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"Lately, my students keep telling me why what we are learning is not important. They ask me when will we use this in the real world?" There's a quick reply to this that I think people won't like, and you would have to be very careful using, but actually makes a more serious than it first appears. One possible answer is: "You're quite right. ...


44

The short answer to your question is: everyone is right. I agree with people here that in many contexts, $0.75$ or $\frac{3}{4}$ would be a more desirable answer than $\frac{45}{60}$. I also agree with several here that when the context is "out of 50 people", an unreduced fraction like $\frac{40}{50}$ makes perfect sense. Certainly, when I write ...


37

In the U.S. Common Core standards, functions are supposed to be introduced in the 8th grade, i.e., around age 13-14. So arguably age 15 is a year or two behind where they ought to be. The standard for the 8th grade says: Understand that a function is a rule that assigns to each input exactly one output. So honestly that really doesn't seem like a hugely ...


37

Math is just as useless as almost any other subject As a math tutor, I've thought about this a lot over the last 15 years or so. Aside from tutoring, I don't use my math education in "the real world". Here's a list of educational requirements that I have also never used in "the real world": American literature (Hemingway, Flannery O'...


30

I would say the standard implementations of the Rational Root Theorem (make a huge list for the sake of making the huge list) indeed feel like a complete waste of time. However, the theorem can sometimes combine nicely with technology tools. I'll provide an example of this. Consider the following algebra puzzle: Solve for $x$: $7x^3 -39x^2+52x+30 = 0$. If ...


28

Coming from the perspective of someone who reteaches this material at the college level, neither the graph perspective nor the list of properties perspective really translate into a deep understanding later. For the former, all you get is an understanding that the graph goes up, but not a lot, and is too vague to really be useful. For the latter, they don't ...


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

One practical reason for choosing a Taylor Series approximation of a function over the function itself is if you are able to compute using only the four arithmetic operations. For example, if you are asked to find the cosine of an angle and the only computing device you have is a four-function calculator, then you can get a good approximation of the cosine ...


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 ...


23

added Oct 6 The reason mixed numbers are found in US education is that mixed numbers are found outside of school in the US, so the children need to learn to understand them. Mixed numbers are found in road signs, cooking recipes, length measurements, and so on. (Denis Nardin commented that mixed numbers are never seen in Italy. Meters, centimeters, and ...


22

As an answer so that I can paste in a picture: I think that the problem with the one on the left is that it is possibly "good enough" that someone might think that it is to scale and that apparent regularities are real. (Well, maybe not one of those "right angles".) In particular, someone might become under the impression that the length ...


21

In real-world applications, the typical case is that the domain is neither implicit in an expression we write down, nor explicitly stated along with the expression. Rather, one uses knowledge of the real world to decide what numbers make sense as inputs to the function. So for example, if $x$ represents an investment, and $f(x)=x-3$ represents the profit on ...


20

The answer to your question depends on the pedagogical goal of the exercise, and what learning outcomes you have identified. It basically comes down to the following question: Is manipulating fractions one of the skills which you are emphasizing in this class? If one of the goals of your class is to get students to more handily work with fractions, then ...


20

I am a GCSE Maths examiner. For a question like this, any correct equivalent decimal, percentage or fraction, whether simplified or not, would receive full marks. It is only specifically if it says in the question that the answer should be simplified that a simplified fraction is required to obtain the final accuracy mark. The reason for this is that the ...


19

There is no value in drawing the figure exactly to scale, but the left-hand figure is inaccurate to the point where it is positively misleading. Since the angle marked 30 degrees is actually drawn greater than 45 degrees, it gives the impression that a is less than 3 (or maybe equal to it, if the student takes the "30 degree" angle as being shown ...


19

Common knowledge The formula $a^2+b^2 = c^2$ is common knowledge and the words for hypotenuse and leg (is "cathetus" not used in English?) are basic mathematical vocabulary. Including these seems a good idea. Connections to other mathematics The notation with AB, CA and BC might be something the students have used or will use in less analytical ...


19

I think the basic answer is that there are all sorts of things that we could write, but don't. We usually leave off things that are redundant, but we can add them back in when convenient. For instance, instead of $1x$ we could also write $1\times 1\times 1\times x$. Or we could write $x + 0$ or even $0x^3 + 0x^2 + x + 0$. We could also write $0yz + 0y^2 + ...


19

An excellent introductory example would be exponential function $\exp(x) = e^x$. By definition, this is the function that is its own derivative, i.e. $\exp'(x) = \exp(x)$. That's all nice and swell from a mathematical stand point, and it makes it easy to prove interesting properties of the function. But how do you actually compute it? The definition above is ...


18

I've never had success with giving a list of applications to such students - because, realistically, we don't use most of the math we teach. For example, I teach early equations in one of my classes, things like "solve the equation $5x = 15$". If asked why this question is "useful", I could give this answer: Well, suppose you were ...


17

I can't cite research to back me up on this, and you will want answers that reference research. But. Most of the differences between groups of boys and groups of girls come from how they've been socialized. Our culture pushes boys one way and girls another. Because of this (and not any inherent differences) I imagine that the girls will enjoy cooperation ...


16

Functions are far broader and more applicable than you give them credit for. Consider the following: Country or state Capital Elevation (in meters) Bolivia Sucre 2783 Ecuador Quito 2763 Colombia Bogata 2619 Eritrea Asmara 2363 Ethiopia Addis Ababa 2362 Mexico Ciudad de Mexico 2216 New Mexico Santa Fe 2152 Wyoming Cheyenne 1856 Colorado Denver 1613 ...


16

How do you teach students about the operator $\sqrt[3]{}$? It's a similar operator in many ways; when dealing with cube roots I try to show them these things: $\sqrt[3]{x}$ asks "which base to the third power gives us $x$?" $f(x) = \sqrt[3]{x}$ is a one-to-one function, and $y=f(x)$ has a graph of a certain shape. $\sqrt[3]{}$ is the inverse ...


15

The American Mathematical Society provides posters promoting awareness of mathematics, its beauty, and applications. Here's a quote from the AMS Posters website: "Students frequently ask when they will use the math we learn in real life, and your posters provide great visuals to support the answers to this question." The AMS also have "...


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 ...


14

They are basic, friendly pieces of pedagogical advice. Most pre-college teaching is very much STILL in this mold. Where we fall down is in high-end universities and graduate schools, where pedagogy is less emphasized in the paradoxical belief that harder material should be learned with worse training methods. Or that smart students don't need/benefit from ...


14

OP: "Refuse to teach without attention." In my role as chair, I attended an instructor's class where he really refused to advance until he was certain the students were all with him, via detailed verbal feedback. The students responded, stopped the presentations and asked questions. I've changed my own teaching as a result of watching how this can ...


14

I agree with @Tommi that creating community is bigger than one activity. I do a number of things at the start of semester, but building community is also in the way I teach, every day of class. (I teach college, currently online via zoom.) You might check out Francis Su's blog, which is related to his book, Mathematics for Human Flourishing. I also found the ...


13

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 ...


13

We give high-schoolers many different explanations of the word "function." Here are a few that are either implied or outright stated at various points in a student's education: A function is an expression in terms of $x$. This is pretty unusual all by itself, but it may appear in conjunction with (2) or (3) (i.e. students may reason that $f(x)$ or ...


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