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.

 ****

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.

University of Louisiana Uses Breakthrough Program to Help Younger Kids Excel in Algebra

This news item appeared in the Market Watch section of the Digital Network edition of the Wall Street Journal.

Nov. 29, 2011, 8:25 a.m. EST

LAFAYETTE, La., Nov. 29, 2011 /PRNewswire via COMTEX/ -- In a world where t-shirts say kids are allergic to algebra and low national test scores seem to confirm, the University of Louisiana runs a unique teaching program using a strategy called Hands-On Equations to train teachers and sponsor algebra competitions where middle-school students love and excel in the subject.

Dr. Peter Sheppard, Associate Professor of Mathematics Education, Department of Curriculum and Instruction at the University of Louisiana at Lafayette, wants kids to experience algebra earlier and be excited about it. Thus, he and the University of Louisiana hold annual H.E.A.T. (Hands-on Exposure to Algebraic Topics) Competitions where over 350 kids from 20 middle schools participate.

"We use Hands-On Equations in our teaching and developed H.E.A.T. as a means for teachers to employ this creative instructional strategy in after-school math clubs since curriculum constraints often limit the time teachers can devote to this learning approach in their regular classrooms," remarked Sheppard.

Sheppard is none too soon using the innovative approach either.

According to the 2011 National Assessment of Educational Progress, which tested 4th and 8th graders in math and reading, national test scores show only 40% of 4th graders and 35% of 8th graders received proficient math scores. Louisiana and other states including California, Tennessee, Alabama and The District of Columbia often rank near the bottom.

"Even in this digital age, kids learn algebra concepts faster and in earlier grades when they use tactile items they can see and touch such as Hands-On Equations. This program uses numbered cubes and pawns as well as a representation of a balance scale to model and solve algebraic equations," said Sheppard.

Teachers using the Hands-On Equations program ( http://www.borenson.com ) are thrilled with the results.

"Middle-school students have a hard time understanding and grasping algebraic concepts," says Heather Olson of Edgar Martin Middle School in Lafayette Parish.

"Since I have been using Hands-On Equations, my students love to solve algebraic equations because the approach is like a game. Forty-three students joined our after-school math club. The students understand the representation of the balance on the board, the pawns for the unknown number, and the number cubes for whole numbers. The students are enjoying solving the algebraic equations, especially the extremely long equations," continued Olson.

The H.E.A.T. project is funded through a grant from the Louisiana Board of Regents, Louisiana Systemic Initiatives Program and the State of Louisiana STEM Goals Office.

The program provides participating teachers with professional development in algebra instruction. The students involved in H.E.A.T. learn using Hands-On Equations materials in after-school programs. Some 50 undergraduate students will assist in grading the student papers during the competition.

Dr. Henry Borenson, the creator of Hands-On Equations, will address teachers and be present at this year's H.E.A.T. Competition Awards on December 7, 2011 at the Student Union Ballroom of the University of Louisiana at Lafayette, from 9 a.m. to 1 p.m. Award ceremonies take place from 12:30 - 1 p.m.

Drs. Sheppard and Borenson are available for media interviews. Email Dr. Sheppard at psheppard@louisiana.edu or call 337-482-1514. Contact Dr. Borenson via Heather Harter at 800-993-6284, 404-925-2840 or Heather@borenson.com

About Borenson and Associates:

Borenson and Associates, Inc. provide the Making Algebra Child's play workshop for teachers with students in grades three through nine. Additionally, they present introductory webinars for parents. Even in the digital age, the hands-on approach to learning algebraic concepts is in high demand, as the company celebrates its 25th year in business. The company is based in Allentown, PA.

For information, go to http://www.Borenson.com

Contact:

Heather Harter

Heather@borenson.com

Phone: 800-993-6284

404-925-2840

SOURCE Borenson and Associates, Inc.

Copyright (C) 2011 PR Newswire. All rights reserved

Hands-On Equations Introductory Webinars

Demystifying the Learning of Algebra

Click here for current free webinar offerings. Invite your friends or colleagues to join you.

Overview of Complimentary Introductory Webinar

This webinar will provide an overview of Hands-On Equations. Participants will gain an idea of how equations are represented and solved and how the concepts are applied to the solution of verbal problems.

We will show how the game pieces are used to represent equations such as:

• 4x + 3 = 3x + 9
• 2(x + 4) = x + 10
• 5x + 2(-x) + 3 = x + 9
• 2x = (-x) + 12

In addition, we will provide a glimpse of Hands-On Equations can assist students in solving verbal problems such as:

• Three times a number, increased by 2, is the same as the number increased by 10. Find the number.
• Eight years from now, Tom will be 4 years older than twice his present age. How old is he now?

By participating in this introductory webinar, participants will understand how Hands-On Equations can be of value to them in introducing their students in grades 3 - 9 to basic algebraic concepts. Full-length webinars and full-day workshops are available to provide educators with in-depth training.
New: Verbal Problems Introductory Webinar...also free. Click on the above link for more information.

Learning the Relational Meaning of the Equal Sign

"Students who exhibit the correct understanding of the equal sign show the greatest achievement in mathematics and persist in fields that require mathematics proficiency like engineering."

- Education Research Report Blogspot, citing the research of Robert and Mary Caprano

As Robert Caprano notes students need to see and experience the relational meaning of the equal sign. Young students are quite familiar with its operational meaning, whereby the equal sign is used to indicate the result of a series of operations, such as 4+3=7. This is indeed a legitimate and pervasive use of the equal sign and one can open any advanced mathematics or science textbook to practically any page to find this use being employed. Hence, the calculator use of the equal sign is not incorrect; it is simply only one use of this sign.

An understanding of the relational meaning of the equal sign, on the other hand, as Mr. Caprano notes. would enable students, to provide the answer of 7 to the problem 4+3+2= __ + 2.

In studies involving more than 2500 students conducted with the assistance of researcher Larry Barber, the results are conclusive: students as early as the 4th grade can solve equations such as 4x+3= 3x+9, thereby demonstrating that a) they understand the relational meaning of the equal sign, b) they can understand the concept of an unknown and c) they can work with equations having unknowns on both sides of the equal sign. This research can be found on www.borenson.com.

A video of an 8-year old solving 4x+5=2x+13 can be found by going to YouTube and searching for Algebra Hands-On Equations.

What is particularly interesting about this approach is that students pick up the relational meaning of the equal sign in only a few lessons. The students EXPERIENCE the new meaning of the sign. They quickly learn that the correct value of the unknown will make both sides have the same value.

For teachers not using Hands-On Equations, I would recommend an approach whereby the students experience the relational use of the equal sign in gradually more complex examples. In other words, the teacher gradually enables students to develop meaning to expressions such as 10= 4 + 6, 7 +3 = 9 + 1, and 10 + 2 = 2 + 5 + 5. Next, the teacher omits any one of the given numbers and asks the class for the missing number. In this manner the student soon learns to correctly answer examples such as 4+3+2= __ + 2.

In summary, there exist sound pedagogical interventions to enable even young students to understand the relational use of the equal sign.

Teaching 4th Graders How to Do Algebra

December 12th, 2010 3:05 pm ET

Joe Sipper, Durham K-12, Examiner

In the spirit of making algebra more concrete and accessible to students, Dr. Henry Borenson invented a method of teaching variables in a concrete method. Hand-on Equations® is actually targeted toward a 4th grade audience, with some classes using it in as early as 3rd grade.

The system uses a structure (or a drawing) that resembles a double pan balance. An algebraic equation is given. The students “set up” the equation on the balance using game pieces and cubes with number values written on them. The numbered cubes literally represent the number written on them. The game pieces are “x”. The center of the balance is the equal sign.

Students begin by taking away equal amounts from each side until the equation is simplified. Then they do the simple arithmetic needed to solve for “x”. Maybe the best part is that students then have a way of checking their solution by seeing if both sides of the balance have an equal value.

Some video demos of elementary students using the device can be seen here.

Concerning the ideas that sparked his invention, Dr. Borenson allows:

"Even before the National Research Council issued a report in 1998 that included the statement, ‘We know from experience that the current school approach to algebra is too abstract and an unmitigated disaster for most students,’ I was already aware of this phenomenon. I wanted to find a way to make the abstract concepts of algebra visual, hands-on, and accessible to students of all ability levels-- and at much younger age. It turns out that the symbols and concepts of algebra related to solving equations such as 4x + 3 = 3x + 9 can be expressed perfectly via objects and actions. Using the Hands-On Equations® approach that I developed, even 3rd and 4th graders can solve equations that many 9th graders find difficult when presented abstractly."

The main idea behind the product is to take the fear out of algebra and allow students to strengthen applicable skills before actually enrolling in the class. Users and researchers swear by the results and hail it as a method for teaching how to think mathematically.

For more information about the teaching model, the Website offers free webinars. The Website also gives information on demonstrations and seminars.

About Joe Sipper

A former science teacher and coach, Joe Sipper also has experience as a content and assessment developer, project director, program manager, strategic planner, presenter, and director of staff of large educational publishing companies. Joe Sipper has experience on more than 20 testing programs across the United States, Puerto Rico, and Chile. Joe Sipper is owner of his own educational consulting company, iJS Education Services . His client list includes Pearson Educational Measurement, Educational Testing Service, the Educational Records Bureau, Education 2020, and several large school districts.

Sharing the Excitement of Hands-On Equations

At the end of the workshop, one participant indicated that she had the materials in her building but had no knowledge of what they were or how to use them. She asked how she could get others excited about HOE. I told her my story and my first experience with HOE. I stated I was so excited about the presentation that I immediately went back and used my individual set with one student during our remediation/enrichment period and invited my principal to come and watch. After my principal saw how quickly my student was picking up on the concepts, he ordered me a class set of 30. I then used my planning period for several weeks to go to different classrooms and model HOE with individual teachers’ classes within my building with the stipulation that they had to remain and watch. After demonstrating HOE in their classrooms, they became excited and wanted the materials as well. I even went to resource classrooms to model the power of the program. She was excited about this suggestion and stated she would start with her principal first and share in a departmental meeting with her peers.
-Instructor Kathryn Dillard

COMMON CORE STATE STANDARDS

General Comments on the Algebra-Related Standards, K - 6

Formulating and disseminating national math standards is an ambitious goal, but also one that can backfire if the standards are not capable of being attained by a large portion of the intended student population.

Before the draft Common Core State Standards are proposed for adoption, they need to be pilot-tested. Failure to conduct this pilot testing has the potential to cause more damage to students than the educational system that is already in place.

For the pilot testing to be effective, it cannot be conducted with teachers who volunteer to participate. Rather, districts and teachers must be selected at random from the inner city, suburban and rural areas and asked to participate so that a reasonable assessment can be made.

Proposed standards should be one that can reasonably be attained by at least 80% of the intended student populations through instructional means which teachers can reasonably be expected to learn and implement.

This pilot testing will determine, for example, if some of the math standards related to algebra in grades K – 6 are overly formalistic and, whereas they may have some value for mathematically gifted students who may have the intention of becoming professional mathematicians, they would further discourage the average and below average student.

For an example, in the 6th grade standard students are expected to see the need for and to understand that through the use of the multiplicative identity and the distributive and commutative laws they can get from y + y + y to 3y as follows: y + y + y = y(1+1+1) = y(3)= 3y. This is only one of many examples where a formalistic approach is used for an obvious result.

Indeed, in the standard for grade 3 the students have been informed that multiplication by a whole number can be considered as repeated addition. Hence, adding the same item three times, namely y, is the same thing as having three of those items, that is 3y.

So that while on the one hand there is an excessive formalism in many of the standards in the area of algebra -- a formalism which is not likely to endear many students to mathematics, but rather cause them to wonder why the subject is so obtuse and dry -- standards that would give students an early and solid foundation for algebra through concrete and pictorial means, is lacking.

For example, in kindergarten or the 1^st grade (and not in grade 6) the students can learn to represent an apple by the letter "a". They can then work with questions such as 2a + 3a (2 apples and 3 apples are 5 apples), so 2a + 3a =5a. Likewise, "g" can represent a grape. Hence students can be asked, “What is the total number of apples and grapes that we have in basket containing: 3a + 2a + 2g – g?” In this manner, young students can become comfortable working with mathematical expressions containing letters.

Later on, in grades 2 or 3, "a" and "g" can represent the cost of an apple and the cost of a grape, respectively and the students can be asked for the total cost of 4a + g (4 apples and one grape) if each apple costs 25 cents and each grape is 5 cents.

Additionally, in the current set of standards it is only in grade 6 that students are asked to use the idea of maintaining equality to solve equations of the form x + p = q and px = q where x, p, and q are all non-negative rational numbers. In other words, at the same grade level the students are being asked to use properties of equality to solve x + 3 = 10 and 4x = 20, they are also expected to solve x + 2/5 = 3/4 and 2/5x = 3/4. It should be clear to the reader that the former are much more simple and obvious than the latter and should therefore be presented several years earlier.

Indeed, using concrete or pictorial methods, it is know that 4th and 5th graders, including inner city minority students, can solve equations such as 4x + 2 = 3x + 10 with unknowns on both sides of the equation. They do so by first transforming the abstract equation into a physical or pictorial representation (the same physical or pictorial icon is used for each x, e.g., 4x is represented by 4 of the icons; the constants can be represented by numbered cubes or boxes) and then physically or pictorially maintaining the balance by removing the same number of icons (x's) from each side of the equation or the same value from the constants.

Students in the 4th and 5th grade can then transfer this learning to solving word problems such as, "Four times a number, increased by 2, is the same as twice the number, increased by 10."

By moving away from the formalism that is suggested in a number of the standards related to algebra and instead replacing those with standards that recommend concrete and pictorial representation and solution of algebraic equations -- an activity that students enjoy doing because they can understand the process and experience success-- the standards will be empowering younger students and laying a foundation for later success in algebra. Otherwise, algebraic equations will continue to be seen as abstract even by students who have strong arithmetic skills.

In summary, this educator believes it would be a grave error to propose the draft core standards for national implementation absent an extensive pilot testing with a broad population of students and teachers. This testing will likely take from one to three years.

Recommending the draft core standards, absent this pilot testing, poses the risk that the attempt to raise the level of mathematics education in the United States via the formalistic and "rigorous" algebra-related standards, may backfire, that is, it may lead to a higher level of failure and frustration then already exists in the mathematics classroom and may therefore result in a very strong back-to-basics movement.