How to resolve the new definition of subtraction and division seen in college algebra?

Question

Here's the foundational thing that irritates me the most when teaching college algebra.

Up through the secondary level, I think that instructors and students are trained to understand subtraction and division in terms of the inverse operation. Focusing on division here, if one asked "Why is $6/2 = 3$?", then one would most likely say it's because:

$$3 \times 2 = 6$$

But in every college-level algebra book I've seen, a different definition is given (and this goes for any texts in remedial elementary algebra, intermediate algebra, college algebra, etc.). Specifically, such books begin with the accepted "properties of real numbers", which are basically a restatement of the axioms for a field. In particular, one of the basic axioms is the existence of inverses: e.g., for multiplication, for any $a \ne 0$, there exists a value $1/a$ such that $a \times 1/a = 1$. (This is already problematic because students at this level are not yet familiar with statements involving existential quantifiers.) Thereafter, division is defined this way: $a/b$ means $a \times 1/b$. Of course, that's exactly what we see for a definition in most abstract algebra texts. But then technically this commits us to justifying "Why is $6/2 = 3$?" with something like the following chain of reasoning from the axioms:

$$6/2 = 6 \times 1/2 = (3 \times 2) \times 1/2 = 3 \times (2 \times 1/2) = 3 \times 1 = 3$$

Which I'm pretty sure no one actually ever does. Rather, they continue to use the secondary-school justification, even though this is technically out-of-synch (although, obviously, provably consistent with) our starting textbook axiom-properties.

Furthermore: When radicals are defined in the college algebra text, then the definition will once again look like the understanding of inverses from secondary-school subtraction and division (so it is additionally irritating to have these definitions and justifications out-of-synch with each other).

In summary: Advantages of the secondary-school definition: (1) it's what students are familiar with, (2) it provides shorter justifications, (3) it better lays the groundwork for the definition of radicals. Advantages of the standard college-algebra definition: (1) it complies with any standard textbook, and (2) it synchronizes with standard abstract algebra definitions.

So I go back-and-forth about this proud nail every semester. It seems like there would be more advantages to redefining subtraction and division as per the customary secondary-school rules, and thus smooth the way for student entry and understanding of the course; but the labor of going off-book and rewriting everything always deters me.

What is the best resolution to this problem?

Answer 1

Here is what the CCSS have to say when division and fractions are being introduced in grade 3:

http://www.corestandards.org/Math/Content/3/introduction/

1. Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding an unknown factor in these situations. For equal-sized group situations, division can require finding the unknown number of groups or the unknown group size. Students use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving single-digit factors. By comparing a variety of solution strategies, students learn the relationship between multiplication and division.

2. Students develop an understanding of fractions, beginning with unit fractions. Students view fractions in general as being built out of unit fractions, and they use fractions along with visual fraction models to represent parts of a whole. Students understand that the size of a fractional part is relative to the size of the whole. For example, 1/2 of the paint in a small bucket could be less paint than 1/3 of the paint in a larger bucket, but 1/3 of a ribbon is longer than 1/5 of the same ribbon because when the ribbon is divided into 3 equal parts, the parts are longer than when the ribbon is divided into 5 equal parts. Students are able to use fractions to represent numbers equal to, less than, and greater than one. They solve problems that involve comparing fractions by using visual fraction models and strategies based on noticing equal numerators or denominators.

In other words, division is (as you suggest) being defined as an inverse to multiplication. Since the CCSS defines multiplication in a way which distinguishes the two factors ($A \times B$ is the number of units $A$ groups, is each group has $B$ units), there are actually TWO distinct definitions of division which must be unified through the commutative property of multiplication:

The "How many groups" definition of division:

We define $A \div B$ to be the number $C$ which solves the multiplication problem $C \times B = A$. For example $12 \div 4 = 3$ since if want a certain number of groups, each containing 4 units, to give us 12 units total, then we must have 3 of those groups.

The "How many units in each group" definition of division:

We alternatively define $A \div B$ to be the number $C$ which solves the multiplication problem $B \times C = A$. For example $12 \div 4 = 3$ since if want 4 groups, each containing a certain number of units, to give us 12 units total, then we must have 3 units in each group.

The definition of $\frac{A}{B}$ of a unit is to take the unit amount, split it into $B$ equal sized parts to obtain the unit fraction $\frac{1}{B}$. Then $\frac{A}{B}$ is defined to be equal to $A$ of these parts of size $\frac{1}{B}$. For instance, the definition of $\frac{4}{3}$ of a pound would be to take $1$ pound, split it into three equal sized pieces each called $\frac{1}{3}$ pound. Then four of these pieces is $\frac{4}{3}$ pound.

According to these definitions, there is no direct link between $A \div B$ and $\frac{A}{B}$. However, we can argue their equality using the definitions, both "intuitively" and more formally.

From an intuitive "how many units in each group" perspective $A \div B$ could be thought of as the answer to the question: "I have A cupcakes, and B people to share them with. What fraction of a cupcake will each person receive?". One way to answer this is to split each cupcake into $B$ parts. Now I can give $\frac{1}{B}$ to each person from each of the $A$ cupcakes, yielding $\frac{A}{B}$ cupcakes for each person. Thus $A \div B = \frac{A}{B}$.

From an intuitive "how many groups" perspective, $A \div B$ could be thought of as the answer to the question : "I have A pounds of flour. It takes B pounds to make one recipe. How many recipes can I make?". One way to answer this question is to split each recipe into $B$ equal parts. Then it takes $1$ pound of flour to make $\frac{1}{B}$ recipes. Since I have $A$ pounds of flour, I can make $\frac{A}{B}$ recipes.

From a more formal/algebraic perspective we might make the following analogous definitions and proofs:

Let $A,B \in \mathbb{R}$ with $B \neq 0$. We define $A \div B$ as the real number $C$ so that $B \times C = A$.

If we were being really formal, I supposed existence and uniqueness of this number would need to be addressed. I have never seen any elementary text address the uniqueness part.

Let $A, B \in \mathbb{R}$ with $B \neq 0$. Define $\frac{A}{B} = AB^{-1}$, where $B^{-1}$ is the multiplicative inverse of $B$.

Note that this corresponds to our intuitive treatment of fractions. $\frac{1}{B}$ was defined as the number such that $B$ of them yields $1$: aka it was defined as the multiplicative inverse of $B$.

Now to check that $A \div B = \frac{A}{B}$, we just need to check that $\frac{A}{B}$ satisfies the definition of the quotient:

$ \begin{align*} B \times \frac{A}{B} &= B \times (A \times \frac{1}{B})\\ &= A \times (B \times \frac{1}{B})\\ &= A \times 1\\ &= A \end{align*} $

This whole discussion will fly way over the head of almost any student though. Ideally the teacher can be aware of these issues, so that they can design tasks which target the intuitive development of these ideas.

There is a tension between the logical development of the ideas, which has a strict progression from the definitions, and the desired end state, in which the intuitions, understandings, and equivalences are so strong that it is easy to forget what originally implied what. You want to build number and operation sense which is so strong that these ideas are all applied intuitively at the subconscious level. I am not sure how to resolve this tension. We want a really big "The following are all equivalent" statements to be living (implicitly) in the mind of each student, but we cannot get there without a logical progression.

Answer 2

But then technically this commits us to justifying "Why is $6/2 = 3$?" with something like the following chain of reasoning from the axioms:

$$6/2 = 6 \times 1/2 = (3 \times 2) \times 1/2 = 3 \times (2 \times 1/2) = 3 \times 1 = 3$$

Which I'm pretty sure no one actually ever does.

I am reminded of Principia Mathematica famously taking hundreds of pages to prove that $1+1=2$. I'm pretty sure no-one actually ever does that either. (Actually, I doubt that anyone at secondary-school level, much level college level, bothers to justify $6/2=3$ at all).

Nevertheless, since this troubles your conscience, the best resolution to an apparent mis-match between an old, familiar, system and a new system is surely to prove their equivalence and then to work in whichever system is most convenient for the task at hand. So if you prove the lemma $a / b = c \iff a = b \times c$ you can then apply your secondary-school justification without any twinges of conscience. And if you explicitly teach the principle of proving equivalence and then working in the more convenient system you're doing your students a big favour. I think it can be argued that that principle is as foundational to mathematical thinking as axiomatisation, and probably more so.

Answer 3

I'm very much with you concerning the problem.

Striving for conceptual understanding and not merely exercising procedure following, I actually do bother with the question to justify why 6/2=3. But maybe one has to clarify more clearly, what 'justify' in school context actually means. To me, it doesn't mean to give a mathematically valid proof as in the Principia Mathematica, but merely an explanation that allows students to connect to other concepts and to model meaning; loosely speaking...

In the case you mention, I see the problem mainly in a change of concept: In middle school - students develop an understanding of inverse operation: They learn to see division as inverse operation of multiplication. In high school or later in college, students then should develop an understanding of inverse element: they learn that there is a neutral element with respect to an operation and the meaning of an inverse element is that its action on (i.e. operation with) the element itself results in the neutral element.

This is the way that I read your line of reasoning. The first equal sign reads as interpretation what 'division' means: multiplication with the inverse element.

In my experience, students often have a lot of trouble with this abstraction: To see as object what they used to know as an operation. Frankly I don't know what the best way is to approach this problem. That might much depend on the local context you encounter. But I do have some strategies as an approach. Said in advance, it anyway just takes time, and patience, and care...

I explicitly discuss different models for the objects and the operations. For example, taking the number line as model for the numbers (objects), what are models for the operations? addition and subtraction might be straight forward, but if you model muliplication as concatenation of "arrows", that lets you explain what $5 \times \frac{2}{3}$ is, but does not work for a calculation like $\frac{8}{7}\times \frac{2}{3}$. That works if you model multiplication as scaling (e.g. I use the following Geogebra-Applet as part of the many ways to explain why the product of two negative numbers yields a positive number as a result). You can model division in a similar way.

Using explicit models I'm usually pretty successful in talking about the distinction of inverse operation and inverse element.

I'm sure there are other ideas and better ways to address this problem with students, I would love to hear more about how what people in this community think about this question.

Answer 4

The problem stems from effectively defining the division operator twice. You should begin by defining division on $R$ as usual:

For all $x, y, z \in R$ where $y\neq 0$, we have $x/y=z$ iff $x=y\cdot z$

So, $6/2=3$ iff $6 = 2\times 3$ by the definition of division on $R$.

From the field axioms, we know that

For all $x\in R$ where $x\neq 0$, there exists $y\in R$ such that $x\cdot y=1$

Note that I do not make use of the division operator here.

Now, we can prove:

For all $a\in R$ and $a\neq 0$, we have $1/a\in R$ and $a\cdot (1/a)=1$

Suppose $a\in R$ and $a\neq 0$.

Then, from the field axioms, there must exist $b\in R$ such that $a\cdot b =1$

Applying the definition of division, we have $b=1/a$

Substituting, we have $a\cdot (1/a)=1$

We conclude as required that:

For all $a\in R$ and $a\neq 0$, we have $1/a\in R$ and $a\cdot (1/a)=1$

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Follow-up some years later...

When does a theorem become an axiom? You might point out that the real numbers themselves can be constructed (i.e. proven to exist) using only the axioms of set theory. Likewise the addition, subtraction, multiplication and division functions. My "definition" of division will correspond to a theorem about the the multiplication and division functions. Likewise, each of the field "axioms" will correspond to a theorem about each of these functions. Now, these proofs consist of thousands of lines of formal proof and are way beyond the scope of any high school and almost every university math course To get on with real analysis, the field axioms are usually taken as a given even in advanced courses.

Answer 5

This isn't every possibility, but a rough sketch of approaches, rank ordered by what I think will help the (weak, given they are taking this course in college) students the most. Which I believe is the more important objective. Not fussy details of math or of what is most interest to the (more sophisticated) instructors. But helping the kids.

1. Avoid dwelling on the theoretical justifications (at all) and pursue a course of familiarization via practice. Yes, the kids need to learn the ability to manipulate symbols (like polynomial long division) as they do numbers, but this is best accomplished by light practice with the numbers, followed by extensive practice with x-containing expressions. Perhaps even juxtaposed (when they get to the symbols at least, OK if you do some all arithmetic drill earlier, but doing a long division number problem, itself, followed by a polynomial one, is an effective way to build confidence, willingness. If you instead make them question things they already are familiar with (subtraction of numbers), or at least think they are, you will derail them. As other have pointed out, one could spend immense detail on 1+1, but this is not the right time for that. I would even wager that it is hard to learn that level of abstract material without some base in the topic first (why it is easier not to have to learn Rudin before conventional calculus.)

2. Teach the definition you (and they) are comfortable with, the conventional one. But then move to significant calculational practice to build familiarity. (The way to learn quantum mechanics is to do particle in the box problems, not axiom-dwelling. In some cases, you really need a sort of familiarity with the materials, before dwelling on definitions or different axioms or notations [like bra ket] are pedagogically manageable. (There's a 1.5, where you glide over it and move to practice very fast.)

3. Teach the book's definition. Then practice. (There's probably a 2.5 where you glide over it and emphasize practice, though.)

4. Do both, then practice. This has the disadvantage of spending way too much time on fussy details and bogging the kids down, when they need familiarization with manipulating x-expressions. Note, I'm not saying little edge cases are irrelevent (like stuff the "divided by zero" hawks watch for), but even there they are better dealt with after or during the course of building basic familiarity/competence. Not pre-emptively. That is too much cognitive load and not progressive enough.