[Numeracy 492] What does equality mean?
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Sat Aug 14 02:04:55 EDT 2010
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After all this discussion about what the equal sign (or equality) means, I find
myself somewhat in a maze. A discussion of equality takes us into a potentially
esoteric realm from the perspective of our students.
Might it be time to attempt to more clearly (and simply!) define terms among
those who teach math?
Michael A. Gyori
Maui International Language School
From: Ladnor Geissinger <ladnor at email.unc.edu>
To: The Math and Numeracy Discussion List <numeracy at nifl.gov>
Sent: Wed, August 11, 2010 10:33:13 AM
Subject: [Numeracy 484] Re: Monday Puzzle
Thanks for pointing out that my definition of numerical expression seems
somewhat circular. Let me amend it as follows:
"A numerical expression (or formula) is a recipe that describes a sequence of
computational steps to be carried out in order to calculate a number."
Students will see instantly that this is not an exotic definition; it is just
like a cake recipe that describes a sequence of measuring, mixing and heating
steps to bake a cake. They will also know that you can't just arbitrarily mix
up the order of doing the steps of a cake recipe and expect to get the desired
Part of my agenda in my note is to make more conscious use of active language
rather than static or existential language in math, and invite the reader to be
actively involved in every step.
A recipe is something intended to be carried out as a sequence of actions to
arrive at a final result.
When I write an EQN: Expr1 = Expr2
I am making a CLAIM (an assertion) that something is true, that the value of
Expr1 and the value of Expr2 are the same number. I'm in effect saying, go
ahead and follow the steps described in Expr1 and then do the same for Expr2 and
at the end you will see that you have gotten the same numbers. If you have some
doubts, consider it a challenge and check it out for yourself.
Notice that taken together, the definitions I gave of what a formula is and what
we mean when we write an EQN, almost immediately make the usual "rules of
algebra" obviously true. Consider for example the addition rule:
CLAIM: If Expr1 = Expr2, then (Expr1) + 5 = (Expr2) +5.
Suppose we evaluate Expr1 and Expr2 and in both cases get the result 7, then of
course when we evaluate (Expr1) +5 and (Expr2) + 5 the very last step in both
cases will be to evaluate (7+5) and so in both cases the final result is 12. Of
course the same is true if instead of 7 we get some number p, and if instead of
5 we add some number c.
So the rules of algebra for transforming a true equation into another true
equation are simple and easy to understand; what requires more effort for
students is to learn a strategy for how to use the rules to solve math
Of course I want to make it as easy and natural as possible for students to make
sense of elementary math -- so that they can quickly get on with the harder, and
many would say more interesting, work of solving applied math problems. That is
the goal of learning mathematical methods, isn't it?
On 8/11/2010 1:05 PM, Mark.Trushkowsky at mail.cuny.edu wrote:
>Calling the statement "This means that the expressions on the left and the right
>of the = sign are the same" a bold faced lie seems a little drastic to me. It
>may be vague and incomplete, but if you simply add the words, "the value of" in
>front of "the expressions", it becomes workable.
>What my post was really about was ways to elicit student misconceptions and
>engage students in addressing their misconceptions and helping them hone their
>own precise definitions - definitions that are expressed in students' own words,
>and which students arrive at through the modeling of a process of what it means
>to evaluate and test the veracity of a mathematical statement.
>Students come to us with many misconceptions about a great many things, in this
>case we are talking about the equal sign. To simply tell students, "We claim
>that if, or when, you EVALUATE the expression on the left and also EVALUATE the
>expression on the right, then the resulting values will be the same number" does
>little to uncover and interact with student preconceptions, and I would argue,
>statements like that one usually lose out to student preconceptions in the end.
> Students will nod and write it down, but it seems more complicated than it
>needs to be and without having the conversation how could students ever be able
>to judge that your statement is more sound than "= means the answer". We can't
>just tell students definitions - we need to explore all definitions and help our
>students understand what a precise mathematical definition is, so that they can
>evalute definitions on their own. Otherwise the easiest one to remember is the
>one that will stick, and the easiest is usually the one they already have in
>This may be a question of a difference in our student population. From the
>level of the definitions you are using, I'm assuming your students are in
>college. I work with a very different population of students, adults who are
>back in school to get their GEDs, many of whom are immigrants. For example,
>what you call a clear and simple description of a numerical expression would not
>work for them. "A numerical expression is a recipe that describes a sequence of
>steps that one should take to evaluate the expression to get an explicit number"
>would sound circular to them, like saying "A lasagna is a recipe that describes
>a sequence of steps that one should take to cook a lasagna".
>In my classes I am trying to get students to talk about math and put it into
>their own words. To me, it is always better if they observe it and say it, then
>if I tell them.
>When I wrote that there was usually one student who can say something like "An =
>sign means that the expression on one side if the same as the expression on the
>other side", I did not mean that was the mathematically precise definition I
>settle for in class. What I meant was this: When I do this activity, I am
>always looking for a student who says something along the lines of things in
>balance. When facilitating a conversation, it is those students who I can use
>to draw out this idea of balance so that other students incorporate it into the
>quest for a precise definition that holds up and is defensible.
>My students, as many of us do I'm sure, have long histories of being in math
>classes where the emphasis was on procedures, rules and definitions. By far the
>most harmful thing that students come out of those experiences with is the sense
>that math is not suppose to make sense, it is to be memorized and then
>forgotten. At the same time as I am teaching them the content, I am teaching
>them what it means to explore through conjecture, analysis and a understanding
>Ladnor Geissinger <ladnor at email.unc.edu>
>Sent by: numeracy-bounces at nifl.gov
>08/09/2010 11:57 PM
>Please respond to
>The Math and Numeracy Discussion List <numeracy at nifl.gov>
> To numeracy at nifl.gov
>Subject [Numeracy 479] Re: Monday Puzzle
>Here are a couple of notes on some comments in the email below about the meaning
>of = as we or our students understand it.
>First, consider this equation which we consider to be correct: EQN 5 - 2 =
>and the STATEMENT "This means that the expressions on the left and right of the
>= sign are the same".
>Now it will seem clear to anyone who hasn't been indoctrinated in inconsistent
>ambiguous shortcut school math language that the STATEMENT is a bald-faced lie
>-- on the left appear the digits 5 and 2 and a minus sign, while on the right
>appear the digits 3 and 6 and a plus sign and a shorthand name for the square
>root function and some parentheses. In other words, the string of symbols that
>make up the numerical expression on the left is nothing like the string of
>symbols that make up the numerical expression on the right --- they certainly
>aren't the same.
>So in order to make clear to our unschooled doubter what WE REALLY INTENDED by
>that EQN, we should say:
>***We claim that if, or when, you EVALUATE the expression on the left and also
>EVALUATE the expression on the right, then the resulting values will be the same
>I think students need to hear that claim stated carefully and in full many
>times; it's a servicable definition of a mathematical equation.
>Students also need to have a clear simple description of what a numerical
>A numerical expression is a recipe that describes a sequence of steps that one
>should take to evaluate the expression to get an explicit number.
>The actual evaluation may be done by someone using pencil and paper, a
>calculator or a computer.
>Final note: Since numerical expressions (also called formulas) are the
>principal objects that appear in math, students should learn how to correctly
>build formulas, and how to read a formula to unravel the time sequence of what
>to do in step 1, then step 2,.etc.. until the evaluation is completed
>correctly. The main tool that lets us prescribe such a time sequence in
>standard one-line mathematical form is the use of paired parentheses. Beginning
>students need to use lots of parentheses. When formulas start getting long and
>we get tired of writing lots of parens, it's really useful to know the
>associative and distributive laws so that we can eliminate some of the parens
>but still be certain that staying within the remaining restrictions will give us
>the correct result.
>For example, the associative law of addition says that two different ways to
>calculate a sum of three numbers will always give the same result:
>for any numbers s,t,u it is true that (s + t) + u = s + (t + u)
>and then we can generalize to a sum of 4, then 5, .etc... summands. Unless you
>point it out to them, students may not realize that a calculator or computer can
>only add TWO numbers at a time.
>Once we are convinced of this law, we may safely write such a sum in shorthand
>form as s+t+u. Next it will be useful to know similar theorems about alternate
>ways to calculate with multiplication, then with powers, etc. , and each of
>these will allow us to simplify our formulas and arrange our computations to be
>made more conveniently.
>At some point we may decide to push this condensation idea even further, so we
>learn and follow some standard set of rules of precedence (one such has the
>acronym PEMDAS) and use formulas written in calculator shorthand. This order of
>operations is NOT an inherently mathematical topic, it is simply a convention to
>allow formulas to be written in very short form with few parens.
>On 8/3/2010 3:09 PM, Mark.Trushkowsky at mail.cuny.eduwrote:
>I have observed something interesting in my students, when it comes to the equal
>sign, and I wonder if others have had similar experiences.
>I've taken to asking my students what they think the equal sign means and while
>there is usually one student can express that it means that the expression on
>one side is the same as the expression on the other side, the vast majority of
>students say something imprecise, like "equals means the answer". If you think
>about it, that makes sense - when using a calculator, one pushes "=" to get the
>answer; problems are often written with an equal sign followed by a blank space,
>implying "the answer" should follow. I know some of us have enough of a
>conceptual understanding of "=" to be versatile in our use of it, but we should
>be conscious of the fact that we need to give our students a consistent and deep
>sense of what we mean, especially considering the misconceptions they often
>bring to class.
>I like to have this conversation with students where first I make a list of all
>of their definitions of the equal sign. Sometimes I have them write it on a
>post-it note and put it on the board. Then I try to provide exception to the
>definitions that students tend to give, which are often mathematically
>imprecise. For example, if a student says "an equal sign means the answer",
>I'll give an examples to help them see that their definition does not always fit
>and that they can strive for one that does always fit. For example, I might ask
>them to consider that definition with "8 + __ = 35". I'll try to keep doing
>that until we have a precise definition (from the students own words) that
>explains exactly what an equal sign means.
>My goal is for this activity is three-fold:
>1) I want to ellicit students misconceptions because that is where I need to
>begin my teaching - I need to know where they are coming from to know what work
>I need to do to help them see the limitation of those misconceptions and help
>them move beyond them
>2) I want to model what it means to "know" or "test" mathematical knowledge
>3) I want students to have a mathematically precise understanding of the equal
>Doing this activity really emphasized for me how important it is to not take for
>granted that students see the same things as we do when they see mathematical
> symbols, and how important it is to have them explain their use of symbols. It
>is also important to help students recognize the need for precision and how to
>go about testing and refining observations to get there.
>Mathematics Staff Developer
>CUNY Adult Literacy and GED Program
>101 W. 31st Street, 7th Floor
>New York, NY 10001
>"Istas, Brooke" <IstasB at cowley.edu>
>Sent by: numeracy-bounces at nifl.gov
>08/03/2010 12:49 PM
>Please respond to
>The Math and Numeracy Discussion List <numeracy at nifl.gov>
>To <numeracy at nifl.gov>
>Subject [Numeracy 457] Re: Monday Puzzle
>I am glad that many of you are contributing to the discussion with your methods
>for approaching these problems. My learners seem to really like these warm-up
>questions or brain teasers. Ladnor made a good point with his comment about the
>“=” symbol. I have seen the misuse of the equals symbols in many classrooms
>not just in adult education but in college/university classes, too (I have even
>been guilty of misusing it myself). Does anyone else have an opinion about the
>misuse of math symbols? Does it create more math confusion? What other math
>symbols do you feel are misused and lead to further math frustration with
>Let’s discuss this!
>Ladnor Geissinger, Emer. Prof. of Mathematics
>Univ. of North Carolina, Chapel Hill, NC 27599 USA
-- Ladnor Geissinger, Emer. Prof. of Mathematics Univ. of North Carolina,
Chapel Hill, NC 27599 USA ladnor at email.unc.edu
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