Includes the
New York
City Modifications
By Bill
Quirk (wgquirk@wgquirk.com)
Here you will find a compact version of
the NCEE "Work Sample & Commentary" sets (WS&C sets) for
elementary school math. We also cover the New York City
modifications
to these sets. In brief, the New York City authors added one set,
changed some work samples for the first two sets, and chose to omit
four
NCEE sets. As you read below, please note that the NYC authors
regularly
missed opportunities to improve the NCEE's product.
NCEE elementary school math is
exemplified
by the TERC K-5 math program, Investigations
in Number, Data, and Space. Widely used in New York City,
TERC
openly claims a "constructivist approach," and uses the phrase
"constructivist
math" to identify their curriculum. For more about TERC
math,
click on TERC Hands-On Math:
The
Truth is in The Details.
The NCEE claims that the student work
samples
found below are "genuine student work,"
demonstrating "standard-setting
performances"
at the end of the fourth grade. Apparently there's one more year
to learn elementary school math. Unfortunately, the Chapter
3 middle school work samples offer no evidence of genuine K-5 math
by the end of the 8th grade.
Direct Access Links to Each WS&C
Set
- Sharing 25
[Includes the NYC modifications]
- Arithmetic
[Includes the NYC modifications]
- 3-D to 2-D
- Patterns
- Pumpkin Activity
- Height Measurement
Statistics
- Two Dice Sums
- Creatures
- Se Hace un Triangulo?
- The Great Fish Dilemma
[Not in the NYC version]
- How Many Handshakes?
- Tangram Dispute
- Feverish Freddy
- Counting on Frank
- School Uniforms Project
[Not in the NYC version]
- Catapult Investigation
[Not in the NYC version]
- The Never Ending Four
- Dream House Project
- Constructing a Polyhedron
[Not
in the NYC version]
- Making a Cube
[Only in the NYC version]
The Work Sample & Commentary Sets for
NCEE Elementary School Math
- Sharing 25
- Task: "Share 25" as
equally as possible
in three different concrete situations.
- How can four friends share 25
balloons "as
equally as possible?"
- How can four friends share $25
"as
equally
as possible?"
- How can four friends share 25
cookies "as
equally as possible?"
- Method: Answers are
shown for
one student. Computation details are missing.
- Comments:
- Constructivist math students
are
expected
to remember many "number fact equivalents," such as 25 ÷ 4 can
be
expressed (for the three contexts) as 6 R1, $6.25,
and
6 ^{1}/_{4}.
- Constructivists wants students
to
develop
number sense and the "meaning" of operations by working
constantly
with "familiar numbers." The "landmark
numbers"
are small whole numbers, multiples of 10, and multiples of 25.
The
"familiar fractions" are proper fractions with denominator equal
to 2, 3, 4, 5, 6, 8, 10 or 12. Yes, 7, 9, and 11 are unfamiliar
denominators.
- Although constructivist math
educators expect
students to remember many "familiar number" facts, they reject the
importance
of remembering single-digit number facts, such as 7 x 8 = 56, and 56
factors
as 7 x 8. Automatic recall in the forward direction is necessary
for later mastery of multidigit computation, operations with fractions,
and comparing fractions. Automatic recall in the reverse
(factoring)
direction is necessary for simplifying fractions. It also orients
the mind to learn about the next level of factoring found in
algebra.
But these are all non-calculator skills, and that explains the
attitude
of the NCTM and NCEE. They believe that such skills are now
obsolete.
- New York City Variation:
Answers
are shown for two students. Computation details are missing.
- NYC Comments:
- Physical objects can't always
be
divided,
but this point isn't mentioned. The balloon problem could have been
used
to make that point clear. Instead both NYC students found a way
around
the problem of the extra balloon. One threw it "up
into the air," and the other made "this
one bigger so
they can all
play with it together."
- One NYC student drew a picture
(an
arrow pointing to drawing of cookie) showing the extra cookie
equally
divided into 4 pie-shaped sections. Inserting the picture in the
sentence,
the student wrote "There
is one left. I (picture inserted) this cookie into 4
shares as there are 4 people." The
fraction ^{1}/_{4 } is never mentioned.
- Arithmetic
- Introductory Comments:
- This is the longest set.
With the exception
of one work sample (computing 13 x 14) found below in the Dream
House set, all student work demonstrating non-calculator multidigit
computation is found here. .
- We are assuming that these
work
samples demonstrate
non-calculator work, but the NCEE instructions never rule out the use
of
a calculator, and many details are omitted. Since constructivist
math educators encourage the regular use of calculators, our assumption
may be wrong.
- The student is regularly asked
to
describe
multiple methods. This allows the student to explain the method,
provide
answers, but omit the complete details. This presentation
approach
is very useful when you have a superficial knowledge of the methods
used.
- The distributive law is used
multiple times
below, but it's never explicitly recognized in NCEE math. The
NCEE
sometimes comments that the student "broke the number apart."
- When we express 87 as (80 + 7)
or
as (8 x
10) + 7, we are expanding 87 using its place value definition.
Such
expansions are used multiple times below, but place value isn't
explicitly
mentioned here or elsewhere in NCEE math. Similar to applications
of the distributive law, place value expansions are vaguely described
as
"breaking numbers apart."
- Handling the need to carry
(compose
relative to place value) and borrow (decompose relative to
place
value) is an essential requirement of any computational
method.
It goes to the heart of our ingenious system for representing numbers
in
terms of powers of 10. But carrying, borrowing, or the general
term,
regrouping, are never mentioned in NCEE math. This is
accomplished
by choosing numbers so that neither carrying nor borrowing is
required,
or by transforming the the problem into an equivalent problem which
requires
neither, or by simply providing the answer, with no mention of the
carrying
or borrowing details.
- Properties of equality, such
as
substitution,
are used frequently in this WS&C set, but the NCEE doesn't list
specific
properties of equality. They simply say that the student should
understand
that "an equality relationship
between two quantities remains the same as long as the same change is
made
to both quantities."
It's
up to the student to decide what they mean by "the
same change."
- The standard algorithms for
multidigit computation
are not demonstrated here or elsewhere in NCEE math. The
standard
algorithms are efficient, accurate, general, and work the same way in
every
case. Their power is best understood with more difficult
computations,
such as 8,756 x 334.26. Once mastered, the standard methods can
be
carried out automatically, without conscious thought. This frees
the mind to learn about the generalization of these ideas found in
algebra.
But this is no problem for the NCEE. They also omit genuine
algebra.
- Task 1: Compute
63
x 46.
- Method: The student
wrote "I
knew that I could add 63 46 times but it would take too much time
so I decided to do it with doubling."
The student then computed 63 + 63,
126 + 126, 252 + 252, 504 + 504, and 1,008 + 1,008 to reach the
intermediate
total of 2,016. Then, realizing that 2,016 corresponded to
32
63's and doubling again would go too far, the student changed
strategies
to take advantage of previously calculated sums. The
student
continued by computing 2,016 + 126, 2,142 + 504, and 2,646 + 252
to get the final total of 2,898. All along the way, the student kept
track of how many 63's corresponded to each intermediate total.
- Comments:
- This method certainly
demonstrates
impressive
thinking, but it requires significant conscious effort to carefully
calculate
the eight sums and write down the "keeping track" details.
Additionally,
it's a special case method, limited to small numbers, with details that
vary depending on the specific numbers involved.
- The need for separate notes to
keep
track
is a major inefficient characteristic of the non-standard alternatives
to the standard algorithms of multidigit computation.
- The NSPS authors say this
sample "was
produced before the class had received any instruction about two digit
by two digit multiplication."
Approaching the end of elementary school, that instruction is
overdue.
The good ideas presented don't require continued demonstration with
numbers
requiring so many steps.
- Task 2.1: Compute
522 - 367
in two different ways.
- Method 1: The
student
added up
from 367 as follows: 3 + 30 + 100 + 22 = 155. A note says: "the
student used landmark numbers to go from 367 to
370
to 400 to 500."
- Method 2: The
student
added 43
to both 367 and 522 to "make the bottom
number
easier to subtract," and then
subtracted
410 from 565.
- Comments:
- Both methods avoid
borrowing. That's
the idea.
- Why add 43?
Consider
367
and note that 4 + 6 = 10 and 3 + 7 = 10. This
method
usually leads to an equivalent subtraction that avoids borrowing.
But one computation is replaced by three, and carrying may be required
for calculating the two sums.
- This borrowing avoidance
strategy doesn't
always work. Try 578 - 385. The magic number is now
25,
the subtraction transforms to 603 - 410, and the need for
borrowing
raises its ugly head. This is a characteristic of constructivist
math programs. Their special case methods often don't work if you
change the numbers.
- With a little experience
(familiarity),
the new "bottom number" can be quickly written down, without thinking
about
carrying. See the pattern?
- The idea of transforming a
problem
into an
equivalent, easier problem is a good strategy, but the transformation
here
significantly increases the effort.
- Task 2.2: Compute
87 x 9 in
two different ways.
- Method 1: The student
wrote "I
used 8 x 9 as a stepping stone and I tacked on a zero."
After writing down 720, the student wrote 7 x 9 = 63, placed the 63,
right-justified
under 720, drew a lines under the 63, and then wrote 783 under the
line.
The NSPS authors observed that the student "broke
the number apart."
- Method 2: The
student
wrote "What
I did is split 80 in half and multiplied it by nine."
In a vertical column the student wrote 40 x 9 = 360, then another 40 x
9 = 360 directly below, then 7 x 9 = 63 directly below that, and
finally 783 for the sum: 360 + 360 + 63.
- Comments:
- Note the non-standard student
language ("stepping
stone", "tacked on", and "split in half"). The NCEE makes no
comment.
- The NCEE uses "broke
the numbers apart" as a blanket
phrase
to cover the two key ideas illustrated here:
- 87 is expanded, relative to
place value, as
(80 + 7).
- (80 + 7) x 9 is rewritten as
(80
x 9) + (7
x 9), using the distributive law.
- 80 x 9 is rewritten as (40 +
40)
x 9 = (40
x 9) + (40 x 9) using the distributive law. The fact that that
substitution
is also involved is never mentioned by anyone.
- Task 3: Demonstrate
multiple ways
to multiply 62 x 85.
- Note: Please click here
to link to the Chapter 1 discussion of this
task.
- Task 4: Four
children are
to "fairly" share 3 bags, with each containing 52 of M & M's.
How much does each child get?
- Method: The student first
drew a four
column table, identifying the columns with 1, 2, 3, and 4 as column
headings.
The student then said "3 times
52 equals ... 150 ...152, 154, 156. And that equals 156. So
since there's four children, I split the 156 and I said, 20, 20, 20,
20."
The student wrote one 20 in each column and said
"I added all the 20's up. 20, 40, 60, 80. . . . Then I said 85,
90,
95, 100." The student wrote
one
5 in each column and said "10,
20, 30, 40. These are tens."
The student wrote one 10 in each column. The student then counted
from 141 to 156, placing a 1 in one column for each number counted,
moving
left to right from columns 1 through 4 and starting again at column 1
until
all 16 1's were placed. At the end of this process each column
contained
20, 5, 10, 1 , 1, 1, 1. The student then chose one column and
added
(in the following sequence) 20 + 10 + 5 + 1 + 1 + 1 + 1 to get 39.
- Comments:
- For twice as much fun, try 312
divided by
8.
- Students at the level should
know
how to divide
a 3-digit number by a 1-digit number.
- The NYC version omits the
preceding method
and substitutes the following two methods:
- NYC Method 1: The student
added 52
+ 52 + 52 and then divided 156 by 4 to get 39. The
computational
details were omitted.
- NYC Method 2: The student
added 52
+ 52 to get 104, and then added 52 to 104 to get 156. The student
then wrote the column:
156 ÷ 4 =
100 ÷ 4 = 25
50 ÷ 4 = 12^{1}/_{2}
6 ÷ 4 = 1^{1}/_{2}
The student then wrote the column:
25
12^{1}/_{2}
1^{1}/_{2}
_{____}
39
- NYC Comments:
- NYC Method 2 converts to
familiar
divisions,
allowing the student to utilize remembered number fact
equivalents.
The two computations, 100 ÷ 4 and 50 ÷ 4, involve the "landmark
numbers" 100 and 50, and the third computation, 6 ÷ 4, is
remembered
as an equivalent form of the "familiar fraction"
^{3}/_{2}.
- The NYC authors say: "the
student broke
156 into 100,
50, and 6 and divided each part by 4." This
explanation hides the use of the distributive law to expand 156
÷
4 as (100 ÷ 4) + (50 ÷ 4) + (6 ÷ 4).
- Students in NYC TERC
classrooms
work constantly
with the numbers in NYC Method 2. They are expected to remember
the
four equivalent values (25, 12^{1}/_{2} , 1^{1}/_{2
},
and 1) for the four familiar expressions (100 ÷ 4, 50 ÷
4,
6 ÷ 4, and ^{1}/_{2} + ^{1}/_{2 }).
[If you find this last statement hard to believe, click on 50
÷ 4 to see an illustration in TERC math.]
- The NYC authors make no
mention of
the incorrect
use of the "=" sign following 156 ÷ 4 in the first column for
Method
2. See Adding
It Up (page 270) for a related comment.
- 3-D to 2-D
- Task 1: Given a
picture
of a 3-D
"cube building," draw the front view.
- Method: The student was
shown
a 14
cube "building," consisting of two 2 by 3 cube layers, topped by the
remaining
two cubes positioned diagonally opposite in corners. The student
drew the correct front view and then belabored the obvious by providing
a lengthy written defense of the drawing.
- Task 2: Given
pictures of
three
3-D "cube buildings," draw the front, side, and top views.
- Method: The student first
"built these shapes with inter-locking cubes" and
then drew the three 2-D views.
- Comment: Too much
busywork,
with pictures,
plastic cubes, drawings, and written opinions.
- Patterns
- Task: The
teacher
asked
"what is a mathematical pattern?"
- Method: The student
gave examples
of "linear patterns" and "non-linear patterns." A "linear
pattern"
example is {9, 1, 9, 1, 9, 1, 9, 1, ....} A
"non-linear
pattern" example is {1, 4, 9, 16, 25, 36, .....}.
- Notes from the NCEE:
- What do they mean by "linear"
and
"non-linear?"
One NCEE note tells us the student "shows
how one quantity determines another in a linear ('repeating')
pattern."
In the next sentence they say "a
pattern can be linear, i.e. 'may or may not repeat itself.'
" Next
we are told that "a pattern
can
'grow or lower itself,' i.e., be non-linear." These are the only
"definitions" offered
for "linear" and "non-linear."
- The NCEE points to "pattern
may or may not
repeat
itself" and "shapes that keep
repeating" as examples that "parts
of the work
provide evidence
of appropriate use of mathematical terms and vocabulary."
- Comments:
- The "definition" of "linear
pattern" should
link to the concept of a straight line. The "linear pattern" examples
offered
here should be described as repeating sequences. The
"non-linear"
examples offered should be described as increasing sequences.
- Notice that {4, 8, 12, 16,
...}
isn't a linear
pattern because it's not "repeating." On the other hand, maybe it
is linear because a linear pattern "may not repeat itself."
- If non-linear ("grow or
lower")
really means
"increasing or decreasing," and non-linear also means "not linear,"
then
any pattern (sequence?) can be classified as repeating, increasing, or
decreasing. Then, since there are sequences that don't fall
into one of these three classes, it follows that such sequences that
don't
qualify as patterns. But {1, 1, 2, 1, 2, 3, 1, 2, 3, 4,
....
} looks like a pattern to us. We must have
misunderstood.
Perhaps they're not using 2-valued logic.
- Pumpkin
Activity
- Task: Estimate the
height, diameter,
circumference, weight, and number of seeds for your group's
pumpkin.
Then measure and count the the number of seeds.
- Method: Eight
groups of
four
students carried out this task. The estimates (guesses) were
recorded
for each student. One set of measurements was recorded for each
group.
Weighing was accomplished by weighing a student, first holding the
pumpkin,
and then without the pumpkin. The "diameter" was measured by placing
two
rulers, vertically and "on the sides," with another ruler balanced on
the
top used to determine the measurement. The "circumference" was measured
by "wrapping a string around the pumpkin" and then measuring the
string.
The "height" was determined via a vertical ruler on "the side" and a
horizontal
ruler on "the top". Seed counts, ranging from 233 to 824,
were
recorded for the eight groups, but the counting method wasn't
mentioned.
The students developed charts and tables to record each group's
estimates
and to show how the eight pumpkins compared for weight, "diameter,"
"circumference,"
"height," and number of seeds.
- Comments:
- The terms "diameter,"
"circumference," and
"height" are used for an irregular object. But a pumpkin doesn't
have a well defined diameter, circumference, or height. This
attempt
to be "real world" leads to misconceptions, such as the need here to
distinguish
between diameter and height.
- One student estimated the
"diameter" as 6 ^{1}/_{2}
inches and the "circumference" as 7 ^{1}/_{2}
inches.
Another student estimated the "diameter" as 7 inches and the
"circumference"
as 9 inches. There's no NCEE comment.
- There's nothing about the key
idea
that circumference
divided by diameter is a constant. On the other hand, this
can't be demonstrated with a pumpkin.
- A note claims that this
example
demonstrates
statistical knowledge, but the students counted every seed in all eight
pumpkins. Keeping the students busy was more important then the
time-saving
concept of a sample.
- Height
Measurement
Statistics
- Task: "Collect
everyone's height measurement in inches, make a line plot with the
data,
and write about what you noticed about the data."
- Method: The
students
developed
a line plot of 24 measured heights. They found the range, median and
identified
an outlier. They also identified "bumps" where "the
line plot rose,"
and and "holes" where "no
one is that height."
- Comments:
- Bumps and holes?
Students
are being
mislead into thinking that such non-standard descriptors are of some
statistical
importance.
- More time-consuming
busywork. Why not
give them the data? They could then find the range, median,
and outliers for multiple data sets in less time. With the
repetition
of multiple examples, they might remember the ideas. There might
be time to also cover the arithmetical mean and mode(s).
- Two Dice
Sums
- Task: List all
possible combinations
for two dice, list all possible sums for two dice, and list all
possible
probabilities for two dice sums. Then play a game involving two
dice
sums.
- Example: The
combinations (1,4),
(4,1), (2,3), and (3,2) are the only combinations for the sum 5.
- Note: This
statement of
the task
and the example are not found in the NSPS notes. The actual notes
say "think of a way to figure
out all
the combinations (sums) for dice rolled together." This
apparent equating of the term "combinations" with the term "sums" is
confusing
and has been avoided here to improve clarity for the reader.
- Method: The student
listed all
possible pairs and then entered the sum for each of the 36 pairs in a 6
x 6 grid provided by the teacher. The student then counted the number
of
distinct occurrences for each possible sum and wrote the eleven
requested
fractions (example: 4/36 for the sum 5). Fractions were not
simplified.
Students observed that 7 was the most likely outcome in the game, with
probability 6/36. This fraction wasn't simplified to 1/6.
- The Game: Each student is
to
draw eleven
circles, placing each circle under one of the eleven column headings
for
the eleven possible sums, 2 through 12. For example, the student
might draw five circles under the sum 7, three under 6, and three under
8. Each time the student rolls the dice, all the student's
circles
corresponding to the two dice sum are filled with an "X". The
first
student who so marks all eleven circles is the winner. Each
student
was asked to "write
the reason why you put the circles where you did."
- Comments:
- A note states "The
activities were spread out over two days." There must be time to fill
out more
tables and charts. It's necessary to carefully record every case
for two dice combinations and two dice sums, and then there are the
lengthy
written justifications for circle placement choices in the game.
The students are kept busy.
- A note tells us that the game
comes from constructivist
math educator, Marilyn Burns. By emphasizing "the winner"
and
subjective arguments defending the choices for placing the eleven
circles,
this game misleads students to think that probability has major
significance for
just one game.
- Creatures
- Task: How many
characters
can you make by combining 4 (different) heads with 4 (different) bodies.
Fully stated, this task called for drawing "four
Halloween characters (a ghost, a witch, a skeleton, and a pumpkin-head
scarecrow)." The
students were then instructed "to
cut the figures into four head and four bodies and staple each set into
small 'flip books.' "
The teacher then asked "How
many characters could you possibly come up with by combining the
different
parts in different ways? Show and explain in detail all the
combinations
you could make."
- Method: The NCEE
says "the student used two
approaches
to solve the problem: a 'flip book'/systematic listing and a
student-created
multiplicative formula." The
student developed 16 drawings, showing each of the four heads combined
with each of the four bodies. The drawings are accompanied
by a lengthy written explanation. The NCEE used bracketed
insertions
to clarify part of the student's statement "the
student made
connections among
different conceptual approaches: 'I know that my [systematic listing]
solution
is correct because there are four monsters, so that means there are
four
bodies [and four heads] and four times four equal sixteen.' "
- Comments:
- The "student-created
multiplicative formula"
should eventually lead to the recognition of the Fundamental
Counting Principle (FCP) and advanced counting techniques, including
permutations
and combinations. But that's never done at any level in
NCEE
math.
- The FCP: If one event can occur in N ways, and a second
event can occur
in M ways, then both events can occur in N x M ways.
- An application of the FCP
quickly produces
36 as the number of combinations for the preceding Two Dice Sums
WS&C
set.
- Why can't the students be
asked
"How many
different ways can each of the four letters A, B, C, and D be combined
with each of the four numbers, 1, 2, 3, and 4? If it must
be
"hands-on," why not use manipulatives already available in the
classroom?
- The NYC version includes more
samples of student
work, but no new ideas.
- Se hace un
triangulo?
- Task: "Suppose
you were given a string that is sixteen inches long. If you cut
or
fold it in any two places, will it always make a triangle?"
- Method: Using "trail
and error" and "string, scissors, and a
ruler,"
the student discovered "if
the two sides add up to more than the bottom side, it will make a
triangle."
- Comment:
Constructivist
math
is hands-on concrete math. This can't be done more simply with a
pencil, paper, and a ruler.
- The Great
Fish
Dilemma [Not in the New York City version]
- Task: "How
many different ways can you put nine fish in two bowls?"
- Method 1: The first
student made
a two-column chart, with "Bowl 1" and "Bowl 2" as column headings, and
then wrote 10 rows. The first row indicated 0 in Bowl 1 and 9 in
Bowl 2. The second row indicated 1 in Bowl 1 and 8 in Bowl
2.
Continuing in this fashion, the last row indicated 9 in Bowl 1 and 0 in
Bowl 2. The student wrote "I
found 10 possibilities," and
then wrote why some possibilities were better than others for the
comfort
of the fish.
- Method 2: The
second
student "used cubes and blocks,
and then
drew pictures."
A note says "the student used
cubes
and blocks to make decisions."
This student modified the
problem by
declaring that "two
of the fish are Simese (sic) Fighting Fish, and the rest are
Neons.
The Simese (sic) Fighting Fish can't be together ever or they kill each
other." The student then drew eight
pictures, each showing two bowls, always with one "S" in each
bowl.
The first two-bowl picture shows the letter "N" repeated 7 times in
bowl
1. The second two-bowl picture shows the letter "N" repeated 6
times
in bowl 1 and one "N" in bowl 2. Continuing in this fashion, the
last picture shows the letter "N" repeated seven times in bowl 2, with
just the letter "S" in bowl 1. This student found 8 possibilities.
- Comments:
- This WS&C set includes
pictures of different
types of fish and both students mention different types, but the fish
are
considered identical for the math of this problem. Put another
way,
both students simply counted the different possiblities for the total
number
of fish in Bowl 1.
- But a "real world" owner of
fish
is likely
to distinguish one fish from another, so the correct answer, using the FCP,
is 2^{9 }for the first student and 2^{8 }for the
second
student.
- Considering how this problem
should be correctly
solved, 3 or 4 fish would be a better example at the elementary school
level. The 9 fish case should wait until exponents and the
FCP are both formally introduced. Neither are ever
discussed
in NCEE math.
- The NCEE is pleased with the
second student's
creative modification of the problem. They don't recognize the
concept
of a well-posed problem with precise conditions that lead to one right
answer.
- This problem was apparently
contributed by
a Vermont source. A note says that "students
could choose to include their work in the Vermont statewide portfolio
assessment."
One wonders if the 2^{9 }solution would be considered wrong in
Vermont.
- How Many
Handshakes?
- Task: How many ways
can
five people
shake hands with each other, shaking every hand just once.
- Method: Students
drew
"bubbles"
and connected each bubble with one line drawn to each other
bubble.
They then counted the lines. Some students noticed a
pattern:
the first person can shake four hands, then the second person can only
shake three hands not previously shaken, then the third person can only
shake two hands not previously shaken, and the fourth person can only
shake
one hand not previously shaken. So the answer is 4 + 3 + 2 + 1.
- Comments:
- The preceding statement of the
method isn't
found in the NSPS document. Although work samples are shown for
several
students, they all say "number 1 shakes 4 people, number 2 shakes 3
people,
etc." or something similar. Of course each person shook 4
hands.
At each point we are interesting in counting the hands that haven't
already
been shaken. It's not clear that any of these students really
understood
this.
- More generally, the answer for
N
people is
found by summing (N-1) + (N-2) + + + 1. The interest then
should
shift to an efficient way to compute such a sum. But, although
examples
other than 5 are shown, the students always carried out the full
extended
addition.
- The FCP
can be
applied
here. There are 5 ways to choose the first hand (first event) and
4 ways to choose the second hand (second event). So there are 20
ways to choose a pair of hands. But this counts each pair twice,
so 20 must be divided by 2 to yield the answer 10.
- For more about the teaching
possibilities
associated with the handshake problem, click on Adding
It Up The discussion, beginning in the middle of page 107,
begins
by noting "this
problem appears often in the literature on problem solving in school
mathematics,
probably because it can be solved in so many ways."
- At the
bottom of page
108, read about busywork when Carl Friedrich Gauss (1777 - 1855) was a
schoolboy. His teacher attempted to occupy the class by asking
the
students to compute the sum of the first N whole numbers, for the case
N = 100. How Gauss quickly produced the solution and the
simplicity
of his idea (see below) should fascinate young children. But this
beautiful opportunity to teach about a math "discovery" was missed. Perhaps the
NSPS authors don't know about the most famous schoolboy discovery in
the
history of mathematics.
- Gauss recognized that
he
could quickly
compute twice the desired sum:
1.
(
1 + 2 + 3
+
4 + 5 + + . . .
.
+ 100) = S
2. + (100 +
99 +
98 + 97 + 96 + + . . .
.. + 1) = S
3.
101 + 101 +101 + 101 + 101 + + . . . . +
101
= 100 x 101 = 2S
Therefore S = (100 x 101) ÷ 2 = 50 x 101 = 5050
- The "Handshake Problem"
reappears
as one of
the WS&C sets for NCEE middle school math. There it's called
"Points and Segments." Points
correspond
to bubbles and segments correspond to lines.
- Tangram
Dispute
- Task: Tracy's
"whole"
parallelogram, formed from two tangram triangles, is compared to
Terri's "small" tangram parallelogram. Terri claims that her
parallelogram
is "half as big" as Tracy's. Tracy thinks that Terri's is
smaller.
They have a dispute.
- Method: The student
physically
demonstrated (via tangram manipulative pieces from multiple tangram
sets)
that the "small parallelogram is
equivalent
to one out of four equal parts of the whole parallelogram."
- Comments:
- Concrete experience relating
plastic triangles
to plastic parallelograms.
- The NCEE suggests that this
example involves
considerable content about fractions. The only evidence is the
student's
statement that "Tracy's
shape was ^{1}/_{4 } not ^{1}/_{2 }
of Terri's shape."
- Feverish
Freddy
- Task: The appraised
value
of lot
A is $88,000. Find the value of 6 other lots relative to the
value
of lot A
- Note: The student is given a
diagram
showing
7 lots. Two lots are shaped as hexagons, 2 as trapezoids, 2
as parallelograms, and one as a triangle.
- Method: The
students
used "pattern
blocks" to find the size (relative to size of lot A) of the other 6
lots.
They found them to be 1/6, 1/3, 1/2, or the same as the size of lot
A.
They used a calculator to find 1/6, 1/3, and 1/2 of 88,000 and to
compute
totals.
- Comments:
- It's assumed, but never
stated,
that the value
of the lot depends only on the size of the lot.
- The student ignored cents
because "I didn't think it was
important." The
NCEE says "the
work provides
evidence of
making sense of what to do with the cents that were left over."
- The student uses letters to
represent lots,
and writes equations such as 3F = A, commenting that 3 F lots equal one
A. The student means that 3 times the area of lot F equals
the area of lot A, but this is never stated in this way.
The
NCEE makes no comment about this lack of precision in language.
They
say that the student demonstrates "Function
and Algebra" concepts by using "letters,
boxes, or other symbols to stand for any number, measured quantity, or
object in simple situations with concrete materials, i.e demonstrates
the
use of a beginning concept of variable."
- Counting
on
Frank
- Task: Test
Three
claims made
in a book. (The book is Counting on Frank)
- Claim 1: It took 11 hours
and
45 minutes
to completely fill a bathroom with water from two running faucets.
- Method:
- Used the school's sink to
determine that one
faucet produces one cubic foot in 40 seconds.
- Deduced that two faucets
produce
one cubic
foot in 20 seconds.
- Assumed that a typical
bathroom
measures 8
x 8 x 8 (unit = feet).
- Used a calculator and a
formula
provided by
the teacher ( L x W x H), to calculate:
- (8 x 8) x 8 = 64 x 8 =
512
(cubic feet)
- 20 x 512 = 10, 240 (seconds)
- 10,240 ÷ 60 = 170
(minutes)
- 170 ÷ 60 is "almost 3
hours"
- Argued for the correctness of
the
process
and concluded "I have proved
him
wrong."
- Comment: Why assume that
a
typical
bathroom faucet produces water at the same rate as the school
faucet?
Could water be also draining out, via the tub, shower, toilet, door, or
windows? There's no statement to the contrary, and this is
promoted
as "real world" math. Finally, why assume such a small
bathroom?
Today's bathrooms can be very large. With some
different
assumptions, you might conclude that the claim could be true.
- Claim 2: If 15 peas were
dropped on
the kitchen floor every day for 8 years, then the peas would reach the
level of the kitchen table top.
- Claim 3: An average ball
point point
will draw a line 2,300 yards long, before running out of ink
- Method for Claims 2 and 3:
Similar
to the method of claim 1, the student used estimates, assumptions,
"experiments,"
personal opinions, and a calculator to "prove him wrong" two more times.
- Comment: Students
shouldn't
be allowed
to go away thinking that such subjective reasoning constitutes a
mathematical
proof.
- School
Uniform Project [Not in the New York City version]
- Task: "The
teacher asked students to come up with a question for a data study that
they would then carry out."
- Method: The work
for
one student
is shown.
- This student conducted a
survey,
asking students
if they would like school uniforms. Three possible color schemes
were offered. The student chose a sample size of 100 students,
dividing
that into 25 students from each of four different grades. The
student
conducted the survey and recorded the results in two tables, one table
for a yes/no vote about having uniforms and the other recording the
choice
of color scheme. These two tables were also presented as bar
graphs.
- Four out of 100 said no to
uniforms, leaving
96 to choose a color. The student wrote 23, 37, and 40 as the
three
totals for the choice of color, and then expressed these as fractions
with
denominator 96 (23/96, 37/96, 40/96).
- Comment: The
presentation
of this project is spread out over 6 pages in the NSPS Volume 1
book.
There are extensive author notes about the complex set of activities
carried
out by the student. But no one noticed that 23 + 37 + 40 = 100,
not
96, and the sum of the three factions is 100/96, not 1.
Apparently
this problem was modified to recognize the student's right to say
"no."
Now it's politically correct, but not mathematically correct.
- Catapult
Investigation [Not in the New York City version]
- Task: "Design
and carry out a test that determines the optimum setting for shooting a
wet sponge the furthest."
- Note: The teacher gave
this
instruction "after each student had
built an
individual catapult according to the teacher's design."
- Method:
- The "student
used a
calculator (as
verified by the teacher) to compute averages" of
three shots, for each of 9 catapult settings. The student listed
the results in a table and "graphed" mean shot length vs.
catapult
setting using graph paper.
- The "graphing" uses a 2-D
coordinate grid,
but in a non-standard manner. The x-axis is marked off in 5 inch
increments, as 0, 5, 10, . . . 105. The y-axis is marked off with
the nine catapult settings, A1, A2, A3, B1, B2, B3, C1, C2,
C3.
The nine (x, y) coordinates, (0, A1), (5, A2), (10, A3), (15, B1), . .
. (35, C3), all fall on the line y = x. The student "graphed" the
9 average three-shot averages by placing a period at the (x, y)
coordinate
where the y-coordinate equals the catapult setting, and the
x-coordinate
equals the average distance for that setting. For example,
there's
a period at the coordinate (A3, 30) to indicate that the average
distance
for catapult A3 was 30 inches.
- Comments:
- The non-standard graphing
method
is a bad
idea at any time, but can be particularly misleading for students in
elementary
school. The 2-D coordinate grid should only be used to represent
the standard coordinate plane, where two perpendicular number lines
provide
a system of coordinates for each point in the plane. The pair (x,
y) should fall on the line y = x if and only if x and y are both real
numbers,
and x = y.
- Why did the teacher need to
verify
that the
student could correctly use a calculator to add three number and divide
the sum by 3? It's not explained.
- The Never
Ending Four
- Task: Investigate
and explain
why the following iterative process eventually leads to four:
- NSPS Note: The
teacher
asked
students to complete a large scale mathematics project chosen from
among
five kinds. This student chose to do a
"Pure Mathematics Investigation."
- The Process:
- Choose any whole number.
- Spell the number.
- Count the number of letters
used
to spell
the number.
- Continue to repeat these three
steps, each
time using the count of step 3 as the number in step 1.
- Note: This explanation of
the
process
is not found in NSPS. The idea is explained though an example.
- Method: Multiple
demonstrations showing
that it works.
- Comments:
- The student attempted to give
a
verbal explanation
as to why this works, but finally pointed to a table for the first 30
whole
numbers. Here's the explanation: The number of letters is
always less than or equal to the value of the number for whole numbers
greater than or equal to 4, and the number of letters needed to spell
any
whole number is greater than or equal to 4, except for one, two, and
six.
It takes 3 letters to spell each of these, and "three" contains 5
letters.
It takes 4 letters to spell five.
- The NCEE classifies this set
as
demonstrating
"Functions and Algebra" because "the
student built (again and again) the non-linear pattern generated by
this
'trick' question."
- This NCEE also classifies this
set
as demonstrating
"Statistics and Probability" because "the
student collects
and organizes
data to answer a question."
- This is a parlor trick, not
"pure
math."
- Dream House
Project
- Task: Given a maximum
budget of
$100,000, a cost of $75 per square foot for "regular room", and cost of
$150 per square foot for "special rooms," design a dream house.
There are several other "requirements" associated with the task.
These include:
- "Rooms and hallways must
have
reasonable
areas."
- "The overall design must
be
convenient
and practical."
- "The number of sides to
your
floor plan
should be limited, to avoid a sprawling, awkward design."
There are several instructions
describing
how to draw the floor plan, record calculations, and draw views.
Students are to meet "once in
a
small peer response group and once with an editing partner."
They "were allowed to use
calculators
to check calculations and for multiplication with multipliers of three
or more digits." - Method:
We have omitted the
"real world"
forms and written defenses of "design decisions." As for the math
in the method, the student multiplied two 2-digit numbers to calculate
the area for multiple rectangular rooms, multiplied by cost per square
foot to determine the cost per room, and then added room costs to get
the
total cost. Calculators were usually used for computations, but
one
student carried out a two digit multiplication as follows:
- "If one of my rooms is 13
ft x
14 ft,
I first do 3 x 4 = 12. So I write the 2 and add the 10 to the 4 x
10 which is 40, but if I add a 10 to the 40, it'll be 50 so I write the
5 on the left side of 2, it's 52. Then I do 3 x 10 equals to
30.
I write the 3 under the 5 on top from 52. Then I do 10 x 10 and I
write the 1 on the left side of 3. Then I add the 52 and
130
together and I got the answer of 182 square feet."
- Comments:
- Note the very loose conditions.
- The NCEE notes: "students
completed
this project
over the course of four weeks."
All that time. What a
shame.
- Constructing
a
Polygon [Not in the New York City version]
- Task: Select a
polyhedron
from choices
on a poster. Use a compass, rubber bands, etc. to build the
polyhedron
of your choice.
- Method: The student
chose to
build a snub icosadodecahedron, which is a "polyhedron
consisting of 80 triangles and 12 pentagons."
The NCEE informs us tht the student
"spent several hours
beyond the
allotted time for this project, including before and after school and
during
lunch recesses."
- Comment: This
student
will be
able to recognize a pentagon.
- Making a
Cube [Only
in the New York City version]
- NYC Task: Draw a
pattern
for making a cube
- NYC Method: The
student
provided
answers, with drawings of two correct patterns and a lengthy written
explanation
about patterns that "don't work."
- NYC Comments:
- This is about improving the
student's ability
to visualize and represent three dimensional objects in two
dimensions.
That's it. There's nothing about surface area or volume.
- There's no explanation of the
methods used
to reach the conclusions given. But a NYC note tells us: "In
order to complete
the task,
students were given access to the following materials: rulers, markers,
wooden cubes, graph paper, construction paper, scissors, paper squares,
paper circles, paper triangles, and tape."
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Copyright
2002-2011 William G. Quirk, Ph.D.