Mar 5, 2013

Mathematics and Language

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Science Math Primary/Secondary Education

Mathematics and Language


Edited by Marsigit

Doug Hainline:

I believe that the language (including the symbolism) that we use to teach, and do, mathematics, is a factor in how easy it is for chldren to learn the subject.

Both our language, and our symbolic notation, have "just growed", although our notation has been subject to a slightly more rational process of selection.

To take a small example: I would much rather discuss geometry with nine-year-olds, by focussing on the interesting and important features of, say, plane figures, than by having them memorize that a triangle with only two sides equal is an isoceles triangle, but one with three sides equal is an equilateral triangle.

I would rather just say, "Let's look at the smallest number of sides we need to make a pen that would hold in a horse.... see, you can't do it in fewer than three.... now let's look at what we can say about three-sided figures ... how they can differ among each other ... what they all have to have in common... " and then go on to discover that all three-siders can have no sides equal, or two sides equal, or three sides equal. And then to look at the angles that we can have with each of these kinds of three-siders.

I think there are many areas in mathematics where, if we were starting over, we could make things much easier for ourselves and our students, if we could choose our vocabulary and notation.

Two more things -- from a longer list - which irritate me as being unnecessary impediments in learning mathematics:

(1) Conflating identities and equations and relations.

(2) Making the exponentiation operator implicit instead of explicit.

I wonder if others have any thoughts on this issue?

David McAdams :

I quite agree that the language of math needs as much attention as the process of math. I discovered that those students who can 'talk math' are the ones who succeed.

I compiled a list of words and phrases that have specific mathematical meanings for pre-algebra, beginning algebra, geometry and intermediate algebra (usually grades 7-10). There are over 2800 words and phrases that a teenager needs to know Vocabulary building activities are an essential part of any math instruction program that wishes to help all students succeed.

Two easy to implement ideas for vocabulary building are math cards and vocabulary sheets. Math cards are index card where students write the definitions of words, formulas to memorize, etc. The students are supposed to review the cards a few times a day. Vocabulary sheets contain words or phrases for which students are to find a definition. I have found that giving no credit for incomplete definitions, or sentence fragments increases the learning the students get from this activity.

One aid to math building activities is All Math Words Dictionary. For more information and discount rates, go to

Helen Mason :

 Doug, I have the same concern. Many math books use or assume a lot of vocabulary that is not necessary to the understanding of the mathematics. It would be useful to have people sit down and discuss how much vocabulary is necessary at what point. The current vocabulary-heavy topics can make it hard for students with learning difficulties or language processing issues to learn.

Marsigit Dr MA :

I also have the same concern; however, in the case of the role of language in learning math, I prefer to facilitate the students in order they are able to translate and to be translated, to produce and to be produced, to construct and to be constructed, to reflect and to be reflected, to evaluate and to be evaluated, to judge and to be judged.

Guangtian Zhu :

To provide another perspective from different culture, I would like to say that such concern would vanish in some other languages. For example, try this question
"how many sides are there in a
边形 (pentagon)"?
B. C. D.
You don't even need to know Chinese to find out the answer to be C (same as the first character in
边形). The name itself tells the students that a pentagon has five sides (-five, -side, -figure). Similarly, the vocabulary such as isoceles triangle and equilateral triangle are translated in Chinese in a straight forward way so that students can recognize the feature of the figures based on the names.

Helen Mason :

That would make things easier, Guangtian. Math and science terms in English tend to come from Latin or Greek roots. A century and half ago, educated people (who were upper class males only) learned those languages so might not have had the problems with the terms that students do today. Of course, there was more concentration on rote learning than on understanding back then.

Marsigit Dr MA :

So language has its own cultural context; the context in which the students are coming from. Therefore, I agree with the effort to employ language (mother tongue) to develop math educ. It leads to prove the fail of global software translator. In Indonesian we called Fraction as Bilangan Pecah. If we translate back into English using that translator, will be Broken Number. Ideot!. So language and mathematics can be contextual.

Helen Mason :

The same science and math terms have been employed for more than a century. The advantage of many of the current terms is that they are similar to those in other related European languages, making it easier to communicate between European cultures. For example:
English equilateral triangle
French triangle équilatéral
Italian triangolo equilatero
Spanish triángulo equilátero

As you can see, English, French, Italian, and Spanish all derive a lot of their words from Latin, which is no longer spoken.

David Reid :

It is not only the teachers who need to watch their vocabulary, and it is not just in the individual words that are used. The explanations in most mathematics textbooks for high school that I have used stink, not to put too fine a point on it. I often feel like doing like the English teacher in "Dead Poets' Society" and ripping out the gobbledygook --or at least what is gobbledygook for teenagers who have grown up with Tweets rather than sentences. Because of this language barrier, many mathematics teachers are tempted to just go for the numbers and equations, giving the students the erroneous idea that mathematics is only composed of numbers and equations. Reasoning seems to get lost in there somewhere.

Doug Hainline :

 I thank all of you very much for these most interesting comments.

Mr McAdams :

I have purchased your dictionary. I hope it does not contain the error made by a UK school dictionary of mathematics in my possession, which on one page defines a "fraction" as a "number smaller than one", and a few pages later defines pi as "a fraction approximately equal to ...."
I believe the names given to numbers and other mathematical concepts by different languages is a very interesting topic. The fact that the main Oriental languages are more logical than the European languages in their counting names -- and Guangtien has added a new bit of information on this subject -- has for about twenty years been adduced as one of the advantages that chldren learning to count in these languages have. ("ten-one, ten-two, ten-three ..." instead of "eleven, twelve, thirteen..." ) although this has recently been called into question, apparently.

Helen, thank you for your list of triangle-names in other European languages. And I don't think it's only kids out toward the end of the Special Needs spectrum who needlessly suffer because of clumsy, unnecessary vocabulary. I think we all do.

I wonder if anyone who knows other languages -- Arabic, or the languages of the Indian sub-continent -- might be persuaded to list some mathematical vocabulary, with literal translations if possible? What is an "equilateral triangle" [my "all-equal three-sider"] in Hindi? In Farsi? What are the names of the first three or four numbers after ten?

On a related, perhaps deeper matter -- I think that words that sound familiar, which have everyday 'meanings', but which are also used in mathematics, can also be problematic. I am particularly thinking about the way we assign a word-sound to this sign: '='. Surely this sign -- " = " -- is used to refer to at least two, maybe three, rather different ideas. "2 + 1 = 3" and "X + X + X = 3X" has one meaning; in "X + 3 = 5" it has another; and in "y = 3X + 2" it has yet another. (Or perhaps not -- I'm not a mathematician. It just seems that way to me, and I find when I am tutoring, that it appears to help clarify things for my tutees if I point out that these are three different kinds of things.)

David Reid :

Doug-- in mathematics, there are primarily two possible meanings to equality (of course, you can make it mean a poodle if you want, but keeping a bit to convention....): intensional and extensional. The intensional meaning is basically that a=b if for all relations, you can substitute a for b or vice-versa, and the sentences retain the same respective truth values. The extensional one is when you treat your entities as sets (or as being in a one-one correspondence with sets which are elements in a structure isomorphic to your structure, etc.), and then the two entities are equal if they have the same elements (or if their sets with which they are in get the idea). (Then you have a separate definition for urelements if you want to include those.) In any case, no, the meaning of equality will be the same for all the examples you gave, just that in elementary mathematics we leave the quantifiers to be implicit. That is the difference between a sentence with variables and a sentence without them.

Doug Hainline :

David, thank you very much for your comment, which I don't really understand completely at first reading. But I'll think about it, as I find mathematical things are often this way. You have to come back and chew over the ideas.

Perhaps I am expressing myself clumsily, and blaming the innocent equals sign for a misdemeanor whose source is elsewhere.

Here's what I'm trying to say: that X^2 -Y^2 is the same thing as (X+Y) x (X-Y) is a fact that is true for all X and all Y provided they are real numbers. (I don't know about other kinds of numbers, like complex numbers. Perhaps it's true for them too.) You can, and should, learn this fact by heart (as well as knowing how to show that it's true.)

But X^2 -9X + 20 = 0, is only a fact for X = -5 and X = -4.

So perhaps it's not the equals sign, but something else, that I am trying to get at. In any case, it's my experience that young people learning mathematics are not taught that there is a difference between these two kinds of sentence (and the same when they come to functions), and I wonder if this is a barrier to learning to be proficient in using them.

David Reid :

 Doug, sorry for being a bit short in my explanations. But your are right that
(which is correct for complex numbers as well)
differs from the sentence
x^2 - 9x + 20 = 0
but not because of a difference in meaning of the equal sign in both. Rather, there is a difference in the quantifiers (such as "for all x" , "there exists an x such that...", ) that we suppress in schools. That is, the difference of squares reads, more fully,
For all x in D, for all y in D, (x+y)(x-y)=x^2-y^2
(where D is your domain: real numbers, complex numbers, whatever)
The second one is , if posed as a question,
There exists an x in D such that x^2 - 9x + 20 = 0. (The question is then to find all x which will satisfy this sentence.)
If it is posed as a fact as you expressed it, the sentence is
For all x, (x in {-5, -4} implies x^2 - 9x + 20 = 0)
So, that is the difference. Teachers in high school suppress these quantifiers in writing the equations because they are "understood" -- or at least that is the way the teacher sees it. The way the student sees it may be different. Teachers suppose, often incorrectly, that the pupils will understand this difference without having it explicitly explained to them.

Doug Hainline :

Got it! I've just been re-reading about universal and existential quantifiers -- oddly enough, in a little booklet on Language in Mathematics published in the early 1960s -- and I see that this makes perfect sense. I am going to experiment with explaining this to my tutees. (I have to scientific data on this, but my subjective impression is that ideas that seem far beyond school mathematics are actually not all that difficult, in their basic form, if introduced properly.)

But .. what about functions? y = 3X + 2 "looks like" one half of a simultaneous equation. Yet I teach my tutees that it expresses a relationship between X and Y, true for an infinite number of X's and Y's (assuming the domain of X is not restricted).

David Reid :

Doug: there is no problem here, both are true. y=3x+2, as a function, is simply
for all x, there exists a y such that y=3x+2.
[To emphasize that it is a function, you can put
for all x there exists a unique y such that y=3x+2, that is,
for all x there exists a y such that y=3x+2 & for all z (z=3x+2 implies z=y)].
Then, as far as simultaneous equations, in schools we are a bit sloppy: whereas we will use two letters to distinguish between the two functions, eg
f(x) = 3x+2 and g(x) = 4x-4
we then sloppily put
y = 3x+2 and y = 4x-4.
It should properly be
y = 3x+2 and z = 4x-4.
or, more fully, to talk about the two functions superimposed on a single set of axes (where really there are two vertical axes, y and z):
for all x in D (there exists a unique y such that y=3x+2 and there exists a unique z such that z=4x-4).
Then, "intersection" is "and", so putting them together,
for all x in D (there exists a unique y such that y=3x+2 and there exists a unique z such that z=4x-4 and y=z).
or more simply
There exists an (x,y) such that y = 3x+2 & y=4x-4.
Existence is not the same thing as finding it; for that one can use the iota notation, which is a "definite descriptor" (there is debate as to whether it can be called a quantifier, but it can be rewritten in terms of more conventional quantifiers)
(Unfortunately, this forum does not take Greek letters, so "iota" below stands for the Greek letter with that name; "iota x" is read as "the unique x such that....")
(x,y) such that y = 3x+2 & y=3x-4.
And yes, since English and other languages use quantifiers on a daily basis, there is no reason why a patient school child cannot learn them.

David McAdams :

I'd like to comment on suggestion of Helen Mason that we reduce the amount of math vocabulary that children have to learn. I say that, given the nature of math, that is possible only to a small degree. All of the vocabulary that exists in math exists for a reason: someone used a word or phrase, borrowed or invented, to express a specific concept. To eliminate a word or phrase would usually erase that concept from math.

The are a number of mathematical synonyms, which have little or no difference in denotation, at least in the middle school and high school settings. Some of these are Abelian and commutative; absolute magnitude and absolute value; accidental sample and convenience sample. Agreeing to use a particular word or phrase would reduce some of the confusions.

However, when one accounts for the multiple forms of English throughout the world, the phrase selection becomes more difficult. For example 'absolute magnitude' is preferred in the United Kingdom, and 'absolute value' is preferred in the U.S.

Even if these differences could be agreed upon, this method would only eliminate a few words or phrases from children's math vocabularies.

Guangtian Zhu's note suggests one way to accomplish this. Many of the words used in English math vocabulary have roots in Greek. To use Mr. Zhu's example, penta- means five. Replacement of the Greek roots would change pentagon to five-gon. One could go a step further and replace -gon with -sides. Then pentagon would become five-sides. And this brings to light a major problem with this methodology.

There are many things, both mathematical and not, with five sides that are not pentagons. Pentagon means specifically, a five sided object with straight sides where the two of the sides meet at each of five nodes. The use of -gon implies everything after the 'five'.

The use of ancient Greek has a specific advantage here. English students really don't care much how the root -gon was used in ancient or modern Greece. The just have to know what it means mathematically. And, it has a very specific, detailed mathematical meaning. So, I say it would be reasonable to rename pentagon into fivegon, but would be unreasonable to call it a fivesides.

Again, I say that reduction of math vocabulary for students in middle and high schools is possible on a small scale, attempting it on a large scale would create more confusion that before.

Doug Hainline :

Thank you very much. This is beginning to become clear to me now.

I have been trying to deal with my tutee's lack of clarity by talking about the equals sign, when actually, I should have been doing a bit of logical analysis. Your explanation of the meaning of functions, and especially of what we should be saying when we have more than one function, was very helpful.

I don't think the idea of universal and existential quantifiers is too difficult for kids doing algebra. I wonder why it's not taken up in mathematics curricula?

David Reid :

 Doug, I am not sure which curricula you are referring to, but I will answer what I know of the US curriculum. (I have worked with other curricula as well, but I am less informed about the history of the curriculum development in those curricula.) As far as I understand it, the reason is historical: back in the 1960's, one decided to bring down some of the foundational work from the universities down to the schools, and set theory was introduced at all levels under the name "New Math". But teachers were not clear on how to teach it, the textbook writers were even worse (so of course the teachers followed the textbooks and exams were set accordingly) in that the ideas became downgraded to exercises that were just as much rote exercises as the ones that they were meant to be a conceptual support for. As a consequence, it didn't really form this support, and, time constraints being what they were, many of the basic skills never got learned, leading to a disastrous drop in skills in the 1970's. So the whole "New Math" was chucked overboard, and the baby was thrown out with the bathwater: now there is very little mention of set theory or logic in the US curriculum; what little there is is pitiful, and again the teachers still don't get taught it much, so how can they properly teach it?

David McAdams, I agree that tinkering with vocabulary in the schools is only possible on a limited scale. One reason that you did not bring up is that it is always a problem when a concept is named one thing at the school level and another at the university level: and there is no way the mathematics community is going to change their vocabulary to appease the school mathematics teachers.

Doug Hainline :

The big weakness of trying to make the mathematics curriculum more 'mathematical' is that the curriculum has to be delivered by the existing set of primary [elementary] school teachers, and in many countries, they are not sufficiently trained in mathematics to do anything like teach even elementary set theory. In fact, I am afraid a significant number of teachers at this level are not very well versed in mathematics, don't like it, even fear it, and this attitude is transmitted to their pupils.

I don't think anything can be done about this in short run. I do believe that the internet is going to make it possible for us to do a MUCH better job of teaching, even with weak teachers.

But it is interesting to speculate about what changes there ought to be, and also about what a more rational language for learning and thinking about mathematics would be.

With respect to language, I would certainly follow the Oriental languages and replace 'eleven' and 'twelve' with 'ten-one' and 'ten-two', as well as making the 'teens' into 'ten-three' , 'ten-four' and so on.

I would call plane figures by the number of the lines that made them up: three-liners, four-liners, five-liners, etc. (I wouldn't use the word 'side' because it's ambiguous: is a 'side' a line or a plane? I'd have 'lines', 'faces' and even 'corners' (instead of 'angles').

Of course this is all silly, because it's not going to happen. But at least we should be aware of the ridiculous, confusing hoops we're needlessly making children leap through, by making them have to learn that a ten-two-facer is a 'dodecahedron'. (And the screaming irony is, in Greek, it's almost as it should be: a two-ten-facer, or, rather, a two-ten-flat-thingie-you-might-sit-on.)

WIth respect to logic and set theory -- I don't believe that, at the elementary level, these subjects should be beyond the grasp of, say, 12-year olds.

At the moment, I think we almost teach anti-logic. I recall my granddaughter coming home with a handout informing her that there were four kinds of triangles: scalene, isosceles, equilateral, and right. After I finished banging my head on the kitchen table, I explained that this was like saying that there were four kinds of people at her school: teachers, boy pupils, girl pupils, and kids who played in the school orchestra.

Marsigit Dr MA :

@ Dough: Still I am interested with your statement "The big weakness of trying to make the mathematics curriculum more 'mathematical' is that the curriculum has to be delivered by the existing set of primary [elementary] school teachers, and in many countries, they are not sufficiently trained in mathematics to do anything like teach even elementary set theory. In fact, I am afraid a significant number of teachers at this level are not very well versed in mathematics, don't like it, even fear it, and this attitude is transmitted to their pupils. "

However, my concerns are reflected, to some extend by reversing your notions, as the following "The big weakness of trying to make the mathematics curriculum more "humanized" are coming from the adults (teachers, educationist, and pure mathematicians). In fact, I am afraid a significant number of outsiders (pure mathematicians) behave inappropriately to intervene primary schools teachers, by employing their very simple logic that if their mathematics is okay, then everything (their teaching) will be okay.

No, teaching learning of mathematics is not just as simple as that idea. For me, to be a primary school mathematics teacher, she/he should not be a pure mathematician.

David Reid :

Dr. Marsigit, or someone who has better parsing skills than me, please clarify the sentence, which apparently contains a typographical error (but I am not sure what it is ): "I am afraid a significant number of outsiders (pure mathematicians) behave inappropriately to intervene primary schools teachers," ("intervene" is not a transitive verb, and if it is simply a case of a missing preposition, I cannot figure out which one would make the sentence make sense.)
By the way, there isn't much danger of a pure mathematician becoming a primary school teacher. In fact, the problem is usually in the other direction, as Doug pointed out, with primary school teachers knowing too little mathematics.

Marsigit Dr MA :

@ David Reid: Thanks for the response. Further I wish to know your ideas about the position of language in intuition or vice verse. Do you have any ideas about the role of language in developing mathematical intuitions? Thank.

David Reid :

Er, thanks for thanking me for the response, but you haven't answered the question I put there.
The role of language in developing mathematical intuitions is a very broad and complex subject, involving the question of the relations of the language centers in the brain with the other parts of the brain. Yes, my studies have included this theme, although I am no expert. In any case, any answer I would give in a short paragraph to such a complex question would be superficial. Perhaps you would like to make your question more specific?

Marsigit Dr MA :

@ David, firstly I wish to thank to Doug for his exposing the relationship between math and language. I think that the role of language in developing mathematical knowledge is a central; because mathematics itself is a language (Wittgenstain). Even now, I, you and others are exchanging the language. I wish to say that the language has its intentional and extensional dimensions ( I am not sure whether the terms I used are similar meaning with yours). By intentional dimension,I mean the deepness; and by extensional is the variety. I am aware that for a certain concept (not only in math) I and others have different terms to express. You have intensively posed and mixed the different level of language.

While, we understand, at a common-sense level, that the main purpose of using the language is for communications. The intentional dimension of communication can be translating and to be translated. If we extend this intentional dimension we may find that the role of language is for constructing the life. So I understand the small part of your life by understanding your notions. Further, this rule should meet to every single people. In sum, I wish to say that the language is the life itself. In fact, there is no similar people in the world; so theoretically, the number of language is the same with the number of human being ever life.

In the case of education, if you respect the existence of individual differences, you may aware that there may many peoples outside you who do not understand much of your notions you produced. Here, the role of language is not just as communication, but also for interaction in order to construct individual life. So the awareness and the study of language of a certain people from a certain context in which you may want to interact, is very-very important. That's why Theresia Nunes was very popular with her Street Mathematics in Brasil.This perspective lead to the study of ethnomathematics, supporting possibly by ethnography.

The last point will give us the space and time to do better for developing (students') mathematical intuitions. More than ninety percent of the younger mathematical knowledge are coming from their intuition. By developing the dimensions of the contextual language, ultimately we may conclude that the students are actually their languages. In the contemporary philosophy, even analytically the language can be define as this world.

So I always happy to read any language among mathematical contexts; not only to read but also to develop them. The implication for this kind of understanding may lead the adults (math teachers) to facilitate their students' learning of math in the perspective of their constructing of their life. In order to facilitate their needs, we as adults, to some extent do not need  always to indicate our determinations but to indicate passionate to wait the emerge of their mathematical intuitions. If you meet those criteria you may arrive at the stage of innovative mathematics teaching.

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