Hands-On Equations App - The Easy Way to Learn Algebra

Dr. Davidovic, founder of Gifted Child Magazine, was very impressed with the Hands-On Equations app. He interviewed Dr. Borenson for the September 2012 issue (available at the App Store). That interview is shown below.


 
"Dr. Borenson has devised an ingenious iPad application called Hands-On Equations that not only makes it exceptionally easy to learn algebra, but makes it fun too."

    - Dr. Alex Davidovic - founder of Gifted Child Magazine and a former International Chess Master  

Don't give up on algebra...

... find a better way to teach it

 

 

 

 

 

By Jennifer Bardsley
Monday, August 20, 2012 | 12:01 am

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  Hands on Equations

A couple of weeks ago Herald columnist James McCusker wrote an article called ""Are we teaching algebra to  the wrong students?" As a former teacher, I would answer McCusker with my sincere belief that no, we are not teaching algebra to the wrong students; we just need to find a better way to teach them.

Just because algebra is a difficult subject known to induce emotional panic does not mean we as a society should give up on teaching advanced math. To give up on algebra would mean to give up on students and their capacity to learn. I passionately believe that all children are capable of learning, and there is no reason that a neurotypical adolescent cannot master algebra.

I think that the answer to the “algebra problem” is equipping parents and teachers with the tools they need to teach algebra in a way that makes sense, and to start teaching algebra to children as young as eightyears old. If you think I am crazy, then you have never seen Hands on Equations, the brainchild of Dr. Henry Borenson.

I cannot talk about Hands on Equations without sounding like I am a paid spokesman for the company, (which I am not). I also can't describe Hands on Equations without getting a bit weepy about it. I guess that's further proof that algebra is fraught with emotion.

Hands on Equations is, quite simply, the way that I wish I had learned algebra. Instead of esoteric rules to memorize, Dr. Borenson has children move around pawns on a yellow balance, so that each algebra problem is solved with manipulatives. Children learn the laws of algebra as “legal moves” that will help them play the game more effectively.

I purchased the Hands on Equations home pack for my son last winter for $34.95. It includes 26 lessons, and so far he has completed 18 of them. We do about two Hands on Equations lessons a month, usually on the weekend accompanied by a bowl of ice cream. If your child can do third-grade math and handle checkers, he or she is ready for Hands on Equations.

Here are two examples of the level of problems my son is learning to solve:

• x + 12 = 2(-x)+ 6.
• Find three consecutive even numbers whose sum is double the third number, increased by 8.

The amazing thing about Hands on Equations is that it makes so much intuitive sense. Before you know it, you are looking at algebraic equations and imagining blue and white pawns moving around in your head. That's a lot better than trying to remember some sort of rule that your algebra teacher wrote on the white board for you to memorize.

I am passionate about Hands on Equations because it is how I wish all children could learn algebra. Please let us not give up on teaching high school students how to do advanced math. Let's just find a better way to teach them.

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Common Core Algebra Standards for Grades 3 & 4: Are they Reasonable? Rethinking Word Problems Using a Letter for the Unknown

- by Henry Borenson, Ed.D.

This article appeared in the NCSM Newsletter, Summer 2012, Volume 42, Number 4, 24-25.

The Common Core State Standards in Mathematics (CCSSM 2010), referred to as Standards here, set ambitious goals for third- and fourth-grade students. Not only are they expected to solve complex word problems—two-step problems at third grade (3.OA.8), multistep problems at fourth grade (4.OA.3)—but also to represent these problems with an equation using a letter for the unknown quantity. The latter is an advanced algebraic skill that usually is not introduced until later grades on international testing in the highest-achieving countries

For example, according to Ginsburg, Leinwald, and Decker (2009), it is not until the fifth grade that students in Hong Kong, Korea, and Singapore are expected to use a letter to represent the unknown. The example they provide is: “John is x years old now. How old will he be after 10 years?” (p. 33).

Even within the Standards, it is not until the sixth grade that students are specifically expected to represent an expression using a letter for the unknown (6.EE.2a). The wording of this Expression and Equations (EE) goal is,

Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation, “Subtract y from 5” as 5 – y.  (p. 43)

Yet the ability to represent an expression with a letter for the unknown is a prerequisite skill to representing a word problem with an equation with a letter for the unknown —something that is expected in the Standards for third- and fourth-graders.

This presents a quandary: The Standards expect third- and fourth-grade students to use a skill that is not specifically taught, if the Standards are followed, until two or three years later in the sixth grade.

Clearly, this seems to be an error in the Standards. But this unsupported expectation in the earlier grades also may present an instructional opportunity.

What makes algebraic equations with letters difficult? According to Ginsburg, Leinwald, and Decker (2009), algebraic notation can be challenging to students because it differs from ordinary numerical notation. For example, whereas 52 is the sum of 50 and 2, 5x is not the sum of 50 and x. Furthermore, students may have trouble with the meaning of 4x + 5y because it appears as though they are being asked to add apples and oranges.

Lam (2002), discussing the elementary math curriculum in the high-performing country of Hong Kong, states, “Algebra occupies a very minor place in the primary curriculum, for the simple reason that algebra involves a certain level of abstraction which should be introduced at a later stage of development, in this case P5 [fifth grade] onwards” (p. 206, emphasis added).

Internal inconsistency between 3.OA.8/4.OA.3 and 6.EE.2a in the Standards is obvious. However, talented third- and fourth-graders can learn how to deal with this level of algebraic abstraction if the abstraction is presented in an understandable way. An effective instructional strategy begins with transforming the word problem into a concrete or pictorial equation and then uses that representation to construct an abstract equation using a letter for the unknown.

The approach outlined here is not currently being used in Hong Kong but is consistent with the pedagogical approach of that country’s schools. According to Lam, “Student learning is expected to progress from the concrete to the abstract” (p. 204). That is the essence of the strategy here.

This strategy also is philosophically consistent with the pedagogical approach used in Singapore, where, according to Cai and colleagues (2011), “The intent of using the ‘model method’ is to provide a smooth transition from working with the unknown in less abstract form to the more abstract use of letters in formal algebra in secondary school” (p. 33).

Transforming a word problem into a concrete or pictorial equation: The Standards do not provide an example of the type of multistep word problem it has in mind for 4.OA.3, so let’s consider the following example:

Tom buys two packs of football cards to add to the six cards a friend gave him. His mother then gives him three packs as a present. Now he has as many cards as Jane who owns two packs and 18 loose cards. If all the packs have the same number of cards, how many cards are in each pack?

Figure 1 shows how this word problem can be concretely represented using the approach of Hands-On Equations (Borenson 2010, Borenson 2009). The pawn represents the number of cards in a pack. A numbered cube represents a number of loose cards. The left side of the balance represents the total number of cards Tom has; the right side shows the number Jane has, with the loose cards represented by the sum of the three cubes. The balance image indicates that both sides have the same value. This equation using concrete objects fully represents the conditions of the problem.

Figure 1. A concrete equation of the multistep word problem.

 We can now assign a value to the pawn and designate it by a letter, namely x. The x represents the unknown number of cards in a single pack. The objects on the balance scale in Figure 1 are additive, as are weights on a scale, thus the concrete representation can be stated algebraically as x + x + 6 + x + x + x = x + x + 18 or 2x + 6 + 3x = 2x + 18. Either of these equations would satisfy the requirement of goal 4.OA.3 in the Standards.

The Standards do not require fourth-grade students to solve the linear equation but simply to find the solution to the multistep word problem. Hands-On Equations students, however, would easily solve the concrete equation algebraically by physically removing, first, two pawns from each side and then a value of 6 from the number cubes on each side (Borenson 2011). Doing so leaves a reduced setup of three pawns on the left and a value of 12 in cubes on the right. Consequently, it is easy to see that each pawn has a value of 4, the number of cards in a pack. Counting 4 for the pawn in the reset physical representation shows that both sides have the same value, namely, 26.

Just as students can represent the above problem concretely, they can also do so pictorially by drawing a picture of the balance scale and drawing shaded triangles and boxed numbers to represent the pawns and the numbered cubes, respectively. Arrows can be used to show the removal of pawns and cubes from each side of the balance.

A study conducted in the spring of 2008 with talented third-graders demonstrated that with appropriate instruction young talented students can make significant progress in pictorially representing and solving two-step and multistep word problems (Borenson 2009). For example, whereas 28 percent of the 195 students in the study solved the above multistep word problem correctly on a pre-test, 88 percent solved it correctly on the post-test. The instruction, using the Hands-On Equations strategy, involved seven lessons working with equations and another six working on representing word problems concretely or pictorially.

Conclusion

Goals 3.OA.8 and 4.OA.3 in the Standards, requiring a student to solve two-step and multistep word problems and to represent the problems using a letter for the unknown, appear to be misplaced, because a prerequisite skill involving a letter for the unknown is goal 6.EE.2. Nonetheless, talented third and fourth graders can—and have been shown to—meet the goals of 3.OA.8 and 4.OA.3, depending on the complexity of the problem. These young students first represent the word problem in a concrete or pictorial equation and then transform that representation into an abstract equation using a letter for the unknown. They then solve the problem algebraically from the concrete or pictorial representation. By using this approach, proceeding from the concrete to the abstract, U.S. students can exceed their age/grade counterparts in high-achieving countries on this goal.

References

Borenson, Henry. “Demystifying the Learning of Algebra Using Clear Language, Visual Icons, and Gestures.” Newsletter of the National Council of Supervisors of Mathematics (NCSM) 41, no. 3 (2011): 24-27.

Borenson, Henry. The Hands-On Equations Introductory Verbal Problems Workbook. Allentown, Pa.: Borenson and Associates, 2010.

Borenson, Henry. Hands-On Equations Research Study: Third-Grade Gifted Students. Allentown, Pa.: Borenson and Associates, 2009.

Cai, Jinfa; Swee Fong Ng; and John C. Moyer. “Developing Students Algebraic Thinking in Earlier Grades: Lessons from China and Singapore.” In Early Algebraization: A Global Dialogue from Multiple Perspectives, edited by J. Cai and E. Knuth, 25-42. Berlin, Germany: Springer, 2011.

Common Core State Standards in Mathematics (CCSSM). Washington, D.C.: Common Core State Standards Initiative, 2010. http://www.corestandards.org/assets/ CCSSI_Math%20Standards.pdf.

Ginsburg, Alan; Steven Leinwald; and Katie Decker. Informing Grades 1- 6 Mathematics Standards Development: What Can Be Learned from High-Performing Hong Kong, Korea, and Singapore. Washington, D.C.: American Institutes of Research, 2009.

Lam, Louisa. “Mathematics Education Reform in Hong Kong.” Paper presented at the 4th International Conference on Mathematics Education into the 21st Century, 2002. http://math.unipa.it/~grim/SiLam.pdf.

This article appeared in the NCSM Newsletter, Summer 2012, Volume 42, Number 4, 24-25. Newsletter of the National Council of Supervisors of Mathematics, Denver, CO. 

Algebra Phobias Vanish with New Hands-On Equations App for iPad from Borenson.com

 

 

 

ALLENTOWN, Pa., April 4, 2012 — Educators say algebra phobias vanish and test scores improve with the new iPad app from Borenson and Associates at Borenson.com. This app, which can be used by students as young as eight years old, literally makes algebra child’s play. The original, physical Hands-On Equations® program has already helped more than a million students and adults.


“Kids don’t have to wait until the eighth grade to begin learning algebra with this new app,” said a jubilant Dr. Henry Borenson, creator of Hands-On Equations. “In six lessons even eight-year-olds can learn to solve algebraic equations normally presented in the eighth or ninth grade.”

“As soon as I used the new Hands-On Equations app, I knew that every student would benefit, especially those who are more visual or tactile,” said Pat Wyman, author and founder of HowToLearn.Com. “This app takes into account the different learning styles that students have,” she added.

In its non-digital form Hands-On Equations uses actual game pieces: pawns, numbered cubes, and a graphic representation of a balance scale. These elements have been transferred into digital form to the app. Younger and older students alike can understand and successfully solve the equations in the app by moving the game pieces with their fingers or with a stylus on the digital screen – and have fun doing it.

Borenson’s mission is to ensure that every child succeeds in algebra and learns to love it.

“I created Hands-On Equations because I know that algebra is the language of mathematics, and helping kids solve basic algebraic equations will make it easier for them to pursue higher-level math and science classes,” said Borenson.

A research study conducted in 2008 with 195 fourth- and fifth-graders in the Broward County, Florida public schools revealed that while only nine percent of the students could solve an algebra equation such as 4x + 3 = 3x + 9 on the pre-test, 80 percent of them solved it correctly on the post-test following six lessons of Hands-On Equations instruction using the actual game pieces.

“It is the carefully thought-out sequence of lessons, along with the use of the game pieces, that enables students to gain a deep understanding of the underlying concepts,” said Borenson. “Purely abstract or symbolic instruction is not helpful for many students,” he added.

According to a 2008 report by the National Mathematics Advisory Panel, many students in middle and high school algebra classes do not understand procedures for transforming equations or why those transformations work.

“Hands-On Equations has been the sole reason our students understand algebra and, as a result, they are no longer afraid of it,” said Cheri Godek, principal of Gotha Middle School in Windermere, Florida. “Students have actually been reduced to tears of joy when they felt that pride in math they had never felt before,” continued Godek.

Many districts are expecting digital technology to make a big difference in student learning. For example, the McAllen Independent School District in Texas recently announced a plan to purchase 25,000 iPads, enough for all of its K-12 students. “It’s about transforming learning; it’s really not about the device,” said Carmen Garcia, director of instructional technology for the district.

The Hands-On Equations app provides another means for districts to provide this powerful algebra teaching method to their students in grades three through nine.

In a review of this app, expert app reviewer Michael Vallez wrote, “Everything about this application is great and is presented in a very simple, straightforward and helpful way. Not only is Hands-On Equations engaging, but it’s a fun way to learn a very intimidating subject. I think Dr. Henry Borenson (inventor) is brilliant – and so is this application.” The full review can be found here.

More information on Hands-On Equations can be found at http://www.borenson.com. In addition to the app, the company offers physical sets of Hands-On Equations for use at home and in school, as well as the Making Algebra Child’s Play  workshop for teachers. The app can be downloaded from the App Store. For a video demo of this app from Borenson click here.

"Why Teach?" Touching Children's Lives: An Algebra Success Story

Kathryn Dillard, Motivational Speaker

What is the purpose of teaching? In this highly motivational and humorous presentation, Kathryn Dillard, a distinguished educator, conveys the bottom-line message: our purpose as educators is to inspire students and to lead them to success. She relates-- and we laugh all along the way-- how she was able to have a dramatic effect on the lives of her low achieving middle school students and inspire them to succeed with algebra using Hands-On Equations. Recorded at the 25th Annual Meeting of the Benjamin Banneker Association in Philadelphia.

****Highly recommended. You will laugh and be inspired.