[Numeracy 482] The equal sign

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Michael Gyori michael_gyori at yahoo.com
Wed Aug 11 15:03:47 EDT 2010

Greetings all,

I so often use the analogy of a scale in discussing equal signs and equations,
especially when I teach algebra. As long as the scales are balanced, there is an
equality (even if of weight in non-mathematical reality because you might have
rice on one side and apples on the other).

Mark is spot on, IMO, in terms of the need to create meaningfulness.  I have
found that so many of my students have learned to "hate" math because they fail
to understand the language that it is and its applicability to real life daily
tasks they are or could be engaged in  Once you build affective barriers to the
subject, the burden becomes twofold.

Michael A. Gyori
Maui International Language School

From: "Mark.Trushkowsky at mail.cuny.edu" <Mark.Trushkowsky at mail.cuny.edu>
To: The Math and Numeracy Discussion List <numeracy at nifl.gov>
Sent: Wed, August 11, 2010 7:05:13 AM
Subject: [Numeracy 480] Re: Monday Puzzle


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
their minds.  

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
of evidence.  


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
expression is.

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.

Ladnor Geissinger

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.

Mark Trushkowsky
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

Hello All!
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!
Brooke Istas

Ladnor Geissinger, Emer. Prof. of Mathematics
Univ. of North Carolina, Chapel Hill, NC 27599 USA

ladnor at email.unc.edu----------------------------------------------------
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