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2

my question is about how to specifically soothe a curious student quibbling about the derivative “paradox” in the same way that I once (detrimentally, with no one to guide me) did. I’m not interested in general ideas of whether we should or shouldn’t push rigour into introductory calculus. Do not soothe them. Encourage them. Cheer them on as they fight this ...


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So many answers, but not yet pointed out is that your proposed proof of the derivative of $x^n$ with respect to $x$ is wrong because it uses the wrong notation. There is a difference between Big-O and little-o (see the formal definitions of Landau notation for details). $ \def\lfrac#1#2{{\large\frac{#1}{#2}}} $ That said, there is a way to make things ...


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Hopefully, they know some basic physics, e. g., that if you release a heavy object, its velocity grows uniformly, and after $t$ seconds it will have velocity $gt$. Now, ask them, if they had to prove this experimentally, how would they go about it? They probably will come up with some experimental design involving stopwatch, then you point out that they are ...


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First off - I 100% agree with Collins here that there's no shortcuts. To really understand how derivatives work, you need to learn the epsilon-delta definition. But it's my sense that you're not really shooting for that here - it sounds like what you want is a way to convince a student, not necessarily a way to prove it to them. The difference can be hard ...


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As an intuitive aid to the calculus I like to explain that infinity is a number so big that it wouldn't make any difference if the you made it bigger. This is really an epsilon-delta argument in a disguise that makes it look friendlier. It can adapted to any limit situation.


2

I am a father of 14 and 17 yo children in France. I have a rudimentary understanding of math through my studies (PhD in physics) and I always used math as a useful toolbox. When my older kid had derivatives (during COVID, which in France was an educatory disaster), he was disappointed because he could not understand what this was after reading his book. It ...


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There is a reason why the binomial theorem and calculus were the subject of study by Newton---they are closely related! There are computational methods that work consistently and then there are proofs that these methods are "correct". This represents two different approaches to mathematics. Many working mathematicians were (in their youth) ...


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Personally I found the calculus is done in two parallel tracks: 1) on how essentially the derivatives and integrals are, and 2) the delta-epsilon language to be rigorous. Sometimes there needs a "translation" in the mind when one tries to understand or compose a proof. I would like introduce "Calculus made easy" by Silvanus P. Thompson, ...


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I do not know your textbooks, but when I first learned about limits in school (I believe in my school system that happened when we were about 15/16 years old, but it has been an awful long time ago, so I'm not sure), the $\delta-\epsilon$ approach was used, with a strong geometric aspect. I cannot remember every having any inclination that there were any ...


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I admit that I'm unable to follow the proof you give as an example in your question, but am I correct when I assume that your question simply wonders how to reconcile $dy/dx$ with the fact that $dx$ approaches zero — and hence is considered zero by your students? Then I'd simply explain the limit operation visually by exploring a curve. This can be done on a ...


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The only way I found to explain this is with infinitesimals. It's not that $dX$ is vanishingly small... its that it is "infintessimally small." My usual approach to explaining all of this is to start with Zeno's paradoxes. His most famous is good enough -- the idea that you can't run to the end of a football field without first running to the ...


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I teach calculus at a community college in the U.S. (2 year college, from which many students transfer to a university). I explain limits from about day two in an informal way ("h gets infinitely close to 0"), and talk about the problems with saying infinitely close (but I keep saying it...). I tell students that our learning journey will match the ...


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This might be opening up another can of works but have you considered introducing infinitesimals? Like imaginary numbers, they are created by appending a new element to the reals, but instead of root -1 the element added is smaller than any positive real number but greater than 0, effectively the "dx" from the integral. Calculus can be consistently ...


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Start with a numerical example. Say you want to find the gradient of the tangent to $y=x^2$ at $x=1$. Obviously the point itself is $(1,1)$. Pick a nearby point, say $x=1.1$. A moment with a calculator shows $y=1.21$ and the gradient of this chord is $0.21/0.1=2.1$. Now pick a closer point, $x=1.01$. We again use the calculator to find $y=1.0201$ and the ...


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We routinely get questions related to pushing more rigor in early calculus. Usually from outstanding students and based on sample of one I like it that way logic. There's a reason why things are the way they are. And that's because most students would get the opposite of a benefit pedagogically by emphasizing increased rigor in early calculus. It's not ...


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There is no royal road to geometry. - Euclid Nor calculus. The essence of calculus thinking is really the limit concept. One needs to wrap one's mind around that. Formally: it's the core technique that defines derivatives and integrals. Poetically: it's the eye-of-the-needle through which you must pass to get to the next level of mathematics. There are many ...


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Three references. Wikipedia has an article, Studies of Waldorf education, which cites various publications, not, however, specifically focused on mathematics education. [Not the same Wikipedia article to which @JoelReyesNoche refers.] For example, (1) A 2012 study compared the reading and math standardized test scores obtained in public Waldorf schools in ...


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You may have better luck finding data in a different area. Many forms of GEO analytics have rich sets of data for things like: income, political, home prices, etc that may be more practical. The very concept of average, median, whatever can be a challenge by State, County, population vice geographic area. If you compute averages by County, what does that say ...


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No, I don't. If you restricted it to price and rating, you could use Amazon. P.s. Also, to Joel, delivery is an average value, like vendor rating.


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For gifted students, you could do them from algebra on. Say starting at grade 9 (1age 14). In some ways, they are very similar to the sorts of applied algebra that are involved in chemistry (but still seem to challenge university students...word problems are hard!) Conceptually even earlier, you could use them, from grad 6 (11). I would just watch out, ...


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A guess animation and real life are opposite? On the real life side, Dan Mayer's collection of 3-Act Tasks provide many examples with video for the first act. You might need to look for the ones that are appropriate to your age group. Note: If you're not familiar with the format of '3-act tasks', you might look at Dan Mayer's TED-talk Mathclass needs a ...


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Not quite "animated", but still very nice: James Tanton Youtube https://www.youtube.com/c/JamesTanton_SquineCosquineTanq/videos He starts with a kind of problem, sometimes applied, but often just non-standard, and then shows mathematician viewpoint on problem solution. You may start discovering with his "Why I became mathematician" https:/...


3

Before listing 'resources', let me start with some official documents (e.g. to clarify the difference of sustainable math and math for sustainability, see xkcd:Sustainable :-). The official UN Site lists the 17 Goals for sustainability; especially Goal 4: Education. Still debating what this all means, I find the paper by Brundiers et. al. (2021, see below ...


3

Personally, the perspective I recommend is kind of the reverse. Despite the linked thread, I find that most of the time, the mathematical term has some reason why it was picked for the mathematical concept in question. It wasn't picked in an arbitrary or malicious fashion. At some point when the practitioner first used it, it seemed like the best English ...


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Since asking the question, I've come across Thomas J. Pfaff's website, Sustainability Math. Plenty of material there, including projects and links to further resources. Contains a blog that refers to current/recent events. I found Pfaff's website when perusing Mathematics of Planet Earth, specifically the list of curriculum materials. (Made this community ...


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Possibly worth looking at is: Earth Algebra: College Algebra With Applications to Environmental Issues by Christopher Schaufele and Nancy Zumoff (1995 1st edition and 1998 2nd edition) See this review and this project/funding document. For what it's worth, in Spring 1993 I interviewed (on campus) for a tenure-track position at the university this was ...


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Perhaps it's the mathematical way of thinking about and approaching problems that is useful, and this way of mathematical thinking can be carried over into many domains of study and work. So while there might not immediately be direct real world applications to parts of it, it's the way of mathematical thinking and approaching problems that is useful. That ...


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Some of us would point to political pressures to evidence higher "success" in terms of increased graduation rates, which wind up pressuring institutions to reduce standards and pass students regardless of whether they've mastered material skills or not. Note that U.S. high school graduation rates have been spiking upwards in recent years, which ...


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