# Tag Info

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

The following response is written with elementary-to-high-school mathematics in mind. A lack of a decent number sense really does encumber making sense of and parsing word problems, as well as the ...
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### Why do we introduce the notion that triangles are "congruent" instead of just saying that they are "the same" or "equal"?

Colloquially, there's a lot of conceptual overlap between all of these terms, but "sameness" is not a well-defined mathematical property. Congruent shapes need not be "the same" or ...

### Explaining why (or whether) zero and one are prime, composite or neither to younger children

"Because we said so" is a bit of a conversation closer, I agree. But "Because some people agreed a long time ago to define it that way so we could have conversations where we all understood each ...
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### Future educators writing nonsense questions

You might try starting this kind of lesson with an assignment where you provide a list of different responses to the prompt "Write a variety of word problems which would require the student to ...
• 9,699
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### Which product of single digits do children usually get wrong?

https://www.theguardian.com/news/datablog/2013/may/31/times-tables-hardest-easiest-children There are links to a dataset in the article. As far as I can tell, this isn't a formal study: But some new ...
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### Explaining the order of negative integers

To my mind, the problem is the word smallest. If you asked me which is smaller, $-1$ or $-9$, I'd ask you to clarify in what sense. Colloquial use of small refers to magnitude rather than ordering. It ...
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### What can (and should) an educator do about ambiguous terms like "triangle", "square", etc?

I'm confused. Are you really going to try to make this sort of distinction when teaching geometric figures to students "around 9-13 years old"? Students that age (and engineers my age -- ...
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### What value is there in requiring students to answer word problems in complete sentences?

Yes, there is mathematical pedagogical value in the usage of complete sentences - but this does not only refer to "answers" and not only to "word problems", but to all parts of the ...
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### What value is there in requiring students to answer word problems in complete sentences?

I do think there is value in expecting students to give answers in correct English. It will certainly help when they start to face longer and less structured questions. However, the example you give ...

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### Is this primarily a "rote computational trick" for multiplication by 9?

Anything that is just a trick leads to students having wrong ideas about what math is. But methods that help students see the patterns can help them learn the multiplication facts, along with getting ...
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### How can I explain why we need proofs to someone who has no experience in mathematical thinking?

More fun than equations are patterns that seem to hold. Put a dot on a circle, connect it to all the other dots (none yet), there is 1 region. Second dot connects to first, two regions. Third dot ...
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### Why do we introduce the notion that triangles are "congruent" instead of just saying that they are "the same" or "equal"?

The smart-aleck answer is that most congruent triangles, or congruent figures more generally, aren't actually "the same" or "equal". Usually when we say two things are "the ...

### Why do some students struggle so much with fractions?

There are many reasons why fractions are so hard for students to learn. Mostly, they're taught gibberish and assessed according to such gibberish. Example 1 You are a 12-year-old student who has ...
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### Is there a virtue to learning how to compute by hand?

I find the ability to estimate calculations quite useful and I think you need to be able do do calculations to estimate them. If you are keeping a grocery budget, I would suggest you should know what ...

### What is the justification to teach the (redundant) use of parentheses in multiplications?

Early in their education (even well into learning algebra), students don't naturally see the structure within algebraic expressions. It takes a bit of mental overhead to see that 5 x 10 + 5 x 8 is the ...
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### Is there a virtue to learning how to compute by hand?

Yes! But the virtue doesn't lie in being able to do the calculation but in gaining a feel for numbers as well as algorithmic thinking. I teach Computer Science freshmen and one of the first things we ...
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### What can (and should) an educator do about ambiguous terms like "triangle", "square", etc?

One encounters exactly the same issue teaching multivariable calculus when one treats integrals over three-dimensional regions and integrals over the surfaces that are their boundaries. In particular ...
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### What is the rationale for distinguishing between proper and improper fractions?

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 ...
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### How to encourage young student to think in unusual ways?

A lot of the answers here are suggesting giving examples where the alternatives you're interested in are necessary (or at least wildly easier than the "rigid" approach). That's a good option,...
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### Should math for elementary teachers content be taught under the direction of the math department?

It seems like a big problem to me. So many people in education are scared of math. You need a course that helps teacher candidates get over their fear. And you need someone who loves math teaching it. ...
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### How to justify formula for area of triangle (or parallelogram)

Remember that this is an explanation, not a proof. You don't need to define precisely how to construct the encompassing rectangle. Just demonstrate drawing an encompassing rectangle and have the 8-...
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### What can (and should) an educator do about ambiguous terms like "triangle", "square", etc?

I think the distinction you are raising is not natural to students at this age. I teach undergraduates and graduate students, not elementary schoolers, but I find that it is not natural for ...
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### What value is there in requiring students to answer word problems in complete sentences?

An example of a problem where phrasing the answer as a sentence might prevent mistakes and encourage understanding is: Alice goes to the store with \$2.00. A gumball costs \$0.80. How many gumballs ...
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### Fun set theory for kids

Hilbert's Hotel is a nice thought experiment for explaining results about cardinality of infinite sets and the aleph numbers. I have also used plastic bags to explain the difference between \$\...
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### Teaching Math for Elementary Teachers

I don't know where you're from, so I'll address this from a North American perspective. If you're thinking research-level mathematics is useful to learning how to deal with 5-year-olds, you might just ...
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