distance problems, including rowing up and downstream! This workshop will also be of interested to teachers of the gifted grades 2 and up, and teachers of low-achieving high school students.
Kira Brennan, age 8 presented the solution below to the above problem:
Kira's solution in her words:
"I solved the problem by drawing a picture of four presents and a person with a doll that has a $3 tag on it in her hands, and put an equal sign next to, and then I drew three presents and a girl holding a $5 doll in her hands.
"When I saw the picture, I saw that each present could be a blue pawn, and the doll could be a block (cube). So I put four blue pawns and a red 3 cube on the left hand side, and three pawns and the 5 cube on the right side. I guessed then that each present costs $2, but I took three pawns off each side anyway, and I could see you have to add $2 to 3 to equal $5 on the other side. Also, I counted 2-4-6-8-11 on the left, and 2-4-6-11 on the right, 11 equals 11, so each present must cost $2. It's harder if you just set up the equation, I think, but it was easy after I drew the picture."
Kira Brennan,
Age 8
(Note from Kira's mom: Kira has not tried verbal problems yet, so it was her idea to draw the picture first. I scanned her first sketch she did quickly to solve the problem, but then she insisted on drawing another one with the entire solution)
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.
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
BROWARDCOUNTY 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
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.
I am fond of simplifying ideas, like making basic algebra (some of the topics normally learned in the 8th or 9th grade) accessible to grade school children.
One of my goals is to convey to students, through their teachers, the high level of excellence they can achieve.
For many elementary teachers, attending our workshops is a liberating and emotional experience. The resulting sense of mathematical power they attain is very exhilarating and they can't wait to provide this experience to their students.
In my library
(Students may use the game pieces on this post-test- time limit: 15 minutes)
