Visual Algebra For The Early Grades

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Visual Algebra for the Early Grades William G. Mandras One possible way to help all students but especially those whose circumstance places them at high risk of failure in mathematics education is to introduce a simplified version of algebra starting in the first grade. To that end this article was written. Mathematical ability, like all human abilities, is normally distributed as evidenced by the following data1 published in a journal of the Mathematical Association of America. F 13.8%

D 18.7%

C 32.4%

B 22.9%

A 12.2%

No. of cases 8319

The mean of a normal distribution (Bell curve) implies that one-half of the student2 population is below average in mathematical ability, that is, 29.5 million students in K12. This number is approximate and is likely higher; it does not take into account the homeless students, the undocumented immigrant students, the published assertion that the mean of the distribution varies widely among the ethnic groups and the studies that show a rapidly declining number of students that receive the educational and psychological benefit of traditional upbringing. This is the existential condition that a teacher encounters when they attempt to teach any level of mathematics in K-12. The first thing a newborn infant will see is the reflective symmetry of the maternal face and later, the reflective symmetry of the entire human form. The reflective symmetry of the human form is a natural concept that is an organic part of human understanding, is ingrained very early in life and implies a one-toone matching of parts. If these assertions are true then visual algebra can be taught starting in the first grade. The concept of one-to-one matching is most likely deduced from the result of a simple experiment that everyone does very early in life: match thumb to thumb, index finger to index finger, and so on to obtain a perfect left to right hand match. A classroom demonstration that one-to-one matching is within the purview of the students; ask anyone to respond if they think the students in the room is greater than the number of chairs. No one will respond. The students need only observe one empty seat or that everyone has a seat and they can say with certainty that the number of chairs is greater or equal to the number of students. There is a more advanced way to determine the facts, count all the chairs and count all the students then compare the two numbers. Counting is advanced and requires prior knowledge whereas one-to-one matching is primitive. Visual Algebra is a method that was prevalent at the dawn of the mathematical sciences. At the University of Alexandria, 310 BCE, geometric methods were used in the solution of the algebraic equations. For the equation: 2 L = 8, each term in the equation was depicted as an area and a geometric diagram was constructed. In the first textbook on mathematics, The Elements, by Euclid, the fourth axiom, “Things which coincide with one another are equal to one another,” a student would conclude that equality requires that the two rectangular areas are equal; therefore length L must coincide with length 4.

×

2

2 L

8

2

4 L Symmetry in algebra is achieved when an equation is reformulated such that the form of the equation is the same on both sides of the equality. Visual algebra does not solve for the unknown value of the variable,

2 but applies the axioms to reformulate and render an equation symmetrical. The solution is then found implicitly by a one-to-one matching of terms.

3 Visual algebra is geometric algebra, with symmetry substituted for geometry. The solution using symmetry: Translational Symmetry: Reflective Symmetry:

2 2 2

× L=8 × L = 2 × 4, × L = 4 × 2,

L = 4, from the one-to-one matching of terms L = 4, from the one-to-one matching of terms

Note: Translational Symmetry will be used to formulate the algebraic equations throughout this article. Algebra is the first level of abstraction beyond arithmetic and requires the use of literals to represent numbers. In an equation, the literals initially have unknown or variable values that can be determined by algebraic means. The literals are generally the letters of an alphabet: A, a, B, b, C, c, X, x, and so on; using capitals letters first and lower case later in parallel with student early learning. Avoid the use of the literals I, i, O, o, and lower case ell l as they are too easily confused with zero and one. One possible way to introduce algebra is to initially have the students’ conduct the finger-matching experiment (see page 1). The pedagogy can then proceed through equations that are already written in translational symmetry form. The first grade students would only be asked to identify and state the unknown value of the variable by the one-to-one matching of terms. As skills develop, the equations can require arithmetic operations to achieve translational symmetry. Examples: Operations that require arithmetic Acronym: TS = Translational symmetry a)

3´ A + 5 = 3´ 2 + 5  Translational symmetry (TS); A = 2 by the one-to-one matching of terms

b)

3 ´ A + 5 = 3 ´ (1+ 2) + 5  Translational symmetry (TS) requires 1 operation; note that A is now 3

c)

3 ´ B + 5 = 3´ (2 + 2) + 2 + 3  TS requires 2 operations; note that B is now 4

d)

3 ´ D + 5 = 3´ (3 ´ 4 ¸ 2) + 7 - 2  TS requires 3 operations; note that D is now 6

e)

3 ´ D + 5 = 3 ´ (6 ¸ 2 + 4 ¸ 2) + 6 - 1  TS requires 4 operations; note that D is now 5

f)

3 ´ X + 5 = 3 ´ (3 ´ 6 ¸ 2 - 10) + 4 +1  TS requires 4 operations; note that X is now –1

g)

3´ A + 5 = 3´ A + 5  Equations, written in translational symmetry form, can be reformulated in many ways to enable the students to acquire both arithmetic and algebraic skills.

Examples: Operations that require algebra and arithmetic to achieve translational symmetry a)

A + A + A + 5 = 3´ 2 + 5 → 3´ A + 5 = 3´ 2 + 5  A = 2 by the one-to-one matching of terms

b)

A + 2 ´ A + 5 = 3 ´ (1+ 2) + 5 → 3 ´ A + 5 = 3 ´ (3) + 5  A is now 3

c)

B + B + B + 5 = 3 ´ (2 + 2) + 2 + 3 ® 3´ B + 5 = 3 ´ (4) + 5  B is now 4

d)

D + D + D + 5 = 3 ´ (3 ´ 4 ¸ 2) + 7 - 2 ® 3´ D + 5 = 3´ (6) + 5  D is now 6

e)

D + 2 ´ D + 5 = 3 ´ (6 ¸ 2 + 4 ¸ 2) + 6 - 1 ® 3´ D + 5 = 3 ´ (5) + 5  D is now 5

f)

X + X + X + 5 = 3 ´ (3 ´ 6 ¸ 2 - 10) + 4 +1 ® 3 ´ X + 5 = 3 ´ (- 1) + 5  X is now –1

Endnotes:

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Grades and Distributions, by Norman E. Rutt – Mathematical Association of America ©1943 http://www.edreform.com/Fast_Facts/K12_Facts/ - Total K-12 Enrollment 59.0 Million