32

To expand on my comment, I found that high school kids like watching YouTube videos. (I mean, they don't have to do any work right? Just sit and listen.) These are a few of my go to channels to pull mathematical ideas from. I try to show them short clips that might motivate them to think abuot math in a different way, not only just "plug it into the formulas....


32

Imagine you are put in jail. You are forced to paint a painting every day for 10 years. You have no choice in the subject: one month you paint dogs, another month you paint horses, another month you paint lampposts. The prison guard verbally chastises you when your painting is not up to their standard. If you doodle something on your own, outside of the ...


16

It's hard to tell what will be the changes in 10 years. Maybe we will all have jacks in our heads. Then again, some things change slower. (Where's my flying car?) Most of the reasons for getting rid of worksheets have been around for 20-40 years and they still have high demand. They still have various advantages. Most importantly, if you look at the ...


14

I would just like to mention that other similar flowcharts have been developed, of varying degrees of generality, which you might consult. Here is one (by Adam Monahan). And another (by Jeremy Higgins): And another (by Enrique Areyan):      


13

When I need to distinguish $f$ and $F$, I choose - "small eff" vs. "big eff" / "capital eff". I find "uppercase/lowercase eff" to be a bit awkward; it doesn't run off the tongue as easily as the other options. However, I might not distinguish them orally every single time. The flow of the lecture would determine when I would choose to emphasise the ...


12

On the contrary, many seem surprisingly impatient when being asked to prove 1+1=4/2, whose proof (with properly delimited deepness) involves nothing beyond and possibly well below most people's working knowledge. The practice of starting students out with trivial arithmetic proofs like proving 1+1=4/2 seems to be pretty common, but I'm very skeptical of it. ...


11

One of the charms of graph theory is that people of all ages often enjoy learning graph theory ideas and tools. One place one can read about these ideas is in the book called For All Practical Purposes (I am a co-author) which has gone through 10 editions. The book was designed for college liberal arts students who might have almost no proficiency with ...


10

You may be interested in Academically Adrift: Limited Learning on College Campuses, Arum and Roksa, 2011. Also summarized in http://www.newyorker.com/arts/critics/atlarge/2011/06/06/110606crat_atlarge_menand . They have a lot of discussion of something called the Collegiate Learning Assessment, which is a standardized test of critical thinking. They find ...


7

A point (say, in $\mathbb{R}^n$) is a vector. Vectors and points are really no different. They are both $n$-tuples in $\mathbb{R}^n$. The difference between two points (in $\mathbb{R}^n$) is a vector, but a vector has no fixed position. Points are positions in space. Vectors are displacements. It makes no sense to add two points, but it does make sense to ...


7

Parallelograms are useful for understanding: Paths taken by light, especially through a layer of a medium with a different refractive coefficient Shear, and related deformations Area = height * width (but not necessarily the product of the sides' lengths) Dot products Surface integrals Paths taken by light are useful for understanding which routes people ...


7

There are many reasons, but the classic explanation is that professors (especially at research universities) are picked (and compensated) for research ability versus teaching efficiency. (In general...caveat hawks.) Probably a secondary reason is that mathematics tends to be a field with a high emphasis on logic and precision. However EDUCATION is more ...


7

It sounds like you're producing high school math material, whereas I'm teaching mostly physics (and a little math) at a community college in California. Hopefully there is enough overlap to make my experience helpful to you. I do a variety of paper and pencil exercises with my students, some of which could be characterized as worksheets. Below is an example ...


6

This does not directly address your question (and so is not an answer). But I consult this textbook when touching upon linear programming when teaching Algorithms, and I find it an impressive, unusually concise (~200 pages) text. Matousek, Jiri, and Bernd Gärtner. Understanding and using linear programming. Springer Science & Business Media, 2007.. "...


6

As a child I found maths extremely boring, the biggest problem was that what I was being taught had no relevance to my life as a child, after all apart from counting and spending pocket money what possible use was it to me? Until I discovered the equation for calculating the optimal size of loud speakers for my bedroom. Finally maths that I could relate to ...


5

I am just now close to completing my first time teaching geometry (at a community college, a course that parallels the typical high school course) Using the fact that parallelograms (and hence rhombuses) have bisecting diagonals makes a short proof for why the line segment midpoint construction works. Other than that, I don't see rhombuses as a big part of ...


5

For a much lower-level topic, consider explaining to beginning algebra students why "like terms" can be combined. On a few occasions, I have resorted to reasoning with students that adding algebraic expressions is like adding quantities with units. [Our curriculum begins with units and geometry before algebra, so this is usually safe ground in my class.] If ...


5

“The Mechanical Universe,” is a critically-acclaimed series of 52 thirty-minute videos covering the basic topics of an introductory university physics course. Each program in the series opens and closes with Caltech Professor David Goodstein providing philosophical, historical and often humorous insight into the subject at hand while lecturing to his ...


5

Worksheets are useful as a fill-in-the blanks forms to sign up for a credit card or for a car registration, but are detrimental in education. They instill the thinking that all information needed to solve a problem is presented on the same worksheet the students need to fill out. They supply information and problems in piecemeal fashion. They discourage ...


4

Here are GeoGebra materials: Graph Theory for Kids, inspired by Joel Hamkins' notes, to which @A.Goodier pointed.                     Four-color challenge.


4

I personally would prefer a textbook recommendation I can download or pick up that is [preferably] not old and does not make trigonometry intimidating to approach (especially one that emphasizes understanding proofs behind properties/theorems). I don't have textbooks to recommend, but I can recommend an approach to doing trigonometry that facilitates ...


4

Maybe a visual approach could supplement your study? There are many such resources available on the web, not in textbooks. E.g., Trig Intuitively:                     Note: the labels show where each item "goes up to." Another: Interactive Unit Circle. Another: Inverse Trig Functions.


4

Schaum's outlines are very practical in general and cheap. Well suited to an older learner. Often the answers are right after the problems versus at the end. And you get all the answers, not the odd/even gyp. Thus suited to self learning. I like this one, overall and own it: https://www.amazon.com/gp/product/0070026505/ref=...


4

I shall post my humble and incomplete list of bad explanations I've given or heard over the years: A function is continuous if you can draw its graph without lifting your pencil As you mentioned this is bad, but, depending on the level of the student, it can be a reason for big or small misunderstandings. At a high school level, this simply ignores the ...


4

I'd look at the books by Dunham ("Journey through genius", "The mathematical universe" are more or less general, he wrote several others) or at the outstanding "Proofs from THE BOOK" by Aigner/Ziegler. On Quora there is a section on "Beautiful mathematics", mostly elementary stuff. Probably a search for "beautiful mathematics" will net a selection of blogs ...


4

I'm having a lot of success right now in that area with a class I'm teaching, in which I have several assignments over the course of the quarter that center around self-directed learning. It takes some scaffolding to make it work well (how do you ask a good mathematical question? What counts as a good answer?) but once I got that done, a lot of the students ...


3

As for 3's, I always found it easy as a student to think it as $3=2+1$. So, in order to multiply by 3, one has to double and then to add what they've found. For instance: $$3\times7=2\times7+7=14+7=21,$$ which is much easier, since douling is carried out relatively easy by most students and addition is, in general, easier than multiplication. Similarly, ...


3

I can think of two books which might provide a roadmap, though you would have to flesh it out a lot depending on your prerequisites and the timeline you propose. Frank Morgan's Real Analysis has a minimum of prereqs and a maximum of topology and series. Now, it doesn't construct the real numbers (they are just infinite series of decimals, if I recall ...


3

The following remarks do little more than amplify on a comment made above by Andreas Blass. A basic use of parallelograms is to represent the sum of vectors in the plane as the diagonal of the parallelogram they determine. Visualizing the action of a linear transformation of the plane can be achieved by examining the parallelogram determined by the ...


3

Many of the students say they don’t like mathematics because they only learn for examinations. How do I show them the true beauty of mathematics? How about testing them on that beautiful content on the examinations? EDIT: And/Or give a wider variety of math assessments that count for grades. Projects, presentations, performances, role playing, games, ...


3

I agree with anjama's comment: give them applied examples. Use something like the free Godot engine to create simple "games" (read: interactive apps) that demonstrate concepts of trigonometry, geometry, and linear algebra. It's really quite easy to get started with Godot if you know some basic programming, and once the kids see a game engine, and see how ...


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