67 votes

Why is it possible to teach real numbers before even rigorously defining them?

Expansion of mathematical knowledge does not unfold in Bourbaki progression. This is true at the level of both societal and individual knowledge. Just as the invention and significant applications of ...
  • 8,313
44 votes

Why is it possible to teach real numbers before even rigorously defining them?

It is possible to teach real numbers in elementary school before even rigorously defining them by using what H. Wu ("The Mis-Education of Mathematics Teachers," Notices of the AMS, vol. 58, no. 3, p. ...
  • 10.6k
40 votes

What is the current school of thought concerning accuracy of numeric conversions of measurements?

The product of two numbers should be given with as many significant digits as the least precise of the numbers multiplied (see https://www.nku.edu/~intsci/sci110/worksheets/...
28 votes

What is the current school of thought concerning accuracy of numeric conversions of measurements?

Here's a joke I like to tell when people could use a reminder about precision vs accuracy: A tour guide at Giza was explaining how the Pyramids were 4507 years old. Someone in the crowd asked: "...
24 votes

What number is the sum of two roots

First of all, $\sqrt{2}+\sqrt{3}$ most certainly is a number. It is a real number, approximately equal to $3.14626$ Perhaps what you're asking is why the sum of two simple radicals isn't also a simple ...
23 votes
Accepted

Should high school teachers say “real numbers” before teaching complex numbers?

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 ...
  • 6,771
23 votes
Accepted

Should an undergraduate math program contain a course on Lebesgue integration?

I think the existing answers understate how much a standard American math major does not see the Lebesgue integral. I'm going to poke around at a variety of college websites to see how they cover this ...
20 votes
Accepted

What number is the sum of two roots

I think G Tony Jacobs's answer is an excellent one, and I admit I don't quite understand what the OP's objection to it is all about. But I am going to take a stab at trying to explicate what I think ...
  • 16.9k
20 votes
Accepted

Inability to work with an arbitrary mathematical object

I'll focus on question 2 from a perspective of "maybe the right thing to think about is: what happens in the students' minds while they read this question?" When you say "Suppose $A⊆R$ is nonempty ...
17 votes

Why is it possible to teach real numbers before even rigorously defining them?

But as far as real numbers were concerned, they were taught very early without any clear definition was even given. Were they? I would say that usually no teacher really speaks of real numbers to ...
  • 569
17 votes

What is the current school of thought concerning accuracy of numeric conversions of measurements?

Just to play the devil's teacher's advocate here: one can make a point that rounding should be generally avoided but measurement uncertainty instead be expressed explicitly. Specifically, rounding ...
15 votes

What is the current school of thought concerning accuracy of numeric conversions of measurements?

When a tutoring student asks me about rounding, I tell them that absent specific instructions from a teacher, common sense should apply. For a conversion, 22 miles isn’t 22.0000 miles, there’s the ...
14 votes

Why is it possible to teach real numbers before even rigorously defining them?

At German universities, one of the first lectures in mathematics is "Analysis 1" which is a kind of "rigorous calculus" and there one always proceeds more or less like this: We start with an ...
  • 2,962
14 votes

Should an undergraduate math program contain a course on Lebesgue integration?

Is it standard for a math undergraduate program to have a course on Lebesgue integration? No (assuming that "have a course" means "require people to take such a course in order to get ...
  • 141
9 votes

Why is it possible to teach real numbers before even rigorously defining them?

Introductory mathematics is not done with formal definitions. This is because most people at the corresponding age (kids) don't know what the words "formal" or "definition" actually means. This only ...
9 votes

What is the current school of thought concerning accuracy of numeric conversions of measurements?

When I was in school, I once got an answer marked as error for having too many digits. IIRC it was in trigonometry and I had just written down as many digits as the calculator displayed. (I was able ...
  • 191
8 votes
Accepted

Why is highschool math so unrigorous?

For the same reason that elementary counting numbers of more than a single digit are explained as ones, tens, hundreds, etc. . The concept of powers and exponents has not yet been developed. Later it ...
8 votes

Should an undergraduate math program contain a course on Lebesgue integration?

Is it standard for a math undergraduate program to have a course on Lebesgue integration? Yes, and I find it bizarre that a university would not have one. Lebesgue integration (or measure theory more ...
  • 189
7 votes

Why is highschool math so unrigorous?

As a consequence of this, we get a lot of definitions that are not really definitions but are more like "defining things into existence", for example this year we learned about the square ...
6 votes

How to teach real analysis?

This comment from MSE by Kimchi Lover is fantastic. Examples every time. Can you think of a target end example result and work the course up to understanding it? If I were teaching it, I'd make the ...
  • 1,625
6 votes

Why is it possible to teach real numbers before even rigorously defining them?

I believe that, unlike many other mathematical concepts (especially more abstract ones), even fairly young people have a basic understanding of distance, including fractions to any degree of units of ...
6 votes

Should high school teachers say “real numbers” before teaching complex numbers?

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 ...
6 votes

Should an undergraduate math program contain a course on Lebesgue integration?

For undergraduate maths students at the University of Oxford, Riemann Integration is mandatory for 'freshmen' (year 1), and Lebesgue Integration is an elective taken by 'sophmores' (year 2). Full ...
  • 161
5 votes

Why is it possible to teach real numbers before even rigorously defining them?

You are confusing definitions and models. A definition of real numbers is a set of axioms they obey. Different such sets exist, but they can be shown to be equivalent. A model of real numbers is a ...
5 votes

Inability to work with an arbitrary mathematical object

Can all of your students correctly explain what the supremum and infimum of a set are? If not, they will be unable to reason about these concepts. It would be like expecting a student to do something ...
  • 2,798
5 votes

Specific Intervention(s) for Middle School 'Place Value' confusion

I always emphasize comparing decimals with the same number of places to my students. Therefore before comparing 0.12 and 0.125, I would first teach them that 0.12 = 0.120. Students can than grasp ...
  • 7,665
5 votes
Accepted

How to practically teach surds?

Chapter 2 of Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory (Edwards, H. M.) is dedicated to surds and their application to Euler's treatment of the problem of whether there ...
5 votes

How to practically teach surds?

In my experience (remedial and college algebra), these topics are always resisted and show poor performance -- very analogously to the topic of fractions (irreducible divisions). One thing I do is ...

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