# Thoughts on teaching calculus to five-year-olds

**doctorow**#1

What would they use it for? Is this a troll post? What would 5 year olds use toys for, or soccer, or paint sets?

I think the loose point of the article was that 5 year olds can learn rather advanced math *if* it's given to them like a toy or a paint set. You aren't supposed to worry about the practical application, it's this kind of thinking that kills the subject.

I probably deserve a thatsthejoke.jpg at this point.

Are young kids capable of abstraction?

Gotta be. From the first brainwaves, to birth, to growth, everything they perceive in any manner must pass through the abstraction until it's settled in the mind. I envisage it as an ever-diminishing curve, where the mass of experience and increasing focus on particular arcs of the 360 of life means that as time passes, we are less and less in abstraction.

Until of course, we get old, and the system starts hiccuping. Then we might reverse the trend.

They're more able to function in abstract than adults, they just don't yet have terms of reference.

**jandrese**#5

They'll use it for the same thing most adults do: Nothing at all.

If a 5 year old can handle this, then great! More power to them. I'm a little skeptical though, symbolic reasoning is one of those things that most kids struggle with for a long time, and 5 is very young. Maybe if we get them started earlier it will be easier, but that's a rather optimistic statement.

**Ethel**#6

Reading is in itself an abstraction, there is nothing concrete to it except making the letters with a stylus (concrete would be hieroglyphics, things that are one to one in symbolism, this is not true of our written word). Reading is an abstract process, and while reading should start after age 5, we start pre-literacy at birth. So what's the point of reading?

Math has real world applications, if only counting the cars on the highway (it is how I drive to keep track of people that may be in my blindspot) and keeping track of pattern flows. Calculus is important in building and considering materials needed. We use math even if we don't abstract it into language (written math). And I consider math a language, one that is neat and precise. Well, as long as it is not illogical numbers or any funny stuff like statistics or trig. Algebra and geometry have important thinking process that are applicable in coding and in understanding genetics and biochemistry. For our kids to understand what we are learning now and expand on it, math is important early.

Cory didn't mention "practical applications"; he asked what they'd use it for, and I'd think that could include play. But that begs the question of *how* they'd use it to play. Using a paintset is a matter of grabbing a brush in one fist and smearing it around to make pictures. What does a 5 year old playing with calculus look like? For example would it be fun for them to maybe draw a graph of df/dx given a graph of x? I'm skeptical but I admit I don't know the age group well enough to have a good intuition about that.

**Hyphen**#8

Bill Nye said that studies show the children who are good at algebra tend to go into sciences. That's a pretty good reason.

**Hyphen**#9

Math is very important, and if they are learning anything related to calculus, then it's probably worth it. Having said that, there's a pretty big difference between reading and math. Reading is using words to represent physical objects (especially at 5). Calculus and even algebra is using an abstraction (X) to represent another abstraction (a number) that you don't know. I had always heard that type of capability didn't exist in young children-- for similar reasons, you can have a toddler pray to God all day long, but he or she doesn't have the abstraction ability to understand what they are praying too. But theories change everyday. I could be wrong.

**micah**#10

Last year my son's kindergarten was doing combinatorics problems nearly identical to the combinatorics problems my cousin was teaching to math and computer science majors at the University of Pennsylvania.

The Ivy League college students were expected to use more sophisticated strategies to arrive at more precise and complete solutions than the kindergarteners (who were effectively brute-forcing the answers), but the questions were essentially the same.

I talked a bit with the kindergarten teacher about it. She had never heard the term "combinatorics" and was a little surprised to hear that the same problems were being taught at such a high level. In her class the point wasn't to get the "right" answer, but for the kids to spend time thinking about the relationships of the numbers, and from my conversations with my son I think it succeeded.

When folks read about kids doing algebra or calculus at age 5 they think back to their high school algebra and calculus courses and assume that kids are learning about slope intercepts, quadratic equations, derivatives and integrals. But just as my son's kindergarten class never heard the word "combinatorics" yet used combinatorics problems to learn more about how numbers work together, there are many ways for very young kids to learn concepts that are core to algebra and calculus without having to memorize formulas or visualize the X and Y axes. And spending time thinking about those concepts can help them tremendously later in life.

Heck, I was doing geometry at age 5 when I had to learn about angles to make the Apple Logo turtle draw cool pictures on the computer screen. I don't think I realized it was "geometry" until I got to middle school, but that doesn't mean it wasn't helpful to my ultimate ability to understand math.

**crashproof**#11

Also conditioning you to prefer Apple over other brands. (...there were no TRS-80s around by the time I was an adult with computer purchasing power.)

**stephen_schenck**#13

"Even if 5-year-olds understand calculus, what would they use it for?"

Calculating the integral for the area under the covers, I assume, to make sure there's no room for monsters.

**sebwiers**#14

I'm guessing kinda like my two year old playing angry birds... only cracking open the dev tools and rebuilding it. Toddlers are pretty adept with computers if you give them the chance.

**crenquis**#15

Teach em molecular biology and gene sequencing instead -- let them direct their own evolution...

**techdeviant**#16

Maybe early learning of any math in a fun way allows for more acceptance later in life. I believe that fear of math is a big problem, so even if a 5 year old doesn't learn calculus as high school students learn it, at least they can learn it is not scary?

**Boundegar**#17

Tuck in the sheet and it becomes a boundary value problem. They won't solve that until at least 6-1/2, and by then it's too late.

**noahdjango**#18

a caveat:

as a child who started in Ann Arbor public schools, we were taught only CEMREL math

this was a curriculum that emphasized mathematics as ideas over number-crunching. which was fine by me. until I moved away in the third grade. I'm not much of a math-oriented person (though I find the concepts interesting.) as such, going from CEMREL to traditional math--overnight, mind you--was crippling. Separating a child from the larger culture is never a good idea. Teach both, but don't set your kid up for an integration failure. It's not fair to them.

That said, I also had these books growing up, which didn't help me with my school work of number-crunching *at all* but they were pretty neat understanding-math-broadly books.

**PaulDavis**#19

calculus or algebra? there is a bit of a difference, and the difference is frequently glossed over in this math-illiterate USA. teaching 5 year old kids stuff that leans towards symbols to represent things : seems good if it can be done right. trying to get them to understand integration and derivatives? seems a bit pointless.

it actually feels just a bit pointless even to adolescents these days. number theory, set theory, boolean logic and algebra, and more ... these are all more appropriate in a world where discrete rather than continuous math is the order of the day.

**AnthonyC**#20

There's a lot of fun to be had learning these things in the right environment. My concern is that the article glosses over how much the adults involved need to know to make it work. Most kindergarten and elementary schools teachers I know stopped taking math after high school. *They* don't understand math at anywhere a deep enough conceptual level to really guide kids to such an understanding. Heck, my 4th grade teacher couldn't divide numbers with decimal points without a calculator (apparently she wasn't good at problem solving that wasn't purely algorithmic, because she knew how to divide fractions and how to convert decimals to fractions).