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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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The structure of statements

in mathematics-

The role of

objects

and

actions

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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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Some examples of

Understanding tasks

with a focus on

formulating & modeling

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designing

• a measure

• a computation

• a mathematical object

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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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Which is a better way to construct aregression line?

For what purpose?

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Some examples of Skills tasks

with a focus on

manipulating

and

transforming

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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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Implicationsfor the

writing of rubricsand

for grading

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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.