# Tag Info

## Hot answers tagged primary-education

57

I'm going to focus on one aspect of the question that I think has not been fully appreciated: How can one best convey to beginners—without algebra—the flipping of denominator fractions... what would convince a novice Many of the ingredients of this answer are already present in some of the other answers to this question, but are rearranged here in a ...

54

"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 other. Does that seem like it would be a good idea?" is both more inclusive and more correct. I don't even think it's that hairy to talk through the FTA with ...

53

To the student who wants to know why he or she should learn addition/multiplication/other mathy thing when they can use a calculator or computer: It is a matter of independence. You will not always have access to your preferred tool. Being able to compare prices in a store without shamefully pulling out a calculator will boost your confidence that you can ...

53

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 "equal" in all respects - they can be rotated differently, or be in different positions, or be different colors, or have different names, or differ ...

51

I couldn't agree more with @Steve's comment. 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 process of exploring solution strategies. It is akin to interpreting a passage written in a not-so-familiar dialect: ...

48

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 multiply 2.3 by 1.4" and have students (perhaps in groups) arrange and rank them by clarity/mistakes/etc. Instead of having students start by writing their ...

46

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 data generated by pupils at Caddington Village School in Bedford sheds light on which multiplications are actually the hardest – and how kids do overall. The ...

43

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 is not true that $-9$ is smaller than $-1$ in magnitude, although it makes sense to say that $-9$ is less than $-1$ although it is bigger in magnitude. The kid'...

40

Questions of this nature were addressed in some part by New Math in the late 50s and 60s. They were also used as a point of criticism by its opponents. Most notable is Morris Kline's book Why Johnny Can't Add in which he writes relevantly in Chapter 1 : Evidently the class is not doing too well and so the teacher tries a simpler question. "Is 7 a number?"...

38

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 -- much, much older) think that a triangle is a triangle. It's a polygon formed by three non-colinear points. A triangle has many ways you can think about it. ...

34

There's a lot in this brief question, and I would like to try to give a brief answer, so I'm going to pick and choose what I respond to (and others might choose different things). Here are the parts of this question I see: Is teaching multiplication as repeated addition problematic? Is the problem with teaching multiplication as repeated addition that it ...

34

33

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 a better feel for what's going on. I'd call this a way to think about 9s. (There are many.) This method shows that you add 10 for each new nine, and then take ...

33

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 connects to first two, there are now 4 regions. It sure looks like the regions are doubling. In fact, they are not. It's a fun problem, and takes you by surprise. ...

29

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 same", we mean that they are not just indistinguishable, but that they are literally the same exact thing. "Equality" means two numbers are the ...

28

In the first place, the impetus to "reform" math education was motivated by politics, not by any serious observed deficit. By coincidence, there was a "new" style in higher mathematics, reflecting the previous 50-60 years assimilation of set theory and rewriting of many things in terms of set theory. But until the "sputnik scare" no one had incentive to ...

27

I think that actually trying to get students at this age to contemplate infinities in a rigorous way is probably ill advised. I do think that exploring counting from both an "ordinal" and a "cardinal" point of view is probably a good idea. Example for a 5 year old: Something you could do is have 20 stuffed animals, only 18 of them wearing hats. You can ...

27

I think the issue fundamentally is about a student's comfort with symbols and notation. A student who relies on a calculator thinks of it as a magical oracle that returns an answer to the question being asked. So, when that same student is asked to solve even the simplest of algebra problems, since the calculator can't answer that problem, the student is ...

27

It's pretty rare these days for anyone to actually do division by hand. Most people reach for a calculator. Given those realities, I would question whether it even makes sense that we spend such a vast amount of time teaching young children even one algorithm for division. Maybe we should postpone it, deemphasize it, or replace long division with a slower or ...

26

As an example of one curriculum, the Common Core standards say that students, "By the end of Grade 3, know from memory all products of two one-digit numbers." (http://www.corestandards.org/Math/Content/3/OA/C/7/) This matches my personal experience, too, for what it's worth, in the 1970's. In the U.S., this would be around age 8 or 9 years old. Now, as ...

25

Personally, I'd say "no." The students who don't write out the steps in their algebra classes appear in calculus thinking that they should be able to write down all the answers without any intermediate steps. I even have some students in Calculus 2 who think that there is some kind of value in not writing down the steps. None of these students can complete ...

24

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 your groceries will cost within $10\%$ before they are rung up. To know that, you need to be able to add and multiply in your head. You don't need many ...

24

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 the word sphere is particularly confusing in this context. (Mathematicians use sphere to mean the the two-dimensional surface; colloquial speech and some ...

22

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 $\varnothing, \{\varnothing\}, \{\varnothing,\{\varnothing\},\{\{\varnothing\}\}\}$ etc. to kids. Let an empty plastic bag represent the empty set. Then a plastic bag ...

22

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 need to do is introducing base 2 as well as number systems working with modulo (two's complement). We also introduce basic circuitry to do addition/...

21

Personally, I refer to this phenomenon as students "submarining" a broken understanding on a particular kind of problem. Example #1: Our in-house elementary algebra textbook, in its first edition, had this problem: If $1.05x = 22.05$, then $x = ?$ Note that the result is the same whether the student correctly divides both sides by 1.05, or incorrectly ...

20

The other answers are good, and/but some of those points are reinforced by a diagnostic that has ever-more been important to me: is intention being communicated? And a different, more linguistic point mentioned in other answers: toleration of ambiguity, allowing context (where intent is adequately clear) to disambiguate. Thus, yes, it is perverse to entrap/...

20

Most of the research on gender and math education is focused on student gender differences. However, a few references can be found that focus on the what differences there may be based on the gender of the teacher. One thing that appears to be common among some of these studies appears to be that perception of student performance varies based on gender of ...

20

Yes. This is also a trick that you can do on your fingers, too. For instance, let's say you wanted to calculate $9\times3$. Hold out your hands and bend your third finger down as shown. So nine fingers are "up" (fingers up, $9$, finger #3 down. (9x3). You have two fingers to the left of the bent finger and seven to the right, indicating the product of $27$...

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