Against Factoring Trees

Nowadays, I'm anti-factoring trees. It's funny, because they're not usually part of the classes I teach, but they've come up a few times recently -- including twice just yesterday (as I write this), when they were included in a new book I received, and then within an hour a student came asking about them (because they were part of a YouTube lecture on reducing radicals she'd been trying to watch).

By factoring trees, I mean the method for producing the unique prime factorization of a number that looks like this:
By which we can conclude (re-sorting the leaf nodes) that 48 = 2^4 * 3. Of course, this is pretty customary, and it's how pretty much everyone I know (including myself) learned how to do it.

But now my primary complaint against them is that they're a nonstandard method of writing mathematical relationships, and most of all, they're a lost opportunity to practice writing equality statements. With all the problems that students have using, writing, and understanding the equality sign, why not use this as an opening to reinforce their meaning -- especially so in a context just like this, where we do not intend to simplify (evaluate) on the right hand side? Why not instead write in a more standard format like this:

(Or whatever your preference is for use of parentheses or exact number of steps.) It highlights all these issues with the meaning of equality signs that we struggle with later on students' behalf, and it avoids using a special one-off writing technique for the singular task of factoring a number. It's likely easier to read for some students (who may have trouble identifying where the leaves of the tree are). It even saves on lines of paper, and is easier to type out in an email or website if you have to do that. This is actually what I do in class when it comes up now. The more mental connections we can make to the "correct" way of writing math, the better.

PEMDAS Poetry, Pt. 1

Like a drunk uncle
Stumbling towards the bathroom door
Mistakes will happen

More anti-PEMDAS proselytizing here.

Comments on Decimal Places

Previously observed -- Operations on powers with the same base effectively shift one place down in the order of operations. Examples:

• Exponentiation will multiply powers; e.g., (x^3)^4 = x^12
• Multiplication will add powers; e.g., x^2 * x^3 = x^5
• Addition does a no-op on powers: e.g., 3x^2 + 5x^2 = 8x^2
  

Note that the same rule generally applies regarding number of places after a decimal point. Examples (working in the other direction):

• Addition does not alter number of places; e.g., 1.2 + 3.4 = 4.6 (1 place)
• Multiplication adds the number of places; e.g., 1.2 * 3.46 = 4.152 (3 places)
• Exponentiation multiplies the number of places; e.g., 1.234^2 = 1.577756 (6 places)
  

And the reason, of course, is that the decimal places themselves represent exponents to the common base 10. Example:

• 1.2 + 3.4
  = (1*10^1 + 2*10^-1) + (3*10^1 + 4*10^-1)
  = (1*10^1 + 3*10^1) + (2*10^-1 + 4*10^-1)
  = 4*10^1 + 6*10^-1
  = 4.6

Since the number of decimal places is dictated by the lowest power, if the addition operation 2*10^-1 + 4*10^-1 doesn't change the power (as above), then neither will it change the number of decimal places. And so on and so forth.

Calculator Equals

On the subject of students not understanding equals signs -- probably not the first time someone pointed this out, but -- How much of this is caused by usage of the equals sign button on a calculator?

It's really a bit malformed, if you think about it. Mathematically what's really happening when you hit that button is a request to "simplify" the numerical expression that you've typed in so far. So perhaps it would be better if the button were labelled "simplify" or "evaluate" -- or maybe a "total" button like on cash registers, or some abbreviation along those lines.

Possibly the malformed understanding of the equals symbol (thinking that a simplified number always goes on the right side) is due to the hundreds and thousands of times that students have used a calculator "=" button by the time the issue matters in algebra?

Related posts:

• Equals or Not Equals
• More Equals Signs

Concrete P-Value Demonstration

I find that students in my statistics class are almost totally bewildered by the logic of hypothesis testing and P-values (for hypotheses based on a population mean), no matter how carefully I try to explain the concepts. Here's an idea for a super-short and simple, concrete demonstration of hypothesis testing. Tell me if you think this would be worth the class time:

1. Start with a hand of four cards: {A, 2, 3, 4}
2. I'll turn my back and secretly do one of two things:
   H₀: Leave the Ace in, or
   H_A: Take the Ace out
   
3. Now shuffle the hand and deal out 3 cards.

Question: Say I get a draw of {2, 3, 4}. What's the chance of this happening if I did not take out the Ace (H₀)? Note that all possible draws would be {{A,2,3}, {A,2,4}, {A,3,4}, {2,3,4}} so the probability of seeing that would be P = f/N = 1/4 = 0.25.

Conclusion: If I draw {2,3,4} then we have some evidence that I did change the deck (H_A) -- because it's unlikely to see that result if I didn't (P = 0.25).

Now -- You can actually demonstrate this and ask the class if they think I left the Ace in or took it out each time. I'd recommend 3 run-throughs: leave it, leave it, then take it out. (In the latter case, also ask: Is it possible that I left the Ace in?) In reality, you should probably hold the cards against the otherwise full box, so it isn't obvious if your hand becomes empty in the take-it-out case. (And otherwise practice the prestidigitation in advance so your handwork doesn't give it away.)

Open Question: Should I actually reveal to the class which one I did each time (for confirmation), or leave that as a mystery (modeling real-world usage)?

The G*d*mn Particle

Another fabulous anecdote from the RJLipton Blog, regarding the Higgs Boson (what's popularly called "The God Particle"):

Higgs himself believes neither the particle nor the mechanism should carry his sole name, and was happy that he, Brout, Englert, and the three authors of another 1964 paper (Gerald Guranik, Carl Hagen, and Tom Kibble) were all awarded the 2010 J.J. Sakurai Prize for this work. He may have gotten his wish, as the popular name “The God Particle” has stuck to the boson. This is the title of a 1993 book by Nobel prize-winning physicist Leon Lederman and science writer Dick Teresi.

According to Higgs, Lederman had wanted to title the book The G*d*mm Particle to emphasize how elusive the boson was. His publisher declined to have a swear word in the title, but thought it fine to use just “God.” However, they could have settled on the Orthodox Jewish practice of writing “G-d” to avoid situations where the fully-written name might be erased or discarded. The title The G-d Particle could then be read with Lederman’s original meaning or not. Higgs is said to join many scientists regretting the “God Particle” name, more from concern over hype than irreverence.

I love this story so much. First, it finally makes sense of that stupid name in a way that eluded me until now. Secondly, it again shows that the "real" existential experience of scientific problem-solving is more generally one of a desperate, teeth-grinding, curse-filled battle (and not so much a dainty and refined observation of museum-like beauty).

Divine grace is a marketing pitch you use to sell something to the public. It's not something seen in the real world, or actual live math work, very much.

On that note, happy holidays from AngryMath! :-)

Dyson Quote

An excellent AngryMath-approved quote from Freeman Dyson:

If science ceases to be a rebellion against authority, then it does not deserve the talents of our brightest children.