Transcript HELP... - mathnetwork

I’m going to be a
mathematics teacher:
Why didn’t I know this before?
Steve Williams
Associate Professor of Mathematics/
Coordinator of Secondary Mathematics Education
Lock Haven University of PA
[email protected]
© 2006 Steve Williams
NCTM Regional, Cleveland, 10-17-08
Goals for Presentation
•
•
•
•
•
To provide participants with some of the
information that pre-service secondary math
teachers state was lacking in the explanations
they received concerning certain mathematical
concepts.
To motivate participants to consider a more
conceptual way of communicating mathematics
(it’s not always the students’ fault).
To motivate participants to consider other
mathematical concepts where we have “shortchanged” our students.
To provide participants with some alternate ways
of viewing certain math topics that are more
conceptual than traditional ways of viewing
them.
To help participants begin to be able to see
mathematical concepts despite being blinded by
the mindless procedures.
NCTM Regional, Cleveland, 10-17-08
Where did this stuff come from?
• Seven years of teaching all levels of
secondary mathematics
• Ten additional years of working with
preservice teachers and talking to
them about potential conceptual
deficiencies or misunderstandings
• Seventeen years of trying to closely
examine as many secondary (and
elementary) concepts in as much
detail as possible
• A sincere interest to have my students
develop that Profound Understanding
of Fundamental Mathematics by
looking at concepts in different or
non-traditional ways
NCTM Regional, Cleveland, 10-17-08
None of this stuff is “new,” or even
difficult...it’s just that most secondary math
majors, and many in-service teachers, have
never been asked to examine many things
conceptually, and certainly not in different
ways.
When shown these simple connections or
concepts, most preservice teachers get upset
with themselves and state that they feel like
“idiots” for not seeing it before.
I just remind them that it isn’t necessarily their
fault. Much of the fault rests on us as their
teachers (and on the curriculum we choose for
training them).
NCTM Regional, Cleveland, 10-17-08
Concept #1
Why the numbers we call the
“perfect squares” are called
such.
NCTM Regional, Cleveland, 10-17-08
What secondary math majors think:
• 36 is called a “perfect square” because
its “square root” is a whole number, 6.
What’s wrong with this?
• It is only a procedural understanding.
• It leads to circular reasoning because they
answer “why do we call raising a number to
the second power ‘squaring’ the number?”
with, “because you are trying to find the
number that you could take the ‘square root’
of to get back to the number that you are
squaring.”
NCTM Regional, Cleveland, 10-17-08
So…what do I do?
Construct all of the rectangular arrangements that
can be made with 24 objects
24 x 1
12 x 2
8x3
6x4
These are all
rectangles...
nothing special
NCTM Regional, Cleveland, 10-17-08
But what happens when we try this with 36?
36 x 1
18 x 2
12 x 3
9x4
These are all nice rectangles; however...the last
rectangle is a special type of rectangle.
6x6
A “perfect” square!
Question: Could we consider all other numbers
“perfect rectangles?”
NCTM Regional, Cleveland, 10-17-08
Concept #2
It is perfectly permissible to
simply “divide the numerators
and divide the denominators”
when dividing fractions.
NCTM Regional, Cleveland, 10-17-08
What secondary math majors think:
• You have to “invert and multiply”
when dividing fractions.
What’s wrong with this?
• It is only a procedural understanding.
• It doesn’t help students understand why
“invert and multiply” actually works.
• You really don’t “have” to.
NCTM Regional, Cleveland, 10-17-08
12 3
Consider

49 7
Most students insist that you must “invert and multiply” in order
to perform this operation.
However, this is simply not true. It is perfectly permissible to
“divide the numerators and divide the denominators,” just like
multiplication of fractions.
12 3 12  3 4
 

49 7 49  7 7
Of course, they always
want to verify it.
12 3 12 7 4 1 4
 
   
49 7 49 3 7 1 7
NCTM Regional, Cleveland, 10-17-08
Certainly, this isn’t always the most efficient way to
perform this operation, but it certainly can always
be done.
Most of the time, we need to get a common denominator.
2 5 2  8 5  3 16 15 16  15 16  15 16
 






3 8 3  8 8  3 24 24 24  24
1
15
Once students get used to this method, they
find that it isn’t any more time consuming
than “invert and multiply” and wonder why no
one ever told them this before.
And, of course, this is also the method that allows us to
justify “inverting and multiplying.”
NCTM Regional, Cleveland, 10-17-08
Concept #3
Why the number 1 not a prime
number.
NCTM Regional, Cleveland, 10-17-08
What secondary math majors think:
1) “Because the definition of a prime number is a number ‘greater
than or equal to 2.’”
2) “I thought it was.”
3) “I think it should be, since the definition of a prime number is any
number whose only divisors are itself and 1.”
What’s wrong with these?
• The first one provides no understanding of
why prime numbers are defined that way.
• The second one is just incorrect.
• The third one lacks an understanding of an
important connection between mathematical
concepts…but it does get them thinking.
NCTM Regional, Cleveland, 10-17-08
What do I do?
I ask my students what the Fundamental Theorem of
Arithmetic says.
Every composite number can be written as the product of
prime numbers in exactly one way (excluding arrangement).
Prime factor 24 : 2 3  3
This is the
only way.
Prime factor 24 if 1 is a prime number :
1 2 3  3
15  23  3
142  23  3
From here, they get it. They have just never been
asked to actually think about it.
NCTM Regional, Cleveland, 10-17-08
Concept #4
The difference between
“square units” and “units
squared.”
NCTM Regional, Cleveland, 10-17-08
What secondary math majors think:
• “square inches” and “inches squared” mean the same thing.
What’s wrong with this?
• It really isn’t correct.
• They think that reading math from left
to right is always correct. They forget
that they often have to “interpret”
what they are reading.
• It doesn’t distinguish between
“adjectives” and “verbs.”
NCTM Regional, Cleveland, 10-17-08
What do Geometry textbooks say?
“Area is measured in ‘square units.’”
From there on out, the texts only use
the symbol in2, leaving the students to read it
as either “inches squared” or “square inches”
and think the phrases are interchangeable,
and even think that “inches squared” is more
correct since they read from left to right.
So, what’s the difference?
NCTM Regional, Cleveland, 10-17-08
2
Consider 5 in as " five inches squared ."
This implies that we should take five inches and “square it”
5 in
5 in
Area = 25 in2
5 in
5 in
But the textbook said that area was measured in
“square units.”
NCTM Regional, Cleveland, 10-17-08
2
Consider 5 in as " five square inches."
This implies that we show five “square inches.”
1 in
1 in
This is what 5 in2 really means!
5 in2 should be read as “five square inches.”
NCTM Regional, Cleveland, 10-17-08
Concept #5
The “definition” of “arithmetic
average (or mean)” is NOT
“the number you get when you
add up all of the numbers and
divide by the number of terms
present.”
NCTM Regional, Cleveland, 10-17-08
What secondary math majors think:
• That this “procedure” is actually the “definition.”
• However, modern textbooks do use this phrase as the
“definition” of the mean.
What’s wrong with this?
• It’s not a definition. The definition of
definition is a “statement of the meaning of a
word” or a “statement expressing the essential
nature of something.”
• At best, it’s a “procedural definition” which
tells us nothing about the “concept” of
“mean.”
• Students don’t know what it means when their
“average” in my course is 82%.
NCTM Regional, Cleveland, 10-17-08
The question remains: What does it mean for a
number to be the “average” of a set of numbers?
Secondary math majors are stunned when they
don’t know the answer to this question. They
claim they have never been asked.
1902 textbook:
“The average of a set of numbers is a number
which can be put in place of them without
changing the sum.”
Think about what this means for
the “average” of 3, 6, and 12.
NCTM Regional, Cleveland, 10-17-08
The average of 3, 6, and 12 is “a number which can be put
in place of 3, 6, and 12 without changing the sum.”
Put one number in place of
each of them so the sum
doesn’t change (i.e.
rearrange the squares you
have so each column has
the same number in
them…but keep the same
number of columns).
There are 7 in
each column, so
the average is 7.
Secondary math majors have never looked at the average like this
before.
NCTM Regional, Cleveland, 10-17-08
My “compromise” definition of
average
The number that represents an
equal distribution of the total
among all elements present.
So…when a student gets an 82% in my
course, it means that 82% is the grade
that they would have received on each
assignment if they had done the exact
same on all of them.
NCTM Regional, Cleveland, 10-17-08
Concept #6
You CAN multiply the bases
when you have am●an.
NCTM Regional, Cleveland, 10-17-08
What secondary math majors think:
a m  a n  a m n
a5  a3  a8
35  33  38
What’s “wrong” with this?
• Of course, there is no real error in
thinking here.
• It just doesn’t take into consideration
one of the most common student
errors: 35●33 = 98
 .
 Of course, students rarely say that a  a  a
5
3
NCTM Regional, Cleveland, 10-17-08
2
8
What do I have my students do to
examine this concept?
I want my students to closely examine the rules and procedures they
have always used; sometimes having them develop “uncommon”
rules and procedures of their own.
I ask my students if it is okay to
multiply the bases together
when presented with am●an.
They always say
“no!”
m n
I remind them that I did not ask
m
n
a a  aa 2
them if am●an = (a●a)m+n. I
asked them if they “could”
12
multiply the bases together. (I 35  37  9 2  96  312
then have to encourage them to
develop their own rule.)
That’s something I
never thought about!

NCTM Regional, Cleveland, 10-17-08

Concept #7
A linear function has a constant
“first difference,” a quadratic
function a constant “second
difference,” and a cubic function a
constant “third difference,” with
respect to their tables of values,
when the independent variable
increases by 1.
NCTM Regional, Cleveland, 10-17-08
What secondary math majors think:
The have no idea what I’m talking about
because they’ve never actually “analyzed” the
table of values of a function.
What’s wrong with this?
• Students only recognize functions by
either their graph or from their
equation.
• It leaves out an entirely different
analytical technique that brings out
some good mathematics.
NCTM Regional, Cleveland, 10-17-08
Examine a table of values for the
linear function y  2 x  5
x
y
diff
0
5
1
7
2
2
9
2
3
11
2
4
13
2
5
15
2
NCTM Regional, Cleveland, 10-17-08
Examine a table of values for the
quadratic function y  3x 2  7 x  5
1st diff
2nd diff
x
y
0
-5
1
5
10
2
21
16
6
3
43
22
6
4
71
28
6
5
105
34
6
NCTM Regional, Cleveland, 10-17-08
Examine a table of values for the
cubic function y  2 x 3  7 x 2  3x  8
1st diff
2nd diff 3rd diff
x
y
0
8
1
2
-6
2
-30
-32
-26
3
-100
-70
-38
-12
4
-220
-120
-50
-12
5
-402
-182
-62
-12
This is usually a completely new way of looking at these functions,
even though we always have students use a table to graph.
NCTM Regional, Cleveland, 10-17-08
Concept #8
The Law of Cosines is a
generalization of the
Pythagorean Theorem...or...the
Pythagorean Theorem is a
special case of the Law of
Cosines.
NCTM Regional, Cleveland, 10-17-08
What secondary math majors think:
The Pythagorean Theorem is used for right
triangles and the Law of Cosines is used for
oblique triangles.
What’s wrong with this?
• This is, of course, true...just not
complete, since the Law of Cosines
can technically be used for both.
NCTM Regional, Cleveland, 10-17-08
The connection is simple and
straightforward...

Law of Cosines :
c
a  b  c  2bc cos 
2
2
b
2
a
Now let   90o
c
a 2  b 2  c 2  2bc cos 90o

b
a
a 2  b 2  c 2  2bc 0
a2  b2  c2
NCTM Regional, Cleveland, 10-17-08
Concept #9
The Power Rule for exponents
is NOT “an exponent raised to
an exponent.”
NCTM Regional, Cleveland, 10-17-08
What secondary math majors think:
x 
3 5
 x15 is the rule for " raising an exponent t o an exponent."
What’s wrong with this?
• Although many textbooks phrase it
this way, this is NOT “raising an
exponent to another exponent.”
• It causes real problems when trying to
relate the rules of exponents to the
rules of logarithms to an inquisitive
student.
NCTM Regional, Cleveland, 10-17-08
Have students read the phrase “raising an
exponent to an exponent” carefully and write down
what it is really saying.
35
x is " raising an exponent t o an exponent."
x  x 243  x15
35
This may seem innocent enough, until you try
connecting these rules to the rules for
logarithms to the inquisitive student.
NCTM Regional, Cleveland, 10-17-08
 
We usually connect log m  n log m to x
n
m
n
 x mn
and say something like "since a logarithm is an
exponent" that this rule of logarithms is similar to
the rule of exponents in which you "multiply
exponents" when "raising an exponent (log) to
another exponent ."
But an inquisitive student commented that the
correlating logarithm should look like this:  log m  ...
n
and they were right.
But log m   n log m
n
NCTM Regional, Cleveland, 10-17-08
The Solution...
Stop phrasing this rule of exponents as,
“raising an exponent to another exponent.”
The rule should be phrased as,
“when you raise a base that has
been raised to an exponent to
another exponent, you multiply the
exponents.”
log mn can then be interpreted as
"finding the logarithm of something
that has already been raised to an exponent."
NCTM Regional, Cleveland, 10-17-08
An interesting can of worms...
TI  83 family
TI  82
log 5
log 5
0.488...
1.397...
2
2
correct, right?
TI  83 family
 
log 5  log 5 not log 52  2 log 5
2
2
TI  82
 
log 5  log 5 2  2 log 5
2
NCTM Regional, Cleveland, 10-17-08
Concept #10
The meaning of the slope of a
line as a “number.”
NCTM Regional, Cleveland, 10-17-08
What secondary math majors think:
Slope is “rise over run” or “the change in y-values
divided by the change in x-values.”
What’s wrong with this?
• They are both very “procedural” definitions.
• This is all students are left with from their
textbooks...any many teachers.
• It leaves out any real conceptual
understanding of one of the most important
Algebra I concepts.
• It abuses a very important understanding of
fractions: as a “number.”
NCTM Regional, Cleveland, 10-17-08
What is slope (as a number)?
Slope: The amount that the dependent variable
changes when the independent variable increases
by one unit.
Consider y  3 x  8
How much does the y-value change when the
x-value changes by 1?
What is the value of this function when x  7 ? 13
3
What is the value of this function when x  8 ? 16
3
What is the value of this function when x  9 ? 19
The slope is 3.
NCTM Regional, Cleveland, 10-17-08
3
Consider y   x  2
4
How much does the y-value change when the
x-value changes by 1?
What is the value of this function when x  5 ?  1.75
 34
What is the value of this function when x  6 ?  2.5
 34
What is the value of this function when x  7 ?  3.25
3
The slope is  .
4
Note that if this were continued for a change of 4 on the
x-value, the y-value would change by -3 (over four and
down 3 rather than down 3 and over 4).
NCTM Regional, Cleveland, 10-17-08
y=(-3/4)x+2
6
dependent variable
4
2
0
-5
0
5
10
-2
-4
-6
independent variable
NCTM Regional, Cleveland, 10-17-08
15
The abuse of fractions
Understand ing slope as only " the change in y  values
divided by the change in x  values" actually forces the
students to only look at the fraction from a " numeral"
perspectiv e and not as a " number" by itself.
NCTM Regional, Cleveland, 10-17-08
Other topics…
• Why the method of “adding two
equations together” works when
solving a system of equations.
• How a radian is defined.
• The difference between a number and
a numeral.
• How the area formulas are derived.
• How the Quadratic Formula is
derived.
• What is x2
• And so many more…
NCTM Regional, Cleveland, 10-17-08
Conclusions
• Secondary mathematics majors (and many
inservice teachers) usually have not been provided
the opportunities to closely examine many of the
basic concepts that they take for granted and will
one day have to teach.
• The traditional curriculum to prepare students to
teach secondary mathematics feeds into this “gap”
in teachers’ knowledge. There is usually no
course that students take in which to discuss these
potential deficiencies.
• We, as their teachers, should admit some of the
responsibility in helping to cultivate these
deficiencies.
NCTM Regional, Cleveland, 10-17-08
Contact Information
Steve Williams
Associate Professor of Mathematics/
Coordinator of Secondary
Mathematics Education
Lock Haven University of PA
[email protected]
NCTM Regional, Cleveland, 10-17-08