# Danielle Stanton

#### University Of Minnesota – Mathematics

It’s the height of wanton hubris to claim that “mathematics is all around us.” Mathematics isn’t all around us, mathematics is only in our heads. To claim that “functions are everywhere” or that “numbers are everywhere” is only to claim that what is going on inside of our heads is actually going on outside of our heads. Belief in mathematical Platonism is anthropocentrism at its worst and is completely absurd.

For example.

Let’s look at $$\sqrt{2}$$. $$\sqrt{2}$$ must exist, right? Like… it has some sort of an existence. What is $$\sqrt{2}$$? One of the things it is, how $$\sqrt{2}$$ arguably came about historically, it’s the length of the hypotenuse of an isosceles right triangle with leg length 1. Okay. How do we know an isosceles right triangle with leg length of 1 has a hypotenuse of length $$\sqrt{2}$$ and not 1.4147? How do we know that? The Pythagorean theorem. How do we know the Pythagorean theorem is true? We know that the Pythagorean theorem is true because of the Euclidean Parallel Postulate. How do we know the Euclidean Parallel Postulate is true? WE DON’T KNOW ITS TRUE! IT’S STIPULATED. We ASSUME it to be true. We take it to be true. Why do we take it to be true? Because it kinda seems like it might be true. When we’re dealing with flat surfaces.

The only reason that we know the hypotenuse of the triangle is $$\sqrt{2}$$ is because of math. Not because we can measure the length with a ruler. Not because a measurement conforms to our sense experience. Not because $$\sqrt{2}$$ exists as an object outside of mathematics. $$\sqrt{2}$$ is just a figment of our imaginations! In order to have that isosceles right triangle, you HAVE to have the Euclidean Parallel Postulate. We HAVE to be working on a flat surface. Otherwise… we don’t have the triangle, we don’t have $$\sqrt{2}$$.

We have to define distance, a figment of our imaginations. We have to define area, also a figment of our imaginations. And then later comes $$\sqrt{2}$$. In the above sense, $$\sqrt{2}$$ doesn’t exist outside of the Pythagorean theorem. (I mean, it does exist outside of the Pythagorean theorem, we can find $$\sqrt{2}$$ other ways… but they’re all math ways.) It’s not like an empirical, “oh. THIS thing right here that I am pointing at is the square root of 2.” It’s all just made up. Number itself has only some sort of quasi-empirical existence. The idea of “number” itself is a map that humans lay on top of the territory of our sense experience. The map is not the territory.

P.S. And to ask students to “discover” mathematical objects as though they exist outside of the minds of mathematicians is completely absurd and unfair.

I wrote these .pdf “lecture” notes Spring 2019 for use in the UMTYMP Math Analysis class I taught that semester. We used Cohen’s PreCalculus with Unit Circle Trigonometry, Fourth Edition. I hewed to the text fairly closely. These documents are a hybrid of a lesson plan, lecture notes, direct instruction, and whole class discussion. I project and scroll through this document throughout class, using it to structure our work over the weekly two-hour class meeting. Pink ink tends be additions to the notes that occurred during class discussion (but not always).

Please use these pdfs as you see fit: in your own classroom, as inspiration for your own teaching, or to be a Judgy McJudgerson. Whatever floats yer Boaty McBoatface. They were created using Notability on an iPad with an Apple Pencil. If you would like the original Notability file(s) to use for your own devious plans (or to just see the gifs actually move), lmk, I’m happy to forward.

Enjoy!

Alison King (1992) Facilitating Elaborative Learning Through Guided Student-Generated Questioning, Educational Psychologist, 27:1, 111-126,

Three Sentence Summary
For people to learn, new information must be hung onto a structure of previously learned information. The authors developed and tested a procedure for using two dozen question stems based on the higher levels of Bloom’s Taxonomy to aid in this knowledge acquisition. Authors found the question frame protocol, which requires students to complete the question frames themselves, effective for their subjects and circumstance (undergraduates’ post-lecture information retrieval).

What’s Good

The question frames:

What is a new example of ______?
How would you use ______ to ______?
What would happen if ______?
What are the strengths and weaknesses of ______?
How does ______ tie in with what we learned before?
Explain why ______.
Explain how ______.
How does ______ affect ______?
What is the meaning of ______?
Why is ______ important?
What is the difference between ______ and ______?
How are ______ and ______ similar?
What is the best ______, and why?
What are some possible solutions for the problem of ______?
Compare ______ and ______ with regard to ______.
How does ______ effect ______?
What do you think causes ______?
Do you agree or disagree with this statement: ______? Support your answer.

What’s Not So Good

Authors assert, “Students who learn the most are those who provide elaborated explanations to others in their group.” I say, “chicken, meet egg.” Are students learning the most because they are providing explanations, or are they able to provide explanations because they have learned the most? If not one direction of causality or the other but rather some combination, which percentage for which?

The study lacks quantitative data. Authors make claims of efficacy of their protocol without including evidence of measurement of their claims so we cannot determine how effective the protocol is.

Here is Frank Quinn’s article “A Revolution in Mathematics? What Really Happened a Century Ago and Why It Matters Today.” Appearing in Notices of the AMS, January 2012

This article articulates clearly, (in a way I wish I could!), some of the confusion/friction I have encountered between the fields of Mathematics and MathEd. It seems to me that so often that when a professional mathematician says “mathematics” they mean something entirely different than when a mathematics educator says “mathematics”. Quinn points to the mathematical revolution occurring in the late 19th and early 20th centuries, a revolution that in his estimation reached the math research community but did not reach the MathEd community as the source of this disconnect. I have been mostly trained as a proto-research mathematician (what Quinn might call a “core user”), not as a math educator. And as such, this part of Quinn’s article resonated in particular for me:

“The new methodology is less accessible to non-users. Old-style definitions, for instance, usually related things to physical experience so many people could connect with them in some way. Users found these connections dysfunctional, and they can derive effective intuition much faster from precise definitions. But modern definitions have to be used to be understood, so they are opaque to nonusers. The drawback here is that nonusers only saw a loss: the old dysfunctionality was invisible, whereas the new opacity is obvious.”

As a proto-research mathematician, mathematical objects do not have any existence outside of our definitions of them, let alone any sort of empirical existence. They are figments of our collective imaginations (that we can use to predict the future, how cool is that?). So trying to “explore”, or “discover”, or develop Vinner’s concept image, or develop an “intuitive understanding” outside of a definition is literally impossible. One cannot understand a thing that does not exist! One may as well seek to learn facts about unicorns. (There are no facts about unicorns. My apologies to Hannah Gadsby). One develops intuition about a mathematical object by winding up the definition and letting it go to see what it does (aka proving theorems about the object). Proof is the method by which we come to know in mathematics. (Though, like “mathematics”, “proof” seems to mean two different things to the two communities). One can certainly motivate a definition empirically or otherwise, but a motivation does not necessarily constitute mathematical knowledge or learning.

This stands in sharp contrast to the mathematics education perspective, in which one tries to develop a concept image almost exclusively through experiment and intuition, rather than deduction. In my estimation there exist certain areas of math that this approach is better suited to than others. Euclidean geometry comes to mind. Which is ironic, considering this is the area of the school curriculum in which precise definitions and deductive proof are most emphasized.

Anyway. These are some of my initial thoughts. Pelican Brief forthcoming.  I hope I’ve piqued somebody’s interest, I’d be curious to hear your thoughts in the comments or on social media.

What follows are my notes on the Hyman Bass article from 2011, “A Vignette of Doing Mathematics…”, about the slices of cakes.

1. His description of his internal process of working on a problem comports with my experience of my internal process of working on a problem. IT’S LIKE HE IS IN MY BRAIN.
2. He seems to imply a salient difference between an expert practicing their discipline and a novice learning a discipline. (Especially when he says “In doing mathematical research, unlike school work, we don’t want to expend great effort trying to solve a problem that has already been solved, (unless our intention is to find a simpler solution or proof). So, once a question we confront resists our first serious efforts, it is wise to consult the literature, or expert colleagues, to find out what is already known about the problem. This is also appropriate in school mathematics if working on an open-ended and long-term mathematical project. Mathematics is a hierarchical subject [NB; hierarchical!], and we don’t want to constantly reinvent the wheel. But of course this means learning to interrogate and learn from the expert knowledge of others.“). This view (expert vs. novice) resonates with me. There exist activities that are appropriate for trained mathematicians to engage in while doing mathematics, while those same activities may or may not be appropriate for a novice in the service of learning mathematics. I’m thinking in particular of the definition sorting activity (of course).
3. The initial problem, “If c cakes are to be equally shared by s students, what is the smallest number, call it p = p(c, s), of cake pieces needed to make this distribution?”, initially seems a good candidate for a middle or high school complex instruction task. Not sure about groupworthy-ness, just because I think I would want to do this problem on my own… at least until I got stuck. Once Bass started talking induction though, I became doubtful that this problem might be effectively used in HS, as induction generally isn’t covered in HS.
4. When Bass says, “Consulting the literature in pursuit of the questions above was the occasion for learning some very interesting mathematics”, I am struck that this is the first time I remember him mentioning that he learned mathematics. He talks here about the “learning” of mathematics to occur when he consulted the literature, not before… not in the “doing” of the mathematics… he doesn’t “learn” math by exploring and proving his conjectures, connecting representations, etc… he “learns” it by reading about what others have done. I searched the article for the word “learn” and Bass mentions mathematicians learn mathematics by doing a google search! When he’s doing his proof thing, he does refer to assertions proven in the paper as “what we have learned”. Interestingly, not “what *I* have learned” but that could very well be a result of the quirky style in which mathematicians tend to write.
5. I thought about our task debrief questions, (what math might students learn, what math do they need to know), for the problem quoted in 3.
What math would they learn by doing this problem?
Unclear… they prove a theorem, so they’d learn that theorem. They might practice GCD, LCM, fraction arithmetic, Euclidean algorithm, depending on solution path.
What math do they need to know?
GCD, LCM, fraction arithmetic, Euclidean algorithm
There’s been this ongoing tension for me about what constitutes math (obvs), what we might mean when we say “learn math”, and how that might differ, if at all, from “know math.” Learning is the process by which we come to know? If I know it, then I don’t need to learn it… the desired result of learning has been achieved if I know it, I know longer need to continue to learn it.
6. What CCSSM content standards might the problem in 3 address? I have concerns that precisely because this is such a rich problem, that its utility for assessment or teaching is limited. (Whatever the heck we mean by teaching, and whatever relationship that might have to learning and knowing. Oh bother). Here’s the sentence frame I want to fill with math *content*, but have been unable to… By completing this problem, students will be able to <blank>. I can put any number of the mathematical practices in there, but I want my students to know the content, that’s priority one. What’s the point of a math activity if we aren’t learning math content?
7. Also, definitions! Bass needs a thing, he defines a thing. No correct or incorrect definition, just choices, then proofs.

Dear Danielle,

I have been tutoring a kid who is trying to up his ACT score (currently at 29 trying to get at least 30). We focus on the “last 20” questions and efficiency since he doesn’t have much time left by the time he gets to those. This was one of the questions at our last session:

The sum of two positive numbers is 151. The lesser number is 19 more than the square root of the greater number. What is the value of the greater number minus the lesser number?
Choices: 19, 66, 85, 91, 121

Being in full system mode, I told him to identify the variables, set up the equations, and began to solve by substitution until I realized that this would take too long. Graphing was not an efficient option either. By looking at the answer choices, it was clear that the numbers must be “nice”. We used a table approach and only considered perfect squares as the greater number until we found the pair that satisfied our equations. So, we used a combination of algebra and guess/check.

I know this particular problem is out of the scope of 8th/9th grade, but should we be doing more with tables and guess/check to prepare them for systems like this one? Or, could you solve it quicker doing something else? Thoughts?

Signed, Somebody Awesome

Dear Somebody Awesome,

If one of my tutoring students and I were looking at that question in the context of ACT prep, I would tell them to skip it and guess (no wrong answer penalty on ACT) as soon as they “realized that this would take too long”. Maybe pick either 85 or 91 because those numbers are close to each other. And then I would make them solve the system anyway because I’m mean.

There are three ways I can think of that would be “efficient” if you have a graphing calculator (which you should definitely have on the ACT!)

1. Use the graphing calculator to solve the system by graphing. FAST. (it took me 1:23, including reading the problem and writing down the system. I had to change the viewing window and use trace)
2. We hate the square root, so solve the system by letting $$\sqrt{y}=z$$, substitute, then solve the quadratic in z. With a calculator it would be quick to get those zeros, I used the table function rather than the quadratic formula. (2:39) If you didn’t want to use a calculator the quadratic isn’t too bad to factor.
3. Set up a new system that’s easier to deal with

$$\begin{cases} x+y=151 \\ y-x=c \end{cases}$$

where c is one of the 5 choices. then by elimination you get $$y=\frac{151-c}{2}$$, use values of c to get y, then back substitute for x, then check against the third equation. This would probably take the longest. (it did. 4:17)

Anyway, the answer is 91, so if a student takes my original advice they’ve got a 50-50 shot and saved themselves a few minutes.

And that problem is a GREAT example of why the standardized testing industry is an unethical mobster (they certainly have a great racket!), and why I decided to never offer test prep services through my tutoring business. You can increase scores and make money hand over fist teaching “tricks” but the whole system is extortion. I’ll do it for already established students, but man. Ew. Gross. My general perspective is, if you know the mathematics then the ACT is a breeze. Content is king.

“I don’t like it, turn it into something I do like” is a fundamental mathematical “habit of mind”. Though as far as I know I don’t think any education researchers include it in their habits of mind lists! (remind me to tell you the firefighter joke, it’s my favorite) “I don’t like it, turn it into something I do like” is why you’d think to let $$\sqrt{y}=z$$, “I don’t like it, turn it into something I do like” is why you’d look at this system

$$\begin{cases} x+y=151 \\ y-x=c \end{cases}$$

instead of the other one. Not more tables, not more guess/check, mathematics education should not be preparation for standardized tests. I think this problem could be taught in a rich way, but not as a timed multiple choice question.

P.S. Regarding testing industry’s status as an extortion racket, check out Defining Promise: Optional Standardized Testing Policies in American College and University Admissions, or the poster summary of their results.

It is an open question whether it is best to introduce formal mathematical definitions before or after the object to be defined has been explored in a classroom setting. The prevailing Math Education injunction is that formal definitions are to appear after exploration (if at all). I tend to take the opposite view, if for no reason other than that these “explorations” are easily botched, from both a mathematical and pedagogic perspective.

In thinking about the sequence of formal definition presentation, it is useful to distinguish between two ideas. There is a difference between “exploring the mathematical object before formalizing the definition” and “motivating a formal definition before writing it down.”

The first one is no good, because mathematical objects have no empirical reality outside of our definitions of them. You can experience a tree without knowing how biologists classify them. You go up to the damn tree and check it out. On the other hand, mathematical objects are figments of our imagination. They exist only in our heads. You can’t go up to a ratio and check it out, you have to imagine it. Absent a formal definition, there is no way to be sure that the thing you’re imagining in your head and the thing I’m imagining in my head are the same thing. And if what I am imagining and what you are imagining are not the same thing then we cannot communicate. So, were you to say “sort these cards into ratios and not ratios” without first saying what a ratio is (that is, without a definition), that would be an impossible task. Without a definition, you can sort the cards however you want. Without a definition, there is no criteria by which to judge correctness. You relegate mathematics to a matter of taste. And then when a student has the wrong “taste” and is corrected, what you have successfully done is positioned the math teacher as the capricious arbiter of truth, rather than teach the student that they are responsible for figuring out what is true by using their own powers of reason. It’s mathematics by fiat, and it is a good way to make obedient workers, but not a good way to make educated citizens. NOTHING LESS THAN THE FATE OF OUR COUNTRY IS AT STAKE.

Now, motivating a definition, on the other hand, is very desirable! Part of what it means to know something in mathematics is to know why that thing is needed, what its purpose is, how it is connected to other math &etc. Why bother even having something like “slope”? Well, we wanna be able to talk about how “tilted” a line is, and compare how tilted one line is to how tilted a different line is. Doing that sort of thing before a formal definition is probably fine sometimes. Do the motivation before and then the thing can be formally defined with little fanfare. We need to be careful to pull off the mathematics properly, but what else is new.

So for me, it’s not really a question of “do they get the formal definition up front, or not?” It’s more a question about whether we are asking our students to do reasonable things, or if we’re asking them to read our minds. Mathematics is a hard enough game without playing “hide the ball” on top of it.

I’m working on quitting books I don’t enjoy.

This is not so easy.

Spring semester 2018 brought with it a long commute by car. I also have a long list of books to read. Presto chango, audiobooks entered into my life.

I have historically found audiobooks to be “cheating” and not REAL reading, so I avoided them. Yuval Noah Harari’s Sapiens changed that, it was delightful. I loved it so much I listened to Homo Deus too, then loved that so much I bought the real books and am currently almost finished reading reading Sapiens. Pick yourself up a copy if you’re so inclined.

Early on in my audiobook experiment I listened to Nassim Taleb’s Fooled By Randomness. At the beginning of the book he forcefully proclaimed (more or less), “I will not consult a library. I will only speak from my own brain. All these other writers that produce books that are just quote after quote? They suck.”

I found this very strange. What a peculiar pronouncement! You will not consult outside sources? How arrogant! (Taleb’s arrogance is part of his charm, however). But now that I have listened to and quit several nonfiction books that were little more than lit reviews of best seller lists, I appreciate Taleb’s policy. Today I quit Quiet. It’s getting easier. These glorified book reports ALL REFERENCE THE SAME DAMN BOOKS. I don’t understand how a writer can have the wherewithal to sit down and write and not put their own ideas into their work. Hey writers, I’m reading your book to find out what YOU think. Not what everybody else thinks, if I wanna know what they think, I’ll read their books. As Taleb might say, get some skin in the game. Or at the very least do us all a favor and stick your references in the endnotes.

Books I quit reading this year:
Cosmos by Carl Sagan (already know all this stuff Carl)
Life Changing Magic of Not Giving a Fuck by Manson (he lost a fan with this one)
Quiet: The Power of Introverts in a World That Can’t Stop Talking by Susan Cain (I loathed this one so much I think I’m renouncing my introvert card)

Should’ve quit but didn’t:
Why Me Want Eat by Krista Scott Dixon (short, so I finished it)
Awaken the Giant Within by Tony Robbins (also short)
The Willpower Instinct by McGonigal (still kicking myself about not quitting this one)

What books have you quit reading lately?

11.8.3
Prove that the ring $$\mathbb{F}_2[x]/<x^3+x+1>$$ is a field, but that $$\mathbb{F}_3[x]/<x^3+x+1>$$ is not a field.

Solution.
Let $$\varphi:\mathbb{F}_2[x] \to \mathbb{F}_2[x]/<x^3+x+1>$$ be the canonical homo. The kernel of $$\varphi$$ is the ideal $$<x^3+x+1>$$. By a bunch of theorems in this section, we know that a polynomial ring over a field modded out by an ideal is a field if and only if $$<x^3+x+1>$$ is a maximal ideal in $$\mathbb{F}_2[x]$$. That’s a maximal ideal only if $$x^3+x+1$$ is monic (and it is, obvs) and irreducible. Okay so all we have to do is show $$x^3+x+1$$ is irreducible. I cheated and used the computer. It is irreducible. But if you wanna show it’s irreducible without running screaming into the loving arms of Wolfram Alpha, it’s NDB. There’s only a couple polynomials of degree 2 and 1 in that ring, so you just do the long division and see that none of them work. You conclude that $$x^3+x+1$$ is irreducible, therefore $$\mathbb{F}_2[x]/<x^3+x+1>$$ is a field.

For $$\mathbb{F}_3[x]/<x^3+x+1>$$, you do the same thing, but it turns out that $$x^3+x+1=(x+2)(x^2+x+2)$$. So we have a reducible polynomial generating the ideal, so the quotient ring $$\mathbb{F}_2[x]/<x^3+x+1>$$ isn’t a field.

Ta-da!

11.8.1
Which principal ideals in $$\mathbb{Z}[x]$$ are maximal ideals?

Solution.
Let $$I=<f(x)>$$ be a principal ideal in $$\mathbb{Z}[x]$$. We’ll show that $$I$$ is not maximal by constructing an ideal $$M$$ such that $$I \subset M \subset \mathbb{Z}[x]$$.

If $$f(x)=0$$, take $$M=<x>$$ and $$I$$ is not maximal.
If $$f(x)=1$$, then $$I=\mathbb{Z}[x]$$ and $$I$$ is not maximal.
If $$f(x)=c \in \mathbb[Z]$$, take $$M=<c,x>$$ then we have $$<c> \subset<c,x> \subset \mathbb{Z}[x]$$, and $$I$$ is not maximal.

Okay, now if we have $$f(x)=f_0+f_1 x+…+f_n x^n$$ then take $$M=<p, f(x)>$$ where $$p$$ is a prime in $$\mathbb{Z}$$ such that $$p$$ and $$f_i$$ are relatively prime for all $$i$$. Then $$<f(x)> \subset <p,f(x)> \subset \mathbb{Z}[x]$$.

Conclude there are no principal ideals in $$\mathbb{Z}[x]$$ that are maximal.