23

Short Answer You should not avoid use of the term real numbers. This is a term-of-art in mathematics, and it is important for students to learn the correct jargon. However, this technical term should be introduced as such—emphasize that "real" in mathematics does not mean the same thing that "real" means in everyday vernacular English. Long Answer ...


19

I think, while teaching, the principal way to judge mathematical language is not whether it's standard, but whether it's effective communication. This difference applies principally to communication that's more substantive than "read an equation out loud" where there's only really one right way and not much opportunity for change - but, even on the small ...


17

In my job, I evaluate university math courses for transfer equivalency on a regular basis. In the US, "Calculus 1" typically refers to single variable differential calculus up to the fundamental theorem of calculus. So the course includes limits, the definition of the derivative, techniques and applications of the derivative including trigonometric and ...


14

You ask "Why isn't the term inequation widely used in English?" The answer, however tautological, is that the term isn't used in English because it isn't used in English. Perhaps more usefully, remember that languages change and evolve as speakers of the language (for example) start using new words, stop using old words, alter pronunciation, and play with ...


13

I don't think there's a lot of educational value to fixating on trying to word definitions in exactly the perfect way. Students have trouble with the notion of a function because it's hard. The way they're going to get a handle on it is by struggling with it, encountering the hard parts of the definition, and finding and eliminating their misconceptions ...


11

This is an example of what is usually called a flowchart proof (or sometimes a flow proof for short). A quick Google search for "flowchart proof" or "flow proof" shows many, many contemporary examples of the form, including a whole genre of YouTube videos teaching this style of presentation. This style of proof has been promoted at various points since the ...


9

From the article: Grade level was a categorical factor consisting of three levels: preschool–kindergarten students (or its equivalent if outside of the United States), 1st‐ to 6th‐grade students, and 7th grade and above. This variable evaluated whether the effectiveness of mathematics game‐based learning varied across different grade levels as to ...


9

There are already some books dedicated to etymological aspects of mathematics. Here are two of them: Origins of Mathematical Words: A Comprehensive Dictionary of Latin, Greek, and Arabic Roots by Anthony Lo Bello (https://www.maa.org/press/maa-reviews/origins-of-mathematical-words) The Words of Mathematics: An Etymological Dictionary of Mathematical Terms ...


8

Part of teaching mathematics is teaching how to communicate mathematics - both how to understand it, and how to write it clearly. Some of that is obviously complementary to teaching students how to solve math problems and understand mathematics; for instance, students who can't understand a question aren't going to be able to answer it correctly. But it's ...


8

The course titles "Calculus 1", "Calculus 2", etc. are not meaningful terms outside of the specific institutions where there are courses with these titles. These are names of classes, and not some internationally decided-upon list of topics or curriculum. The actual content of a class called "Calculus 1" might vary quite a lot from one institution to ...


7

For 95% of high school students, this sort of thing is of no interest. But: The 5% do need to be served well and helped to achieve their potential. The 95% may find such things confusing if they are never explained, so it makes sense to offer them at least some brief explanation. Even the 5% are in no position at this point to understand fully what is meant ...


7

I very much appreciate these questions. I am in favor of a very strict style; or in other words, helping students to avoid errors and confusions in the standard language as much and as early as possible. The main thing that sticks in my mind is that the longer a bad practice persists, the "stickier" it is, and the harder it is to fix later on. Comparing (a)...


6

Wolfram calls it the "point slope form", you just have $y_0$ on the rhs rather than lhs.


6

It is of course important to give your students a good understanding of the customs and conventions that are commonly used in mathematics, but I think that creativity, imagination and flexibility of mind are just as important - as is the courage to think 'crazy' thoughts, such as "what is $\sqrt{-1}?$", "is infinity always the same?" and so on. Maths is in ...


6

I recall having been taught different classes of numbers (in maths at school) way before we were introduced to complex numbers. Main reason was to distinguish natural numbers integers rational numbers finally ... real numbers It's reasonable to teach students things like the coverage of numbers on the number line: Why are real numbers continuous while ...


5

In Germany, we already do this. A function is introduced as an unambiguous mapping in 7th grade (~13 years). While I don't have any data on this, I doubt that German students do significantly better due to this choice of words.


5

I personally use the following terminology: A relation $R \subset A \times B$ is said to be single-valued if $(a,b_1) \in R$ and $(a,b_2) \in R$ implies $b_1 = b_2$. A relation $R \subset A \times B$ is said to be total if for all $a \in A$ there exists $b$ in $B$ with $(a,b) \in R$. A relation which is both single valued and total is a function.


5

This reminds me of my question “Amplitude” of Tan and Cot functions which referenced a non-standard use of the word amplitude. Math has a language, one that should make communication on this topic pretty clear. When discussing notation or language, I frequently resort to the example "You are on the phone with a friend, pre facetime, audio only. As you ...


4

Regarding all possible numbers on the continuous number line, in my opinion you're overthinking this. The vast majority of students aren't concerned with what the numbers are called, but how to solve the problems they have for homework and on tests. As for the use of real being unnecessary in the absence of knowing about complex numbers, I disagree. For them,...


4

Here's something that the students might easily grasp and that could also be entertaining: What's "natural" about the natural numbers? What's "rational" about the rational numbers? What's "real" about the real numbers? Every one of these different sets of numbers is a mathematical idealization of something encountered in the "real" world. None of them is ...


3

Several Pearson calculus textbooks, such as Calculus: Graphical, Numerical, Algebraic for high school and Calculus and Analytic Geometry for university use what you show for the point-slope form. The latter puts the $y_0$ term first, but I believe the former puts it last as you do.


3

Since "bigger" and "smaller" are ambiguous, it is best to avoid them, as you mention. The methods you mention seem reasonable, though I am not native English speaker. Someone may be able to refer to a credible source or official standard, but regardless, you can only know how someone else understands the terms by asking them. So you might as well do that ...


3

Calculus 1 is Differential Calculus. You start off by learning how to find limits of Algebraic functions, then you learn how to derive every function you learned in High School Algebra. Calculus 2 is Integral Calculus. You learn how to find the area under a curve and between two curves, which are solved using integrals. You will also learn the various ...


3

In the Yoruba Language, we use the word òǹkà to refer to tokens for representing numbers. Thus, it would correspond to the word numeral. The word òǹkà literally translates as 'that which is used for reckoning.' The prefix on (the n is a nasal) means thing, and the other part, ka, means to reckon, count, calculate, etc. On the other hand the word number ...


2

Your premise is not entirely true. We do distinguish between an equation and an identity in English sometimes, but most of the time we use 'equation' for both. 'Identity' is only really used on 'special occasions'. In particular, students are only likely to come across the term in the context of 'trig identities' such as $\sin^2(x)+\cos^2(x)=1$. There are no ...


2

In my opinion if some numbers are "imaginary" it doesn't mean they don't exist. It is needed to distinguish somehow between the two kinds of numbers, the "real" that we easily see im our daily life and that make sense to everyone, and the "imaginary" numbers (or complex in general) that are there in life, real of course with no doubt, but they're not easy ...


2

It's your job to teach standard notation and terminology, and to correct your students if they get it wrong. If they get it wrong repeatedly, it's your job to correct them repeatedly. If a student comes out of an algebra class saying "x two" for $x^2$, then that student has been done an enormous disservice. They will be marked as mathematically illiterate ...


2

I've found terminology differs as well, but this is how I think of the phrase "by a factor of" for $af(x)$ If $0<|a|<1$ then it is a vertical "squeeze" to get a factor of $a$ between the new and old $y$ values If $|a|>1$ then it is a vertical "stretch" to get a factor of $a$... And then a point $(x,y)$ becomes $(x,ay)$. So $a$ is the new scale or ...


1

I'll take the dissenting opinion: don't use the phrase "Real Number" until you're prepared to teach what an imaginary number is. "Today in science, we're going to be using an optical microscope to look at culture slides." ... which immediately begs the question from any remotely curious student: "Wait, so there are non-optical microscopes?" And you ...


1

Definition of slope of a line, given two known points on the line, say $(x_2,y_2), \;(x_1, y_1)$, where $m$ is the slope of the line: $\dfrac {y_2 - y_1}{x_2-x_1} = m$. Note that this is equivalent to the equation: $(y_2-y_1) = m(x_2-x_1)$, which bears close resemblance to the next form of an equation. Once one calculates slope, or it is given, one can ...


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