How the NCEE Limits Middle
School Math
Includes the
New
York City Modifications
Here you will find a compact version of
the
NCEE "Work Sample & Commentary" sets (WS&C sets) for
middle
school math. We will also cover the New York City modifications
to
these sets. In brief, the New York City authors provided
different
work samples for the first two sets and added work samples for two
other
sets. The NYC modifications improved the first set.
Direct Access Links to Each NCEE
Middle School
Math WS&C Set
- Pieces of String
[Includes the NYC modifications]
- Dart Board
[Includes the NYC modifications]
- Science Fair
- Cubes
[Includes the NYC modifications]
- Points and Segments
[Includes the NYC modifications]
- Locker Lunacy
- Who is Best?
- Probability Booth
- Logic Puzzle
- Scaling Project
- A New Look at a Budget
- Candle Life
The Work Sample & Commentary Sets for
NCEE Middle School Math
- Pieces of
String
- Task:"A
length of string that is 180 cm long is cut into 3 pieces.
The second piece is 25% longer than the first, and the third piece is
25%
shorter than the first. How long is each piece?"
- Method: The student
guessed the
length of first piece as 100 cm. Then the length of the other two
pieces would be 75 cm and 125 cm, and the total length of the string
would
be 300 cm. The student then observed that 100, the guessed length
of the first piece, is 1/3 of 300. Hence the actual length of the
first piece must be 1/3 of 180, or 60 cm, and the length of the other
two
pieces must be 45 cm and 75 cm.
- Comments:
- This problem is easily solved
by
setting up
and solving a linear equation. If Let x equals the length of the
first piece piece, then x + (x + .25x) + (x - .25x) = 3x =
180. There's an NSPS comment that recognizes this fact, but
the authors appear pleased that "the
work does not show a standard approach that a student of algebra might
take."
- Notice that the guessing
strategy
only works
with carefully chosen numbers. Constructivist math
educators
frequently offer such contrived problems to show that real math isn't
really
necessary.
- NYC Version: The
NYC
authors
modified this problem, changing the total length to 160 cm, the second
piece percent to 40% longer, and the third piece percent to 20%
shorter.
The NYC authors omitted the NCEE method and offered the following two
alternatives:
- NYC Method 1: The
student solved
the equation x +1.4x + .8x = 160. The numbers work out
nicely,
yielding lengths 50 cm, 70 cm , and 40 cm.
- NYC Method 2:
This student
avoided working with decimal by saying there are 10 pieces in the first
(big) piece, 14 pieces in the second (big) piece, and 8 pieces in the
third
(big) piece. This yields a total of 32 (small) pieces in the 160
cm total length. The student divided 160 by 32 to get 5 cm
as the length of each small piece and then multiplied by 10, 14, and 8
to get the length of the three big pieces.
- NYC Comments:
- This NYC variation is still
easily
solved
using "guess and check." The obvious initial guess (50)
works.
The success of Method 2 depends on the nice numbers.
- Method 1 does demonstrate
important middle
school math: using properties of equality to solve a linear
equation.
NCEE math doesn't include a similar demonstration.
- Dart Board
- Task: "Design
a dart board that has four regions with the following features:
Score value
Probability %
100
points
10%
50
points
20%
25
points
30%
10
points
40%
The dart board may be any shape
(circle,
square, etc.) and must have an area between 1,000 sq. cm and 3,000 sq.
cm Assume the probability is proportional to the area of
the
region."
- Method:
- The student drew 4 concentric
circles, indicating
that the area of the inner circle and three donut-shaped regions,
moving
progressively outward, should be 10%, 20%, 30%, and 40% of the total
area
contained in the outer circle.
- Using A = (3.14) r^{2 }
as the
formula for the area of a circle, the student used guess and check to
choose
20 cm. as the radius of the outer circle. Equivalently, the
student
chose 1,256 sq. cm as the total area. The student then found:
- The radius of the inner
circle
by solving
(3.14) r^{2} = 125.6.
- The outer radius of the 20%
region by solving
(3.14) r^{2} = 3 x (125.6).
- The outer radius of the 30%
region by solving
(3.14) r^{2} = 6 x (125.6).
- Comments:
- There's no mention of the
calculation methods.
At the middle school level it's safe to assume that a calculator was
used.
- There's no mention of
approximating pi.
Pi equals 3.14 for these students.
- Decimals outside the range .00
to
.99 are
not mentioned in NCEE middle school math.
- This is considered a problem
in
probability,
but it's only about using the formula for the area of a circle.
- The problem is not well
posed. That
is, the conditions given don't lead to one correct solution. The
NCEE notes that an acceptable solution would be "a
choice of dart board as a 100 cm x 20 cm rectangle, divided along the
length
in 10, 20, 30, and 40 cm segments."
Thus, a 4th grade solution
is acceptable.
- NYC Version: The NYC
authors
omitted
the NCEE method and offered the following substitutes:
- NYC Method 1: The
student
drew a 10
by 10 grid with 100 component 10 sq. cm boxes. The student then
used
four different colors to mark ten 100 point boxes, twenty 50 point
boxes,
thirty 25 point boxes, and forty 10 point boxes.
- NYC Method 2:
Similar
to NYC
Method 1. Used 10 congruent rhombi.
- NYC Comment: These are
4th grade solutions.
- Science Fair
- Task: A
rectangular
floor
must be divided into 3 rectangular rooms for a science fair.
Room 1 is for a school with 1000 students. Room 2 is for a school with
600 students. Room 3 is for a school with 400 students.
Express
the size of each room, first as a fraction of the total floor,
and
then as a percent of the total floor. If the total floor costs
$300,
find the cost of each room.
- Method: The student
wrote three
fractions with denominator 2000, simplified to 1/2, 3/10, and 1/5, and
then expressed these fractions as 50%, 30%, and 20%. The student
recognized the corresponding decimal equivalents and multiplied each
decimal
by 300 to get the cost per room.
- Comments:
- This is an appropriate problem
at
the end
of the 4th grade, not at the end of the 8th grade.
- The NCEE says that this work
sample
illustrates standard-setting
performance
for computing accurately with arithmetic operations on rational
numbers.But
there's no illustration of the addition, subtraction, multiplication,
or
division of fractions. Simplifying fractions doesn't qualify as
operating
with fractions.
- Such overreaching claims are
typical in the
NSPS.
- Cubes
- Task: Find the
volume and
surface area of a cube. Then determine if a claim is true.
- "What is the volume of a
cube
whose edges
each measure 3 centimeters?"
- "What is the surface area
of a
cube whose
edges each measure 3 centimeters?"
- "A student named Eddie
says, 'No
matter what size the cube is, the number you get when you calculate its
surface area is always twice as big as the number you get when you
calculate
its volume.' Is Eddie correct? Show how you know."
- Method:
- The student computed the
volume as
3 x 3 x
3 = 27 and the surface area as 6 x (3 x 3) = 54.
- To test Eddie's claim the
student
tried three
other edge lengths, but didn't find another cube example with surface
area
double the volume. The NCEE says "the
students gave several examples to counter Eddie's claim. Not only
is Eddie's claim not true for all cubes, it is false for most cubes."
- Comments:
- Tasks 1 and 2 are basic 5th
grade
math.
- Note the constructivist
reasoning
method for
task 3. The students tried several cases, while looking for a
pattern.
But no pattern was found and that ended the discussion. These
students
don't know another way to reason.
- No one mentioned the algebraic
formulation
of Eddie's conjecture. It's equivalent to saying that the
equation
2x^{3 }= 6x^{2 } is true for all x greater than
zero.
But, if x is greater than zero, both sides can be divided
by
2x^{2} to yield the answer 3.
- The students and the NCEE
didn't
notice that
Eddie said "always," or perhaps they don't know that one
counterexample
is sufficient. It's an example of how they have redefined
mathematical
logic.
- Once the students determines
that
the claim
isn't generally true, the focus should shift to finding other patterns
involving volume and surface area. No other patterns are
discussed
anywhere in NCEE math.
- NYC Version: Three
students tested
Eddie's claim by trying seven other edge lengths. More drawings
and
written explanations, but no new ideas.
- Points and
Segments
- Task: "How
many segments are needed to connect 5 points? 6 points? 8
points?
10 points? 30 points? 100 points? n points?"
- Method: The students
looked
for a pattern.
They sketched and counted for cases 2, 3, 4, 5, 6, 7, 8, 9, 10,
and
30. Finally, one student said "then
I found this ((n ÷ 2) - .5) x n as the formula and
it
worked." The
NCEE commented: "the
student's phrase, "Then I found this," leaves the reader to wonder,
'How'."
- NYC Version: The
NYC
authors
revised the method, omitting the "leaves the reader to wonder"
statement.
They want to explain "How" the student discovered the formula.
For
cases 5, 6, and 8, the NYC student "noticed" that the number of points
times the number of segments equals "double
the answer."
- Comments:
- This problem is a variation of
the
"Handshake
Problem" found in the NCEE elementary school math examples. Here
it's
points and segments. There it was bubbles and lines.
Both problems required finding the number of combinations of 5 elements
taken 2 at a time. The formula given above yields the number of
combinations
of n elements taken 2 at a time.
- The NCEE never acknowledges
that
the student
"discovered" an important general formula. More generally,
advanced
counting methods, such as permutations and combinations, aren't covered
in NCEE math. Why not? These topics are quite accessible
for
middle school students. All that's required is prior mastery of
fractions.
Ooops! That may be the problem. But the "wonder
how" comment gives us a hint that the NCEE may recognize the
limitations
of pattern recognition and discovery learning. Perhaps they
are conceding the stretch necessary to sell the idea that the student
really
discovered the formula for the number of combinations of n objects
taken
2 at a time. If so, they also know that no one will buy
student
discovery of more general counting formulas.
- Locker
Lunacy
- Task: "Analyze
a scenario in which school lockers are alternately opened and closed in
a particular manner." Lockers,
numbered 1 to 100, are all opened. Then lockers with numbers
divisible
by 2 are all closed. Then lockers with numbers divisible by 3 are
"reversed" (closed if already opened, or opened if currently
closed).
If this "reverse" pattern continues for N = 4, 5, ... 100, which
lockers
will remain open?
- Method: The NCEE
explains that "Students notice
experimentally
that those lockers whose numbers are perfect squares remain open at the
end. It is only on reflection (asking 'Why?') that they realize
the
relationship between the perfect squares and their odd number of
factors."
- Comment: NCEE
notes refer
to "a long while of
experimenting"
and "the student created a
large
table of values." Much
busywork and more pattern recognition, but calculators save time.
- Who is Best?
- Task: "Determine
which of three golfers is the 'best' chipper." If three golfers each
chips 10
golf balls onto a green, decide "who
is closest and most consistent" based
on a statistical analysis of "distance
from the pin" data. Measured
in inches, here's the data:
- Rick: 40,
60,
100, 120,
312, 320, 152, 105, 95, 46
- Mike: 52,
76,
184, 288,
230, 120, 64, 60, 88, 188
- Sarah: 84, 99, 130, 135,
200, 165, 120,
129, 135, 152
- Method: For each of these
three sets
of ten measured distances, the student found the lower extreme, lower
quartile,
median, upper quartile, and upper extreme. The student then
developed
three box plots. Sarah was voted "most consistent" because her
measurements
had the smallest inter-quartile range (upper quartile minus lower
quartile).
In spite of an extended discussion, no golfer was voted "who is
closest."
- Comments:
- The consistency part is fine
for
7th grade
math.
- The failure to award "closest"
is
puzzling.
Two of Rick's chips, at 40 and 46 inches, are closer than any chip
for
the other two golfers. Looks like a winner! But
Rick's
chips were considered flukes. He was penalized for lack of
consistency.
But wasn't that the other award?
- Probability
Booth
- Task: "Design
an activity that uses multiple probability events and that would
attract
lots of players."
- Method: The
student's
activity
consisted of first predicting a coin toss and then rolling an 11 or
12,
using a pair of dice. The student wrote "the
probability of winning is 1 in 24 because it's 1 in 2 (^{1}/_{2})
to win the coin toss and 1 in 12 (^{1}/_{12})
for the roll of the dice."
The student proposed to "charge
$1 to play and give $10 to a winner."
The student wrote "Out of
every
24 players one would win. My profit out of 24 players would be
$14." The NCEE complimented the
student for "realizing that
his ^{1}/_{24}
probability implied that he could expect one winner per 24 games
played."
- Comments:
- It appears that both the
student
and the NCEE
believe that exactly one win will occur every 24 games.
- Used here, but not explicitly
recognized by
the NCEE: If A and B are independent events, with probabilities p
and q, then the probability that both events occur is p x q.
- The NCTM and NCEE claim that
they
emphasize
probability for all the K-12 years, but this is as difficult as it gets
in the 38 math-related WS&C sets found in the three volumes of the
NCEE's New Standards Performance Standards.
- Logic Puzzle
- Task: Given 7
equations
involving
10 variables, where each variable represents a unique digit
between
0 and 9, find the value of all 10 variables.
- The Equations:
- g + g + g = d
- j + e = j
- g^{2} = d
- b + g = d
- f - b = c
- i/h = a (h > a)
- a x c = a
- Method: The student
recognized
zero as the additive identify and one as the multiplicative
identity.
Here's a taste: j + e = j, therefore e = 0. Since 3g
= d, g^{2} = d, and e already occupies the zero slot, it
follows
that g = 3 and d = 9. You get the idea.
- Comment: The
student
solved a
system of equations, by using substitution, properties of equality, and
knowledge of identity elements. If done correctly, this problem
qualifies
as genuine algebra. Unfortunately, because of the "unique digit
between
0 and 9" constraint, it can be solved using "guess and check."
- Scaling
Project
- Task: "Design
and build a scale model enlarging or shrinking an everyday object using
a ratio of 1:10 (or specify a different ratio) for each dimension."
- Method: The student
built a 10
to 1 enlargement of a LEGO^{ }piece. The materials used
included
rectangular "tagboard," toilet paper rolls, glue, and red spray paint.
The student applied standard volume formulas for rectangular prisms and
cylinders. The student assumed that pi = 3.14.
- Comment: A less time
consuming variation
may be an acceptable 6th grade activity.
- A New Look
on
a Budget
- Task: "Determine
the cost of redecorating your room. You must carpet the
room, paint two coats, and use wallpaper in some way. Draw to
scale,
on graph paper, each wall, including windows and doors."
- Method:
- The student "measured
the walls in yards and inches," and then converted "yards to inches and
found the total inches."
Then "I
multiplied on my calculator the length and height in inches to find the
square inches of each wall. I divided the area by 144 to find the
square feet." The
student carried out similar calculations for the floor, ceiling, and
woodwork.
The student divided by 9 to convert to square yards for the floor.
- Given coverage information,
the
student used
the calculator to determine how much paint, wallpaper, and carpet to
buy,
and then calculated total costs for all materials and supplies.
- The student developed all the
required drawings.
- Comment: This is
4th
grade math.
With the "real world' aspects and required drawings, there's too much
time-consuming
busywork.
- Candle Life
- Task: "Determine
the relationship between the volume of a container and the length of
time
a candle will continue to burn when covered by it." Experiment with several
containers.
Graph the data and "choose the
line that best fits your data."
- Method: The student:
- Measured the volume of 12
different glass
containers by filling them with water and then pouring the water into a
measuring cup.
- Placed each container over the
burning candle
and used a stop watch to determine the time the candle would continue
to
burn. This was done 3 times for each container.
- Selected the "intermediate
value"
from each
of the 3 trials and used a coordinate plane to graph container volume
vs.
flame extinguish time.
- "Drew a line of 'best fit'
and
identified
two points off of it."
- Found the equation of the
straight
line joining
these points.
- Comment: Finding the
equation
of a
straight line, when given the coordinates of two points on the
line,
is a very important middle school topic. Too bad more time wasn't
devoted to equations of straight lines and less time to pseudo science.
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Copyright
2002-2011 William
G. Quirk,
Ph.D.