O Tannenbaum, How Lovely Are Your Primes

Students love these problems. They're neat, there's one possible solution, and Hey! - they look like trees.

No need to discuss prime factorization, because a) students don't understand those terms, b) what students really want to know is how to solve the problem, and c) it's much easier (and cuter) to say, "Just make a factor tree.”

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What we have effectively done here is reduced one of the most fundamental theorems in all of mathematics to a Christmas tree ornament. This isn't the first time we've done this. We define prime numbers for students, we tell them the value of pi, we calculate the sum of the triangle's angles for them. We sell all the secrets, reveal all the magic, and spoil all the endings. 

Granted, we've all tried to play it up big in the classroom in an attempt to generate some enthusiasm. We soon discover that not all students are as interested in significant mathematics as we are. I'm not saying we should attempt to surprise students for the sake of enthusiasm levels. Besides, students who are struggling to understand something may not feel any sense of surprise. Significant mathematical surprise requires solid intuition - enough to appreciate the differences between intuition, expectation, and fact. But if I can get a "Wow, that's kinda cool!" or a "I didn't expect that!" from a student, I'll take it.

So here's my attempt to return to students some opportunity for discovery:

1. Make a list of the first few primes: 2, 3, 5, 7, 11. Give students a composite number, say 6. 
   
2. Now ask, "Can you make the number 6 using multiplication and only numbers in this list? You can use a number more than once."
3. When they come back with a solution, point out the obvious. "You didn't use any other numbers? Only primes?"
4. Give a few more to try: 12, 14, 20, 45, 28. "See if these numbers are possible." Or ask students to volunteer composite numbers between 1 and 50.
5. Ask, "What other composite numbers do you think you can make?"

What are students wondering now? Are they thinking about which numbers are possible? At this point do they expect that every composite number can be written as a product of primes in a unique way? Do they understand that they have just stumbled upon the Fundamental Theorem of Arithmetic that was first proven by mathematician Carl Gauss in 1801? I can't expect surprise, or even appreciation of this discovery. But students are making conjectures and testing them as thoroughly as they can, and that's worthwhile.

George Pólya gave ten commandments to teachers. This seems fitting:

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9. Do not give away your whole secret at once – let the students guess before you tell it – let them find out by themselves as much as is feasible.

I'll agree that a factor tree is an efficient tool for grinding out prime factorization. However, it is unfortunate to see absolutely no connection between the tool and the discovery. Why not wait until students see the need for such a tool? When discussing ways to reach the number 48, I have seen students start with factors 6 and 8, then represent these using primes. These students needed a way to organize and keep track of this process and were able to invent their own tree-like structure. Save the trees for art class and let your students do some real thinking. You'll be helping to spread the joy of discovery.