The "Deep Question Box" (DQB) thread

The DQB is meant to expand your philosophical outlook on math. When we exercise higher-order thinking, we will better understand the thing it is we are learning.

The only DQB rule is that every student is required to submit at least one "deep question" into the box every quarter. If for whatever reason you can't put your question in the box, send me a private message on this forum.

I may answer a select number of questions out loud in class, but I will post them all in this thread so everyone can see them. They will be both from 2nd and 6th period. If I struggle to answer one of them, I will leave it blank and allow others to chime in and try and answer them for bonus rewards.

_________________
"Give a man a fish and you feed him for a day. Teach a man to fish and you feed him for a lifetime." - Anne Thackeray Ritchie

It is more worthwhile to teach someone to do something, than to do something for them.

Rather then telling them the answer and the formula how can we explain math conceptually to someone who is confused about an equation and needs help understanding the problem?

Quinton R. wrote:

Rather then telling them the answer and the formula how can we explain math conceptually to someone who is confused about an equation and needs help understanding the problem?

For teaching in general:

Know who your audience is, and let your audience know who you are. Come to a mutual understanding that you won't give them the answer, and that uneducated guessing will amount to nothing as well. Any guesses they have must be substantiated with reasoning, and anything you do to guide them will be in question form alone. Shoot, even think about the way I might ask a question to the class. Instead of saying: "What kind of shape is this?" I might ask: "What do we have to do to find out what kind of shape this is?" It helps stress the idea that there is an undertaking process for which the knowledge must be obtained.

Also, reinforce vocabulary/common sentence structures over and over and over again. The more you emphasize some terms, slow it down, and point as you go, the better something will be retained. Remember, we remember 50% of what we see and hear, but break those apart and you're looking at 20-30%.

And I'll talk about this one further down, but math is difficult for a lot of people because they treat it like its own language that they try to decode the numbers and letters as numbers and letters, rather than as words and complete sentences. As I've said often, CPCTC is one of the biggest "no duh" concepts in math that you'll ever learn, but people try and treat it like such a foreign concept that they're making it out to be much more difficult than it really is. If you strive to memorize something rather than just take a step back and make logical sense of it, I guarantee you'll have trouble. That's not the way that something is taught, and in turn it shouldn't be the way that something is learned.



Specific math examples:

In Geometry, everything always best works with pictures. You might have noticed that the basis of everything has either appeared in triangles or circles, and you build from there (they both have their different branches thereafter). As you probably well know, formulas didn't pop up out of thin air; they exist for a reason.

Take the "area of a rectangle/kite/parallelogram/trapezoid" formulas that we did earlier in the year on poster paper. Notice how every formula can be found by understanding something about triangles. And, of course, triangles are just half of a rectangle when you think about it. This is why two-column proofs are so important (so is CPCTC), because when you can prove triangles congruent within something, you're already halfway to where you need to be. And trust me, congruent triangles exist all over the place. So do similar triangles, I suppose.

Not everything is easily discoverable, so sometimes it takes somebody to teach you the methods of discovery for you to then understand it for yourself. However, building one thing upon another is always the best strategy when it comes right down to it. That's why your book layers chapters in a specific way, to make sure that you scaffold your learning process to make use of what you just learned and apply that to the next thing.

As for Algebra II concepts (things like logarithms, permutations, matrices, sequences, conic sections, inverses, and rates), things make best sense when they're not looked at like numbers, but rather spelled out into plain English. I've been tutoring somebody in that course all year at Burlingame High School, and she gets a clear explanation when she's looking away from the paper trying to decode the letters and numbers on paper like it's math, and when she hears it in plain English it feels more comfortable. It's tough, like translating from another language... but it comes with enough practice.

I'll give you a fair Algebra II example by teaching you something right now that you'll learn very late next year:

An arithmetic series is a list of terms added together that share a common difference. For instance, 3+5+7+9+11+13 (common difference of '2'). This is called a finite arithmetic series because has a stopping point (not infinite), as it has only six terms (the first term in the series is '3' and the last term is '13'). There is a formula to find out what the sum of the series is without having to enter every value in your calculator. The formula is as follows:



Where S_n is the sum, n is the number of terms, a₁ is the first term, and a_n is the last term. Let's look at the previous example and try to see if we get the correct number by doing this:

{3, 5, 7, 9, 11, 13}

n = 6 (number of terms)
a₁ = 3 (first term)
a_n = 13 (last term)

S₆ = 6/2 * (3 + 13)
S₆ = 3 * (16)
S₆ = 48

You can obviously double-check this by adding 3+5+6+7+11+13, and you'll get the same value out.

There, we just did a problem and that's all great, but now we're sitting here wondering where the heck the formula came from. And honestly, I didn't know until I had to teach it to this student.

Here's what I got out of it... look at the list again:

{3, 5, 7, 9, 11, 13}

We know they all are two units apart, so that means that if I add together the first and last terms (3 + 13 = 16), that will be the same as adding together the second and second-to-last terms (5 + 11 = 16), and likewise it will be the same as adding together the two middle terms (7 + 9 = 16). So in the end, it's like adding 16 + 16 + 16, or 3*16. How does the 3 fit in? Because we're adding 3 pairs of terms together, and they all come out as the same amount. If you look back at the formula:



You can see that n/2 is counting how many pairs exist in the series (like the way I added them since they all come out to the same amount). Then you choose two values to add together to that amount, like the first and last term, because they are plenty easy to find.

That's a rough and long example. It's just something to get you into the mindset. Formulas aren't always easy explained, and furthermore they can be extremely difficult to discover. But I have found that the more you can extract out of a formula, the better off you'll be because you can make sense of it, rather than having it "make sense of you."

I probably could have explained this all much better, but I hope that was a valid initial attempt!