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Why are elementary students taught to represent one and a half as 1 1/2 rather than 1 + 1/2?

This mode of expression seems standard throughout at least North America. I think it is bad pedagogy for a couple of reasons:

First, doing it the 'correct' way would give students tonnes of personal experience equating the English word 'and' with the mathematical symbol '+'. This is a good thing, since it fosters the notion that mathematical statements (or in this case expressions) have tangible meaning. Students who understand what a half is do well with understanding what 3 and a half is, and I expect that representing it as 3 + 1/2 can do a great deal to cement the 'true meaning' of addition. Consider that the previous experience of students at this age is dominated by calculating 8+4 either counting 9-10-11-12 or by rote, neither of which is all that connected to the physical reality of addition.

Second, students will reach a point where they are expected to abide by the convention that ab represents a * b. Strong students will do OK with this other than a few early mistakes during an adjustment period. But students struggling in math, especially those experiencing phobia or anxiety around the subject, will have little chance but to understand this shift in notation as yet another in a seeming unending string of indications that what's expected from them in math class is entirely arbitrary, changes from one teacher to another, and is some sort of arcane magic. The horrible thing is that in this case they're correct to interpret it this way!

I have to be missing something. What advantages does the current scheme provide?

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  • $\begingroup$ This is not meant to be an answer, but I think this is mostly due to reality. In reality, you want to write 4 1/2 fast because you have e.g. 140 characters for the ad for your new appartment or the length of a screw is 5 1/8 inches, and 5.125 means nothing compared to 5 1/8 for the layman. Dividing 1 by 8 is much easier to deal with to have a picture than dividing 125 by 1000. Because computationally this (5 1/8) convention is horrible. $\endgroup$
    – Patrick Da Silva
    Feb 23, 2014 at 5:09
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    $\begingroup$ The current scheme has inertia on its side. You aren't likely to get a hundred years or so of printing changed. $\endgroup$
    – John Habert
    Feb 23, 2014 at 5:10
  • $\begingroup$ I had considered this prospect, but I hope there's something better. I wouldn't expect or even necessarily want that a recipe call for 2 + 1/2 cups of flour, but curriculum setters have lots of opportunity to weigh consequences of their materials and have obviously settled on this method. $\endgroup$
    – ColinK
    Feb 23, 2014 at 5:15
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    $\begingroup$ As a teacher, all my upper students (college level classes) get to hear my take on mixed numbers and why they go against math notation they see everywhere else. Having taught future elementary school teachers, they have heard the same. But part of the problem is the US doesn't employ mathematicians/mathematical people to teach in elementary schools where the bad habits are learned. $\endgroup$
    – John Habert
    Feb 23, 2014 at 5:22
  • $\begingroup$ John, the US (and other governments) does educate people to teach in elementary schools, and this convention is prescribed to those educators. Curriculum guides are extensively researched and meticulously assembled, and this convention is a conscious decision. $\endgroup$
    – ColinK
    Feb 23, 2014 at 11:58

5 Answers 5

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Subtraction.

The mixed fraction form makes subtraction, and possibly addition, easier to parse.

$$2\frac{1}{2} - 1\frac{1}{4}$$ is easier to read, write, and understand at this stage than either $2+\frac{1}{2} - (1+\frac{1}{4})$ or $2+\frac{1}{2} - 1-\frac{1}{4}$. Students at this age have not encountered the distributive property, and they may have trouble attaching the '$-$' to the $\frac{1}{4}$.

Negative values are similarly challenging.

I don't think that this is a good enough reason, but it's a rationale that occurred to me.

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In my opinion, there is no good reason to write "mixed numbers" without a "$+$" sign. When I teach college students or tutor high school students and notice that notation in their work, I insist that they retire that habit for both of the two reasons that you mention in your question.

Furthermore, it fosters the sense that an expression like $\frac{17}{5}$ is not a "real" fraction and that you always need to perform division to write it as $3 + \frac{2}{5}$. This is detrimental, as it obscures the obvious fact that $$ 5 \cdot \frac{17}{5} = 17 $$ by making it look like the less obvious $$ 5 \cdot \left( 3 + \frac{2}{5} \right) = 17. $$

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Well since we're being pedantic, $1 + \frac 1 2 \text{ grams } \ne \frac 3 2 \text{ grams } = \left(1 + \frac 1 2\right) \text{ grams }$. Also, the expression $1 + \frac 1 2$ is not the same as the expression $1 \frac 1 2$, the first is an addition of 2 rational values and the second is just a single rational value written differently. In algorithmic formal logic this sort difference can be quite significant.

Since this is an opinion question, I'll vote for the actual culprit being the common omission of the multiplication sign, which leads to plenty of other ambiguities.

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  • $\begingroup$ Very good point. However, it is also conventional to refer to both the (limiting) process and a number/function with $\sum_{i=1}^\infty$ or $\int$ $\endgroup$
    – Karl Kroningfeld
    Feb 24, 2014 at 0:00
  • $\begingroup$ I'm afraid I don't follow. Can you provide an example? $\endgroup$
    – DanielV
    Feb 24, 2014 at 1:47
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Discounting practical aspects and tradition, as mentioned in the comments, there are no advantages.

On the other hand, ambiguity is simply just a fact of life. Even within Mathematics.

http://en.wikipedia.org/wiki/Ambiguity

http://www.xamuel.com/ambiguous-math/

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Here's one (small) advantage: the mixed number $$-3\frac{1}{2}$$ implicitly means $$-\left(3+\frac{1}{2}\right),$$ so the notation $-3 \frac{1}{2}$ allows you to omit a pair of brackets.

(For what it's worth, I don't really think this is enough to justify the notation, and your second objection is imo a very strong one.)

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