"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.
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.
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
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 fromy + 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 1st 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 of4a + 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 isonly 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 and2/5x = 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.
The Day2 Hands-On Equations Verbal Problems Workshop or webinar is designed to show teachers of Hands-On Equations how to present verbal problems to their students. In the example below a gifted student going into the 6th grade solves a consecutive integer problem which most likely would be a challenge to many Algebra 1 students.
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 Hands-On Equations for the SMART Board 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.
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 SMART Board application. Hands-On Equations for the SMART Board is available for purchase via download and very soon via disk.
Cost: for one teacher $125. License for five teachers: $500
Borenson and Associates, Inc. 800-993-6294 info@borenson.com
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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.
