Algebra for Secondary Teachers: Engaging Students to Develop Symbolic Meaning

One of the earliest numeration systems was the tally system or what we like to call the WYSIWYG (what-you-see-is-what-you-get) system. In this system, simple tally marks represented a number. For example, suppose you had seven bags of grain. In the tally system, the person would represent their possessions as |||||||. As you can imagine, this would be fine if you had little, but if you had many possessions, it would quickly become cumbersome. The Egyptians developed a way of eliminating the need for tally marks by inventing new symbols to represent groups of objects. In this case, the Egyptians grouped by ten (probably due to the fact that people had ten fingers and when counting on fingers, to keep track, they would need to begin again on the first hand after counting to ten). Once they collected ten of something, they created a new symbol to represent a bundle of ten. Once ten of those bundles were gathered, they created another symbol to represent ten bundles of ten and so on (see Figure 2.1.2). This way, to write a numeral, they would never need to have any more than nine of any symbol. They used the following symbols to represent various powers of ten.

With each different symbol, the Egyptians could now write larger numbers using a more condensed collection of symbols than in a tally system. For example, suppose we wish to express the number 31,254. In our Hindu-Arabic system, we allow each space to represent a different power of the base (ten in this case). Thus, \(31,254=3\cdot 10^4+1\cdot 10^3+2\cdot 10^2+5\cdot 10^1 +4\) expressed in expanded base-ten notation. The Egyptians, on the other hand, did not use the spatial location of the symbols to convey the specific power of the base, but rather they created a different symbol for each power (size bundle). Therefore, the number 31,254 would be written as

Although this system was a great improvement over the tally system of numeration, it doesn’t take long to see a major drawback. Since each new power of the base requires a new symbol, expressing larger and larger numbers becomes cumbersome. In order to express any given positive integer, the Egyptian system requires an infinite number of symbols.

So how do we improve on the problem of needing to create new symbols for each power of ten (or any base for that matter)? The ancient Babylonians developed a place-value system for representing any positive integer using a finite number of symbols. The Babylonians used the position of the symbol or collection of symbols to represent the power of the base (in this case a base sixty system which meant that in any single position, the symbols could total at most fifty-nine). Although they bundled in groups of sixty, they had only two symbols along with a place holder to be used for when there were no bundles of a certain power of the base (see Figure 2.1.4). The early Babylonians did not use the place holder, but the confusion caused by not being able to indicated easily when there were no bundles of a certain size necessitated the creation of a symbol to "hold an open space". The Babylonians did not have a fully functioning zero, but the place holder provided a way to indicate no bundles of a specific size by its location in the string of symbols.

To illustrate, consider the expanded base sixty notation as written using our familiar Hindu-Arabic numerals. Here we have \(39,605=11\cdot 60^2+0\cdot 60 +5\text{.}\) And so the early Babylonians would write this number as

The problem was in how this numeral could be interpreted. Did this collection of symbols mean \(11\cdot 60+5=655\) or \(11\cdot 60^2+0\cdot 60+5=39,605\text{?}\)

To make this expression less ambiguous, the placeholder was used to express the meaning more clearly. Thus the later Babylonian system would write the intended number as

As we shared before, although the use of the placeholder serves the same role as our zero in the Hindu-Arabic system, the Babylonians did not recognize the concept of zero as being a number representing nothing.

The ancient Mayan civilization were probably the first to fully recognize the concept of zero. The Mayans also used a place-value numeration system. In this case, their grouping number was twenty. They actually had a pseudo base twenty system since their place values progressed as: \(1, 20, 18\cdot 20, 18\cdot 20^2, 18\cdot 20^3, \ldots\text{.}\) The reason for the deviation from \(20^2\) was due to their calendar that had 18 months with 20 days per month (\(18\cdot 20=360\)). Like the Babylonians, the Mayans had very few symbols. The difference is that the Mayans had a symbol for zero (or nothing) represented by an empty oyster shell. This was different from the Babylonians in that the Mayan’s symbol was not just a placeholder, but rather was a symbol representing nothingness. The symbols are given below.

Like the Babylonians, the position of the collection of symbols determined the power of the base. However, unlike the Babylonians, the Mayan base is not exactly uniform. Instead of a base twenty system, they used a pseudo-base twenty system. The reason for this is that the base was tied to their calendar year. The Mayan calendar contained 18 months each containing 20 days. The result was a 360-day year. This can be seen below by the replacement of the normal place value, \(20^2\text{,}\) with the place value of \(18\cdot 20\text{.}\) After this, the place values simply increase the power of 20 by one for each successive place. The Mayans wrote the numerals vertically with the higher powers of the base above the lower powers and the symbol for zero was used as a placeholder. Within a single place value location, the unit dots would be written above the horizontal bars that each represented five. For example, the number \(21,717=3\cdot 18\cdot 20^2+0\cdot 18\cdot 20+5\cdot 20+17\text{,}\) in Hindu-Arabic would be written as

To this point, civilizations had focused mainly on representing the natural or counting numbers, \(\mathbb{N}=\left\{1,2,3,4,\ldots\right\}\text{.}\) The Mayans extended this system to include zero \(\left\{0,1,2,3,4,\ldots\right\}\text{,}\) often referred to as the whole numbers. Some texts include 0 in the set of natural numbers and refer to the counting numbers as the positive integers, denoted \(\mathbb{Z}^+\text{.}\) Thus, the use of \(\mathbb{Z}^+\) seems to be less ambiguous since zero is neither positive nor negative and so we will use \(\mathbb{Z}^+=\left\{1,2,3,4,\ldots\right\}\text{.}\)

With the advancement of a place-value system, cultures evolved to use different symbols for each face value (i.e. the symbols used to represent the number of any particular sized bundle). These symbols were dependent on the base (or bundle size) that was used. For example, in our current commonly used base system, (base ten), we have ten symbols since once we get ten of anything, we bundle them and we create another bundle of the next place value space. For base six, there would be six symbols. For base, there would be three symbols, and so on. Therefore, in base ten, we have the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The symbols used here are the Hindu-Arabic numerals (not to be confused with Arabic numerals) used on many Middle Eastern countries (see Figure 2.1.6). Both use a base-ten place-value system.

An interesting fact is that even though Arabic is written right-to-left, the numerals are still written left-to-right. This is due to the fact that the symbols were adopted from India where language is written left-to-right. This means that while reading text, you would read right-to-left until reaching a numeral and then switch to left-to-right for the number and then switch back to right-to-left as you continue to read the text. Again, this shows how mathematics and language are both culturally connected. Below (Figure 2.1.7) is a picture of a license plate from Iraq from 2025. This would be read as 656874 in our Hindu-Arabic symbols. Iraq was in the process to switching to the use of the Western Hindu-Arabic numerals at the time and this is an older plate that had not yet been replaced.

Again, one of the reasons we do not want children to simply recognize a collection of symbols as representing a single number is that it masks the meaning behind how our numeration system is expressed. We must first expect children from an early age to form bundles of varying sizes and count how many of each there are before then translating this into symbols used to represent the number. Jerome Bruner, the noted learning theorist, suggested that children must go through three main phases in order to understand representations: Enactive, Iconic, and finally Symbolic. In the enactive stage, the students "do" something to act out a process or concept. The iconic stage consists of students producing images that are raw representations of the process or concept. These representations usually lead to limitations requiring a refinement into more abstract symbolic representations. By experiencing these stages in order, meaning is attached to the symbols so that students can understand what they are symbolizing as opposed to just memorizing without meaning. Kaput, Blanton, and Moreno’s (2008) model is an extension of this theory and we will discuss their mechanisms for transition among these phases later when we look at algebraic concepts. For now, we can see an example of an elementary teacher engaging Kindergartners in Bruner’s theory in the video in Figure 2.1.16.

You may have noticed the children in the previous video excerpt "folding back" on their prior expereinces as they began to make connections between the "Ten Friends" and "One Friends" in physical sense and the numerals they have already experienced in their daily lives. So what does this mean for the students in Abby’s class from the earlier vignette Activity 2.1.1 at the beginning of this section? To wrap up this subsection, consider the following revisit to Abby’s class.

As time went on, the need arose to represent the concept of debt. Since cultures had created the whole numbers and a way to represent them, they again chose to extend the current number system to accommodate the need to represent quantities that were the "opposite" of the positive integers. Thus, the accepted set of numbers became the integers, denoted \(\mathbb{Z}=\left\{\ldots,-3,-2,-1,0,1,2,3,\ldots\right\}\text{.}\)

However, operations on these negative numbers were still looked at with skepticism. For example, what does it mean to multiply two negative numbers? Adding a negative to a positive had some meaning since the debt model would mean that some of the positive value would be cancelled by the negative values. Multiplying a negative by a positive value also had some meaning since if we owe three people $10, then it makes sense that \(3 \cdot -10=-30\text{.}\) But what about multiplying two negative numbers? Consider the video in Figure 2.1.17.

As convincing as our pile-and-hole model may seem from a physical manipulative perspective, it should not seem too surprising that negative numbers had not been widely accepted for many years since reasoning by analogy can give us insight, but it hardly constitutes abstract proof. Mathematicians were still very skeptical of negative numbers even in the 1700s (this is around the time of the founding of the USA). This is really not that long ago. In fact, in 1770 Euler used basic properties of the number system to show the product of two negatives is positive. His proof is really not that involved and can be followed by most middle school students familiar with the properties of additive and multiplicative identities, additive inverses (i.e. "opposites"), the distributive property of multiplication over addition, and the zero product property (see Theorem 2.1.18).

Theorem 2.1.18. Euler’s Proof that the Product of Two Negatives Is Positive.

The product \(-1 \cdot -1=1\text{.}\)

Proof.

Consider that \(\left(-1\right) \cdot \left(-1\right)=\left(-1\right) \cdot \left(-1\right)+0\text{,}\) since \(0\) is the additive identity. This now leads to the fact that \(\left(-1\right) \cdot \left(-1\right)+0=\left(-1\right) \cdot \left(-1\right)+\left(-1\right)+1\) since \(-1\) and \(1\) are additive inverses. But, \(\left(-1\right) \cdot \left(-1\right)+\left(-1\right)+1= \left(-1\right) \cdot \left(-1\right)+\left(-1\right) \cdot \left(1\right)+1\) as \(1\) is the multiplicative identity. The distributive property now allows us to factor out \(-1\) from the first two terms of this expression giving \(\left(-1\right) \cdot \left[-1+1\right]+1\text{.}\) Since \(-1+1=0\text{,}\) this leads to \(\left(-1\right) \cdot \left[-1+1\right]+1=\left(-1\right) \cdot 0+1\text{.}\) Finally, since the product of any number and \(0\) is \(0\text{,}\) this shows that \(\left(-1\right) \cdot 0+1=0+1\) and as \(0\) is the additive idenity, we get \(0+1=1\text{.}\) Since this line of reasoning began with \(\left(-1\right) \cdot \left(-1\right)\text{,}\) we have shown that \(\left(-1\right) \cdot \left(-1\right)=1\text{.}\)

Eventually, there was a need to represent quantities that were smaller than a whole unit (for a fun cartoon see Figure 2.1.19 below) and so the rational numbers again extended the known set of numbers. To the Greeks, all quantities could be expressed as the quotient of two integers and so the rational numbers, denoted \(\mathbb{Q}=\left\{\frac{a}{b} \mid a,b\in \mathbb{Z} \right\}\text{,}\) came to be the accepted system of numbers.

When Pythagoras’ work suggested that the hypotenuse of a right isosceles triangle with legs of length 1 could not be represented as the quotient of two integers (see Figure 2.1.20) and therefore, his work was considered heresy. According to Pythagoras, if \(a\) and \(b\) are the lengths of the legs of a right triangle, then their relationship to the length, \(c\text{,}\) of the hypotenuse was, \(a^2+b^2=c^2\text{.}\) So if \(a=b=1\text{,}\) then this would say that \(c^2=1^2+1^2=1+1=2 \Rightarrow c=\sqrt{2}\text{.}\) However, \(\sqrt{2}\) would then need to be a rational number if all numbers must be rational (according to the Greeks at the time). The problem is that if we make this assumption, we get a contradiction (see Theorem 2.1.21).

Theorem 2.1.21. Irrationality of \(\sqrt{2}\).

The \(\sqrt{2}\) is not rational.

Proof.

(By contradiction) Suppose \(\sqrt{2}\) is rational. Then there exist \(a,b \in \mathbb{Z}\) such that \(\sqrt{2}=\frac{a}{b}\) where \(a\) and \(b\) have no common divisors except 1 (i.e. \(\frac{a}{b}\) is in lowest terms). Then \(2=\frac{a^2}{b^2}\) and thus \(a^2=2b^2\text{.}\) Since \(2 \vert a^2\text{,}\) this implies \(a^2\) is even. If \(a^2\) is even, then \(a\) must also be even because if \(a\) were odd then \(a^2\) would be odd as well.

Therefore, let \(a=2k\) for some \(k \in \mathbb{Z}\text{.}\) By substitution, \(\left(2k\right)^2=2b^2\) and so \(4k^2=2b^2\text{.}\) Dividing by 2 we get \(2k^2=b^2\) and thus \(b^2\) is even (has 2 as a factor) and therefore \(b\) is also even. But \(a\) and \(b\) were assumed to have no common divisors except 1. Since \(2 \vert a\) and \(2 \vert b\) we have a contradiction and so \(\sqrt{2}\) must be irrational.

Note that this proof can be replicated for \(\sqrt{p}\) for any prime, \(p\text{.}\)

Eventually, these irrational numbers such as \(\sqrt{2}\) became accepted by the mathematical community and thus the current number system of the time was again extended to include both rational and irrational numbers. This set of numbers formed by the union both of these sets is called the real numbers, denoted \(\mathbb{R}\text{.}\) Other well-known irrational numbers include transcendentals such as \(\pi\) and \(e\text{.}\)