Suppose for a moment a parent shows a child images of 10,000 blue cows. Yep. No typos. Ten. Thousand. Blue. Cows. We are talking blue. Like the cow pictured at the left. 10,000 is a healthy number of cows. This would amount to showing the child **one blue cow every second for roughly 2 hours, 45 minutes**. The child might conclude, given this overwhelming evidence, every cow is blue. We could really rock the child's world by introducing an image of a *white cow* or a brown cow or a **black cow**. This counterexample would stimulate the child to re-evaluate his or her conception about what makes a cow. The hope would be that the parent would help the child understand better the definition of "cow." What makes a cow? Four legs? Not necessarily, the cow could be an amputee. An udder? Not necessarily - couldn't the cow be a bull? Horns? A tail? A particular set of adenine, guanine, thymine, and cytosine? Do we need to be that specific? We collectively have a definition we use for the animal "cow," and it's often based on experience. As you are reading, you may have even conjured up, in your mind's eye, a picture of a cow or two. Specifically, that definition of "cow" may vary according to the context in which we are operating.

Take the above situation and replace every word "cow" with "**mathematical example**." If developing students with strong mathematical understanding is the goal, we must be wary of how to model for students how to behave when the white cow or brown cow or black cow comes along.

My experience with teachers and students tells me identifying the domain of a function is often tricky business for students. What types of functions does a typical Algebra 2 book examine when looking at the domain of a function? They look at lines, parabolas, and cubics for sure. A student sees the domain of a run-of-the-mill linear function and a run-of-the-mill cubic function are the same. They see the domain for a parabolic function is also the same. The student's mind attempts to search for some sort of pattern. The student may formulate misconceptions that do not generalize. It is a challenging task for the teacher to develop a robust understanding of domain in students. We hope the student understands the notion of a function, input values, binary operations that are not closed in the real number system, etc.

I was helping a former student yesterday with some material for an upcoming College Algebra exam. We came across the following problem.

There are many things to like about this question. The student must have a pretty solid understanding of what linear functions look like. The student must understand A(t) does not mean the product of quantities A and t. The student must recognize the quantity t/3 can be rewritten as (1/3)*t by leveraging the distributive property to combine the like terms 4t and t/3. The student must also recognize 8 can be rewritten as 8*t^0 to explain why 8 is not a like term with the others. Our work for the problem is below.

No trouble, run of the mill example (blue cow).

Now for the brown cow.

What a GREAT QUESTION. <*Trumpets herald from the heavens*>

Here's some initial work on the problem, which looks an awful lot like the student that does not see the problem for what it is - something different.

As a teacher, I was thinking of how to leverage other connected ideas to help the student deepen his understanding of linear functions. Heck, we even graphed it in Geogebra and the silicon genie confirmed the student's suspicions about linearity with a picture.

I put all that other stuff in the sheet (the slider and the point A whose x-coordinate is governed by the value of the slider) after the fact. The student was convinced the function was linear and was ready to move on.

My background knowledge from modern algebra and rings and binary operations and all that jazz let me see the problem for what it was. I was trying to think through how to meet the student at his level to help him develop an understanding that would allow him to identify functions that may appear linear but have potential domain issues. One could argue the function is "*linear*" everywhere except at x = 0. I asked the student about the operations he saw going on within the original function statement.

Multiplication by x, the difference between 2 and the quantity 3/x, and the quotient of 3 and x. The following exchange ensued.

**Mr. A:** "What do you get when you add two real numbers?"

**Student:** "A real number."

**Mr. A:** "What happens when you subtract two real numbers? What do you get?"

**Student:** "A real number."

**Mr. A:** "What happens when you multiply two real numbers?"

**Student:** "You get a real number."

**Mr. A:** "So what do you think about division?"

**Student:** "You get a real number."

**Mr. A (with poker face):** "Is what you said always true? Will there ever be a case where you divide two real numbers and you do not obtain a real number as a result?"

Then we did a little work to back it up.

I even went so far as to write this statement on the board:

The student immediately questioned this claim, since we had worked this problem earlier:

We had mutually agreed the domain of t in the function above would be numbers greater than or equal to zero, since it wouldn't make sense for the variable t to take on negative values, as this would correspond to times before the billing cycle began. From the perspective of polynomials, however, the function C(t) = 0.1t + 11 has a domain of all real numbers. We choose to impose the restriction that t must be strictly greater than or equal to zero in the context of the problem. The student was drawing from a previous experience in trying to understand the domain issue we faced. The fact we disallowed negative values appeared to be a contradiction to the student because the purple sentence I wrote contained the phrase, "does not allow."

My thoughts turned to polynomials and how I could help the student move forward. I asked the student, "What is the definition of a polynomial?" My hope was that he would respond with something about the operations allowed on the variable. No such luck. After a long pause, I said, "Well, let's do what any sensible person would do. Let's look it up."

This definition didn't help unmuddy the waters. I am a firm believer in not clicking on the first link in Google immediately and helping the student reason through which links to use. After some digging, we eventually clicked the first link and came across this definition:

Indirectly, this definition states the operations addition, subtraction, and multiplication on the variable are permitted. I wanted the student to also recognize there are different classifications into which functions could fall, like rational functions or radical functions.

We used the examples towards the bottom to address different possibilities for powers that may lead to division by a variable or taking the root of a variable. This rich discussion took roughly forty-five minutes to run its course, with some pauses for questioning, thinking, and doing some algebra. I finished our discussion of this problem with some other looks this problem might present.

I hope you enjoyed this article. I would argue one of the fundamental purposes of schooling in mathematics should be to help students develop a rich understanding and to know how to behave when a cow of a different color comes along.