In my last post, I mentioned my top ten list of 'problems'. I decided to call it my top ten 'issues' because I wanted to frame them in the positive - not what's wrong, but how to do it right. However, this is still a list of the top ten problems, because most of this is done wrong in most classrooms.
And here they are:
1. If you're going to teach math, you need to enjoy it.*
The best way to help kids learn to read is to read to them, lots of wonderful stories, so you can hook them on it. The best way to help kids learn math is to make it a game (see #3), or to make dozens of games out of it. Accessible mysteries. Number stories. Hook them on thinking. Get them so intrigued, they'll be willing to really sweat.
2. If you�re going to teach math, you need to know it deeply, and you need to keep learning.
Read Liping Ma. Arithmetic is deeper than you knew (see #6). Every mathematical subject you might teach is connected to many, many others. Heck, I'm still learning about multiplication myself. Over at Axioms to Teach By, I said (back in September), "You don't want the product to be 'the same kind of thing'. ... 5 students per row times 8 rows is 40 students. So I have students/row * rows = students." Owen disagreed with me, and Burt's comment on my last post got me re-reading that discussion. I think Owen and I may both be right, but I have no idea how to use a compass and straightedge to multiply. I'm looking forward to playing with that. I think it will give me new insight.
3. Games are to math as books are to reading. Let the kids play games (or make up their own games) instead of "doing math", and they might learn more math.
Denise's game that's worth 1000 worksheets (addition war and its variations) is one place to start. And Pam Sorooshian has this to say about dice. Learn to play games: Set, Blink, Quarto, Blokus, Chess, Nim, Connect Four.
4. Students are willing to do the deep work necessary to learn math if and only if they�re enjoying it.
Which means that grades and coercion are really destructive. Maybe more so than in any other subject. People need to feel safe to take the risks that really learning math requires. Read Joe at For the Love of Learning. (Maybe you'll get to read him here soon.) I'm not sure if this is true in other cultures. Students in Japan seem to be very stressed from many accounts I read; they also do some great problem-solving lessons.
5. Math is not facts (times tables) and procedures (long division), although those are a part of it; more deeply, math is about concepts, connections, patterns. It can be a game, a language, an art form. Everything is connected, often in surprising and beautiful ways.
My favorite math ed quote of all time comes from Marilyn Burns: "The secret key to mathematics is pattern."
U.S. classrooms are way too focused on procedure in math. It's hard for any one teacher to break away from that, because the students come to expect it, and are likely to rebel if asked to really think. (See The Teaching Gap, by James Stigler.)
See George Hart for the artform. I don't know who to recommend for the language angle. Any recommendations?
6. Math is not arithmetic, although arithmetic is a part of it. (And even arithmetic has its deep side.)
Little kids can learn about infinity, geometry, probability, patterns, symmetry, tiling, map colorings, tangrams, ... And they can do arithmetic in another base to play games with the meaning of place value. (I wrote about base eight here, and base three here.)
7. Math itself is the authority - not the curriculum, not the teacher, not the standards committee.
Read Math Mojo � you can�t want kids to do it the way you do. You have to be fearless, and you need to see the connections.
8. Real mathematicians ask why.
If you�re trying to memorize it, you�re probably being pushed to learn something that hasn�t built up meaning for you. See Julie Brenna's article on Memorizing Math Facts. Yes, eventually you want to have the times tables memorized, just like you want to know words by sight. But the path there can be full of delicious entertainment. Learn your multiplications as a meditation, as part of the games you play, ...
Just like little kids, who ask why a thousand times a day, mathematicians ask why. Why are there only 5 Platonic (regular) solids? Why does a quadratic (y=x2), which gives a U-shaped parabola as its graph, have the same sort of U-shaped graph after you add a straight line equation (y=2x+1) to it? (A question asked and answered by James Tanton in this video.) Why does the anti-derivative give you area? Why does dividing by a fraction make something bigger? Why is the parallel postulate so much more complicated than the 4 postulates before it?
9. Earlier is not better.
The schools are pushing academics earlier and earlier. That's not a good idea. If young people learn to read when they're ready for it, they enjoy reading. They read more and more; they get better and better at it; reading serves them well. (See Peter Gray's recent post on this.) The same can happen with math. Daniel Greenberg, working at a Sudbury school (democratic schools, where kids do not have enforced lessons) taught a group of 9 to 12 year olds all of arithmetic in 20 hours. They were ready and eager, and that's all it took.
In 1929, L.P. Benezet, superintendent of schools in Manchester, New Hampshire, believed that waiting until later would help children learn math more effectively. The experiment he conducted, waiting until 5th or 6th grade to offer formal arithmetic lessons, was very successful. (His report was published in the Journal of the NEA. Although some people disagree about the success of this experiment, there is nothing published which contradicts his evidence. I'd like to find more information about how this project ended.)
10. Textbooks are trouble. Corollary: The one doing the work is the one doing the learning. (Is it the text and the teacher, or is it the student?)
Hmm, this shouldn't be last, but when I look at the list, they all seem important. I guess this isn't a well-ordered domain. ;^) Textbook Free: Kicking the Habit is an article by Chris Shore on getting away from using a textbook. (After clicking the link, click on 'Articles'.) I've been duly inspired, and will report in the fall about how it goes for me in my classes to teach without a textbook. See dy/dan on being less helpful (so the students will learn more).
31. Multiplication is not (just) repeated addition, it�s much richer than that.
Wait. I said that already... (I warned you, it's just not in my top ten.)
What do you see as the biggest issues or problems in math education?
____
*I know, top ten lists are supposed to start at number ten to keep the suspense up. But the suspense is gone - I already told you my top two in my last post. And I can't help it, I just have to start at the top.
And here they are:
1. If you're going to teach math, you need to enjoy it.*
The best way to help kids learn to read is to read to them, lots of wonderful stories, so you can hook them on it. The best way to help kids learn math is to make it a game (see #3), or to make dozens of games out of it. Accessible mysteries. Number stories. Hook them on thinking. Get them so intrigued, they'll be willing to really sweat.
2. If you�re going to teach math, you need to know it deeply, and you need to keep learning.
Read Liping Ma. Arithmetic is deeper than you knew (see #6). Every mathematical subject you might teach is connected to many, many others. Heck, I'm still learning about multiplication myself. Over at Axioms to Teach By, I said (back in September), "You don't want the product to be 'the same kind of thing'. ... 5 students per row times 8 rows is 40 students. So I have students/row * rows = students." Owen disagreed with me, and Burt's comment on my last post got me re-reading that discussion. I think Owen and I may both be right, but I have no idea how to use a compass and straightedge to multiply. I'm looking forward to playing with that. I think it will give me new insight.
3. Games are to math as books are to reading. Let the kids play games (or make up their own games) instead of "doing math", and they might learn more math.
Denise's game that's worth 1000 worksheets (addition war and its variations) is one place to start. And Pam Sorooshian has this to say about dice. Learn to play games: Set, Blink, Quarto, Blokus, Chess, Nim, Connect Four.
4. Students are willing to do the deep work necessary to learn math if and only if they�re enjoying it.
Which means that grades and coercion are really destructive. Maybe more so than in any other subject. People need to feel safe to take the risks that really learning math requires. Read Joe at For the Love of Learning. (Maybe you'll get to read him here soon.) I'm not sure if this is true in other cultures. Students in Japan seem to be very stressed from many accounts I read; they also do some great problem-solving lessons.
5. Math is not facts (times tables) and procedures (long division), although those are a part of it; more deeply, math is about concepts, connections, patterns. It can be a game, a language, an art form. Everything is connected, often in surprising and beautiful ways.
My favorite math ed quote of all time comes from Marilyn Burns: "The secret key to mathematics is pattern."
U.S. classrooms are way too focused on procedure in math. It's hard for any one teacher to break away from that, because the students come to expect it, and are likely to rebel if asked to really think. (See The Teaching Gap, by James Stigler.)
See George Hart for the artform. I don't know who to recommend for the language angle. Any recommendations?
6. Math is not arithmetic, although arithmetic is a part of it. (And even arithmetic has its deep side.)
Little kids can learn about infinity, geometry, probability, patterns, symmetry, tiling, map colorings, tangrams, ... And they can do arithmetic in another base to play games with the meaning of place value. (I wrote about base eight here, and base three here.)
7. Math itself is the authority - not the curriculum, not the teacher, not the standards committee.
Read Math Mojo � you can�t want kids to do it the way you do. You have to be fearless, and you need to see the connections.
8. Real mathematicians ask why.
If you�re trying to memorize it, you�re probably being pushed to learn something that hasn�t built up meaning for you. See Julie Brenna's article on Memorizing Math Facts. Yes, eventually you want to have the times tables memorized, just like you want to know words by sight. But the path there can be full of delicious entertainment. Learn your multiplications as a meditation, as part of the games you play, ...
Just like little kids, who ask why a thousand times a day, mathematicians ask why. Why are there only 5 Platonic (regular) solids? Why does a quadratic (y=x2), which gives a U-shaped parabola as its graph, have the same sort of U-shaped graph after you add a straight line equation (y=2x+1) to it? (A question asked and answered by James Tanton in this video.) Why does the anti-derivative give you area? Why does dividing by a fraction make something bigger? Why is the parallel postulate so much more complicated than the 4 postulates before it?
9. Earlier is not better.
The schools are pushing academics earlier and earlier. That's not a good idea. If young people learn to read when they're ready for it, they enjoy reading. They read more and more; they get better and better at it; reading serves them well. (See Peter Gray's recent post on this.) The same can happen with math. Daniel Greenberg, working at a Sudbury school (democratic schools, where kids do not have enforced lessons) taught a group of 9 to 12 year olds all of arithmetic in 20 hours. They were ready and eager, and that's all it took.
In 1929, L.P. Benezet, superintendent of schools in Manchester, New Hampshire, believed that waiting until later would help children learn math more effectively. The experiment he conducted, waiting until 5th or 6th grade to offer formal arithmetic lessons, was very successful. (His report was published in the Journal of the NEA. Although some people disagree about the success of this experiment, there is nothing published which contradicts his evidence. I'd like to find more information about how this project ended.)
10. Textbooks are trouble. Corollary: The one doing the work is the one doing the learning. (Is it the text and the teacher, or is it the student?)
Hmm, this shouldn't be last, but when I look at the list, they all seem important. I guess this isn't a well-ordered domain. ;^) Textbook Free: Kicking the Habit is an article by Chris Shore on getting away from using a textbook. (After clicking the link, click on 'Articles'.) I've been duly inspired, and will report in the fall about how it goes for me in my classes to teach without a textbook. See dy/dan on being less helpful (so the students will learn more).
31. Multiplication is not (just) repeated addition, it�s much richer than that.
Wait. I said that already... (I warned you, it's just not in my top ten.)
What do you see as the biggest issues or problems in math education?
____
*I know, top ten lists are supposed to start at number ten to keep the suspense up. But the suspense is gone - I already told you my top two in my last post. And I can't help it, I just have to start at the top.
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