You're thinking about it in a good way. Consider that you already know the relationship between the price of watermelons and oranges, so you don't need to have two separate variables. 240 = 48 times some price (let's call it x), plus 36 times some other price, which we know is exactly twice the first price, so we can call it 2x. That gives us:
48x + 36(2x) = 240.
48x + 72x = 240.
120x = 240.
x = 2.
So the watermelons are $2 and those are exactly half the cost of a bag of oranges, so the oranges are $4 a bag.
Now let's make sure that answer is correct by plugging in the numbers and seeing if it gives us a true equation.
You know, you're probably right. Most parents these days would probably say something like "Well of course, honey, if you identify as a non-driver, I wouldn't dream of invalidating your identity by actually teaching you to drive."
Yes. And the people who have learned math while simultaneously swallowing the cultural bullshit are incentivized to keep it going so they can feel "smart," instead of admitting that anyone of average intelligence with enough persistence and a decent teacher can learn and understand mathematics, including calculus, complex statistics, and at least basic number theory.
I like your argument, but — how far does this kind of argument extend? For example: Should everyone be able – at least to some degree – to see the physical universe as known by physics? Here's a possible test for threshold physics-think actualization: I hand you a cheap, simple (by 2023 standards) telescope, and ask you to show me a sequence of phenomena with it that make the universality of classical gravitation at least plausible.
I'm not seeing the parallel. Can you point out to me some times in ordinary life (like splitting a dinner check or one of the many other examples I might have used -- verifying that a discount was properly applied, evaluating which cell phone plan is cheaper, comparing insurance rates that are based on different formulas, calculating how much carpet is needed in a room, etc.) that knowledge of basic physics is called on but ignorance is excused, praised, or applauded because it's cool to be "not a physics person" ?
If I've misunderstood how you scope "ordinary life" – sorry.
But perhaps —
Ordinary life might include being able to reason about the safety of one's freeway driving first-person-physically. Or, ordinary life might include being able to assess an authority claim in light of one's first-person experience of substantiating a sector of exact science.
I'm still not sure I understand, sorry. I took a semester of physics, and aside from being able to use some of the concepts and formulas to answer questions about driving, I'm not understanding what you mean. What part of the basic physics I studied should be informing my driving decisions?
Aug 26, 2023·edited Aug 26, 2023Liked by Holly MathNerd
Unless new research has come up in the 35 years since I was in college demonstrating otherwise, the part of your brain responsible for language is the same part of your brain responsible for math. IOW, if a person can speak a language as complicated as English, there's zero reason they can't be competent at least through high school algebra.
When I was a cop, I used to work a security job at a high school in Houston that was about 95% Latino. I knew a lot of the kids by sight and name, and talked to them quite a lot. "I'm not a math person" was a common refrain, even 30 years ago.
"But you speak English AND Spanish fluently?!?!?"
"Yeah, so?"
"Math is just another language. You learned English and Spanish, now learn the language of numbers."
No idea how many, if any at all, took that to heart. I am hopeful that some did. "How many licks to get to the center of a Tootsie Pop?" The world may never know.
IMO, the 2 most important mathematical concepts to learn are modus ponens and why affirming the consequent is a logical fallacy that only looks like modus ponens. While these concepts are only formally presented in a course on logic, they are woven throughout all of mathematics education beyond simple arithmetic.
a -> b
a
∴ b
I am all for a formal course in propositional logic being taught as early as 8th or 9th grade.
This is going to be harder to teach than ever, the more institutions lie to us. We are *constantly* presented with statements in the form of a -> b that are bullshit. If Trump becomes POTUS, he'll become a dictator who starts World War 3 and throws all LGBT folks in concentration camps. If everyone gets vaccinated, COVID will go away. I wouldn't blame any kid who found this an exceedingly difficult concept to grasp, especially after the last few years.
Aug 26, 2023·edited Aug 26, 2023Liked by Holly MathNerd
No doubt. Any useful course on formal logic also must address the utility of assumptions and provisional assumptions. IOW, judgment must also be taught.
I fear you are correct, though. We can't have people exercising independent thought/judgment.
If one derives a contradiction in one's formal argument, one must go back and re-evaluate one's assumptions, because at least one of them is wrong.
A non-trivial part of formal logic, and why I think it's so important it be taught, is the difference between validity and soundness. A sound argument is necessarily valid, but validity doesn't automatically confer soundness. Valid + unsound = floating abstractions.
I have plenty of experience with students who use this very excuse, but I find that with a little explanation, they get it. If I had a nickel for every time I heard, "Why can't my teacher explain it like this?"
Poor instruction is endemic.
Another issue is emotional. Actually I think that emotional issues impede academic performance more than any other cause. Those are rough to deal with effectively.
And, it drives me nuts when I hear students say, "I'm a visual learner." Or, "I'm an auditory learner." Or something equally inane. And I wonder what idiot teacher or counsellor told them that and caused an ossification of their identity. I have to continually tell these students not to put themselves in a box, that all of that crap is on a continuum and therefore is crap, and that they can learn in multiple ways. I mix up letters and numbers all the time, but I don't consider myself lysdexic. But of course, when I was a kid, that wasn't a thing.
I had a professor in community college who said that the secret to success in math is persistence. I think that's true of life generally.
Excellent piece, Holly. I tried making this point many, many times to colleagues at several schools I've taught at over the years. I've had some success, but I've never really been able to convince people that so-called "non-math people" are confessing to a form of illiteracy, and not with some degree of embarrassment and desire to not be illiterate, but with a sort of "well, them's the breaks" kind of attitude.
Interestingly, I've only really encountered opposition within a math department at a community college (which says something about higher ed). It's a somewhat long story, but the upshot is that I encountered an Emeritus Professor who advocated for turning a significant portion of basic college math classes into instruction for how to interface with technology rather than on learning and practicing essential mathematical skills (ChatGPT was fairly new at the time, and he wasn't familiar with it). However, it did lead me to try and steel-man his position and I'm curious what you think of it, and how you'd respond:
Maybe knowledge of mathematics is passing out of the required skill set for everyday life. We've seen other bodies of knowledge that used to be common become relegated to a smaller subset of specialists as technology has advanced (an example might be animal husbandry or the use of farm tools). Perhaps that's what happening to math, as large language models continue to improve and interfacing with computers becomes more and more seamless, is mathematics going to be important to leading a productive, flourishing life in tomorrow's technological society?
I think you're answer (and mine) is summarized in your statement that: "...the logic and discipline imposed by mathematics makes a person better at critical thinking."
I think this is going to be a point that we (those who argue for the value of mathematics in education) will need to continue to flesh out and argue very strongly in the coming years because we are going to continue to see people being comfortable, even proud of their innumeracy. And with the tools for doing math problems without thought becoming more and more accessible what remains of the shame and embarrassment is going to continue to wither away.
ChatGPT told me that 4 was prime less than a month ago. I'm not remotely impressed with its ability to do math. I asked it for a solution to the Twin Primes Conjecture and instead of a summary of the recent work it confidently spit out bullshit. We're a long way away from it being reliable. But if/when we eventually get there, people are still going to need to know math well enough to spot errors, from the merely financial to the life threatening. A computer can't be held responsible for a .1 error that causes a medication dose that turns out to be 10x too high and kill someone. A human has to. That human has to know math. This same paradigm will apply to a lot of things.
Besides, the Chinese are already kicking our asses. Their kids do algebra much earlier. Hell, the rest of the world is kicking our asses. We really can't afford to set our standards and expectations any lower than we already have.
So, I'm not very familiar with ChatGPT and the like, but from what I understand, the likelihood that they will spout out BS goes up rapidly the more obscure the knowledge they are asked about. But I don't think this is inherent to the tool. I think these things will continue to improve and probably quite rapidly. Maybe this limitation is never going to get worked out of the technology, but let's give it the benefit of the doubt and say that fairly soon LLMs will be a highly reliable technology that can generate answers to mathematical questions from abstract verbal prompts, images, etc. I would still argue that this doesn't negate the value of math education for exactly the reasons you indicate: getting to the right answer of hard questions requires mental discipline and rigorous adherence to rules of logic and reasoning. Being handed the answer is like me having a machine launch a baseball into the left field bleachers and saying, "Wow, I'm as good as a major-league hitter!" No matter how good the tools get at giving answers they are not going to replace the real, individual thinking a person will need to do the flourish in their life (and it's a dystopian future for those who allow the tools to replace their own thinking). I think this is the thing case we're going to need to make: math education isn't just about getting the correct answers to math problems, it's about training your mind to operate in a specific way that has application far beyond the scope of math.
LLM stands for "large language model." It's not *thinking*. It's running sophisticated algorithms to guess what to say based on its training data, which is the internet. It will get better, but I don't think it's going to get a *lot* better with math, because deep mathematical insight isn't on the internet in the way that literary analysis is. It's not going to get good at writing proofs for the same reason it's already good at writing poetry and episodes of Star Trek.
Hehe, this kind of makes me want to go see all the different ways I can break it. I think we're largely seeing the same thing here: the tool will be better at providing "answers" if there's a large body of similar "problems" for it to look at. It starts to break when you ask it questions without a large amount of information to draw on (but from what I hear this doesn't stop it from lying its ass off complete with fake references). Forget about asking it to generate new knowledge.
What I was getting at is that I think soon, if not already, it will be able to handle most of the mathematical tasks that "non-math people" are likely to give it (dividing checks, calculating interest, determining rates of consumption, etc). I think you're arguing that it won't be all the tasks an average person could ask, which is fair. I'm making a slightly different case that even this new could do all of those things perfectly, this wouldn't be an argument that education in mathematics was no longer needed for most people. Would you agree?
Of course, but I don't agree with the premise. Calculator apps can do arithmetic perfectly, but the person who puts the problem in still has to understand math well enough to put problems in correctly. It is extremely dangerous to expect technology to get to the point where pharmacists and nurses don't have to be expected to understand math, because they're the ones who will be plugging in patient weights and whatever else dosages are based on. The premise implies that a state of affairs is possible and likely to occur where it would be reasonable to allow our level of mathematical illiteracy to get even worse, AND to allow ordinary citizens working non-tech-god jobs to become just as mathematically illiterate as everyone else under the new, lower standards. I reject that premise. That state of affairs is by definition impossible because a human being must take responsibility and that human being must understand math well enough to do so.
Is the number between any twin primes ever a power of two?
Its answer was gobblegook that included the line "the only even power of 2 is 2 itself."
Proving that the number that falls between twin primes is never a power of 2 took me less than an hour, writing it out by hand, slowly and with my usual lack of confidence. It would take most people who are better at number theory than I am about ten minutes, or less. Chat GPT cannot do it at all.
Aug 28, 2023·edited Aug 28, 2023Liked by Holly MathNerd
That is very interesting, and yeah it's spouting bullshit. I think I may take on a side project in which I put all of my homework questions from my Algebra and Physics classes into and see what it gives me. My sense is that it will do better than what it gave you because the kinds of questions I'll be throwing at it have been solved thousands of times on the internet, but I am really curious now.
Let me make the issue I'm struggling with a little more concrete, just to see what your take on it is. Sorry for the long story, but it was a session in a community college PD conference given by a veteran professor and it made me despair. The guy had been teaching math since the 80s, so he oversaw the introduction of graphing calculators into his calculus classes. He was one of the people pushing for it and a lot of not-very-nice things to say about the nay-sayers. For his session, he was talking about how he has used the Wolfram Alpha software in the a college algebra class that is typically the last class that the non-business, and non-stem majors take (nursing students in particular take this class). He had made a significant portion of his class about how to interface with Wolfram Alpha. My sense was around half of his course was going over the algebraic methods (quadratic equations, rational expressions, exponentials, logarithms, and systems of linear equations are the big categories of algebraic equations for this class), and half was learning how to make Wolfram Alpha solve the problems.
I teach a similar class in which I have them use Wolfram Alpha for exactly one topic: systems of equations with more than two variables (only because course objective explicitly states that I have to use technology for this topic). I was more than a little surprised by their approach, because while there's nothing wrong with knowing how to use a tool like Wolfram Alpha, I was amazed that he would devote so much time to its use, especially when those algebraic topics do require both clear explanations of the process and a lot of practice. Wolfram Alpha sort of short circuits the practice in my view. What kept going through my mind was, "Just because the tool can do it for them, doesn't mean they don't need to know how to do it."
He had a hand-wavy explanation saying that there was no evidence that mastering mathematical algorithms led to success in other areas, whatever that meant. This was about a month after I had first heard of ChatGPT, and I was thinking, "soon enough you'll just be able to take a picture of the question and tell ChatGPT to give you the answer." Someone brought this up saying she had switched from using Wolfram Alpha to ChatGPT for all of the things he had brought up in his talk.
To sum up, it wasn't clear to me after listening to his talk what these two people thought the purpose teaching math to these majors was. My sense is that it they viewed the purpose of the math course as being able to get the right answers to the questions in the textbook. Therefore it was a completely acceptable change to the math education to take a significant amount of time to teach students how to use programs that make it easier to get the answers. The problem is that I think very soon the tool is not going to require very much instruction, so what is the point of math education then? They can easily get all the right answers from the textbook. So do they all pass?
I found this whole experience very unsettling because the thrust seemed to be going in the direction of "oh, just let the tool solve the problem," which it can probably do because it's from a widely available textbook. If people are taught that this is math, then we've got a huge problem because they've never been required to think mathematically (at least in the college math class). My visceral response was, "I don't care if the damn program can solve all the problems perfectly, that's not why we teach them math!" Which is the sentiment I've been trying to express here.
What do you think? Am I Chicken Little? Do you think that the products of these "math" classes are going to get filtered out in other ways? Am I overestimating what LLMs are capable of? Do you think the view I saw here is one that's going to gain traction or are people going reject it?
I immediately thought of baking, the attitude of "I'm just not a ____ person." Many, many years ago a bunch of us in my office job were being trained to use a new program. Lots of people were confused, as people don't handle change will. The instructor, in an attempt to assuage insecurity, asked, kindly:
"What happens when you're baking and you follow a recipe?"
"It turns out bad," one woman humbly responded.
Cute line, which got a laugh from the room, but inaccurate. (I had an internal "actually" moment, even though I was years from pastry school.) When you follow instructions it comes out as desired because baking is first and foremost math and science. Understanding the how and why is necessary to perfect or replicate a recipe, and as you say, not easy. A couple years ago I translated into painstaking detail a souffle recipe for a nonbaker friend--in a wheelchair--and I'm proud of both of us that it mostly turned out good. (Laziness kept it from perfection.) But writing it was a slog. You need the nuts and bolts, THEN you can be creative and toss flour in the air like pixie dust. Now, bad recipes exist; I am skeptical of home baker bloggers without professional education and industry experience. Granting that the author is competent the laws of physics and chemistry are consistent. So when "it turns out bad" happens it's likely user error. One of the best things about immutable rules is that you learn to troubleshoot. The rules require that there is solution, even if you haven't yet found it. Applied knowledge is produced.
The skills you say come from learning a hard skill like math or science translate anywhere, even on a broad, personal level. The concept of mis-en-place made me a better person. Mis-en-place is literally "put in place", meaning have everything ready when production begins. No searching for things (or reading the recipe!!) *as you go*. The recipe has been read all the way through at least once. All ingredients, tools, hardware are out on the table or counter when you start. "Oops, I forgot one thing" does not happen. (In pastry school our chefs taught us that if you screw up in culinary you can still fix, alter, or re-purpose. If you screw up in baking you start over from square one.) I've never been a total slob but for sure an eager, easily distracted scatterbrain. Baking made me a much better organizer. I tolerate far less mess in my living spaces. I've slowly become more structured, though still far from where I want to be. But I am measurably an improved, less inefficient person.
You can have a growth mindset or a fixed mindset. With a fixed mindset you think things like, "I'm not a math person" (and will never be one because that 'fact' cannot change). With a growth mindset you think things like, "This is hard, but with enough study and asking for help I can figure it out."
I think some people may have growth mindsets on all subjects, but some others have a mix depending on the subject.
For some reason society has decided that we are "fixed" on Math skills and can be a math person or not. I never viewed people who were not good at math as "stupid" at it. I knew they could figure it out with time, patience, and study. But so many did not seem to want to do it. I enjoyed it, because I thought of them as puzzles, like Sudoku. To me it was a game. With enjoyment it wasn't difficult to do the work.
But that is not to say it wasn't challenging. I remember in college Calculus we were give two class periods to get an assignment done. After class on Monday, I would get to working on all of the problems (that would be due in 2 classes from then, Friday). I would go into the next class, Wednesday, with my problems that I couldn't figure out. The teacher would ask if anyone had any problems. So few people raised their hands. You could tell people were not working on their homework yet (procrastination). Because I worked ahead I asked questions on all of my tough areas. It was sometimes like a private tutoring session since no one else had issues they wanted help on. I would wait and see, then raise my hand if no one else was. We were all given the opportunity to get help but so few chose to take up the offer.
Thank you so much -- in one essay you hit on two of my pedagogic pet peeves!
When I was in my 20s (so, grad school into my first professional job) you heard this saying all the time: “I’m not a numbers person -- I’m a *people* person.” It managed to turn what should have been a rueful or embarrassing admission about one’s substandard education or lack of any drive for self-improvement not just to a neutral but a *positive* statement: I’m not like those soulless number-crunchers and I’m proud of it! And on at least one occasion I snapped and spit out exactly as you did: “If you couldn’t read you wouldn’t announce to the whole world that you were a ‘people person, not a words person’.”
The other peeve is the “you’re so talented...I don’t have any of [that kind of] talent.” My daughter, now 29, is a graphic artist and designer. She often sits in public places sketching, and invariably someone approaches, looks at her work, and says some variant of “you have such talent” in the same way they would comment on the color of her eyes or the blouse she was wearing. She is too polite to ever say anything less mild than “Oh, do you like it?” But it inwardly enrages me, because that kid from the age of eight had a pencil in her hand every waking moment she wasn’t being forced to do something else. She *still* spends time every day practicing new techniques in addition to the 8-10 hours she is doing commercial work. The way she became so talented was she worked at it in the same way a professional athlete works at speed, agility, and strength.
I've drawn Bret and Heather a couple of times. That one was the first. The best one is in their possession, having been given to them for Christmas a few years ago. Did you recognize James Lindsay in the second one?
I was wondering. I knew it wasnt Josh! That one's tougher because he has the cup covering his mouth and his lips/mouth are so distinctive. I don't recognize him as much by the eyes, though I suspect you dial in to the eyes alot. You have great friends. I appreciate the opportunities to plug in with smart, creative, and thoughtful people who are (along with you) moving the needle on the most important and existential issues of the day (and also helping see the colors, nuance, and value informed by the more "mundane" day-to-day).
In a softball quiz at the start of last year's fall semester, students were asked to subtract eight from negative six, recalls Jessica Babcock. “I graded a whole bunch of papers in a row. No two papers had the same answer, and none of them were correct.”
Thank You so very much for the "Excuses vs Explanations" bit, mega Thanks!
. . . my personal note, there are people, not as extreme as "Rainman" who poses unique abilities in varying degrees, I have been blessed with special powers over hardware ( it was said that I could repair Apple][ computers by the laying on of hands . . ) and in all too many other areas, serious deficiencies.
I agree with everything you wrote. Especially the bit about teachable skills, and the difference between explanations and excuses.
But the funny part is, that I actually was that person who didn't want to learn to drive in high school. In my defense, I didn't make the excuse that I wasn't a "driving person." I just didn't like driving. I still don't. I took driver's ed in high school because I had to, got my driver's license, and let it lapse. In the mean time, I actually did travel a lot, including taking trains all over Germany, and studying in the then USSR, without being able to drive.
Then, when I got my first assignment as an officer in the Air Force, they told me I had to be able to drive, so I refreshed my driver's ed and got my license.
You're thinking about it in a good way. Consider that you already know the relationship between the price of watermelons and oranges, so you don't need to have two separate variables. 240 = 48 times some price (let's call it x), plus 36 times some other price, which we know is exactly twice the first price, so we can call it 2x. That gives us:
48x + 36(2x) = 240.
48x + 72x = 240.
120x = 240.
x = 2.
So the watermelons are $2 and those are exactly half the cost of a bag of oranges, so the oranges are $4 a bag.
Now let's make sure that answer is correct by plugging in the numbers and seeing if it gives us a true equation.
(48 * $2) + (36 * $4) = $240
$96 + $144 = $240
Correct!
"This imagined parental response probably sounds reasonable to you."
What scares me is how many--perhaps the majority-- parents do not perceive that as reasonable.
You know, you're probably right. Most parents these days would probably say something like "Well of course, honey, if you identify as a non-driver, I wouldn't dream of invalidating your identity by actually teaching you to drive."
You're so, so correct. The analogy to what it would be like if people bragged about verbal illiteracy is exactly, 100 percent, the same thing.
It's not an "analogy". It is that very thing itself.
Yes. And the people who have learned math while simultaneously swallowing the cultural bullshit are incentivized to keep it going so they can feel "smart," instead of admitting that anyone of average intelligence with enough persistence and a decent teacher can learn and understand mathematics, including calculus, complex statistics, and at least basic number theory.
I like your argument, but — how far does this kind of argument extend? For example: Should everyone be able – at least to some degree – to see the physical universe as known by physics? Here's a possible test for threshold physics-think actualization: I hand you a cheap, simple (by 2023 standards) telescope, and ask you to show me a sequence of phenomena with it that make the universality of classical gravitation at least plausible.
I'm not seeing the parallel. Can you point out to me some times in ordinary life (like splitting a dinner check or one of the many other examples I might have used -- verifying that a discount was properly applied, evaluating which cell phone plan is cheaper, comparing insurance rates that are based on different formulas, calculating how much carpet is needed in a room, etc.) that knowledge of basic physics is called on but ignorance is excused, praised, or applauded because it's cool to be "not a physics person" ?
If I've misunderstood how you scope "ordinary life" – sorry.
But perhaps —
Ordinary life might include being able to reason about the safety of one's freeway driving first-person-physically. Or, ordinary life might include being able to assess an authority claim in light of one's first-person experience of substantiating a sector of exact science.
I'm still not sure I understand, sorry. I took a semester of physics, and aside from being able to use some of the concepts and formulas to answer questions about driving, I'm not understanding what you mean. What part of the basic physics I studied should be informing my driving decisions?
Unless new research has come up in the 35 years since I was in college demonstrating otherwise, the part of your brain responsible for language is the same part of your brain responsible for math. IOW, if a person can speak a language as complicated as English, there's zero reason they can't be competent at least through high school algebra.
When I was a cop, I used to work a security job at a high school in Houston that was about 95% Latino. I knew a lot of the kids by sight and name, and talked to them quite a lot. "I'm not a math person" was a common refrain, even 30 years ago.
"But you speak English AND Spanish fluently?!?!?"
"Yeah, so?"
"Math is just another language. You learned English and Spanish, now learn the language of numbers."
No idea how many, if any at all, took that to heart. I am hopeful that some did. "How many licks to get to the center of a Tootsie Pop?" The world may never know.
Exactly. It's a language, and moreso the higher one goes with it, which is why common core is such a tragedy.
IMO, the 2 most important mathematical concepts to learn are modus ponens and why affirming the consequent is a logical fallacy that only looks like modus ponens. While these concepts are only formally presented in a course on logic, they are woven throughout all of mathematics education beyond simple arithmetic.
a -> b
a
∴ b
I am all for a formal course in propositional logic being taught as early as 8th or 9th grade.
This is going to be harder to teach than ever, the more institutions lie to us. We are *constantly* presented with statements in the form of a -> b that are bullshit. If Trump becomes POTUS, he'll become a dictator who starts World War 3 and throws all LGBT folks in concentration camps. If everyone gets vaccinated, COVID will go away. I wouldn't blame any kid who found this an exceedingly difficult concept to grasp, especially after the last few years.
No doubt. Any useful course on formal logic also must address the utility of assumptions and provisional assumptions. IOW, judgment must also be taught.
I fear you are correct, though. We can't have people exercising independent thought/judgment.
If one derives a contradiction in one's formal argument, one must go back and re-evaluate one's assumptions, because at least one of them is wrong.
A non-trivial part of formal logic, and why I think it's so important it be taught, is the difference between validity and soundness. A sound argument is necessarily valid, but validity doesn't automatically confer soundness. Valid + unsound = floating abstractions.
Great article, Holly. All true.
I have plenty of experience with students who use this very excuse, but I find that with a little explanation, they get it. If I had a nickel for every time I heard, "Why can't my teacher explain it like this?"
Poor instruction is endemic.
Another issue is emotional. Actually I think that emotional issues impede academic performance more than any other cause. Those are rough to deal with effectively.
And, it drives me nuts when I hear students say, "I'm a visual learner." Or, "I'm an auditory learner." Or something equally inane. And I wonder what idiot teacher or counsellor told them that and caused an ossification of their identity. I have to continually tell these students not to put themselves in a box, that all of that crap is on a continuum and therefore is crap, and that they can learn in multiple ways. I mix up letters and numbers all the time, but I don't consider myself lysdexic. But of course, when I was a kid, that wasn't a thing.
I had a professor in community college who said that the secret to success in math is persistence. I think that's true of life generally.
Excellent piece, Holly. I tried making this point many, many times to colleagues at several schools I've taught at over the years. I've had some success, but I've never really been able to convince people that so-called "non-math people" are confessing to a form of illiteracy, and not with some degree of embarrassment and desire to not be illiterate, but with a sort of "well, them's the breaks" kind of attitude.
Interestingly, I've only really encountered opposition within a math department at a community college (which says something about higher ed). It's a somewhat long story, but the upshot is that I encountered an Emeritus Professor who advocated for turning a significant portion of basic college math classes into instruction for how to interface with technology rather than on learning and practicing essential mathematical skills (ChatGPT was fairly new at the time, and he wasn't familiar with it). However, it did lead me to try and steel-man his position and I'm curious what you think of it, and how you'd respond:
Maybe knowledge of mathematics is passing out of the required skill set for everyday life. We've seen other bodies of knowledge that used to be common become relegated to a smaller subset of specialists as technology has advanced (an example might be animal husbandry or the use of farm tools). Perhaps that's what happening to math, as large language models continue to improve and interfacing with computers becomes more and more seamless, is mathematics going to be important to leading a productive, flourishing life in tomorrow's technological society?
I think you're answer (and mine) is summarized in your statement that: "...the logic and discipline imposed by mathematics makes a person better at critical thinking."
I think this is going to be a point that we (those who argue for the value of mathematics in education) will need to continue to flesh out and argue very strongly in the coming years because we are going to continue to see people being comfortable, even proud of their innumeracy. And with the tools for doing math problems without thought becoming more and more accessible what remains of the shame and embarrassment is going to continue to wither away.
ChatGPT told me that 4 was prime less than a month ago. I'm not remotely impressed with its ability to do math. I asked it for a solution to the Twin Primes Conjecture and instead of a summary of the recent work it confidently spit out bullshit. We're a long way away from it being reliable. But if/when we eventually get there, people are still going to need to know math well enough to spot errors, from the merely financial to the life threatening. A computer can't be held responsible for a .1 error that causes a medication dose that turns out to be 10x too high and kill someone. A human has to. That human has to know math. This same paradigm will apply to a lot of things.
Besides, the Chinese are already kicking our asses. Their kids do algebra much earlier. Hell, the rest of the world is kicking our asses. We really can't afford to set our standards and expectations any lower than we already have.
So, I'm not very familiar with ChatGPT and the like, but from what I understand, the likelihood that they will spout out BS goes up rapidly the more obscure the knowledge they are asked about. But I don't think this is inherent to the tool. I think these things will continue to improve and probably quite rapidly. Maybe this limitation is never going to get worked out of the technology, but let's give it the benefit of the doubt and say that fairly soon LLMs will be a highly reliable technology that can generate answers to mathematical questions from abstract verbal prompts, images, etc. I would still argue that this doesn't negate the value of math education for exactly the reasons you indicate: getting to the right answer of hard questions requires mental discipline and rigorous adherence to rules of logic and reasoning. Being handed the answer is like me having a machine launch a baseball into the left field bleachers and saying, "Wow, I'm as good as a major-league hitter!" No matter how good the tools get at giving answers they are not going to replace the real, individual thinking a person will need to do the flourish in their life (and it's a dystopian future for those who allow the tools to replace their own thinking). I think this is the thing case we're going to need to make: math education isn't just about getting the correct answers to math problems, it's about training your mind to operate in a specific way that has application far beyond the scope of math.
LLM stands for "large language model." It's not *thinking*. It's running sophisticated algorithms to guess what to say based on its training data, which is the internet. It will get better, but I don't think it's going to get a *lot* better with math, because deep mathematical insight isn't on the internet in the way that literary analysis is. It's not going to get good at writing proofs for the same reason it's already good at writing poetry and episodes of Star Trek.
Hehe, this kind of makes me want to go see all the different ways I can break it. I think we're largely seeing the same thing here: the tool will be better at providing "answers" if there's a large body of similar "problems" for it to look at. It starts to break when you ask it questions without a large amount of information to draw on (but from what I hear this doesn't stop it from lying its ass off complete with fake references). Forget about asking it to generate new knowledge.
What I was getting at is that I think soon, if not already, it will be able to handle most of the mathematical tasks that "non-math people" are likely to give it (dividing checks, calculating interest, determining rates of consumption, etc). I think you're arguing that it won't be all the tasks an average person could ask, which is fair. I'm making a slightly different case that even this new could do all of those things perfectly, this wouldn't be an argument that education in mathematics was no longer needed for most people. Would you agree?
Of course, but I don't agree with the premise. Calculator apps can do arithmetic perfectly, but the person who puts the problem in still has to understand math well enough to put problems in correctly. It is extremely dangerous to expect technology to get to the point where pharmacists and nurses don't have to be expected to understand math, because they're the ones who will be plugging in patient weights and whatever else dosages are based on. The premise implies that a state of affairs is possible and likely to occur where it would be reasonable to allow our level of mathematical illiteracy to get even worse, AND to allow ordinary citizens working non-tech-god jobs to become just as mathematically illiterate as everyone else under the new, lower standards. I reject that premise. That state of affairs is by definition impossible because a human being must take responsibility and that human being must understand math well enough to do so.
This is how dumb it is:
Just now I asked:
Is the number between any twin primes ever a power of two?
Its answer was gobblegook that included the line "the only even power of 2 is 2 itself."
Proving that the number that falls between twin primes is never a power of 2 took me less than an hour, writing it out by hand, slowly and with my usual lack of confidence. It would take most people who are better at number theory than I am about ten minutes, or less. Chat GPT cannot do it at all.
That is very interesting, and yeah it's spouting bullshit. I think I may take on a side project in which I put all of my homework questions from my Algebra and Physics classes into and see what it gives me. My sense is that it will do better than what it gave you because the kinds of questions I'll be throwing at it have been solved thousands of times on the internet, but I am really curious now.
Let me make the issue I'm struggling with a little more concrete, just to see what your take on it is. Sorry for the long story, but it was a session in a community college PD conference given by a veteran professor and it made me despair. The guy had been teaching math since the 80s, so he oversaw the introduction of graphing calculators into his calculus classes. He was one of the people pushing for it and a lot of not-very-nice things to say about the nay-sayers. For his session, he was talking about how he has used the Wolfram Alpha software in the a college algebra class that is typically the last class that the non-business, and non-stem majors take (nursing students in particular take this class). He had made a significant portion of his class about how to interface with Wolfram Alpha. My sense was around half of his course was going over the algebraic methods (quadratic equations, rational expressions, exponentials, logarithms, and systems of linear equations are the big categories of algebraic equations for this class), and half was learning how to make Wolfram Alpha solve the problems.
I teach a similar class in which I have them use Wolfram Alpha for exactly one topic: systems of equations with more than two variables (only because course objective explicitly states that I have to use technology for this topic). I was more than a little surprised by their approach, because while there's nothing wrong with knowing how to use a tool like Wolfram Alpha, I was amazed that he would devote so much time to its use, especially when those algebraic topics do require both clear explanations of the process and a lot of practice. Wolfram Alpha sort of short circuits the practice in my view. What kept going through my mind was, "Just because the tool can do it for them, doesn't mean they don't need to know how to do it."
He had a hand-wavy explanation saying that there was no evidence that mastering mathematical algorithms led to success in other areas, whatever that meant. This was about a month after I had first heard of ChatGPT, and I was thinking, "soon enough you'll just be able to take a picture of the question and tell ChatGPT to give you the answer." Someone brought this up saying she had switched from using Wolfram Alpha to ChatGPT for all of the things he had brought up in his talk.
To sum up, it wasn't clear to me after listening to his talk what these two people thought the purpose teaching math to these majors was. My sense is that it they viewed the purpose of the math course as being able to get the right answers to the questions in the textbook. Therefore it was a completely acceptable change to the math education to take a significant amount of time to teach students how to use programs that make it easier to get the answers. The problem is that I think very soon the tool is not going to require very much instruction, so what is the point of math education then? They can easily get all the right answers from the textbook. So do they all pass?
I found this whole experience very unsettling because the thrust seemed to be going in the direction of "oh, just let the tool solve the problem," which it can probably do because it's from a widely available textbook. If people are taught that this is math, then we've got a huge problem because they've never been required to think mathematically (at least in the college math class). My visceral response was, "I don't care if the damn program can solve all the problems perfectly, that's not why we teach them math!" Which is the sentiment I've been trying to express here.
What do you think? Am I Chicken Little? Do you think that the products of these "math" classes are going to get filtered out in other ways? Am I overestimating what LLMs are capable of? Do you think the view I saw here is one that's going to gain traction or are people going reject it?
Whew...sorry for the ramble.
I immediately thought of baking, the attitude of "I'm just not a ____ person." Many, many years ago a bunch of us in my office job were being trained to use a new program. Lots of people were confused, as people don't handle change will. The instructor, in an attempt to assuage insecurity, asked, kindly:
"What happens when you're baking and you follow a recipe?"
"It turns out bad," one woman humbly responded.
Cute line, which got a laugh from the room, but inaccurate. (I had an internal "actually" moment, even though I was years from pastry school.) When you follow instructions it comes out as desired because baking is first and foremost math and science. Understanding the how and why is necessary to perfect or replicate a recipe, and as you say, not easy. A couple years ago I translated into painstaking detail a souffle recipe for a nonbaker friend--in a wheelchair--and I'm proud of both of us that it mostly turned out good. (Laziness kept it from perfection.) But writing it was a slog. You need the nuts and bolts, THEN you can be creative and toss flour in the air like pixie dust. Now, bad recipes exist; I am skeptical of home baker bloggers without professional education and industry experience. Granting that the author is competent the laws of physics and chemistry are consistent. So when "it turns out bad" happens it's likely user error. One of the best things about immutable rules is that you learn to troubleshoot. The rules require that there is solution, even if you haven't yet found it. Applied knowledge is produced.
The skills you say come from learning a hard skill like math or science translate anywhere, even on a broad, personal level. The concept of mis-en-place made me a better person. Mis-en-place is literally "put in place", meaning have everything ready when production begins. No searching for things (or reading the recipe!!) *as you go*. The recipe has been read all the way through at least once. All ingredients, tools, hardware are out on the table or counter when you start. "Oops, I forgot one thing" does not happen. (In pastry school our chefs taught us that if you screw up in culinary you can still fix, alter, or re-purpose. If you screw up in baking you start over from square one.) I've never been a total slob but for sure an eager, easily distracted scatterbrain. Baking made me a much better organizer. I tolerate far less mess in my living spaces. I've slowly become more structured, though still far from where I want to be. But I am measurably an improved, less inefficient person.
You can have a growth mindset or a fixed mindset. With a fixed mindset you think things like, "I'm not a math person" (and will never be one because that 'fact' cannot change). With a growth mindset you think things like, "This is hard, but with enough study and asking for help I can figure it out."
I think some people may have growth mindsets on all subjects, but some others have a mix depending on the subject.
For some reason society has decided that we are "fixed" on Math skills and can be a math person or not. I never viewed people who were not good at math as "stupid" at it. I knew they could figure it out with time, patience, and study. But so many did not seem to want to do it. I enjoyed it, because I thought of them as puzzles, like Sudoku. To me it was a game. With enjoyment it wasn't difficult to do the work.
But that is not to say it wasn't challenging. I remember in college Calculus we were give two class periods to get an assignment done. After class on Monday, I would get to working on all of the problems (that would be due in 2 classes from then, Friday). I would go into the next class, Wednesday, with my problems that I couldn't figure out. The teacher would ask if anyone had any problems. So few people raised their hands. You could tell people were not working on their homework yet (procrastination). Because I worked ahead I asked questions on all of my tough areas. It was sometimes like a private tutoring session since no one else had issues they wanted help on. I would wait and see, then raise my hand if no one else was. We were all given the opportunity to get help but so few chose to take up the offer.
Thank you so much -- in one essay you hit on two of my pedagogic pet peeves!
When I was in my 20s (so, grad school into my first professional job) you heard this saying all the time: “I’m not a numbers person -- I’m a *people* person.” It managed to turn what should have been a rueful or embarrassing admission about one’s substandard education or lack of any drive for self-improvement not just to a neutral but a *positive* statement: I’m not like those soulless number-crunchers and I’m proud of it! And on at least one occasion I snapped and spit out exactly as you did: “If you couldn’t read you wouldn’t announce to the whole world that you were a ‘people person, not a words person’.”
The other peeve is the “you’re so talented...I don’t have any of [that kind of] talent.” My daughter, now 29, is a graphic artist and designer. She often sits in public places sketching, and invariably someone approaches, looks at her work, and says some variant of “you have such talent” in the same way they would comment on the color of her eyes or the blouse she was wearing. She is too polite to ever say anything less mild than “Oh, do you like it?” But it inwardly enrages me, because that kid from the age of eight had a pencil in her hand every waking moment she wasn’t being forced to do something else. She *still* spends time every day practicing new techniques in addition to the 8-10 hours she is doing commercial work. The way she became so talented was she worked at it in the same way a professional athlete works at speed, agility, and strength.
Nice Brett and Heather drawing!
I've drawn Bret and Heather a couple of times. That one was the first. The best one is in their possession, having been given to them for Christmas a few years ago. Did you recognize James Lindsay in the second one?
I was wondering. I knew it wasnt Josh! That one's tougher because he has the cup covering his mouth and his lips/mouth are so distinctive. I don't recognize him as much by the eyes, though I suspect you dial in to the eyes alot. You have great friends. I appreciate the opportunities to plug in with smart, creative, and thoughtful people who are (along with you) moving the needle on the most important and existential issues of the day (and also helping see the colors, nuance, and value informed by the more "mundane" day-to-day).
It seems that wuflu plus current school math instruction is failing miserably
https://www.joannejacobs.com/post/math-disaster-in-college-would-be-stem-majors-can-t-add-1-2-1-3
Quote:
In a softball quiz at the start of last year's fall semester, students were asked to subtract eight from negative six, recalls Jessica Babcock. “I graded a whole bunch of papers in a row. No two papers had the same answer, and none of them were correct.”
Thank You so very much for the "Excuses vs Explanations" bit, mega Thanks!
. . . my personal note, there are people, not as extreme as "Rainman" who poses unique abilities in varying degrees, I have been blessed with special powers over hardware ( it was said that I could repair Apple][ computers by the laying on of hands . . ) and in all too many other areas, serious deficiencies.
oops!
I agree with everything you wrote. Especially the bit about teachable skills, and the difference between explanations and excuses.
But the funny part is, that I actually was that person who didn't want to learn to drive in high school. In my defense, I didn't make the excuse that I wasn't a "driving person." I just didn't like driving. I still don't. I took driver's ed in high school because I had to, got my driver's license, and let it lapse. In the mean time, I actually did travel a lot, including taking trains all over Germany, and studying in the then USSR, without being able to drive.
Then, when I got my first assignment as an officer in the Air Force, they told me I had to be able to drive, so I refreshed my driver's ed and got my license.