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As math teachers, sometimes we may wish that we could enter our students' minds to see how they think. In early years, while children are learning language, they build the concept of the word greater in relation to the concept of volume, shown visually in the shape below: $1 > 3$.

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In fact if we rename the greater than sign to be more than, then I believe it will be more harmonious with the language.

Alternatively, to avoid conflict with negative numbers, we could say more positive. For example, read $ -2 > -5 $ as "$-2$ is more positive than $-5$." On the line of integers it would be clearer with this language that $-2$ is closer to the positive numbers than $-5$.

My question is: How deep and meaningful could this language alteration change children's concept of the inequality?

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    $\begingroup$ Language is not consistent and not designed to reflect mathematical/physical concepts. The simple word more can have very different mathematical/physical meanings: number, volume, mass, worth, …. Students have to learn to relate the word to the appropriate concept depending on context. The word greater has similar problems. But none of them relates to the negative numbers. $-2$ is greater than $-5$ and $-2$ is more than $-5$. By renaming, you would change one semantic field for another one but gain nothing significant. $\endgroup$ – Toscho Nov 10 '14 at 13:37
  • $\begingroup$ somewhat related, i like to use more left for less than and more right for greater than, this helps with confusion on negative numbers and helps students to think about numbers in terms of a number line $\endgroup$ – celeriko Nov 10 '14 at 19:35
  • $\begingroup$ +1, very good question. $\endgroup$ – Joel Reyes Noche Nov 11 '14 at 0:16
  • $\begingroup$ I had never even thought of the possibility of thinking that the $1$'s in $1+1+1$ could be smaller than $1$, but I guess it's possible if they never knew better! Reminds me of those biggest-number jokes and how someone used to write a 2 on the board below a 1 and say that 2 is higher than 1. Children and the uninitiated say the funniest and at times most creative things. $\endgroup$ – Vandermonde Nov 17 '15 at 22:24
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Talking about your proposed language alteration with your students would make more of a difference than just using it. You could help them see that math is partly about defining relationships carefully, and that language, with all its richness, isn't so good at that.

In calculus, when I ask where the derivative is greater, and we're looking at negative rates of change, I know there will be a problem in translation from words to math, so I point it out. (With positive slopes, greater means steeper, with negative slopes, greater ends up meaning less steep. It takes some getting used to.)

Asking students how they would describe things is more important than changing the vocabulary ourselves. Take a student's term, and make it a class term.

(Of course, if you use non-standard terms it's important to remind students of the official term that goes with your non-standard term, so they don't go out thinking the rutabaga of 9 is 3. See The Number Devil for more fun non-standard terms!)

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    $\begingroup$ +1 for the rutabaga of 9, although I also appreciate the wisdom in the rest of your answer ;) $\endgroup$ – Rodrigo A. Pérez Nov 11 '14 at 2:59
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    $\begingroup$ I liked making numbers hop. $\endgroup$ – Joe Z. Nov 12 '14 at 19:02
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The answer to the topic question: No.

Toscho should upgrade his comment to an answer. The name of the '>' sign is used in too many places to be replaced without causing confusion elsewhere.

But there is a more important point I want to make: One thing can have more comparable properties as you have demonstrated and Toscho commented. Size of circle vs count of the circles. Height of a person vs her age, or weight. Make sure you define the property you want to compare first. Then you can get another comparative adjectives (3 yrs older, 1 inch taller etc.) to help you explain what the greater means.

When you decide to work with numbers later, keep only numbers! No circles, no distractions. You may want to show the numbers as moves along the number line. Greater than to the right, less than (or whatever you find suitable) to the left. Note that the movement has always direction, this way the sign (+/-) comes into the mind as necessity.

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It is a good habit, in speaking or writing, to get used to saying "more positive" or "more negative" in situations where confusion may arise. However, students should also be taught that $<$ has a completely standard meaning, so that they can read books and listen to people who are not careful about this.

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Determining how mathematics applies to a situation (how to mathematize a situation) often requires measurement. To determine whether one thing is greater than another thing, first those things must be identified as quantities. And then they need to be compared somehow; this can involve a measurement process that results in coming up with a number to represent the quantity in some appropriate system of units.

In the OP example given, we could compare total diameters. We could compare individual diameters. We could compare total area or individual areas. We could compare hue. We could compare a measure of curvature. I'm sure there are a lot of other things to compare.

The point is, identifying quantities and measuring them is an important part of mathematical thinking that, among other things, helps to put mathematics in context. This should be a process students become used to engaging in, else their mathematical knowledge is trapped in specifically-worded problems.

In short, the real solution here is to get students to become aware of quantity and measurement and to ask questions if the quantities under comparison are not obvious. And perhaps specify their own basis for comparison if one is not handed to them by an authority figure.

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