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One of my math professors sugggested adding a formal logic class to early childhood education.
One of my math professors told us that when he started elementary school they tried starting maths classes with logic and combinatorics, because they were most essential maths and in principle could be experienced by children by seeing, feeling etc. He said it was a stupid approach. I say he turned out a math professor, so maybe it worked.
People who want school to be practical scare me.
yes, practical skills change year to year.
what’s important is to learn to learn.
I’m genuinely curious why, if this is serious. I feel like adulting badly needs to be taught better. I’m nearing mid twenties and still get so confused at a lot of adult things, especially government shit, because it’s just so much to figure out for the first time.
It’s definitely important to teach math and science and language, and to teach people how to do their own research, and think, and learn, etc. But are you saying practical skills shouldn’t also be taught?
I interpreted it as a criticism of those who think there’s no point to learning something if there isn’t an immediately-obvious application for that knowledge. Like those who say, “What’s the point of learning history? I’m not going to become a historian,” as if learning needs to have a clear end-goal or else it’s useless. Or those who think it’s pointless to learn to play an instrument because you’re not going to become a famous musician. It’s a mentality that ties in with capitalism, where if you’re not being productive, you have no use.
A well-rounded education should equip students with skills they can apply independently no matter what they do. Learning history provides context for the world we live in, why it is the way it is, and can inform us on how to move forward. Learning to play an instrument builds new connections in the brain, strengthens fine motor skills, and (in the case of reading music) how to move information between abstract concepts and a tangible form.
These skills provide benefits to people that can be built upon in the future. They may not have immediate usage to a student, but they create a foundation upon which a student can reach higher as they progress in life. Not every lesson is practical in the moment, but that doesn’t mean it can’t have value to a growing mind.
If anyone taught you how to do your taxes at school age I bet you’d forgotten all about it by the time you needed it
As OP said, what’s important is to learn to learn
It is sad that the general population is unable to see learning math as good in of itself. Not everything must be solely “practical.”
Math education is basically a Time waster designed to justify hierarchies, it’s tangentially related to math but not really in purpose, there’s just numbers involved
Not really. Not everyone enjoys advanced mathematics the same way not everyone enjoys english literature or engineering, or arts and crafts. People have different interests, aptitudes, and skills. That’s how the world works.
they still should learn them.
you need to know how the world works a bit to be a good citizen capable of critical thinking.
I’m the guy in the background saying “go back to teaching Euclid and proof in schools”, as the real point was to teach logical deduction from established facts.
Logic puzzles should be applied in more classrooms. Start with simple problems in elementary school, and progress to more challenging ones as students grow. Critical thinking needs to start early.
If you are talking about school curriculum, nearly the entire population will keep not learning it as long as it doesn’t have some practical application so people can understand WTF the teacher is talking about.
Citation needed.
Seriously, though, that’s not what the research is showing. Peter Liljedahl’s research, for example, supports that a very effective way to teach mathematics is by having students actually think about math, instead of just passively receiving info dumps (as is common in most traditional math classes). See Building Thinking Classrooms for details but, in short, it’s a method of getting students playing with math concepts for almost the entire class time every day.
No “practical applications” needed. Counterintuitive, but it’s a highly effective practice.
What’s core to practical applications working is student motivation, and practical applications are one way to induce motivation. But it’s often not the best option, especially for inherently abstract skills.
Peter Liljedahl
So… From the publications, looks like he uses problem solving, not “having students actually think about math”.
You want students think about what exactly if you don’t give them an application?
Anyway, thanks, I’m listing his work as evidence supporting my claim.
Anyway, thanks, I’m listing his work as evidence supporting my claim.
Remembering this for next time I clearly don’t understand something. lol
If by “practical application” you mean “motivation for learning the skill”, which is I think the way you’re using it, then yes. But that’s not the usual definition in math education, and not what most people mean by it.
Like, for example, to introduce quadratics, a good progression might be to challenge students to build a table of values and graphs for x², then x² + 3, then graph x² – 5 without a table of values, then 2x² vs. 5x² vs. ½x², –x², etc.
And if you have a Thinking Classroom, every student in the class is working on figuring out that progression collaboratively in small groups. The teacher guides students to discover the math themselves through a series of examples, and mostly interacts with the students by asking questions, never giving them the answers.
That’s not “a practical application of quadratics”—at least not in the usual definition—that’s a learning activity sequence (paired with a set of interrelated pedagogical practices).
A good, practical application of quadratics is more like a Dan Meyer “3 Act Math” lesson on predicting the trajectory of a basketball shot. Also cool, good teaching. But not a great way to introduce quadratics.
(P.S. Yes, I use and like em dashes. I’m not a robot.)
To be honest, i’m not sure what you want.
Like, if i was the student, i think i would be extremely confused from this lesson. I would not know what you want from me. I have had my fair share of teachers trying to get me to “just think about something and figure stuff out myself” which mostly amounted to me sitting there in classroom, staring into the air, confused about what the task is, and mostly waiting till the hour is over.
My brain works differently. When i learn something, before i even start caring about what the topic is, i ask why I’m learning this; and i need to have a proper reason to learn something. The reason needs to be strong enough, and is only strong enough if it is derived from some other, stronger reason. For example, i learned maths because i understood how important it is to grasp the universal, those things that cannot be taken away from us. I grew up in a kinda abusive household, and my mother had a habit of taking away the things that were most precious to me, so i clinged on to maths because i knew that maths was eternal and not dependent on the whims of my mother. That is a clear, practical reason. Maths gives me mental stability, like a skeleton gives stability to the body. It does not shake nor break; for it’s eternal.
Now, if you want me to play around with polynomials, idk what i would do.
Typically, when i learn something, i want to know why but also how to learn something. Especially, to express it in an analogy, my brain is like the C programming language. I need to reserve memory manually, it does not happen automatically, and i need to know how much space will be needed beforehand, in other words i need to have a clear understanding of how big a topic will be before i actually start learning it. When i have no idea what i’m getting myself into, then i don’t get into it, because my brain is very very very (i hope i have made this clear enough) bad at learning many small incremental pieces of knowledge. In fact, it’s similar to if you had to put on your jacket, leave the building, go through the cold icy air into the neighboring building each time you want to get yourself a glass of water. Needless to say, you will not drink a lot of water. You will dehydrate. Obviously you would put yourself a large bottle of water into your room, for which you only have to leave the building once. The same applies to me and learning. I have to take very few, appropriately sized portions of knowledge into me at once. Not many many small ones.
I don’t have time to get into the full 13 (? iirc) steps of Liljedahl’s Thinking Classrooms approach, but it’s exactly designed to meet the needs of students like you. Since highlights:
- Students are randomly assigned to a new group of 3 daily
- All students work on vertical whiteboards, or equivalents
- The teacher presents a math task that starts easy-ish, but requires some work/thought to figure out
- If 30% of students in the room understand the task, then it will quickly trickle between groups
- The teacher circles exemplars of great thinking; students are not allowed to erase these until the next debrief
- The teacher regularly cycles back to get students to explain their work to the class, showcasing and explaining the bits the teacher circled
- Start over with a more advanced task/“next step”
It’s an incredibly effective teaching method for secondary math. And there’s clear motivation every step of the way for what you’re doing and why it matters.
And the teacher only explains about 5-10% of the material; everything else is explained by the students as the carefully curated progression of activities guides them through discovering the math themselves.
motivation for learning the skill
I mean motivation for why somebody cares about the idea at all, but I think that is less strict so yes. A hole in theory or something emerging from an activity are perfectly fine. But there has to be something there.
Not sure why you’re getting downvoted. Having practical applications for higher math makes that shit stick like glue when otherwise it would get forgotten immediately after the test.
They’re getting downvoted because there’s a lot of esoterical people demanding that we learn stuff either for the joy of it (which many are not at all having btw) or because it “purifies character” or sth.
Practical applications are felt like an impurity to that.
Apparently knowing people learn differently and that mathematicians are a tiny minority is neoliberal…
Welcome to the Fediverse, I guess.
As the experts say: “Use it or lose it.”
Maths education is pointlessly overcomplicated. We need to simplify and streamline it. And also add in more practical real-world examples.
Especially when you’re forced to use a complicated method to do basic calculations with. People should be allowed to learn different ways to get to the same answer.
I’m with the pink guy, fight me
I’m always wary of the idea learning should be “practical”. You never know when something will matter and there is an intrinsic value in learning for learnings sake.
Learning needs to be tangible, but I’m not sure it necessitates practicality.
Just look at Boolean mathematics. It was developed 100 years before it had a use (digital logic)
The hidden Factor here is coercion, if you don’t go to school the cops will literally show up at your door eventually. In light of that it’s completely reasonable for the people who have no choice but to be there to ask what purpose it serves
Sure, but learning tends to be easier when there’s a practical application to the things you’re learning
That kinda breaks down in practice, though. Math is hard for a lot of students. Adding an extra layer of domain-specific application on top of an already confusing topic just makes it worse.
Like, we need polynomials for huge swathes of higher-level math. My favourite application of polynomials is that most continuous functions can be approximated by a Taylor series, which makes some functions that are otherwise impossible to calculate a derivative or integral trivially easy. It’s elegant, beautiful, and deeply practical.
And completely useless for a grade 8 student learning about polynomials for the first time.
Sure, there’s lower-hanging fruit for practical uses for polynomials, but they’re either similarly abstract (albeit simpler) or contrived. Ain’t nobody making a sandbox with length (3x + 5) and width (2x – 7), eh?
I could go on. At length.
Point being, yes, practical applications are better. BUT (and this is a big but) only when there are simple practical applications.
Instead, recent math education research supports teaching fluency through playing with math concepts and exploring things in many ways: symbolically, graphically, forwards and backwards, extending iteratively with increasing complexity, etc. This helps students develop intuition for math concepts and deeper understanding. Then, and only then, teach the standard algorithms and methods, as students will appreciate the efficiency of the tool and understand what they’re doing and why they’re doing it.
Thank you for listening to my TED Talk.
Polynomials:
They exist because they are efficient to compute. Computers do well with basic arithmetic operations like addition (+) and multiplication (*). The polynomial functions are simply those that you can construct from those two operations, and constant numbers.
Like consider a polynomial like f(x) = 5x^3 + 3x^2 + 2x + 7
What it really says is
f(x) = 5*x*x*x + 3*x*x + 2*x + 7and here you can see how it’s all built from + and *.This is why polynomials are useful. Because computers have an easy time calculating them. And all modern mathematics is done on computers. All the engineering uses computer simulations, and we want these simulations to run fast on computer hardware, so we make it easy for computer hardware to do. That is why we’re using polynomials wherever we can.
That is how you explain polynomials to 8th graders. No taylor series / calculus needed.
If you want to be really fancy you can show the taylor series of the sine and cosine function as a polynomial and how to compute it on a computer. Gives some pretty graphs, is simple and fun.
Just tell them that polynomials can be used to computer
sinandcosfunctions without going into the details of why that works first.
Edit: Just to clarify this: Yes i think that explaining why students should learn stuff is extremely important. In fact i tend to say that the only thing that you really have to do is to motivate the students to learn; then the learning happens by itself.
However note that giving esoteric abstract playful descriptions of things in my opinion does not motivate people to learn stuff. That just makes them go “huh, neat but useless”. Giving real world practical examples fulfills exactly the purpose of giving students a reason to learn stuff. Because seeing how one can solve real problems with the tools, one learns to value the tools.
Yes, examples like that are good, of course. But, frankly, abstract examples like that won’t do much to motivate the students who need the most help to get motivated learning math.
I like to interject little anecdotes like that, too. One of my “go tos” to “why are quadratics useful” goes something like “Well, they come up a fair bit, so I could give you some examples—and I will, as we with through the unit, but the real reason we teach quadratics is because they’re the simplest non-linear function. This is the first steps into looking at functions that aren’t a straight line. And the tools you use to work with quadratics are super important for understanding all the really cool functions you get to learn on the next couple of years…”
That’s basically your example, but one step lower and more directly applicable to students, imho. The Taylor Series thing I usually only drop in grade 11/12 (pre)calculus classes, mostly as a hook for the math nerds that they have really cool things to look forward to learning in post secondary. It’s a terrible application to use to try to motivate learning about polynomials for a student who couldn’t care less, lol.
Really, we need to intermix all approaches, depending on the students in the class. At private prep schools, leaning into academic needs works well. In a non-academic math stream, both your example and my examples will go over like a lead balloon.
But, regardless, motivating students to be excited for math, and the excitement of finally figuring out a tricky concept/problem? That’s what we need more of.
I’m always for the feckless hippie over the neoliberal sellout tbh.
Except a lot of those switched to antiscience antivaxers, and some bridged from there to facists. While I do also prefer hippies, antiknowledge and antiscience types scare me.
What about feckless hippies that sold out to the neoliberals?
or the commie that runs out of martyric protest
new words: martyric
heard it here first folks
I mean, that’s the sellout part
It’s because they ran out of fecks
