Critical thinking vs education: Teaching kids math without "correct" answers


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I’m not sure what’s meant by a math problem with no one answer. I mean, quadratic roots can be one, or two real or complex numbers, but the roots definitely can be found.

I fell in love with math after getting on effective treatment for ADHD which allowed me the ability to finally focus two seconds on something that isn’t immediately riveting.

In high school, while the teacher would go through their preferred method for solving a given probem ten times, I would come up with multiple equivalent processes for doing the same thing. I’d prove them, and re-teach the methods I came up with when half the class came to me not getting the standard method the teacher failed to get across.

I probably should have been in more challenging math classes, but I enjoyed teaching, and had the determination to make sure other students I helped actually understood the math instead of just letting them copy my answers (which always made me feel dirty. Sticking to my guns, I passed up making a lot of potential friends because I wouldn’t give them a freebie on my own work.)


Math class is bollocks and could easily be incorporated into other subjects. Modern education is hamstrung by the compartmentization of its core “subjects”. This fundamentally deprives traditional “core subjects” of the rich interconnectedness that actually permeates the world. We should be teaching math in science, history, music, and art classes. Numeracy and history is amazing. Music theory is math.

Skills practiced in isolation lack meaning. I mean, if you are studying a new language, what is more effective: reading a dictionary or sitting down with someone and having a conversation? Math, like language, is a tool for understanding; but in isolation it is just dull.


Teaching kids maths without an “s”.

edit: oops I missed the “ing” of “Teaching” originally.


And would the next thing be money instead of monies? Unless you’re emphasise funds coming in from multiple sources.

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When did we brainwash kids into thinking that math was about getting an answer?

When they invented mass education. Simple arithmetic calculation has been the model for mass education from the beginning. It’s only through the rebellion of teachers and students that mass education has ever included humane values.

Many of us now in mid-life went to school with teachers who were trained in the 1970s, a high-water mark for progressive, humanistic teaching – including ideas like constructionism – under the influence of the social rebellions of the late 60s and early 70s.

We’re a long way from that era, politically, and crudely instrumentalist ideas about education dominate.


Looking at the actual work, it looked to me the confusion was the set of instructions. The second run looked like to me, the students finally understood what the teacher was trying to get–the reasoning behind the answer.

The answer could be different and was not the desired result, rather the reasoning for the result was the true answer. Dang, I barely understand what I just typed. I’ll try to be clear, I think the student’s didn’t understand what the teacher wanted. What I think the teacher wanted was the reasoning for the answer, where the answer could be anything as long as you can make a good reason for it.


Correct answers in math can be useful.

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Maths are an illusion. A very useful fiction for parsing the world we interact with. A good way to use one data set, if that’s what one wants to do. The function of math, to me at least, is making the natural world usable/manipulable to us thumbs-wielding tool users (for better or worse).

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People who suggest this never seem to elaborate on which particular mathematic they want to keep.


Hooray for New Math!


There are many problems with no one answer.

For example, I remember a professor in college throwing this out: “how many ping pong balls would it take to fill this lecture hall?”

Some people shouted out things like “a million”, “a billion”, but he was trying to get us to think abou the problem, not simply guess.

“How big is a ping pong ball? About an inch in diameter. ABOUT how many in a cubic foot, then? 12 is close to 10 so 1,000 to a fair approximation. Don’t worry too much about how they’re packed.”

“I’m about 6 feet tall, the ceiling is about three times my height, so how high is the ceiling? About 18 feet, call it 20. The floor tiles are about a foot square, so how wide and long is the room?” About 60 by 100 feet.

So 20 x 60 is about 1,000 (we over-estimated the height and under-estimated the width). Multiply that by 100 and we get 100,000 cubic feet. Each cubic foot holds about 1,000 ping pong balls, so we estimate about 100 million balls.

That’s certainly not the exact answer but thinking about a problem like that helps one figure out when something just doesn’t make any sense. I’ve see many “cool inventions” where a simple estimate blows holes in the claims… “Let’s put spring loaded generators in roadways in front of turnpike toll booths. The cars driving over them will provide all the power for them…”
Or “solar freakin’ roadways…”, which seem to have recently been popularized on the web. Being able to estimate, not getting any one particular “correct” answer, but within an order of magnitude or so is a very useful skill.

In the article, being able to rank “unlikely” “50% chance” and “will I go to the beach?” is a good skill to have so one can visualize and discuss a problem.

Beyond all that, Kurt Gödel’s Incompleteness Theorems mathematically demonstrated that some problems can provably have an answer that can’t be determined, some statements are provably both true and false, and other mind blowing concepts. Douglas Hofstadter’s excellent book Gödel’s, Escher, Bach; an Eternal Golden Braid provides a wonderful explanation and contemplation of number theory and it’s limitation.


It’s a problem about how we deal with mathematical concepts, not an arithmetic problem. In short, the question was “Put these cards with numbers, statements, pie charts and words on a line from 0 to 1”. Given that some of them were fuzzy (is “likely” a 75% chance? Is it higher than “you will watch TV today?”), there is no one correct order - and the point of the problem was to make the pupils think about what probabilities (and their numeric forms) mean.

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Maybe our daughter’s kindergarten teacher is an example of bucking the trend. I was at a school party. All the kids had gathered around the teacher as she prepared to draw a name out of a box for a raffle.
She asked them, “Won’t it be cool to win?”
“Yes!”, they all replied.
“Are you probably going to win?”
“No!”, with the same amount of enthusiasm.

There was certainly a correct numerical answer to the probability that any one of them was going to win, but they didn’t need to know it to prepare emotionally for the outcome.


I’m pretty sure what you’re thinking of is what people who know math refer to as “arithmetic.” I fully agree that giving examples of statistics in real life, or math in basic physics, in math class would help most people understand it, but I don’t think we can give up on teaching is as a subject. If we want a pool of people in our society who are capable of doing things like designing the computers you’re using to write that post, we need to at least give a solid shot at educating everyone in basic math, so we can get it to stick on some people. It would be awesome if some significant percentage of our society actually applied quantitative thinking to their political thought, but I think that might be unrealistic, even if we did improve our math education.


Curious if you would share how you treated your ADHD.
I have terrible ADHD and since expatting myself have had a nightmare finding Addreall at an affordable price, the state doesn’t consider it an official treatment list for ADHD, the private price is about $200US/month which in turn makes finding good paying steady work difficult to find and my current certification classes very difficult in a different language.
Before anyone says Ritalin(on the state pharmacy list) I will add that taking Ritalin in effective doses makes me so sensitive and irritable I feel like I am packed inside a box of broken glass. Any sound or touch irritant as well as bright light or clashing colors is nearly painful on Ritalin not to mention the feeling that you are a walking ghost, adderall just makes me clear and smart.

This is a bit of my followup on women in STEM jobs (see the article about Google’s miserable employment rate of women), but THIS to me is IT - the reason so many people men or women - get sick of math and science in high school. And in my experience working with computer science and engineering, very very few of my colleagues can do the math required to calculate the tip on their restaurant tab, much less any math in their job. They use the computers these days for calculating. Of course they understand how to set up the problems, that is the real skill they learned. But what if in school we taught that instead of focusing so much on the answers and showing work - like, why NOT let the kids use computers and calculators to calculate the answers if the real skill is to figure out the right approach to the problem? Why not have this taught as part of a creative real world exercise of building a structure or a circuit board or setting up an actual experiment and then figuring out whether the answer is significant?

The one-eyed prejudice of maths presentation didn’t bother me so much; either it suited my learning style or I was smart enough to make do.

But what utterly shat me to tears was the glacial pace of the curriculum, and the steadfast refusal to give me a fast track.

The one remotely significant bit of catering to my ability I recall being conceded in primary school was an extra drawing class. I wasn’t thrilled.

It was busywork and childminding for the most part. My whole time in school felt like detention.

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I’ve had good experiences with cognitive behavioral therapy (in conjunction with Ritalin, though, I should add).

The trouble is that the behavioral therapy really helps to have a therapist to at least get you started on it and that’s even less likely to be covered by insurance than Adderall. Because the health care system is whack.


This story reminded me of something that happened to my daughter last year in grade 3. She had a probability math test on which one of the questions was “You will go to sleep tonight a) definitely not, b)probably not c)likely d)definitely” Fuled, I suspect, by epic stories from Daddy about all nighters pulled studying for exams in university she choose c. The teachers marking matrix had d and he marked it wrong. She argued the point but he said her mark stood.

I think the lesson she learned had little to do with probability. But my husband did let her stay up all night the following weekend so she could be satisfied that she was right.

Talking to some of the other parents I now know my daughter wasn’t the only one to argue an answer on that test although each child seemed to have issues with a different question. Props to the teacher in this story for recognizing that different experiences may lend logic to different answer in this kind of probability.