50

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 ...


46

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 ...


44

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'...


36

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?"...


31

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 ...


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 ...


26

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 ...


24

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 (back in the 1970's). Now, as someone who teaches community college remedial math, ...


23

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 ...


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 ...


19

First, an answer to your title question: Is it advisable to avoid teaching “multiplication as repeated addition”? No, one should not avoid teaching multiplication in this way. There are other ways to teach it, carrying their own advantages and disadvantages, but such is the way that multiplication is defined for the natural numbers, and this (also ...


18

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/...


17

I realize that your question is about the $100 \times 100$ table: But since you ask about approaching the multiplication table in ways other than by rote learning alone, I thought I would leave you with a list of problems I generated based on the $10 \times 10$ table. [Edit 5/9/14: You can find some of the problems below in an informal paper of mine; the ...


17

I remember being excited about the following at a young age. If you add consecutive numbers you get triangle numbers. Triangle numbers are fun. If you put two consecutive triangle numbers together you get a square number. You can also make a square number by adding the next odd number.


16

You might want to read Kline (1973). I haven't read the book, but according to Wikipedia, In 1973, Morris Kline published his critical book Why Johnny Can't Add: the Failure of the New Math. It explains the desire to be relevant with mathematics representing something more modern than traditional topics. He says certain advocates of the new topics "...


16

The reason we teach strategies for multi-digit operations is (in a large part) because students are learning to manipulate the symbol system we use to represent numbers, and they need to see that there is meaning behind these strings of digits. That understanding carries through not just for subtraction, but for anywhere they'll use multi-digit numbers. To ...


16

Here are a few comments, and then an attempt at two succinct answers. In particular, I will try to answer this using a measurement interpretation, and then again with an equal sharing interpretation. I prefer the former, but include the latter for completeness. Comments: Some of the key terms in unpacking this are measurement, equal sharing, and missing ...


16

How about: Numbers go the other way, too (negative) You can cut numbers in half, forever What if you cut a number into three pieces? 1 million is a thousand thousands (100 is ten tens) If you don't know the number, call it a letter (or name it :) ) You can add letters together too What if you cut triangles in half? Rectangles? Can you make a box from ...


15

I have been keeping up with the Canadian mathematics education battles from the last few years, especially in how the media discusses it. In fact, my last graduate assignment had everything to do with this issue. I know that the "Math Wars" in Canada and in the United States are closely related. The debate has to do with a "traditional" vs, a "reform" ...


15

People who ask why calculators don't resolve the (very low-level) issues well enough are quite accurate in their skepticism, I think, which makes it hard to present an "absolutist" defense of alternatives. For that matter, seriously (!), why is it ok to use Hindu-Arabic numerals and the weird associated algorithms, rather than "honest" manipulations of hash-...


14

There is some relevant research (and bibliography) in this paper: "Improving basic multiplication fact recall for primary school students" (Wong and Evans, 2007) http://link.springer.com/article/10.1007/BF03217451 For me the most striking thing is how low the scores reported in their Figure 2 are (36.89/60 at best), even after interventions that they ...


14

I think you'll find some of what you want on Berkley mathematician H.H. Wu's homepage. More precisely, see: Pre-Algebra (pdf) and Introduction to School Algebra (pdf). Note: I mentioned the same homepage (and the two pdf textbooks) in an earlier MESE post here; I would have just re-posted this as a comment, but I believe it is the actual answer to your ...


14

You will probably not find precisely what you're looking for if you want to compare one algorithm to another for effectiveness. The reason is, the problem that was identified in the 70's (Erlwanger, 1973) was not about the unsuitability of the algorithm itself. It could better be thought of as a problem of focusing on algorithm. Erlwanger (1973) ...


14

One reason is that mathematics was not handed down by the gods fully formed and unambiguous. It is a human construction over a very long time and mathematical notation even more so. Any time a notation doesn't "work" for all possible contexts, it's an opportunity for us to talk about this human side of mathematics, and about the pros and cons of notation ...


14

A sketch of one idea. I think it's probably better spread over a couple of days. Day one: Start them counting, from zero, out loud to you. Write the numbers on the board as they go. Zero (0), one (1), two (2), ... , ten (10). Stop here. Prompt a discussion about what happened - how is the most recent number different than all of the previous numbers? I ...


14

I find this diagram helpful when relating the two: It comes from a model curriculum unit on Rates and Ratios for 6th graders (you can see them all here after registering), and I have found this particular graphic very helpful with math content professional development with 6th grade math teachers.


14

I have no experience teaching fractions, but I think moving away from using the divide symbol makes things easier. It doesn't get used at university level (but exponents start being used, so there are still two notations). I would do $$3\div\frac{2}{7} = \frac{3}{\frac{2}{7}} = \frac{3}{\frac{2}{7}} \times \frac{7}{7} = \frac{3\times 7}{\frac{2}{7}\...


14

It seems to me that what you asked wasn't really right. -9 is the "lowest one-digit integer" but (at least it can reasonably argued) 0 is the smallest. Maybe making this difference explicit would clear confusion: "big" negative numbers are a long way less than zero and therefore the lowest.


14

Note (Feb 2018): There is an alleged "Chinese math problem" (see, e.g., WaPo article) going around about the second example problem below (cited to Reusser 1988, but can be found in Reusser 1986, as I've tweeted here). Interesting that it has gone viral without anyone having sourced it. This study can be easily replicated, and has been: with multiple ...


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