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The Assessment of
Mathematical
Understanding
&
Skills
–
Both Necessary & Neither One Sufficient
Judah L. Schwartz
Visiting Professor of Education
Research Professor of Physics & Astronomy
Tufts University
&
Emeritus Professor of Engineering Science &
Education, MIT
Emeritus Professor of Education, Harvard
1
The structure of statements
in
mathematics-
The role of
objects
and
actions
2
Typical mathematical objects
encountered in pre-university
education include
number & quantity
e.g. integers, rationals, reals,
measures of mass, length, time, etc
shape & space
e.g. lines, polygons, circles,
conic sections, etc.
patterns & functions
e.g. linear, quadratic, power, rational,
transcendental, etc.
arrangements
e.g., permutations, combinations, graphs,
trees, etc.
3
Assessing understanding
Understanding is largely a matter of
formulating a problem or modeling
and then mathematizing a situation
4
In the case of understanding tasks, this
means that problem solvers must be
asked to
•
choose an appropriate
mathematical object and then
shape it to represent the essential
elements of the situation being
mathematized.
• derive some set of consequences of
their mathematization of the
situation
[i.e., by manipulating or
transforming their models in
some way[
• so that they may then make
inferences and draw conclusions
about their models and
mathematizations.
5
Assessing skills
Skill is largely a matter of being able
to move nimbly [e.g, by manipulating
and/or transforming] among equivalent
representations [almost exclusively
with symbols]
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Assessments of both understanding
and skill need to include opportunities
for problem solvers
• to make inferences about their
actions,
• draw conclusions about the
reasonableness/appropriateness of
their results and
• modify, if necessary, what they
have done.
Thus we see the cyclical (and vector) nature of
problem solving.
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Understanding tasks should include
opportunities to see
• Modeling/formulating
• Manipulating/transforming
• Inferring/drawing conclusions
on the part of those doing the task
Modeling
&
Formulating
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Manipulating
&
Transforming
Inferring
&
Drawing
conclusions
Skills tasks should include opportunities to see
• Manipulating/transforming
• Inferring/drawing conclusions
on the part of those doing the task
Manipulating
&
Transforming
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Inferring
&
Drawing
conclusions
This implies that
• understanding tasks should
have 3-tuple grades
and that
• skills tasks should have 2-tuple
grades.
10
Some examples of
Understanding tasks
with a focus on
formulating & modeling
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designing
• a measure
• a computation
• a mathematical object
12
designing a measure“ - Ness”
tasks
Perceptually available stimuli –
problem can be posed for the
youngest ages but allows for
extension to increasingly
sophisticated students
13
CC
F
A
A
E
B
B
D
G
H
1.
Given the figures above, devise a definition for
square-ness. Arrange the rectangles in order of
square-ness. Given any two rectangles, can you
draw another rectangle that has an intermediate
value of square-ness?
2.
Write a formula which expresses your measure
of square-ness. You may introduce any labels
and definitions you like and use all the
mathematical language you care to.
3.
Use a ruler to measure any lengths you may need
to use in your formula. Calculate a numerical
value for the square-ness of each rectangle. (You
may use a calculator.)
4.
What other measures of square-ness can you
devise? What are the advantages and
disadvantages of each method?
14
J
Interesting extensions include
(but are not limited to)
defining square-ness for a
collection of parallelograms
and
defining square-ness for closedconvex curves
15
Smoothness of spheres
Consider several “spheres” – a ping-pong ball,
an orange, a basketball, the earth.
Devise a measure of “sphere-ness” that allows
you to order these “spheres” (and any other
collection of spheres) in order of their
“sphere-ness”.
16
This is a practical problem in the
manufacture of ball-bearings which in
turn affects the manufacture of bearings
for rotating machinery such as
centrifuges, motors, etc.
17
Mount Everest – 8,850 meters above
sea level
Marianas trench – 10,900 meters
below sea level
Mean radius of earth – 6,378 km –
6,378,000 meters
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Smoothness of surfaces
Devise a measure of smoothness for a
“planar” surface.
[Another practical application[
Here is a function of time
How smooth is it?
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…but smoothness isn’t always
obvious!
etc., etc….
All the horizontal lengths on the
“staircase” and all the vertical lengths
on the “staircase” always add up to the
sum of the lengths of the two legs of
the triangle .
But if we continue the sequence the
“staircase” approaches the hypotenuse
as closely as we want .
Is the hypotenuse “smooth ?”
Is the “staircase” smooth”?
20
Classic example of designing a measure
Body-mass index =
Weight (in Kilograms)
Height (in meters) x Height (in meters)
Body Mass Index
Weight status
<18.5
between 18.5 and 24.9
between 25.0 and 29.0
>30.0
underweight
normal
overweight
obese
Why is this a good measure ?
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designing a computation–
Fermi tasks
On the difference between an estimate
and an approximation
Estimates are approximate computations
that draw upon the students’ knowledge of
the magnitude of “benchmark” quantities
in the world around them such as the
height of a person is about 1.5 to 2 meters
(and not 15 to 20 meters), the weight
(mass) of a liter of milk is about 1 kg (and
not 100 gm or 10 kg) etc.
22
Approximations are computations made
with numbers that are rounded. The
degree of roundedness is determined by
the students’ purpose in making the
approximation and the desired precision
of the computation.
39.67 x 421.8 is approximately equal to 16000 for
some purposes –
it is approximately equal to 16733 for other purposes –
and it is equal to 16732.806 for still other purposes
N.B. if 39.67 and 421.8 are measured numbers then the most one can say with
certainty is that their product is between
16728.71375and 16736.89875
This is because 39.67 is greater than 39.665 and less than 39.675 and 421.8 is
greater than 421.75 and less than 421.85.
23
We estimate
Numbers
e.g., How many pianos are there in Tel Aviv?
Mass (weight)
e.g., How much does a piece of paper weigh?
Length
e.g., How long a line can you write with a ball point pen?
Area
e.g., What is the surface area of a kitchen sponge?
Volume
e.g., What is the volume of a human being?
Time
e.g., How long does it take you to eat your own weight in food?
Derived quantities such as speed, density, etc.
e.g., How fast does you hair grow (in km/hr)?
Answering any of these questions involves designing a
computation that concatenates the multiplication (or division)
of a series of quantitative benchmarks and standard conversion
factors.
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designing a mathematical object
example generation
Here are two shapes.
Which has the larger area? the larger perimeter?
Is it always true that the shape with the larger area has
the larger perimeter? Why or why not?
Consider the shape with the larger area. Can you draw
a shape that has a larger area but a smaller perimeter?
Consider the shape with the smaller area . Can you
draw a shape that has a smaller area but a larger perimeter?
Consider the shape with the larger perimeter. Can you
draw a shape that has a larger perimeter but a smaller area?
Consider the shape with the smaller perimeter. Can you
draw a shape that has a smaller perimeter but a larger area?
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Between-ness questions
Arithmetic
Here are two subtraction problems
-
52
29
-
74
48
Make up a problem whose answer lies
between the answers to these two problems.
How many such problems can you make
up? How do you know?
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Algebra
Here are two quadratic functions
1 + x2
and
19 – x2
Make up a quadratic function that,
for every value of x, is larger than or equal to
the smaller of these two functions AND is
smaller than or equal to the larger of these
two functions.
What can you say about how many such
quadratic functions there may be?
Could there be a linear function that, for
every value of x, is larger than or equal to
the smaller of these two functions AND is
smaller than or equal to the larger of these
two functions? Why or why not?
27
Which is a better way to construct a
regression line?
For what purpose?
28
Some examples of Skills tasks
with a focus on
manipulating
and
transforming
29
“Show that” problems
Build a sequence of allowed
transformations between
x=2
and
4(x + 3(x +2(x +1))) = 104
How many such sequences can you build?
As ordinarily posed, the problem of solving a
linear equation has a unique solution.
Here the student is asked to devise a possible
chain of intermediate equivalent equations.
There is not a unique such chain.
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“Broken Calculator” problems
Place value
Single-digit number facts
Non-uniqueness of computational procedures
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With only the 0, 1, + and – functioning,
make the calculator display 1970.
In leading digit mode, compute 34 x 567.
Compute 987 + 654 with the + key disabled.
With the 0, 2, 4, 6, 8 keys, how many
different ways can you construct an
even number?
32
Fragmented Arithmetic problems
Here is a subtraction problem that was partially erased
8
___
_
___
7
_______________
5
___
1. Can you fill in a possible set of missing digits?
[The missing digits need not be the same as one
another.]
2. How many possible answers are there? What are
they?
3. How do you know you found all the possible
answers?
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And here is a multiplication problem that was partially
erased.
1
___

___
________________
9
___
1. Can you fill in a possible set of missing digits?
[The missing digits need not be the same as one another.]
2. How many possible answers are there? What are they?
3. How do you know you found all the possible answers?
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Write a comparison of functions
whose solution set has
•
•
•
•
•
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no elements
exactly one element
exactly two elements
a finite number (>2) of elements
an infinite number of elements
As a specific example write an
equation or inequality whose solution
set is
•
•
•
•
•
empty
x=1
x = 1 or x = 2
x = 1 or x = 2 or x = 3
x  1 and x  3
In each of these cases, how many
possible correct answers are there?
How do you know?
36
Implications
for the
writing of rubrics
and
for grading
37
Each Understanding task should have
3 grades
• formulating & modeling
• manipulating & transforming
• inferring & drawing conclusions
<f/m, m/t, i/dc >
38
Each Skills task should have 2 grades
• manipulating & transforming
• inferring & drawing conclusions
> m/t, i/dc <
39
Performance on the separate
dimensions of a task should not be
aggregated
A grade of
>5/5 ,1/5 ,3/5 <
is not equivalent to
a grade of
>1/5 ,5/5 ,3/5 <
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Just as it makes little sense to
aggregate grades across problem
dimensions…
…rubrics for understanding tasks
should consider performance on each
of the three dimensions of
performance separately
and
…rubrics for skills tasks should
consider performance on each of the
two dimensions of performance separately.
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