# Tag Info

25

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 ...

24

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 ...

15

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 ...

11

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 ...

9

It's a bad idea. (1) It's not that special. He can get a game from chess very easily from anyone. An individual session with you is not high value, not best use of time. (2) It will raise hackles. I would suggest instead introducing some recreations that are less familiar instead. Playing Hex for instance is an idea. The value is much higher than chess, ...

7

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 ...

7

A once per week ten minute session is objectively not using up that much of the kid's time. In addition, he's not even as much turned off on it as you are. Also, I think you will find that speed increases with volume, so it becomes nonlinear. None of this is to say that I love the particular drill. Sounds too easy for a smart 7yo. Just that it's not so ...

6

EDIT Here are the CCSS for grade 6. http://www.corestandards.org/Math/Content/6/NS/B/4/ At this age, perhaps she is just expected to "find the LCM of two numbers less than or equal to 12". They do not specify a method, but I would still advocate using the definition (as presented below). I would start by making sure she knows, and can compute ...

6

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 ...

4

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 ...

4

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 ...

4

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 ...

4

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 ...

4

I suggest holding office hours, both one-on-one, and possibly in small groups, to help students with the assignments, or just to clarify the material presented in the lectures. Of course the prof may hold such office hours, but since you are giving tutorials, it seems natural, and might serve as a lower-stakes alternative for those afraid to visit the prof. ...

4

“ I know it has a lot of relations with the mathematics, with the logic, many applications in computer science, game theory and so on.“ Do you know these specific relationships and are you prepared to introduce/explain these things in the context of the game? If so, that sounds like it should take quite a bit of preparation on your part to teach a complex ...

4

Copyright is related to works, not ideas or facts. To quote the Oxford Learners Dictionaries: if a person or an organization holds the copyright on a piece of writing, music, etc., they are the only people who have the legal right to publish, broadcast, perform it, etc., and other people must ask their permission to use it or any part of it The important ...

3

I'm not entirely sure if it fits the teaching style you described, but it at least partially resembles the discovery method. Classrooms that adhere strictly to the discovery method typically have little direction provided by the teacher, whereas you seem to be facilitating the discussion more. I'm not familiar with any specific research on the discovery ...

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 ...

2

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 ...

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 ...

2

The information and handbooks at SIAM Mathworks Modelling Challenge pretty helpful. One handbook introduces the reader to the modeling process and the second one introduces the reader to the communication of computation related to modeling (including ODEs via spreadsheets--you could probably generalize what they discuss to solve the logistic equation to ...

2

Use of color definitely can be helpful in learning mathematics. Presentation matters. I have given up on texts using an awful font: if learning the idea requires 95% of my brainpower, and deciphering the bad presentation uses up 10% of my brainpower, it makes the difference between learning and not learning. That said, using color intelligently requires ...

1

I would recommend checking in with the MODULE$(S^2)$ project (Mathematics Of Doing, Understanding, Learning and Educating for Secondary Schools). A few notes: This project is currently developing materials for Geometry, Algebra, Statistics, and Modeling. Obviously, it is the last of these that would be of interest for you. You can find links to some ...

1

I think the spiral approach describes the part of the lesson design that you are focusing on here. I won't deny you the opportunity to practice with Google Scholar, but you should have no trouble finding research exploring both its strengths and weaknesses in mathematics education.

1

You can't. Mathematics is hard, and its beauty must be earned by hard work. Think about how you might have realized the beauty of maths. You most probably have thought very hard about a problem, and when you finally put the pieces together, your reward was a feeling of beauty. This reward then made you more likely to put in effort and get more rewards, which ...

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