I don’t think he’s arguing against mental math. I think he’s arguing against teachers defaulting to those kids lucky enough to have stumbled across some mental math skills. We either need to teach mental math explicitly, or teach the math we do teach in a way that doesn’t assume it.
And I am going to argue that basic Algebra, at least, is part and parcel with basic numeracy. Numeracy is not just the ability to perform memorized procedures quickly–it’s understanding how numbers work. A little algebra really helps in the understanding of the three properties, and the insight that math is about relationship and proportion, not just numbers.
Sure then. Just like music training (having just listened to my daughter practice piano) is part scales over and over, and part music theory, and part playing real songs. We do need to teach/train mental math explicitly, not just assume it. That’s scales.
To clarify, I absolutely agree this is an important and central question, but I was just asking if it was the question you guys are talking about or not. It seemed like some comments I was seeing were presupposing we’re all talking about what every kid should know, while other comments I was seeing were presupposing we’re talking about what should be available to every kid. So I asked–which one is it you are all talking about?
Today I noticed that my kid multiplied a large number by six by first multiplying it by three, then doubling the result. He wasn’t taught to do this, he says, he just figured out it should work and finds it easier. As it happens, that’s also how I would multiply a large number by six.
Is this how everybody does it? I’m never sure about things like that.
Should kids be taught this as a specific technique? Or as an alternative to just multiplying through by six? Or should they just be taught to multiply straightforwardly by six, letting the ones who like to play around and have good number sense (and are lazy like me and prefer not to memorize that many facts) stumble onto the “easier” way I just described?
I feel like these questions have relevance to the thread but I know they’re slightly off center of topic. Treat them as you will in light of that fact.
Adults benefit from math period. The mental aspect is immaterial.
If you see something on the rack for $85 and saw that there was a 15% off sticker stuck on, how would you go about calculating that? How would the general population approach that?
Would you be able to figure it out instantly? Would you at least knew how to set up the problem before punching it into a calculator? Would you just fudge it and assume it was $15 off, more or less? Would you not even bother?
The idea is to get as many kids as possible out of the “not bother” category and into the “at least know how to set it up” category.
Ultimately, what difference does it make if someone can do it in their head or if they’re doing a little side calculation? The problem gets solved either way. The concept of how to find a percent is successfully conveyed. Who cares if it’s mentally done or otherwise?
Because the concept of “how to find a percent” is not what we want to teach. What we want to teach is what a percent IS. This matters not just when adjusting a price, but when watching the news, evaluating a new salary, comparing two numbers . . .percent change is everywhere.
If you know what a percent is, there are a dozen different ways, some mental, some not, to find 85-15%, with different degrees of precision–and different circumstances call for different degrees of precision. If the person knows the concept, they can find the method, or be very rapidly taught any number of methods.
I don’t agree that there is a difference between a mental way and a non-mental way. There are different ways for sure - like Frylock’s kid showed by multiplying by 6 - but there’s nothing intrinsically mental about one way vs another.
Understanding what percent is, what fractions are, proportions, compound interest even … these all fall under not needing to set up algebra problems (yes, I understand that the 15% off requires a two step process) or using trig let alone calc. By 7th grade students have been taught this stuff. It’s pre-algebra tops. The math that most adults use, to the degree they use math at all, (again including many who get advanced education even including in many sciences) has been covered before High School.
Covered but not mastered. I’d argue that mastering that level of math is what counts as basic numeracy, being able to use basic math as a matter of course just like we use reading as a matter of course. The level of math skills that most college educated adults have is akin to reading by sounding out each word and avoiding reading things as a result. They took High School algebra and geometry, they passed the test, maybe even got a good grade, and they never applied it to anything outside of those specific classes. Pretty much it is gone within a few years.
Of course I appreciate the ways of thinking those subjects have the potential to teach even if the specifics are never remembered. I appreciate the need to expose many to more in order to have the relatively few ready when they decide to study advanced subjects that require the knowledge and skills.
It’s again just that I disagree with ITR over how we are failing most of these kids. The big issue is not that we are not getting more intellectually excited about the wide world of math; the big issue is that the vast majority of students end up as adults not able to quickly estimate what 15% of $85 is and avoid applying math as a basic tool of life and learning.
Today’s NYT has something to say. The article agrees that the educational system is failing to excite kids about math but thinks the need is for more applying of the math in the educational system. Real world applications best yet.
Let me chime in - BG: I was a pretty normal kid for the mid 60s - unfortunately 4th grade hit me hard. I moved from one school system that taught adding, subtraction, multiplication and division in 3d grade to one that also taught fractions. When I started school it was the dreaded review last years math for the first 2 weeks. Of course I managed just fine in the addition, subtraction, multiplication and division then I hit the wall, and the wall hit me. I sat there looking at a page of stuff I had no idea about and was told to finish the work on the page. The teacher cut out of the room leaving all us little sprogs to work on the page she left us with, which obviously sat blank on my desk when she came back. So I got yelled at for refusing to do my work - my whole whimpered ‘I don’t know what it is’ was responded to with a rough you did it last year, you need to do the work. This went on for a week with various amounts of time sitting in the idiot seat as punishment until Friday came around and I was sent to the office. My mom came in, and finally the problem was discovered - that I really DIDN’T know how to do the work. Unfortunately this fucked me up with math for the rest of elementary and middle school - until 10th grade. It wasn’t that I didn’t manage to learn anything, but I was so traumatized for being punished for something I honestly didn’t know, I was eternally afraid of being punished for not knowing it fast enough, or good enough.
And I will point out that of all the math I learned, through algebra/trig and geometry - once I stopped working as a machinist, and switched over to accounting I pretty much ended using any math I learned in HS - accounting is very different and now you barely have to learn anything with all the programs available you are more data entry than anything else. Hell, I watched someone doing some computer aided machining, and all they did was plug in numbers into the machine, there was no learning the touch of working a lathe. I could train a 10 year old kid to plug in numbers and feed stock in.
What you say is true, but there is a bit of a skill ceiling depending on how much memorization you’re going to do. I can do algebra and basic geometry relatively quickly, but I’m capped below a lot of my peers because I never bothered to put some things like 30/60/90 triangles in my head. Why? Because you can derive the damn thing from an equilateral! So while many of my peers have a bunch of special cases memorized making them lightning quick, I’m there drawing a triangle (either literally or mentally) and solving for sides.
This doesn’t take too long, but it gives them a distinct advantage when it comes to tightly timed tests (like the GRE and such) where that extra 15-20 seconds can make a difference. However, I don’t think you can really say they have any greater understanding of this level of geometry compared to me. Nor are they more well equipped to solve a problem involving basic triangle geometry.
School has a silly number of special cases, I recently learned about “work rate” problems, which I’ve learned there’s a special form for.
So there’s this special form for this type of problem, apparently 1/x = 1/a + 1/b.
My intuition? This class of problem is a special case of the “velocity” function x(t) = x_0 + vt + at^2. However, in this case rather than, say, meters/second, we can value our function as “jobs remaining/unit of time”. Where x_0 is 1 job left and x(t) is 0 jobs left, where the “velocity” subtracts away how much of the job is left until it’s complete.
So 0 jobs = 1job - (worker A’s rate + worker B’s rate)t + 0job/hr^2 t^2
=> A’s rate + B’s rate = 1/t jobs
Which is actually said “special case”. I still solved the problem. I identified what’s going on, set up a valid scenario, and solved it. But I went through significantly more reasoning and steps than someone who goes “oh… this is a work rate problem” and parrots the special case.
This was a huge problem for me in high school, everything we learned was super nebulous, and often felt taught backwards in many ways. Like learning the quadratic formula well before completing the square. Or learning foil as a Thing™ while mostly ignoring how it’s a sort of just a really special case of the distributive property. Things are given as facts and formulas with no real context, and (as with foil/distributivity) in some cases it’s up to you to puzzle out that they even ARE related.
Now obviously you can go bonkers with my logic. I’m not saying we should start kids off trying to derive physics from first principles, or introduce rings before we talk about adding integers. No integral calc just to teach the area of a circle. But in my experience, there’s an odd amount of emphasis on special cases and solving things under artificial time constraints that don’t necessarily measure the sort of understanding they purport to. Mental math can be taught, but being taught how to solve a class of problem quickly and in your head isn’t always an inherent good.
