[Numeracy 479] Re: Monday Puzzle
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Mon Aug 9 23:35:34 EDT 2010
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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
> 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
> *"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>
> <numeracy at nifl.gov>
> [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
ladnor at email.unc.edu
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