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==Math Teaching== | ==Math Teaching== | ||
*[https://maa.org/sites/default/files/pdf/devlin/LockhartsLament.pdf "A Mathematician's Lament",] Lockhart. | *[https://maa.org/sites/default/files/pdf/devlin/LockhartsLament.pdf "A Mathematician's Lament",] Lockhart. | ||
| − | ::A musician wakes from a terrible nightmare. In his dream he finds himself in a society where | + | ::A musician wakes from a terrible nightmare. In his dream he finds himself in a society where music education has been made mandatory. “We are helping our students become more competitive in an increasingly sound-filled world.” Educators, school systems, and the state are put in charge of this vital project. Studies are commissioned, committees are formed, and decisions are made— all without the advice or participation of a single working musician or composer. ::Since musicians are known to set down their ideas in the form of sheet music, these curious black dots and lines must constitute the “language of music.” It is imperative that students become fluent in this language if they are to attain any degree of musical competence; indeed, it would be ludicrous to expect a child to sing a song or play an instrument without having a thorough grounding in music notation and theory. Playing and listening to music, let alone composing an original piece, are considered very advanced topics and are generally put off until college, and more often graduate school. |
| − | music education has been made mandatory. “We are helping our students become more | ||
| − | competitive in an increasingly sound-filled world.” Educators, school systems, and the state are | ||
| − | put in charge of this vital project. Studies are commissioned, committees are formed, and | ||
| − | decisions are made— all without the advice or participation of a single working musician or | ||
| − | composer. | ||
| − | ::Since musicians are known to set down their ideas in the form of sheet music, these curious | ||
| − | black dots and lines must constitute the “language of music.” It is imperative that students | ||
| − | become fluent in this language if they are to attain any degree of musical competence; indeed, it | ||
| − | would be ludicrous to expect a child to sing a song or play an instrument without having a | ||
| − | thorough grounding in music notation and theory. Playing and listening to music, let alone | ||
| − | composing an original piece, are considered very advanced topics and are generally put off until | ||
| − | college, and more often graduate school. | ||
| − | ::“In seventh | + | ::“In seventh grade we mostly study colors and applicators.” They showed me a worksheet. On one side were swatches of color with blank spaces next to them. They were told to write in the names. “I like painting,” one of them remarked, “they tell me what to do and I do it. It’s easy!” After class I spoke with the teacher. “So your students don’t actually do any painting?” I asked. “Well, next year they take Pre-Paint-by-Numbers. That prepares them for the main Paint-by-Numbers sequence in high school. So they’ll get to use what they’ve learned here and apply it to real-life painting situations— dipping the brush into paint, wiping it off, stuff like that. |
| − | grade we mostly study colors and applicators.” They showed me a worksheet. On one side were | ||
| − | swatches of color with blank spaces next to them. They were told to write in the names. “I like | ||
| − | painting,” one of them remarked, “they tell me what to do and I do it. It’s easy!” | ||
| − | After class I spoke with the teacher. “So your students don’t actually do any painting?” I | ||
| − | asked. “Well, next year they take Pre-Paint-by-Numbers. That prepares them for the main | ||
| − | Paint-by-Numbers sequence in high school. So they’ll get to use what they’ve learned here and | ||
| − | apply it to real-life painting situations— dipping the brush into paint, wiping it off, stuff like that. | ||
| − | ::If I had to design a mechanism for the express purpose of destroying a child’s natural | + | ::If I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done— I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education. |
| − | curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being | ||
| − | done— I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education. | ||
| − | ::Many a graduate student has come | + | ::Many a graduate student has come to grief when they discover, after a decade of being told they were “good at math,” that in fact they have no real mathematical talent and are just very good at following directions. |
| − | to grief when they discover, after a decade of being told they were “good at math,” that in fact | ||
| − | they have no real mathematical talent and are just very good at following directions. | ||
| − | ::How many people actually use any of this “practical math” they | + | ::How many people actually use any of this “practical math” they supposedly learn in school? Do you think carpenters are out there using trigonometry? How many adults remember how to divide fractions, or solve a quadratic equation? Obviously the current practical training program isn’t working, and for good reason: it is excruciatingly boring, and nobody ever uses it anyway. |
| − | supposedly learn in school? Do you think carpenters are out there | ||
| − | using trigonometry? How many adults remember how to divide | ||
| − | fractions, or solve a quadratic equation? Obviously the current | ||
| − | practical training program isn’t working, and for good reason: it is | ||
| − | excruciatingly boring, and nobody ever uses it anyway. | ||
| − | + | ::dDo you really think kids even want something that is relevant to their daily lives? You think something practical like compound interest is going to get them excited? People enjoy fantasy, and that is just what mathematics can provide— a relief from daily life, an anodyne to the practical workaday world. | |
| − | You think something practical like compound interest is going to get them excited? People | ||
| − | enjoy fantasy, and that is just what mathematics can provide— a relief from daily life, an | ||
| − | anodyne to the practical workaday world. | ||
| − | It may be true that you have to be able to read in order to fill out | + | ::It may be true that you have to be able to read in order to fill out forms at the DMV, but that’s not why we teach children to read. We teach them to read for the higher purpose of allowing them access to beautiful and meaningful ideas. Not only would it be cruel to teach reading in such a way— to force third graders to fill out purchase orders and tax forms— it wouldn’t work! We learn things because they interest us now, not because they might be useful later. But this is exactly what we are asking children to do with math. |
| − | forms at the DMV, but that’s not why we teach children to read. We | ||
| − | teach them to read for the higher purpose of allowing them access to | ||
| − | beautiful and meaningful ideas. Not only would it be cruel to teach | ||
| − | reading in such a way— to force third graders to fill out purchase | ||
| − | orders and tax forms— it wouldn’t work! We learn things because | ||
| − | they interest us now, not because they might be useful later. But this | ||
| − | is exactly what we are asking children to do with math. | ||
| − | We never hear a student | + | ::We never hear a student saying, “I wanted to see if it could make any sense to raise a number to a negative power, and I found that you get a really neat pattern if you choose it to mean the reciprocal.” Instead we have teachers and textbooks presenting the “negative exponent rule” as a fait d’accompli with no mention of the aesthetics behind this choice, or even that it is a choice. |
| − | saying, “I wanted to see if it could make any sense to raise a number to a negative power, and I | ||
| − | found that you get a really neat pattern if you choose it to mean the reciprocal.” Instead we have | ||
| − | teachers and textbooks presenting the “negative exponent rule” as a fait d’accompli with no | ||
| − | mention of the aesthetics behind this choice, or even that it is a choice. | ||
| − | No mathematician in the world would bother making these | + | ::No mathematician in the world would bother making these senseless distinctions: 2 1/2 is a “mixed number,” while 5/2 is an “improper fraction.” They’re equal for crying out loud. They are the same exact numbers, and have the same exact properties. Who uses such words outside of fourth grade? |
| − | senseless distinctions: 2 1/2 is a “mixed number,” while 5/2 is an “improper fraction.” They’re | ||
| − | equal for crying out loud. They are the same exact numbers, and have the same exact properties. | ||
| − | Who uses such words outside of fourth grade? | ||
| − | ::Even if we agree that a basic | + | ::Even if we agree that a basic common vocabulary for mathematics is valuable, this isn’t it. How sad that fifth-graders are taught to say “quadrilateral” instead of “four-sided shape,” but are never given a reason to use words like “conjecture,” and “counterexample.” High school students must learn to use the secant function, ‘sec x,’ as an abbreviation for the reciprocal of the cosine function, ‘1 / cos x,’ (a definition with as much intellectual weight as the decision to use ‘&’ in place of “and.” ) |
| − | common vocabulary for mathematics is valuable, this isn’t it. How sad that fifth-graders are | ||
| − | taught to say “quadrilateral” instead of “four-sided shape,” but are never given a reason to use | ||
| − | words like “conjecture,” and “counterexample.” High school students must learn to use the | ||
| − | secant function, ‘sec x,’ as an abbreviation for the reciprocal of the cosine function, ‘1 / cos x,’ | ||
| − | (a definition with as much intellectual weight as the decision to use ‘&’ in place of “and.” ) | ||
| − | What is happening is the systematic undermining of the student’s intuition. A proof, that is, | + | ::What is happening is the systematic undermining of the student’s intuition. A proof, that is, a mathematical argument, is a work of fiction, a poem. Its goal is to satisfy. A beautiful proof should explain, and it should explain clearly, deeply, and elegantly. A well-written, well-crafted argument should feel like a splash of cool water, and be a beacon of light— it should refresh the spirit and illuminate the mind. And it should be charming. |
| − | a mathematical argument, is a work of fiction, a poem. Its goal is to satisfy. A beautiful proof | ||
| − | should explain, and it should explain clearly, deeply, and elegantly. A well-written, well-crafted | ||
| − | argument should feel like a splash of cool water, and be a beacon of light— it should refresh the | ||
| − | spirit and illuminate the mind. And it should be charming. | ||
| − | CALCULUS. This course will explore the mathematics of motion, and the best ways to bury it | + | ::CALCULUS. This course will explore the mathematics of motion, and the best ways to bury it under a mountain of unnecessary formalism. Despite being an introduction to both the differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be discarded in favor of the more sophisticated function-based approach developed as a response to various analytic crises which do not really apply in this setting, and which will of course not be mentioned. To be taken again in college, verbatim. |
| − | under a mountain of unnecessary formalism. Despite being an introduction to both the | ||
| − | differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be | ||
| − | discarded in favor of the more sophisticated function-based approach developed as a response to | ||
| − | various analytic crises which do not really apply in this setting, and which will of course not be | ||
| − | mentioned. To be taken again in college, verbatim. | ||
==Office Hours== | ==Office Hours== | ||
Latest revision as of 10:29, 16 March 2024
- A good 2021 Twitter Thread on exam questions, including hte idea of asking students for questions.
Contents
Equation Sheets
One of my 7th graders had a good idea for something to write on the 1-page equation sheet I allow them for tests. She wrote,
"You've got this!" That's a good use of space.
Math Teaching
- "A Mathematician's Lament", Lockhart.
- A musician wakes from a terrible nightmare. In his dream he finds himself in a society where music education has been made mandatory. “We are helping our students become more competitive in an increasingly sound-filled world.” Educators, school systems, and the state are put in charge of this vital project. Studies are commissioned, committees are formed, and decisions are made— all without the advice or participation of a single working musician or composer. ::Since musicians are known to set down their ideas in the form of sheet music, these curious black dots and lines must constitute the “language of music.” It is imperative that students become fluent in this language if they are to attain any degree of musical competence; indeed, it would be ludicrous to expect a child to sing a song or play an instrument without having a thorough grounding in music notation and theory. Playing and listening to music, let alone composing an original piece, are considered very advanced topics and are generally put off until college, and more often graduate school.
- “In seventh grade we mostly study colors and applicators.” They showed me a worksheet. On one side were swatches of color with blank spaces next to them. They were told to write in the names. “I like painting,” one of them remarked, “they tell me what to do and I do it. It’s easy!” After class I spoke with the teacher. “So your students don’t actually do any painting?” I asked. “Well, next year they take Pre-Paint-by-Numbers. That prepares them for the main Paint-by-Numbers sequence in high school. So they’ll get to use what they’ve learned here and apply it to real-life painting situations— dipping the brush into paint, wiping it off, stuff like that.
- If I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done— I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.
- Many a graduate student has come to grief when they discover, after a decade of being told they were “good at math,” that in fact they have no real mathematical talent and are just very good at following directions.
- How many people actually use any of this “practical math” they supposedly learn in school? Do you think carpenters are out there using trigonometry? How many adults remember how to divide fractions, or solve a quadratic equation? Obviously the current practical training program isn’t working, and for good reason: it is excruciatingly boring, and nobody ever uses it anyway.
- dDo you really think kids even want something that is relevant to their daily lives? You think something practical like compound interest is going to get them excited? People enjoy fantasy, and that is just what mathematics can provide— a relief from daily life, an anodyne to the practical workaday world.
- It may be true that you have to be able to read in order to fill out forms at the DMV, but that’s not why we teach children to read. We teach them to read for the higher purpose of allowing them access to beautiful and meaningful ideas. Not only would it be cruel to teach reading in such a way— to force third graders to fill out purchase orders and tax forms— it wouldn’t work! We learn things because they interest us now, not because they might be useful later. But this is exactly what we are asking children to do with math.
- We never hear a student saying, “I wanted to see if it could make any sense to raise a number to a negative power, and I found that you get a really neat pattern if you choose it to mean the reciprocal.” Instead we have teachers and textbooks presenting the “negative exponent rule” as a fait d’accompli with no mention of the aesthetics behind this choice, or even that it is a choice.
- No mathematician in the world would bother making these senseless distinctions: 2 1/2 is a “mixed number,” while 5/2 is an “improper fraction.” They’re equal for crying out loud. They are the same exact numbers, and have the same exact properties. Who uses such words outside of fourth grade?
- Even if we agree that a basic common vocabulary for mathematics is valuable, this isn’t it. How sad that fifth-graders are taught to say “quadrilateral” instead of “four-sided shape,” but are never given a reason to use words like “conjecture,” and “counterexample.” High school students must learn to use the secant function, ‘sec x,’ as an abbreviation for the reciprocal of the cosine function, ‘1 / cos x,’ (a definition with as much intellectual weight as the decision to use ‘&’ in place of “and.” )
- What is happening is the systematic undermining of the student’s intuition. A proof, that is, a mathematical argument, is a work of fiction, a poem. Its goal is to satisfy. A beautiful proof should explain, and it should explain clearly, deeply, and elegantly. A well-written, well-crafted argument should feel like a splash of cool water, and be a beacon of light— it should refresh the spirit and illuminate the mind. And it should be charming.
- CALCULUS. This course will explore the mathematics of motion, and the best ways to bury it under a mountain of unnecessary formalism. Despite being an introduction to both the differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be discarded in favor of the more sophisticated function-based approach developed as a response to various analytic crises which do not really apply in this setting, and which will of course not be mentioned. To be taken again in college, verbatim.
Office Hours
- Students need to realize that it is especially important both for the BOTTOM and the TOP students to come to office hours. For my 7th graders now, and for college, MBA, and PhD students. We teach to the middle. Office hours are for special stuff.
What I've Learned about Teaching 7th Grade Math
- For Zoom office hours, a document camera is necessary.
- The students need to be told what "show your work" means.
- The students need to be forced to use lots of paper in doing homework.
- The students need to be forced to use looseleaf binders and keep them neat and organized.
- The students need monthly tests.
- Competitions are often useful.
- Slow students need one-on-one tutoring.
- Python coding is fun for them and helpful for teaching them.
Participation
- Casey Wichman: I made a wise decision to allow my "econ & policy" students make up participation points by submitting econ memes that relate to class. Grading has never been more fun.
Reading Books
Ben Smith
@BenDSmith_CA Replying to @drandrewmking I took a class on Isaiah where the professor required us to read through Isaiah 7 times. 4 times in 3 sittings or less, 2 times is 2 sittings or less, and 1 time in 1 sitting. Great experience.
Whiteboards
On chalkboards v. whiteboards, Twitter (2022).
