March 4, 2014 12:23 AM Subscribe

Why playing with algebraic and calculus concepts—rather than doing arithmetic drills—may be a better way to introduce children to math.

posted by Pudhoho (65 comments total) 50 users marked this as a favorite

posted by Pudhoho (65 comments total) 50 users marked this as a favorite

Since I started taking calculus in college, I have frequently wondered why we don't teach the position/velocity/acceleration relations to younger kids. I remember being very young and thinking that there must be a way to determine how far a car had traveled by merely looking at the speedometer. It's not exactly a lofty concept, and many kids would probably be able to understand it if they were exposed to it.

posted by triceryclops at 12:31 AM on March 4 [7 favorites]

posted by triceryclops at 12:31 AM on March 4 [7 favorites]

Pretty sure every type of thinking whether cognitive or creative revolves around patterns.

posted by Apocryphon at 1:22 AM on March 4 [4 favorites]

posted by Apocryphon at 1:22 AM on March 4 [4 favorites]

As the article mentions, the reason math education is structured the way it is is historical; to some extent people assume, I guess, that the order in which *people* across time and cultures, came upon certain mathematical ideas is a measure of their difficulty. Calculus came after algebra, so it must be harder, right?

But this is wrong-headed. Historical development of mathematics says more about what problems those cultures were trying to solve than their difficulty. Discovery of some concepts from calculus predate some concepts from algebra.

Also, it seems like people often mix up teaching *math* and teaching *algorithms*. Look at the picture above the headline in the article. The blackboard shows both: the stacked numbers with a line under them represent working through a particular addition algorithm, but the statements nearer to the top are mathematical expressions. This is another historical accident, because math was used in very practical situations, so teaching about math separately from algorithms was unnecessary.

This separation, though, has implications for teaching math: algorithms are difficult for children, with limited attention span, to sit and work though. But even if a young child does not have the ability to grasp an particular algorithm, they may have the ability to grasp the general idea.

posted by Philosopher Dirtbike at 1:22 AM on March 4 [8 favorites]

But this is wrong-headed. Historical development of mathematics says more about what problems those cultures were trying to solve than their difficulty. Discovery of some concepts from calculus predate some concepts from algebra.

Also, it seems like people often mix up teaching *math* and teaching *algorithms*. Look at the picture above the headline in the article. The blackboard shows both: the stacked numbers with a line under them represent working through a particular addition algorithm, but the statements nearer to the top are mathematical expressions. This is another historical accident, because math was used in very practical situations, so teaching about math separately from algorithms was unnecessary.

This separation, though, has implications for teaching math: algorithms are difficult for children, with limited attention span, to sit and work though. But even if a young child does not have the ability to grasp an particular algorithm, they may have the ability to grasp the general idea.

posted by Philosopher Dirtbike at 1:22 AM on March 4 [8 favorites]

Arithmetic for cashiers, please.

posted by InsertNiftyNameHere at 1:29 AM on March 4 [1 favorite]

posted by InsertNiftyNameHere at 1:29 AM on March 4 [1 favorite]

InsertNiftyNameHere: I'm surprised, because I'm also in the US and I don't think I've ever had a cashier look at me funny for offering an odd assortment of coins-- except when I occasionally screw it up. Maybe this is because a large fraction of the establishments in the area are cash-only, so the cashiers are more experienced.

posted by alexei at 1:32 AM on March 4 [1 favorite]

posted by alexei at 1:32 AM on March 4 [1 favorite]

alexei: "*InsertNiftyNameHere: I'm surprised, because I'm also in the US and I don't think I've ever had a cashier look at me funny for offering an odd assortment of coins-- except when I occasionally screw it up. Maybe this is because a large fraction of the establishments in the area are cash-only, so the cashiers are more experienced.*"

I've no idea, really, but I will be the first to acknowledge that the area I live in is what I call an "intellectual wasteland." And as my history here on Mefi shows, I'm not even very smart. All I know is 20 years ago the cashiers wouldn't bat an eye and would know exactly what my intent was. Nowadays? They've no clue.

posted by InsertNiftyNameHere at 1:38 AM on March 4

I've no idea, really, but I will be the first to acknowledge that the area I live in is what I call an "intellectual wasteland." And as my history here on Mefi shows, I'm not even very smart. All I know is 20 years ago the cashiers wouldn't bat an eye and would know exactly what my intent was. Nowadays? They've no clue.

posted by InsertNiftyNameHere at 1:38 AM on March 4

And what's more, the other day I was at the grocery store and the checker was unable to identify a portabello mushroom!

posted by You Can't Tip a Buick at 1:41 AM on March 4 [33 favorites]

posted by You Can't Tip a Buick at 1:41 AM on March 4 [33 favorites]

You Can't Tip a Buick: "*And what's more, the other day I was at the grocery store and the checker was unable to identify a portabello mushroom!*"

Did you get the correct SKU code?

posted by InsertNiftyNameHere at 1:53 AM on March 4 [4 favorites]

Did you get the correct SKU code?

posted by InsertNiftyNameHere at 1:53 AM on March 4 [4 favorites]

And what's more, that Buick doesn't even know what to do when I give it a twenty and tell it to keep the change!

posted by DoctorFedora at 1:53 AM on March 4

posted by DoctorFedora at 1:53 AM on March 4

[Plenty to discuss here it seems re: math in early education — maybe don't limit the agenda for this thread to one person's gripe with America's retail workforce, thanks.]

posted by goodnewsfortheinsane at 2:01 AM on March 4 [3 favorites]

posted by goodnewsfortheinsane at 2:01 AM on March 4 [3 favorites]

I managed to pass calculus, integrals etc. in highschool, somehow. I've forgotten the technicalities of how to

The thing I wonder about, in regards to this, is the tax system. Income tax brackets. Every country (I assume) has got them. They're the cause of constant political conflict.

But why do we need them? Did no politician ever writing income tax law know about algebra? Can't we have the entire income tax system defined by an equation that scales the rate by income, rather than these pre-defined brackets?

posted by Jimbob at 2:09 AM on March 4 [6 favorites]

I remember my 2nd yr calculus class being reamed out by the prof for being unable to follow his proofs and solve problems because our algebra was lacking. This was in the 80's, and I heard they teach university calc a little differently now.

posted by klarck at 3:07 AM on March 4

posted by klarck at 3:07 AM on March 4

What happened to the anecdata that a calculus professor found a strong correlation over years that the students who solved a first-day algebra quiz the fastest would get the highest grades in his calculus class? When we teach algorithms instead of harder-to-test-for concepts, are we doing a favor to students' futures?

posted by persona at 3:11 AM on March 4

posted by persona at 3:11 AM on March 4

I read this article sitting on a couch next to my four year old nephew who is teaching himself algebra with Dragonbox on my iPad. He's woken me up early every morning for the past week because he's so excited about playing the game. Clearly the concepts of algebra are accessible to toddlers, especially when presented without numbers (and with cute icons and fun sounds).

posted by tractorfeed at 3:27 AM on March 4 [4 favorites]

posted by tractorfeed at 3:27 AM on March 4 [4 favorites]

The new method there is how you teach cashiers to make change, by the way (more or less).

If you are charged $32.47 for some goods and you give the cashier a $50, they give you back

3 pennies to get to $32.50

2 quarters to get to $33

2 $1 bills to get to $35

1 $5 bill to get to $40

1 $10 bill to get to $50.

So the difference between the amount paid and the amount owed is the sum of the change given which is .03 + .5 + 2 + 5 + 10 = 17.53.

Just because an algorithm is different doesn't mean it's not an algorithm.

posted by Elementary Penguin at 3:28 AM on March 4 [6 favorites]

Regarding an

*income tax system defined by an equation that scales the rate by income*

You could build into that algorithm an income curve so you would automatically scale your algorithm according to the current state of wealth.

Set a target for inequality and ad that as a factor.

Hey billionaire, you want to pay less tax? Reduce this inequality factor and your tax bill will drop by x.

Hmm....

posted by Just this guy, y'know at 3:39 AM on March 4 [4 favorites]

You could build into that algorithm an income curve so you would automatically scale your algorithm according to the current state of wealth.

Set a target for inequality and ad that as a factor.

Hey billionaire, you want to pay less tax? Reduce this inequality factor and your tax bill will drop by x.

Hmm....

posted by Just this guy, y'know at 3:39 AM on March 4 [4 favorites]

When I was in kindergarten and grade 1, we had a program in the school whereby older students would read books with us. I think it was called Reading Buddies. Anyway. Our buddies were all in grade 5, and when I was in grade 1 my best friend and I saw the homework of one of our buddies. It *looked* like math, but it was full of letter? WTF?

So we went to our teacher and asked about it. She was very much like pTerry's Susan; if you don't tell kids they don't know how to do something, they'll just go ahead and do it. So she taught us algebra.

Every single one of us picked it up like lint. Math education is broken; somehow every math teacher I had sucked the life and enjoyment out of what*should* have been a fascinating subject and critical life skill.

Don't even get me started on trig though.

posted by feckless fecal fear mongering at 4:31 AM on March 4 [11 favorites]

So we went to our teacher and asked about it. She was very much like pTerry's Susan; if you don't tell kids they don't know how to do something, they'll just go ahead and do it. So she taught us algebra.

Every single one of us picked it up like lint. Math education is broken; somehow every math teacher I had sucked the life and enjoyment out of what

Don't even get me started on trig though.

posted by feckless fecal fear mongering at 4:31 AM on March 4 [11 favorites]

I can get behind this. I was a gifted student in elementary school, but I struggled with mental math and general arithmetic. Once I hit junior high and started learning algebra, I was in heaven! Math became fun and satisfying. My major weakness was fractions, and I will never forget the 'mean' educator who asked that I come in on a day off so we could go over them. I grumbled through it, but came out the other side with a much stronger grasp on working with fractions. Suddenly I was a bit of a whiz and getting straight As, winning math competitions and looking at a future in engineering.

I won't forget you, M.L. That was a huge turning point in my life, not only in terms of understanding mathematical principles, but in appreciating that I*could* in fact learn something I wasn't already 'good' at. I just needed a little shove. I also won't forget the balogna sandwiches he'd offer me on my lunchtime detentions. It's the little things. I wish I'd given him the respect he deserved from the start.

posted by Violet Femme at 4:34 AM on March 4 [9 favorites]

I won't forget you, M.L. That was a huge turning point in my life, not only in terms of understanding mathematical principles, but in appreciating that I

posted by Violet Femme at 4:34 AM on March 4 [9 favorites]

Interesting idea there, Just this guy, including an inequality factor. I've toyed with making income tax scale with maximum income actually.^{*}

We choose a maximum tax rate alpha and a minimum income for taxation beta, and find the maximum income gamma from income data. We then tax an income of x at the rate min(0,k*log(x/beta)) where k = alpha/log(gamma/beta), which has several nice properties : (a) the maximum income earner pays tax rate alpha, (b) anyone earning beta or less pays nothing, and ..

(c) if Alice earns y and Bob earns z, then the difference between Alice and Bob's tax rate is k * log(y/z), meaning larger incomes always pay proportionally more.

In particular, any company that acquires another company always winds up paying more tax, harming investors, and any company that splits itself always pays less tax, benefiting investors, so investors have a strong incentive to ensure that companies are only as big as they need to be. Also, any market collusion should now become tax evasion because the companies acted as if they were a larger company that should pay more tax.

If we've serious income inequality, then by (c) income taxes fall more heavily on the wealthy, and larger companies, and the government's revenues decline. If the government wishes to bolster revenues, increasing alpha or decreasing beta both hit all existing payers in differently multiplicative ways, one inside and one outside the logarithm, and decreasing beta actually creates more tax payers who maybe protest.

^{*} I personally favor taxing only companies and extremely wealthy individuals btw. Ideally, the Sixteenth Amendment itself should restrict direct taxation to organizations and the top 5% of income earners. If you've either restricted taxation that much, or provide de facto free accountants like all European countries, then your taxation system could easily be described by an equation without anyone needing to even learn any mathematics.

posted by jeffburdges at 5:03 AM on March 4 [4 favorites]

We choose a maximum tax rate alpha and a minimum income for taxation beta, and find the maximum income gamma from income data. We then tax an income of x at the rate min(0,k*log(x/beta)) where k = alpha/log(gamma/beta), which has several nice properties : (a) the maximum income earner pays tax rate alpha, (b) anyone earning beta or less pays nothing, and ..

(c) if Alice earns y and Bob earns z, then the difference between Alice and Bob's tax rate is k * log(y/z), meaning larger incomes always pay proportionally more.

In particular, any company that acquires another company always winds up paying more tax, harming investors, and any company that splits itself always pays less tax, benefiting investors, so investors have a strong incentive to ensure that companies are only as big as they need to be. Also, any market collusion should now become tax evasion because the companies acted as if they were a larger company that should pay more tax.

If we've serious income inequality, then by (c) income taxes fall more heavily on the wealthy, and larger companies, and the government's revenues decline. If the government wishes to bolster revenues, increasing alpha or decreasing beta both hit all existing payers in differently multiplicative ways, one inside and one outside the logarithm, and decreasing beta actually creates more tax payers who maybe protest.

posted by jeffburdges at 5:03 AM on March 4 [4 favorites]

So many times articles like this feel like they're arguing against a strawman curriculum taken from a 19th century school house. Teaching young children algebra concepts? That's something we're actually doing right now. They usually avoid variables and instead use little unfilled boxes. I imagine it's not precisely what the woman profiled would prefer, but it's closer than the hundreds of arithmetic problems that the article seems to think we're having kids do. I'm sure there are places doing that, but I don't remember much of that from my math education 20 years ago either.

posted by Bulgaroktonos at 5:10 AM on March 4 [11 favorites]

posted by Bulgaroktonos at 5:10 AM on March 4 [11 favorites]

We could maybe use music, programming, basic electronics, etc. to teach basic mathematics, surely folks have tried. I learned simple programming and electronics extremely young, which helped me enormously.

posted by jeffburdges at 5:15 AM on March 4

posted by jeffburdges at 5:15 AM on March 4

What Bulgaroktonos said: Kids are totally doing algebra in kindergarten right now. We just had a post about how confusing kids' math homework is for their parents (the homework used "number sentence" to mean "equation" iirc). The biggest reason to not completely overhaul elementary school math is that parents who aren't in class won't understand the new methods, and so won't be able to help with homework.

posted by subdee at 5:38 AM on March 4 [5 favorites]

posted by subdee at 5:38 AM on March 4 [5 favorites]

It varies considerably by region and school, Bulgaroktonos. The observation in the article about how we teach relating to our expectations for students' life trajectories aligns with my experiences, for example, in that the schools I attended in less affluent communities focused more on rote memorization and drill, and the schools I attended in more affluent communities focused more on creativity, patterns, and problem solving (this being in the US where school funding is usually through local property taxes, curricula and pedagogy have historically been more locally determined (though are becoming more centralized), and both varied quite widely across communities). My impression (purely from anecdotes, not actual data, though) is that, internationally, there is also some correlation between type of primary school mathematics curriculum in countries and relative affluence or role in the global economy?

Anyway, my current Canadian university students' early math experiences (so, from a decade ago) largely seem to align with the description in the article. From what I've seen of newer core curriculum standards out of the US, that's changing somewhat, but slowly and unevenly.

(Part of the difficulty being that teaching in the manner described in the article requires more creativity and flexibility from the teacher, in ways that current in-service teachers may not have been well prepared for by their own more rote-focused mathematics education, and in ways that are specifically opposed by current trends around more structured classrooms, standardized testing, and the particular ways in which teacher "accountability" is being increasingly assessed in the US. But that's perhaps a whole different discussion.)

posted by eviemath at 5:39 AM on March 4 [2 favorites]

Anyway, my current Canadian university students' early math experiences (so, from a decade ago) largely seem to align with the description in the article. From what I've seen of newer core curriculum standards out of the US, that's changing somewhat, but slowly and unevenly.

(Part of the difficulty being that teaching in the manner described in the article requires more creativity and flexibility from the teacher, in ways that current in-service teachers may not have been well prepared for by their own more rote-focused mathematics education, and in ways that are specifically opposed by current trends around more structured classrooms, standardized testing, and the particular ways in which teacher "accountability" is being increasingly assessed in the US. But that's perhaps a whole different discussion.)

posted by eviemath at 5:39 AM on March 4 [2 favorites]

This is definitely done in elementary school, though you assume constant velocity. I remember doing mixing problems in school as well.

The article seems to be extrapolating wildly from math circles and while I don't doubt there's something to be learned from math circles in terms of teaching math generally (I gather that's what this woman's research is trying to do), they're ridiculously not representative. They're often populated by the sort of kids who go on to get non-zero scores on the Putnam, which is only a subset of the 'good' math majors.

posted by hoyland at 5:40 AM on March 4

(That said, having grown up in a math circle probably turns you into the sort of person who can do Putnam problems.)

posted by hoyland at 5:42 AM on March 4

posted by hoyland at 5:42 AM on March 4

I don't think you have to go back that far. I'm not

It makes sense to me that you could push many of the interesting concepts (and algorithms!) down to much younger ages, and if that is how math is now being taught I am impressed and happy for those students.

posted by Dip Flash at 5:52 AM on March 4 [2 favorites]

I think this kind of play/story based approach would work much better than the old system for most subjects. I'm not sure how much primary education's moved on since I was there about 15 years ago, but I remember having to do a lot of learning by rote and drills in all of the subjects (I'm not sure how much this was my school, it sounds like Bulgarktonos didn't have the same experiences, but the rural primary school in England I went to still had some fairly Victorian practices). When Maria Droujkova says “Calculations kids are forced to do are often so developmentally inappropriate, the experience amounts to torture,” in the article, I have to agree with her without any hyperbole. I think the same is true of writing the same letter over and over or learning lists of dates or flags. I also think that the approach stifles academic curiosity, and pushes people away from higher education. The only two lessons boys would consistently enjoy were P.E and woodwork, which seems like a massive shame (I think girls tended to do better, but that's a different conversation).

I guess this sort of idea has gained more traction away from maths and hard sciences, as perhaps it's more obvious how to apply playful things to school subjects, but that doesn't mean it's not possible for maths and science. I'm doing my second physics degree at the moment, so I use a lot of maths on a daily basis, but until I was about 15 or so and started making the connections myself between maths and my interests, I never enjoyed it. I'd always thought maths was just learning times tables and doing long division, when I wanted to be at home perfecting my sandpit's irrigation system fed from the garden hose, or seeing what the biggest self supporting structure was I could make from sofa cushions, or using my mums sewing elastic to launch apples into next doors garden. I understand that those things aren't possible with a big class of kids, but I think in my school at least, there was no effort to meet anyone half way.

posted by Ned G at 5:58 AM on March 4

I guess this sort of idea has gained more traction away from maths and hard sciences, as perhaps it's more obvious how to apply playful things to school subjects, but that doesn't mean it's not possible for maths and science. I'm doing my second physics degree at the moment, so I use a lot of maths on a daily basis, but until I was about 15 or so and started making the connections myself between maths and my interests, I never enjoyed it. I'd always thought maths was just learning times tables and doing long division, when I wanted to be at home perfecting my sandpit's irrigation system fed from the garden hose, or seeing what the biggest self supporting structure was I could make from sofa cushions, or using my mums sewing elastic to launch apples into next doors garden. I understand that those things aren't possible with a big class of kids, but I think in my school at least, there was no effort to meet anyone half way.

posted by Ned G at 5:58 AM on March 4

Interesting. My (admittedly limited) experience with a large urban school district, one that serves primarily underprivileged kids, is that they're quick to jump on whatever the latest pedagogical bandwagon is, although they're typically terrible at training teachers to implement it. Of course, that's a large city that has plenty of money to spend on education, especially top down improvements like buying new curricula, even if the students don't come from affluent backgrounds.

Obviously, you're right that it does vary considerably by district or even classroom within the same school; my objection is mostly to the breathless tone of these articles, as if they're talking about a brand new discovery when the truth is that they're usually talking about a technique that's been around for years.

posted by Bulgaroktonos at 6:02 AM on March 4

20 years ago, or so, I graduated from HS where I was in AP math and AP physics and had taken several years of "advanced" algebra and calculus.

I got a job as a cashier and giving change really threw me for a loop. It's a skill seperate entirely from the math involved - especially doing it while the grill on fire, and there is a line 8 miles long and so on.

Also, it's hilarious, but there is a portion of the ASVAB that has like 120 arithmetic problems to be solved in 60 seconds or whatever. Nothing hard, just like 3+4 or 6-5. I got to that portion of the test and freaked the fuck out. I didn't know what to do with a problem that didn't have an X or a Y or some shit telling me what to solve for. I had to mentally add X's to the problems as I did them (3x+5x=8x) in order to do them because, jeez.

I don't know what this has to do with the subject, except you generally only get good at things you do everyday.

posted by Pogo_Fuzzybutt at 6:04 AM on March 4 [1 favorite]

In my every day adult life, I reman grateful that I was drilled at an early age in the rote memorisation of times tables, addition, and subtraction. Nothing has been more useful to me than the ability to do simple math problems quickly and accurately and without pen and paper.

posted by three blind mice at 6:31 AM on March 4 [3 favorites]

posted by three blind mice at 6:31 AM on March 4 [3 favorites]

3 pennies to get to $32.50

2 quarters to get to $33

2 $1 bills to get to $35

1 $5 bill to get to $40

1 $10 bill to get to $50.

So the difference between the amount paid and the amount owed is the sum of the change given which is .03 + .5 + 2 + 5 + 10 = 17.53.

Just yesterday on Facebook, one of my friends reposted this from Victoria Jackson's page. Apparently VJ isn't a fan of Common Core (which I know next to nothing about so have no real opinion on it); and though my friend doesn't align politically with her she reposted it because she couldn't make head nor tail of the problem. Which I understand, because just looking at it without a quick explanation could be confusing. So I explained it the same way Elementary Penguin did - using money as an example and counting

I think for a lot of people (and children) counting up is easier and more intuitive than straight subtraction and this is definitely how I do math in my head - by using little mental shortcuts like this. But I spent all of my school years not understanding math and with the deeply ingrained message that girls are bad at math, which kind of made me give up before I'd even started, even though I was testing high enough in all the aptitude tests to be placed in the gifted student programs. It wasn't until college that I found a math professor who took the time to explain simple concepts to me in a way I could grasp that I actually was surprised to find out that I really

posted by triggerfinger at 6:36 AM on March 4 [3 favorites]

I don't know if this is the reason, but if you have a tax rate that scales with income, then there is a diminishing marginal return on every additional dollar earned, whereas with the brackets that only applies when you cross the income threshold for the next bracket.

posted by ultraviolet catastrophe at 7:02 AM on March 4

I think it does the same thing, it's just a curve instead of a step.

posted by Just this guy, y'know at 7:06 AM on March 4

posted by Just this guy, y'know at 7:06 AM on March 4

Except the brackets end at some point, whereas the curve will continue forever, so that your 5 millionth dollar is taxed at 50 percent (or whatever the rate is), but your 10 millionth dollar is taxed at 99.99 percent, so there's really no point in earning more than that. Under the current structure, your 5 millionth dollar is taxed at 39.6 percent, and so is your 10 millionth dollar.

posted by ultraviolet catastrophe at 7:10 AM on March 4

posted by ultraviolet catastrophe at 7:10 AM on March 4

Also, the curve assumes that bracket-stepping is the political issue; it's not. The political issue is the amount paid, and that will be true regardless of how it's calculated.

posted by aaronetc at 7:16 AM on March 4

posted by aaronetc at 7:16 AM on March 4

The solution would be to do some sort of logistic curve to determine the percentage, so that it could never get higher than, say, 40%. That way you could have the lower rate be something like -10%, too, and get negative income tax in for free!

posted by Elementary Penguin at 7:26 AM on March 4 [2 favorites]

posted by Elementary Penguin at 7:26 AM on March 4 [2 favorites]

Hearing proofs described as rote breaks my poor mathematician's heart! But doesn't surprise me. What would have been my first introduction to "proofs" if I hadn't had some more interesting classroom and extracurricular math experiences at a young age was those soulless lists of triangle similarity theorems (Side-Angle-Side, Angle-Side-Angle, etc.) that passed for "proof" at the time.

Re: NedG and Bulgaroktonos' comments, these innovations do come in waves, which borrow heavily from the last time the same general idea came around, yeah. It's influenced by how different philosophies of education interact with dominant strains in social policy at the time, as well as continuing advances in the cognitive science/psychology of learning and child development, combined into different learning theories. Not emphasized in that last link, but, based on some keywords mentioned in the article, something that the researchers profiled seem to be aligned with, are humanism and connectivism (basically: the idea that learning is fundamentally or at least primarily a social rather than individual activity).

Ideally we would adopt evidence-based pedagogies, but this is complicated, or so I've been told informally by folks with more knowledge of the area than me, by the fact that greater student-teacher interaction (smaller class sizes, more personal attention) has a strong positive correlation with student learning, regardless of the pedagogy used. So many studies tend to show positive results for the new method being tried, regardless of which educational philosophy it follows, because the instructors at least, if not the students themselves, tend to receive more attention, more training, and more support as compared to the status quo group. Which means that if students are either going to learn basic math knowledge or not based on factors outside the control of individual classroom teachers and their choice (if they have one) of pedagogy and curriculum to use, it may be more useful to ask what additional effects choice of pedagogy/curriculum has, in the long run? One of the arguments for humanist education relates to how students come to see themselves in relation to society. As you note, NedG, this has received more attention in the arts and humanities, eg. in the work of Freire, bell hooks' series on education, and in the pages of publications such as Radical Teacher. But as Bulgaroktonos notes, there is a small history of similar considerations in math and the sciences, from Bertrand Russell and friends' experiments in the education of their children, through R.L. Moore George Pólya and Paul Lockhart through current initiatives like SENCER.

If folks are interested in philosophy of math education similar to that described in the article, some books I'd recommend include:

Measurement by Lockhart

Radical Equations: Civil Rights from Mississippi to the Algebra Project by Robert Moses and Charles Cobb

To Teach: The Journey in Comics by William Ayers and Ryan Alexander-Tanner

posted by eviemath at 7:32 AM on March 4 [18 favorites]

Flagged as bloody fantastic, eviemath.

posted by feckless fecal fear mongering at 7:37 AM on March 4 [1 favorite]

posted by feckless fecal fear mongering at 7:37 AM on March 4 [1 favorite]

Math team was the most helpful thing I did in school to prepare me for higher maths.

posted by zscore at 7:40 AM on March 4 [2 favorites]

posted by zscore at 7:40 AM on March 4 [2 favorites]

My (admittedly uninformed) opinion about this is that some people are just late math bloomers, but if you aren't getting the level of abstraction that is expected of you at a certain age, you get tracked into the Mindless Drudgery math class and given up on, whereas if you waited a year and tried the abstraction again, you'd get it. Enough people who are "bad at math" get it as adults that I think sometimes you just need to wait until you are more intellectually mature. It also helps to have some extrinsic motivation, which can be lacking in K-12 math classes.

posted by Elementary Penguin at 7:45 AM on March 4 [6 favorites]

Sadly, it was totally by rote, working through prescribed steps in unison as a group.* If nothing else, it was evidence that no matter how cool something is, bad teaching can ruin it.

* My actual guess, looking back, is that my high school trig teacher (aka "coach") understood the material about as well as we did, and was just focused on following the book's lesson plans.

posted by Dip Flash at 8:07 AM on March 4 [2 favorites]

Going off what Bulgaroktonos said (and we share a LOT of opinions on this), I feel like the popular understanding of current math education is dichotomized into articles like this about how we need to be teaching kids more abstract concepts so they can love math on the one hand and on the other hand people bitching about how the Common Core makes no sense and it's not how THEY learned math and what are we even TEACHING kids these days? Sometimes it's the same people saying both things!

You get a lot of people who say "I hated math and my teachers were shitty" who then get upset when the way we teach math now doesn't look like they way they learned it. Teachers ARE doing this stuff even in kindergarten and first grade. I've done algebra stuff with second graders, many of whom were way behind in most areas. The variables included stars and smiley faces as well as letters but it was the same idea.

One of the reasons I sometimes stay away from internet education discussions is because you get people who don't know much about contemporary math education talking about how we SHOULD be doing what we ARE doing as well as getting people bitching about how it doesn't make sense even though they also thought their teachers were terrible.

posted by Mrs. Pterodactyl at 8:08 AM on March 4 [5 favorites]

You get a lot of people who say "I hated math and my teachers were shitty" who then get upset when the way we teach math now doesn't look like they way they learned it. Teachers ARE doing this stuff even in kindergarten and first grade. I've done algebra stuff with second graders, many of whom were way behind in most areas. The variables included stars and smiley faces as well as letters but it was the same idea.

One of the reasons I sometimes stay away from internet education discussions is because you get people who don't know much about contemporary math education talking about how we SHOULD be doing what we ARE doing as well as getting people bitching about how it doesn't make sense even though they also thought their teachers were terrible.

posted by Mrs. Pterodactyl at 8:08 AM on March 4 [5 favorites]

This would also seem like a barrier to changing the way, anything, but especially math, is taught. My mother was a elementary school teacher for 30 years. She's been retired long enough, and spent the last years of her career in a small Christian school that isn't on the forefront of pedagogical techniques. If she had been asked to teach students using algebra concepts, it would have been a real struggle since she hadn't done algebra in four decades. If she had been asked to teach using calculus concepts, she probably would have failed because she never learned them at all. I think things like this are good, but introduction of them is going to be slow, if only because we need to replace/retrain teachers who learned to teach in older ways.

posted by Bulgaroktonos at 8:18 AM on March 4

Meanwhile (also from the Atlantic): BABYS CANT READ

No Duh Atlantic.

posted by Potomac Avenue at 8:35 AM on March 4

No Duh Atlantic.

posted by Potomac Avenue at 8:35 AM on March 4

The maximum amount of time kids will be allowed to play with math concepts will be until the year standardized testing starts.

The only fun thing I remember about math is that the greater-than sign was drawn as an alligators mouth and you have to remember that the alligator wants to eat the bigger number.

posted by shothotbot at 8:39 AM on March 4 [2 favorites]

The only fun thing I remember about math is that the greater-than sign was drawn as an alligators mouth and you have to remember that the alligator wants to eat the bigger number.

posted by shothotbot at 8:39 AM on March 4 [2 favorites]

I already did exactly that with logarithmic function up above using finiteness of the set of taxpayers, Elementary Penguin, adding a negative tax for poor people works easily enough. Imho, we should tax by an equation to constrain congress' ability to manipulate the tax law.

I prefer the logarithm over the logistic curve because my (c) says any two tax payers with the same income ratio have equal differences between their tax rates. A logistic curve eventually says "meh all rich people are the same" while a logarithmic curve says "if you made yourself twice as rich then your ratio still increases by pay this fixed amount". Your logistic suggestion still lets congress place the tax burden on the poor and rich almost equally, which my logarithmic suggest prevents. And the logarithmic model penalizes companies for growing excessively large without penalizing legitimate benefits from economies of scale that exceed some small fixed quantity.

posted by jeffburdges at 8:50 AM on March 4 [3 favorites]

I prefer the logarithm over the logistic curve because my (c) says any two tax payers with the same income ratio have equal differences between their tax rates. A logistic curve eventually says "meh all rich people are the same" while a logarithmic curve says "if you made yourself twice as rich then your ratio still increases by pay this fixed amount". Your logistic suggestion still lets congress place the tax burden on the poor and rich almost equally, which my logarithmic suggest prevents. And the logarithmic model penalizes companies for growing excessively large without penalizing legitimate benefits from economies of scale that exceed some small fixed quantity.

posted by jeffburdges at 8:50 AM on March 4 [3 favorites]

Pretty sure every type of thinking whether cognitive or creative revolves around patterns.

No. Math is about abstraction. Every type of thinking revolves around abstraction.

One of the first cognitive skills a child learns is that an object continues to exist even if they can't see it (e.g. playing peek a boo). This is the fundamental abstraction of math, that objects can have a quality that can be imagined separately from their physical existence.

posted by charlie don't surf at 9:06 AM on March 4 [4 favorites]

Most people confuse arithmetic with the broader field of mathematics. Many mathematicians I know are poor at arithmetic, mainly because it doesn't interest them (although some it interests intensely). Mathematical concepts should be taught from the beginning, not just rote memorization. How arithmetic works is as important as learning the tables so you can do it in your head quickly. Learning elementary versions of counting theory, group theory, algebra, and geometry from the very beginning would enhance children's critical thinking and logical abilities.

posted by Mental Wimp at 9:29 AM on March 4 [1 favorite]

posted by Mental Wimp at 9:29 AM on March 4 [1 favorite]

I can do fairly quick arithmetic add, sub and mult. Division is generally harder for me.

I loved (well... liked... ) algebra. I wanted to pursue the second year of it, but we had to take geometry. It may have been the teacher, but dear god...

Axioms, Proofs. I get it, I mean, ok, now that I'm older and have looked at the world around me and studied shit on my own, I understand why... But... Dear god it felt so pointless.

That geometry class made me hate "math" and killed any desire I had to go any further, which is a real shame, because I think I would've loved Algebra 2, and Trig.

And of course, I'm fascinated by some concepts like functional programming and thus functions and function composition, maps, etc...

And then I'm fascinated by manifolds and topology. I love physics, and string theory and higher dimensions and folding and knots and all that shit just... blows my mind.

It's a real shame, how one teacher can mean the difference in a person's life. Contrast my experience with Violet Femme's above to see what I mean.

I don't think I ever would've been a successful theoretical physicist. The ideas, I love. But it would be nice to have just a tad better understanding of these things.

posted by symbioid at 11:20 AM on March 4

I loved (well... liked... ) algebra. I wanted to pursue the second year of it, but we had to take geometry. It may have been the teacher, but dear god...

Axioms, Proofs. I get it, I mean, ok, now that I'm older and have looked at the world around me and studied shit on my own, I understand why... But... Dear god it felt so pointless.

That geometry class made me hate "math" and killed any desire I had to go any further, which is a real shame, because I think I would've loved Algebra 2, and Trig.

And of course, I'm fascinated by some concepts like functional programming and thus functions and function composition, maps, etc...

And then I'm fascinated by manifolds and topology. I love physics, and string theory and higher dimensions and folding and knots and all that shit just... blows my mind.

It's a real shame, how one teacher can mean the difference in a person's life. Contrast my experience with Violet Femme's above to see what I mean.

I don't think I ever would've been a successful theoretical physicist. The ideas, I love. But it would be nice to have just a tad better understanding of these things.

posted by symbioid at 11:20 AM on March 4

Oh - I guess my point is: make this shit MEAN something to kids. Give them a reason to know it isn't "pointless". I don't know how you make Axioms fun, and that is an essential thing if you're really going to go into math. Maybe experiment with letting them make their own explorations of axioms and proofs (hands on, as the article seems to be implying, or what I gather from the comments since I didn't rtfa).

posted by symbioid at 11:21 AM on March 4

posted by symbioid at 11:21 AM on March 4

I hated math all the way up until I got to geometry, which is when I got my first "A" in the subject. Proofs and axioms were soooo much more interesting to me than computation was.

For me, the abstract reasoning about the nature of the universe was much cooler and deeper than all the stuff that might actually apply to my (boring) day-to-day life.

I don't think my geometry teacher was particularly excellent, but it probably helped that I read a bunch of science fiction around that time which exposed me to some concepts that helped me appreciate how deep the significance of those unprovable axioms is: Clifton Fadiman's anthology "Fantasia Mathematica" and Isaac Asimov's collection "Nine Tomorrows," about Moebius strips and Klein bottles and the four color theorem, about fourth dimensions and tesseracts and geometry on the surface of a sphere or a "hypersphere."

Maybe the math curriculum could include a few awesome SF short stories and pop-science articles? I also remember liking Martin Gardener's "Mathematical Games" essays. And "The I Hate Mathematics Book" which is full of little puzzles, games, paradoxes, etc using probability theory and number theory and so on. And funny cartoons.

I think math education could probably include that and the kind of toys and games mentioned by the article, as well as some times tables drills and long division. Why choose? Why not do both?

In conclusion: not everybody hates geometry. And "Fantasia Mathematica" is an awesome book which everyone should read.

posted by OnceUponATime at 11:41 AM on March 4 [2 favorites]

For me, the abstract reasoning about the nature of the universe was much cooler and deeper than all the stuff that might actually apply to my (boring) day-to-day life.

I don't think my geometry teacher was particularly excellent, but it probably helped that I read a bunch of science fiction around that time which exposed me to some concepts that helped me appreciate how deep the significance of those unprovable axioms is: Clifton Fadiman's anthology "Fantasia Mathematica" and Isaac Asimov's collection "Nine Tomorrows," about Moebius strips and Klein bottles and the four color theorem, about fourth dimensions and tesseracts and geometry on the surface of a sphere or a "hypersphere."

Maybe the math curriculum could include a few awesome SF short stories and pop-science articles? I also remember liking Martin Gardener's "Mathematical Games" essays. And "The I Hate Mathematics Book" which is full of little puzzles, games, paradoxes, etc using probability theory and number theory and so on. And funny cartoons.

I think math education could probably include that and the kind of toys and games mentioned by the article, as well as some times tables drills and long division. Why choose? Why not do both?

In conclusion: not everybody hates geometry. And "Fantasia Mathematica" is an awesome book which everyone should read.

posted by OnceUponATime at 11:41 AM on March 4 [2 favorites]

When I got a job as a cashier my freshman year of college I was taught that method of making change and I was just amazed at how easy it was. I had always been bad at math in school and could barely do math in my head.

I still remember the look on the head cashier's face when I first attempted to make change by subtracting the way I'd been taught to do it in school: on paper and carrying the #s. I'm sure she was thinking "what do they teach in schools these days?"

posted by interplanetjanet at 11:54 AM on March 4

I still remember the look on the head cashier's face when I first attempted to make change by subtracting the way I'd been taught to do it in school: on paper and carrying the #s. I'm sure she was thinking "what do they teach in schools these days?"

posted by interplanetjanet at 11:54 AM on March 4

Literally LOLed.For example, in a group learning about the properties of rhombuses, an artistically inclined person might prefer to draw a rhombus, a programmer might code one,a philosopher might discuss the essence of rhombi, and an origami master might fold a paper rhombus.

posted by Flunkie at 3:17 PM on March 4

I think I remember my parents (boomers) being surprised that my generation was taught calculus starting in high school, instead waiting until college. They were used to high school math covering subjects like geometry, trigonometry, and maybe an introduction to algebra. The same with probability and stats; we were doing intro level stuff fairly soon in high school while they were exposed to these areas in college courses for the first time. Some of Dad's late 60s/early 70s college math textbooks had chapters introducing the same material I was getting at high school. Only in intro level courses, though.

We were also offered two discrete math courses (matrices, transformations, matrix multiplying, "systems" of two equations, etc) in high school, which was new at the time. So new that it was reintroduced to us every year as our cohort was split and combined with other students who hadn't had it in their curriculum the year before. The mechanical, symbol manipulating nature of it always made me wonder if it could be introduced much earlier; maybe even in middle school.

posted by ceribus peribus at 3:23 PM on March 4

We were also offered two discrete math courses (matrices, transformations, matrix multiplying, "systems" of two equations, etc) in high school, which was new at the time. So new that it was reintroduced to us every year as our cohort was split and combined with other students who hadn't had it in their curriculum the year before. The mechanical, symbol manipulating nature of it always made me wonder if it could be introduced much earlier; maybe even in middle school.

posted by ceribus peribus at 3:23 PM on March 4

How our 1,000-year-old math curriculum cheats America's kids: By hiding math's great masterpieces from students' view, we deny them the beauty of the subject.

posted by homunculus at 6:34 PM on March 4

posted by homunculus at 6:34 PM on March 4

This reminds me of the Penny Arcade comic from 2 weeks ago.

posted by kimberussell at 7:31 PM on March 4

posted by kimberussell at 7:31 PM on March 4

Cookie Shapes! Avoiding math to have a relaxing Saturday with friends.

posted by homunculus at 7:46 PM on March 4

posted by homunculus at 7:46 PM on March 4

Yes. This. I think the most important aspect of education is to engage the student's intellect. Everyone is imbued with curiosity until it is beaten out of them by a system of rote memorization of cold facts (3+4=7, George Washington died in 1799, the universe started with a big bang, humans are primates). Snagging this curiosity in whatever subject you want to teach as early as possible is the key to a lifelong learner. It is so easy to fail at this if you are a designer of curricula or a teacher. We needed motivated, high quality educators if we want our society to function well.

posted by Mental Wimp at 9:51 AM on March 5

>Literally LOLed.

I LOLed too.

You remind me of long ago when I was a freshman in Art School, taking Design 101. The professor was a Chinese fellow named Hsu. He gave us intensely difficult assignments with short notice, starting with Day 1: create 100 4x4 inch cards, each uniquely inscribed with two non-parallel lines edge to edge. Make them interesting. Assignment due in about 48 hours, at the next class session. At the next class, we'd get another assignment, create 100 4x4 inch cards, each with two unique curves. Even the most desultory effort took hours, just to create 4x4 cards to his exact specifications.

Then after about 2 weeks of this intense work, he gave an assignment I will never forget. Create 100 pairs of 4x4 cards. Join each pair of cards without glue in a manner that they cannot be pulled apart in any direction. Folding permitted. Most people used the same strategy as me, folding edges. It was nearly impossible to come up with 100 unique sets of folded cards, after a few examples, they were repeats of the same idea. I spent hours and only came up with about 20. Most students were unable to produce more than a few designs. At the class session, everyone sheepishly assembled their few cards on their desks, already aware of their abject failure, then Prof. Hsu went around and pulled everyone's cards apart. A few of mine held together, but he teased most of the cards apart in one direction or the other. He went around from desk to desk, pulling apart each student's cards, and briefly explaining how each design was a failure.

Then he suddenly seemed to grow tired of this exercise. He went in front of the class and said he would show us how it was done. He took two sheets of 4x4 paper, placed them in his palm, and crumpled them into a little ball. He tossed it on the desk. Then he took another pair of cards, and crumpled them into another ball. Then another. After creating about 5 of these little wads, he held them up and asserted they were each unique, and then he showed how you could not pull them apart without unfolding them completely.

Then he launched into a tirade about how stupid we were, and not just that, how stupid

I never forgot this lesson. But I have forgotten everything else in that class that happened after it.

posted by charlie don't surf at 1:11 PM on March 5 [5 favorites]

This reminds me of something I haven't thought about forever which also proves your point.

My family had a shitty car with a shittier speedometer such that when driving to my grandparents on the interstate my dad was never quite sure if he was speeding or not. My mom was always convinced he was going too fast; he was convinced that she just wasn't use to driving in a car that rattled when you got going to a certain speed.

I basically learned to do math pre-kindergarten basically because they bothered to explain why they were timing the distance between mile markers to settle that argument.

Even after we replaced that car, I was probably in my teens before I stopped checking our speed on long family road trips.

posted by MCMikeNamara at 8:47 AM on March 6

Your anecdote is totally consistent with the idea that young children can be taught the concepts behind the mathematics from the start. The fact that we don't teach them is probably why so many kids are turned off to the maths before they actually ever learn any. I tried to teach my kids concepts from the time they first exhibited curiosity about things. Neither became mathematicians, but they never had a hard time with their schoolwork in that area either. I do believe they became better thinkers, because of it, but that may be confirmation bias on my part.

posted by Mental Wimp at 10:19 AM on March 6 [1 favorite]

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posted by polymodus at 12:27 AM on March 4 [5 favorites]