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“High school students’ algebra experience should enable them to create and use tabular, symbolic, graphical, and verbal representations…” So begins the opening paragraph below the heading, “Understand patterns, relations, and functions” on page 297 of *Principles and Standards for School Mathematics*.

These four modes of representation collectively form the “Rule of Four” in mathematics education. The Harvard Calculus Consortium – the group credited with formulating the Rule of Four – lists the following as one of their guiding principles: “Where appropriate, topics should be presented geometrically, numerically, analytically, and verbally.” The idea behind the Rule of Four is that students learn in different ways. As suggested by NCTM, all students should learn various modes of representation, but each student typically has an innate strength in one of these four areas. To ensure that the greatest number of students gain mathematical understanding, it is important to hit all four types of representations. In addition, it is important for students to develop faculty with all four types of representations.

Read what Tom Reardon, the video teacher in Workshop 5 Part I, says about teaching to various learning styles:

**Transcript from Tom Reardon**

These students who we consider poor math students can do these things if you give them enough different ways to look at it. One way doesn’t work; another way might work … you’ve got to try to come up with several different representations so that the kids who don’t understand one way can pick up on another way.

Students learn in various ways, and the article “Focusing on Learners” by Epsilon Learning Systems succinctly describes three classic “learning modalities”: auditory, visual, and kinesthetic. Typically, there is an additional distinction made between textually-based and graphically-based visual learners.

- Auditory learners learn best through hearing, and particularly through the spoken word. They process linguistic information through listening and they benefit from lectures, spoken instructions, talking things out, and keywords on diagrams.
- Visual-textual learners process information best when in the form of written language. They need to see it written down, and they benefit from textbooks and class notes.
- Visual-graphical learners learn best when information is presented in a graphical format or in pictures. They benefit from visual aids, diagrams, and images.
- Kinesthetic learners learn best by manipulating physical objects, acting it out, and other hands-on activities. They process information through physical sensation and action – touch, movement, body language, and gestures.(Definitions taken from “Focusing on Learners,” http://epsilonlearning.com/learners.html. Epsilon Learning Systems, 2003.)

Because students have various learning styles, preferences, and dispositions, teachers who use the Rule of Four facilitate learning for all students by providing several presentations of the same concept. Students who best understand visual presentations may benefit from material that is presented geometrically (on a coordinate plane, perhaps, or using some type of diagram). Students with fluency in computation may benefit from a numerical approach (using a formula, or organizing results in a table to identify a pattern). Students who can think abstractly may benefit from analytical explanations involving symbols and algebraic notation. Students who prefer to read or hear an explanation may be best served by a verbal description. And all students, regardless of their dispositions for learning, will benefit from seeing various representations and identifying the interconnections among them.

As the old adage states, a picture is worth a thousand words. To visual learners this statement is especially true. For those who are not primarily visual learners, diagrams, graphs, coordinate planes, and other figures may still add clarity.

Because visual representations appeal to everyone, not just visual learners, the first representation mentioned in the Rule of Four is the geometric representation. A synonym might be “graphic representation.” This mode uses pictures, diagrams, and models to help make abstract concepts more concrete.

As a first step in understanding trinomial factoring, it is often helpful to present examples using numbers, rather than variables, to students. For instance, you may want to present the multiplication of 12 x 14 using base ten blocks (which later become known as algebra tiles, when x is used instead of 10). The multiplication can be shown as follows:

The large red square indicates 10^{2}, or 100; the green sticks indicate 10 each, and there are six of them, for a total of 60; and the small yellow squares indicate 1 each, and there are eight of them. The total of this multiplication, as shown by the graphic, is 100 + 6(10) + 8 = 168.

An example using numbers makes it easier for students to see the connections to trinomials. Algebra students can factor the polynomial x^{2} + 6x + 8 into a product of binomials using a similar process. A geometric representation that works well for developing student understanding uses a rectangle with side lengths equal to the factors:

Students can generate this representation using algebra tiles. In fact, they may use algebra tiles to factor this polynomial directly. By choosing one large red square to represent x^{2}, six sticks to represent 6x, and eight small yellow squares to represent 8, students can arrange them in any way, as long as they form a rectangle with no gaps. Students then can find the factors by looking at the dimensions of the rectangle. For instance, a student may arrange the algebra tiles (below, left) into the rectangle shown (below, right):

Although not organized like the previous example, this rectangle also represents the product of binomials – the width of the rectangle is (x + 2), and the height is (x + 4).

Tom Reardon recognizes that algebra tiles are a helpful alternative for students who have difficulty with numeric or symbolic representations.

**Transcript from Tom Reardon**

One student in particular today really latched onto the algebra tiles. She has a hard time with basic number facts. She doesn’t really know multiplication tables or even addition and subtraction … But with the algebra tiles, since they are just geometric in shape and deal with areas, and then you just count to get the answer, she’s able to multiply binomials. Then, today we were able to reverse that process and have her factor trinomials with this.

Algebra tiles are a wonderful manipulative for the algebra classroom. Kinesthetic learners, who process information through sensation and action, develop understanding by manipulating algebra tiles. The graphing calculator is a powerful tool for generating visual representations. This is especially important for visual graphical learners, who learn best when mathematical ideas are presented with images and diagrams.

After using algebra tiles to show that x^{2} + 6x + 8 = (x + 2)(x + 4), students in Tom Reardon’s class used the graphing calculator to show that these two expressions are, in fact, equivalent. By entering both functions into a graphing calculator and graphing them simultaneously, students see that the second function is graphed directly on top of the first function, implying that they are equivalent.

Y1 = x^{2} + 6x + 8Y2 = (x + 2)(x + 4) |

In the video for Workshop 5 Part II, Sarah Wallick uses a visual representation to describe the probability of getting heads on the n^{th} coin toss, given that heads hadn’t been obtained on any of the (n – 1) previous tosses. Her visual representation involves a square in which she and the students shaded half of the area at each step to show the probability that another tails was obtained; the remaining unshaded area represents the probability of obtaining heads on that toss. (Click here to view a PowerPoint presentation of Sarah’s drawing. Once the PowerPoint file is open, hit the “Page Down” key to progress from slide to slide, which simulates the coin tosses.)

Students in Sarah’s classroom also determined the probability of getting heads on the n^{th} toss experimentally. Each student flipped a coin and recorded the number of tosses until heads was obtained. The composite results were then displayed in a bar graph like the one shown below. As Sarah explains, some of her students need this type of representation to attain conceptual understanding.

**Transcript from Sarah Wallick**

Some kids interpret numbers, while other kids need a picture, I wanted them to have a visual representation [so] that they could say, “Oh, yes, I can see that most of the time we got heads on the first toss and then the second most frequent was two tosses.

**Reflection:**

Choose one topic from the algebra curriculum that is difficult to teach and difficult to learn. (Choose your topic carefully; as you will refer to this topic for all four reflection questions in this section.) How might you present this topic using geometric or graphic representations?

In the real world, data is presented numerically more often than in any other form. That’s because data is initially generated as a collection of raw numbers – the amount of soda teenagers consume, the exam scores of the students in an algebra class, the number of pickup trucks on Interstate 287, or the number of coin tosses that occur before getting heads. Data of all types typically arrives in raw form, as numbers. Only with manipulation does it become a graph, picture, algebraic equation, or paragraph.

Students need to understand numbers in their raw form. Otherwise, they will not have the flexibility to present them accurately in other arrangements. Further, some students are adept at interpreting numbers and noticing patterns among them, and for these students it is important that they have the opportunity to view numbers in their most basic state.

With regard to the Rule of Four, using a numerical representation refers to displaying data in an organized fashion, possibly as an ordered list or in a table. The table below, for instance, could be used to verify that x^{2} + 6x + 8 = (x + 2)(x + 4) – for every value of x, the values of x^{2} + 6x + 8and (x + 2)(x + 4) are equal.

Tables are useful in presenting information because they make patterns and relationships explicit. Imagine a student who had used algebra tiles to factor various polynomials and then used a graphing calculator to identify the x-intercepts and y-intercept. An advanced student might anticipate a relationship among the numbers and, therefore, might look for a relationship between factored form and the x-intercepts. But for a student who doesn’t expect such a relationship, repeatedly factoring polynomials and finding the x-intercepts may be tedious rather than illuminating; for that student, having the same information organized in the table below could be more enlightening.

As a result of a numeric representation in table form, students may see several patterns:

- The y coordinate of the y-intercept is equal to the constant term in the trinomial; that is, if the polynomial is x
^{2}+ bx + c, the y-intercept occurs at (0, c). - The x coordinates of the x-intercepts are equal to the opposite of the constant terms when the trinomial is written in factored form; that is, if the polynomial can be expressed as (x + m)(x + n), the x-interceptsoccur at (-m, 0) and (-n, 0).
- If a trinomial x
^{2}+ bx + c can be written as (x + m)(x + n), then b = m + n and c = m × n.

The presentation of this information in tabular form makes it possible to discover relationships that might not be revealed otherwise.

When determining the probability of heads on the n^{th} toss of a coin, some students will likely expect that each probability is equal to 1/2 the probability of the previous toss. For students who do not suspect that relationship, however, the following simple two row table will make the relationship slightly more evident:

Numerical representations lay the foundation for the abstract thinking required in algebra, as Tom Reardon learned. Hear what he had to say:

Listen to audio clip of teacher Tom Reardon |

**Transcript from Tom Reardon**

When I gave them a trinomial that didn’t factor [x

^{2}+ 6x + 4] and asked them to try to factor it, they still tried to do that, and then they were able to understand why you couldn’t factor it. They’d be able to say, “Well, the only numbers that multiply to give you 6 don’t add up to 4, and so we can’t factor it.” And to me, that was – I mean, I would love that at any level algebra class. And then when I gave them the expression x^{2}+ 6x + C and said find all the integer values for C that makes this factorable, that’s a pretty abstract question for algebra students. And they tackled it and went after it and came up with some great solutions and great explanations, and even surprised me by thinking of one I didn’t think of.

**Reflection:**

Consider the topic you selected previously that is difficult to teach and difficult to learn. How might you present this topic using a numeric representation?

The use of symbolic notation is so fundamental to mathematics that it is often the only representation used in traditional classrooms. To use only algebraic representations seems to put undue strain on students who have difficulty with symbolic manipulation. As Tom Reardon notes, “If you are only teaching paper and pencil, I can see your kids getting turned off.”

Of course, the importance of algebraic representations should not be diminished. The language of algebra is helpful in expressing situations mathematically. Algebraic notation makes it possible to express thoughts that are cumbersome in words. Consider the verbal description of the Pythagorean theorem: “In a right triangle, the square of the hypotenuse is equal to the sum of the squares of other two sides.” That statement is not nearly as clear or concise as a^{2} + b^{2} = c^{2}.

Geometric and tabular representations are useful in demonstrating that x^{2} + 6x + 8 = (x + 2)(x + 4). However, the distributive property of multiplication (over addition) makes the same point.

For a student who thinks abstractly, this representation may be the most meaningful. In addition, using the distributive property in conjunction with those other representations creates a powerful connection. Students who have difficulty with symbolic notation may benefit from seeing these representations side by side.

Toggling between various representations is important in making mathematical relationships explicit. As shown below, a tabular representation can help students consider the probability of heads occurring on the n^{th} coin toss:

An algebraic representation (in the form of a Now-Next equation) provided in tandem with this table gives meaning to the symbolism:

Next = 1/2 Now

– or –

Next = Now ÷ 2

For both of these equations, the start value is 1/2. The explicit form, when seen in partnership with the table and the NOW NEXT equation, gives meaning to the power function that models this situation:

y = 1 × (1/2;)x

As a final step, students can link the algebraic representation to a numeric representation by verifying that each of these equations accurately models the situation by substituting appropriate values. For instance, substituting Now = 1/4 (the Now value of the second toss), the Now-Next equation yields:

Next = 1/2 × 1/4

This is the probability on the third toss. This verifies that the Now-Next equation yields the correct Next value, since the next value after 1/4 in the table is 1/8. (In general, however, students should check more than one value before they conclude that an equation is valid.)

Likewise, substituting x = 3 into the explicit equation yields:

y = 1 × (1/2)^{3}

y = 1/8

y = 1/8

This is the probability for the third toss, in agreement with the value in the table and the value obtained from the Now-Next equation.

**Reflection: **

Again consider the topic you selected that is difficult to teach and difficult to learn. How might you present this topic using an algebraic representation?

Verbal representations occur in two common forms: spoken and written. Teachers use both daily. Through discussion and explanations, teachers attempt to help their students gain understanding by engaging them in discourse about mathematics. Through textbooks, handouts, and other printed media, teachers attempt to develop understanding by having their students read about mathematics.

At the beginning of Tom Reardon’s class on factoring trinomials, he asked students to describe “what we mean to factor something.” One student replied, “To write as a product.” He asked another student, “Would you happen to know another phrase for writing as a product?” That student said, “A multiplication problem,” to which he shot back, “What other word can you use?” “Factor,” she replied.

This continuing questioning technique is one of the strategies Tom uses to ensure his students understand mathematical vocabulary. Reading mathematical textbooks and participating in classroom discourse require that students have the vocabulary necessary to correctly explain the material.

Visual-textual learners learn best when presented with written descriptions. They benefit from the use of textbooks, class notes, and online resources. On the other hand, auditory learners learn best through spoken-word explanations. They use listening to process information, and they benefit from verbal instructions, lectures, and classroom discussions.

For both visual textual and auditory students, algebra mnemonics such as FOIL may be helpful, whether written out, as shown below, or described to students verbally. This mnemonic stands for “first, outer, inner, last” and refers to the process for multiplying binomials using the distributive property.

F = first |
Take the product of the first term in each set of parentheses. |

O = outer |
Take the product of the first term in the first parentheses and the second term in the second parentheses. |

I = inner |
Take the product of the second term in the first parentheses and the first term in the second parentheses. |

L = last |
Take the product of the second term in each set of parentheses. |

Using FOIL to multiply (x + 2)(x + 4), yields:

Fx^{2}

O4x

I2x

L8

Combining like terms (O and I) yields the trinomial x^{2} + 6x + 8.

In addition to multiplying factors of trinomials, the FOIL method also can be useful for multiplying integers. For instance, the standard algorithm for multiplying 12 x 14 would look something like the following:

12

__x14__

__8__

40

20

100

__168__

40

20

Using FOIL, however, the same multiplication could be perceived as a product of two binomials, namely (10 + 2)(10 + 4). The result of FOIL, then, would be

F10^{2}= 100

O4(10) = 40

I2(10) = 20

L8

When these four pieces are added, the result is again 168.

It is often helpful for students to see the FOIL method used in both circumstances, once with variables and once with numbers. Because the example with numbers includes a more tangible example and uses multiplication with which they are comfortable, it is often more meaningful than seeing only algebraic examples.

As a classroom teacher, it is important to remember that a picture is worth a thousand words, but for a visual textual or auditory learner, it is equally important to use verbal descriptions and explanations. As a checklist for ensuring that you are helping these students, keep the following important points in mind:

- When possible, use color, bold, or italics to distinguish important elements in the text.
- Distribute class notes or handouts.
- Address the class so that all students can see you clearly. For complete understanding, it is essential that students see your facial expressions and body language in addition to hearing the words you speak.
- Provide for some quiet self-study time away from noise and distractions, e.g., in the library.

**Reflection:**

Again consider the topic you selected that is difficult to teach and difficult to learn. How might you present this topic using a verbal representation?