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Transcript of 1 The Assessment of Mathematical Understanding & Skills – Both Necessary & Neither One Sufficient...
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The Assessment of Mathematical Understanding & Skills–Both Necessary & Neither One Sufficient
Judah L. SchwartzVisiting Professor of EducationResearch Professor of Physics & AstronomyTufts University &Emeritus Professor of Engineering Science &Education, MITEmeritus Professor of Education, Harvard
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Typical mathematical objectsencountered in pre-universityeducation 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.
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Assessing understanding
Understanding is largely a matter of formulating a problem or modeling
and then mathematizing a situation
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In the case of understanding tasks, thismeans that problem solvers must beasked 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 makeinferences and draw conclusions
about their models andmathematizations.
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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 understandingand skill need to include opportunitiesfor 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
Manipulating&
Transforming
Inferring&
Drawing conclusions
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Skills tasks should include opportunities to see • Manipulating/transforming
• Inferring/drawing conclusions
on the part of those doing the task
Manipulating&
Transforming
Inferring&
Drawing conclusions
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This implies that
• understanding tasks should
have 3-tuple grades
and that
• skills tasks should have 2-tuple
grades.
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designing a measure - “Ness”tasks
Perceptually available stimuli –
problem can be posed for the
youngest ages but allows for
extension to increasingly
sophisticated students
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1. Given the figures above, devise a definition forsquare-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 youdevise? What are the advantages and
disadvantages of each method?
A
B
A
C
A
B
C C
D
E
F
G H J
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Interesting extensions include
(but are not limited to) defining square-ness for a
collection of parallelograms
and
defining square-ness for closed-
convex curves
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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”.
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This is a practical problem in the manufacture of ball-bearings which in turn affects the manufacture of bearings for rotating machinery such ascentrifuges, motors, etc.
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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!
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?”
etc., etc.…
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Classic example of designing a measure
Body-mass index =
Weight (in Kilograms)
Height (in meters) x Height (in meters)
Body Mass Index Weight status
<18.5underweight
between 18.5 and 24.9normal
between 25.0 and 29.0overweight
>30.0obese
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.
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Approximations are computations madewith numbers that are rounded. The degree of roundedness is determined by the students’ purpose in making theapproximation and the desired precisionof 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
16728.71375 and 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.
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We estimate Numberse.g., How many pianos are there in Tel Aviv?
Mass (weight)e.g., How much does a piece of paper weigh?
Lengthe.g., How long a line can you write with a ball point pen?
Areae.g., What is the surface area of a kitchen sponge?
Volumee.g., What is the volume of a human being?
Timee.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 generationHere 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 74
- 29 - 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?
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“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?
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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 possibleanswers?
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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 functionswhose solution set has
• no elements
• exactly one element
• exactly two elements
• a finite number (>2) of elements
• an infinite number of elements
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As a specific example write anequation or inequality whose solutionset 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?
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Each Understanding task should have 3 grades
• formulating & modeling
• manipulating & transforming
• inferring & drawing conclusions
<f/m, m/t, i/dc >
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Each Skills task should have 2 grades
• manipulating & transforming
• inferring & drawing conclusions
>m/t, i/dc<
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Performance on the separate dimensions of a task should not be
aggregated
A grade of
< 3/5, 1/5, 5/5>
is not equivalent to
a grade of
< 3/5, 5/5, 1/5>
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Just as it makes little sense to aggregate grades across problemdimensions…
…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.