Questions tagged [arithmetic-operations]

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58
votes
15answers
7k views

Is there a virtue to learning how to compute by hand?

I have been professionally tutoring a wide range of students (from elementary school through graduate school) for many years. Most of them are from the United States. I generally focus on helping my ...
33
votes
17answers
9k views

Dividing by zero

I was having a discussion with a friend and fellow mathematics teacher the other day when the topic of dividing by zero came up. She is the department head and had this in a questionnaire she gave to ...
28
votes
1answer
5k views

Which product of single digits do children usually get wrong?

(I was inspired by the comments in this answer to ask this question.) I have some multiplication table cards from Kumon that have a list of commonly mistaken multiplications: $7\times 8, 4\times 8, 11\...
23
votes
7answers
4k views

Repeated addition: standard notation?

My daughter showed me the picture below, which came from 9GAG. It shows a question on an exam asking the student to "use the repeated addition strategy to solve: 5 x 3." The student answered "5+5+5" ...
20
votes
1answer
586 views

Has Benezet's teaching experiment ever been reproduced?

In the 1930's, Louis Bénézet, a superintendent of several schools in New Hampshire made the interesting experiment of teaching no formal arithmetic until grade 6: In the fall of 1929 I made up my ...
16
votes
3answers
246 views

How do I help a 6-year-old understand the meaning of her sums

I am tutoring a Grade 2 girl in arithmetic. She has demonstrated an ability to add two-digit numbers with carrying. For example: $$\;\;14\\ +27\\ =41$$ I asked her to write this out horizontally, ...
13
votes
4answers
401 views

Analyzing an answer to the following problem: Give meaning to $\frac{4}{5} + \frac{2}{3}$

Case: Exam Problem given to student at university: Give a problem/context illustrating the operation $\frac{4}{5} + \frac{2}{3}$ Answer by student: Anna and Beatrice buy flowers for grandpa for his ...
13
votes
3answers
736 views

Why is modulo not an elementary operation?

Like it is just as primitive as makes intuitive sense in a way similar to division. I feel like teaching 4th graders the modulo function would complement learning division extremely well, as it helps ...
12
votes
8answers
12k views

Proof of why BODMAS (or BIDMAS) works?

In my first full-time teaching post, it is very likely that I'll need to be teaching a small amount of GCSE Mathematics to students retaking it. One thing that has been bugging me is that I can't seem ...
12
votes
13answers
2k views

Different ways to multiply decimals

I am an undergraduate secondary math education major. In $2$ weeks I have to give a Number Talk in my math ed class on the problem "$3.9$ times $7.5$". I need to come up with as many different ...
11
votes
3answers
2k views

Third Grade Question -- This makes no sense to me

Third grade grandchild had this for homework. Can someone explain the intent here?
11
votes
1answer
1k views

Method for teaching factorization

A while back I stumbled on teacher's website that advocated a different way to teach factorization. Rather than jumping straight to factorization practice, the teacher first had their student's ...
11
votes
2answers
151 views

Visual questions for 6th graders

I'm tutoring a 6th grader in math at the moment and because she never has a ton of homework I like to give her some interesting extra problems to do. It seems she really enjoyed a problem I showed ...
10
votes
6answers
7k views

Is short division taught these days and if not, why not?

tl;dr I'm interested in opinions on short division. Below I discuss my experience dealing with young children and long division versus short division. For those that don't know of it, wikiHow has a ...
10
votes
3answers
891 views

What is the pedagogical justification and history for using mnemonics to teach order of operations?

There was previously a question/rant here on MESE about why so many are still using the PEMDAS/BODMAS/BIDMAS/BEDMAS mnemonics to teach order of operations. The question was deleted (still viewable by ...
10
votes
2answers
189 views

What is a good math "curriculum" for a mathematically precocious 4-year old?

I am parent to a 4-year-old son who is mathematically precocious. An example of what I mean (since I'm sure guys like Gauss were proving theorems at 4): He multiplies and divides small numbers easily,...
9
votes
12answers
3k views

A PEMDAS issue request for explanation

This question made the rounds recently - $8÷2(2+2)=?$ Now, I glanced at this, answered "1" and then saw the full article printed in the New York Times, The Math Equation That Tried to Stump the ...
9
votes
2answers
201 views

Generating system of equations with unique solutions

I have a similar problem addressed in System of Equations Generator. What I need is an automatic way of generating a system of equations with unique solutions, but the equations are not exclusively ...
9
votes
3answers
161 views

Learning operator priorities by drawing trees

As far as I know (and here I am refering to my own math education), operator priorities of $+$, $-$, $\cdot$, $\div$, power and paranthesis are taught via some simple phrases like "pointy" operators (...
8
votes
5answers
1k views

Exponents with Negative Base; with or without Parentheses

How can I convincingly and mathematically explain the reason behind difference between $(-1)^2$ and $-1^2$? I used to add "negation" to the order of operations, in the same row as multiplication and ...
8
votes
3answers
612 views

Subtraction using Addition (Austrian Method), is it useful to learn this method instead of the usual "borrow" method?

I came across this method to perform subtraction using addition and not using the "borrow" concept, apparently because it is harder to learn it that way. Video - https://www.youtube.com/watch?v=...
7
votes
6answers
1k views

Is there any other procedure to find the square root?

If no calculator is allowed, and we want to find the square root of a square number if it is large and analyzing to prime factors is hard, how can one proceed? For example, what to do if the number ...
7
votes
4answers
339 views

Teaching arithmetic operations ($+ - \times \div$) to a 3 year old

What is the best way to teach the standard arithmetic operations ($+ - \times \div$) to a 3 year old child? Also: How we do know if the child has really understood it? can a kid be 3 years old ...
6
votes
6answers
507 views

How to teach multiplication between integers for the first time

Some teachers teach multiplication between integers with the following rules: plus with plus gives plus plus with minus gives minus etc. So for example in order to deal with the multiplication $2\...
6
votes
1answer
165 views

Is a clear distinction made between signs and operators?

This question about FOIL, comments and answers made me think about the two roles of $-$: as a sign and as an operator. This struck me because the title "Why in the FOIL Method the terms are taken ...
5
votes
4answers
847 views

How to correct visualization of mathematical expressions?

This happens a lot when I try to explain the commutative property, mostly in elementary grade levels. I say 2 + 3 = ? and then the student usually replies with 5. Albeit they're not wrong, it's not ...
4
votes
1answer
747 views

Parse out 2/3 of 30 minus 11

Note - I am not a math teacher; I am seeking an answer from a math teacher. While helping my son with his 5th grade homework today we had one answer wrong. I'm not sure I agree with the book and ...
4
votes
2answers
360 views

What's the word for addition and subtraction without borrowing or carrying over?

Is it regrouping? Upon googling it seems regrouping is borrowing or carrying over collectively. What's the word for not borrowing and carrying over? It's supposedly to train mental computation. ...
3
votes
3answers
374 views

Why in the FOIL Method the terms are taken with their signs?

That was the most boring title I could choose but in all honesty, it is what the question is. Here is a reminder of the FOIL method that is used for multiplying two binomials. For example, to multiply ...
3
votes
4answers
225 views

How to naturally encounter the properties of identity, commutativity, associativity, and distributivity (to define rings)?

(Cross posted at MSE: https://math.stackexchange.com/questions/3742948/how-did-we-isolate-the-properties-of-identity-commutativity-associativity-and) In elementary school, I remember learning about ...
3
votes
3answers
210 views

Teaching three-year-old subtraction using the number line

I am aware of questions such as this one. On the other hand, I still believe that teaching a bright three year old subtraction is possible. He counts from $0$ to $100$ and backwards from $10$ to $-10$,...
2
votes
4answers
506 views

At what age are most children able to convert between rational fractions and decimals?

At what age are most children able/taught to convert between rational fractions and decimals? For example Convert 0.25 to a fraction consisting only of whole numbers. What is 3/4 expressed in ...
2
votes
3answers
145 views

Is there a name for this method of column addition and subtraction?

Suppose I want to subtract 46 from 52. Instead of the borrowing method, I can use this method: \begin{array}{r} & 5 & 2\\ -\!\!\!\!\!\!& 4 & 6 \\ \hline & & -4 \\ +\!\!\!\!\!\!&...
2
votes
2answers
189 views

Introductory exercise for the addition of large natural numbers

I'm starting a repetition with my students in 5th grade after they learned in elementary school how to sum up larger natural numbers (also 5- to 6-digits) by writing down that calculation. As ...
2
votes
1answer
75 views

How to use these actions words for subtraction?

Do the following sentences express 5-2? a. 5 fewer 2 is 3 b. 2 fewer than 5 is 3 c. 5 less 2 is 3 d. 5 gave 2 is 3 I also saw online that "shared" can also mean "...
1
vote
1answer
196 views

Did Americans, before new math came in to schools, really say, 'three from two is nine carry the one", instead of borrowing ten from the tens column?

Tom Lehrer claims and the audience seems to agree with him that the 'old way', before new math to do subtraction was to say, for example, 'three from two is nine carry the one'. I never heard of this ...
1
vote
2answers
234 views

Workbooks for advanced high school math topics

I'm looking for advanced workbooks and exercises for working in class (math high school/undergraduate level) covering the following topics (or some of them): Logic and sets (propositional calculus, ...
0
votes
1answer
169 views

Viewing arithmetical operations as processes-possibly wrong and detrimental to long term math performance of the students [duplicate]

I think that the standard practice in the first grades when addition (or other operation) is taught as a "process" may be not so good. I always wondered why so many children lose interest in math ...
0
votes
1answer
124 views

How to present the order of factors and summands for the usual multiplication procedure

In the following multiplication example, $$\begin{align} 34\;& \\\underline{\times\;\; 7\;}& \end{align}$$ first one would multiply the units digits, producing the partial product $28$ as ...