# [Numeracy 480] Re: Monday Puzzle

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Mark.Trushkowsky at mail.cuny.edu Mark.Trushkowsky at mail.cuny.edu
Wed Aug 11 13:05:13 EDT 2010

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.

Mark

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[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
= sqrt(3+6)
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 number.***
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.

-----------------------------------------------------------------------------------------------------------------------
On 8/3/2010 3:09 PM, Mark.Trushkowsky at mail.cuny.edu wrote:

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 sign

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
646-344-7301

"Istas, Brooke" <IstasB at cowley.edu>
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08/03/2010 12:49 PM

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[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 learners?

Let’s discuss this!
Brooke Istas

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