I mp le me n t i n g t he C o mmo n C o r e :
I t ’s A ll Ab o u t P r ac t i c e
( S t an d ar d s ) !
The 56th Annual Conference
on the Teaching of Mathematics
April 9, 2013
Dr. Donald Porzio
Mathematics Faculty
Illinois Mathematics and Science Academy
1500 Sullivan Rd., Aurora, IL 60506
630-907-5966
dporzio@imsa.edu
http://staff.imsa.edu/~dporzio/
(ICTM President 2011-13)
3 lbs times 2 units away from lower cord equals 6.
1 lb times 4 units away from cord plus 2 lbs times 1 unit away from lower cord
also equals 6, so the sides of the lower bar balance.
6 total lbs times 2 units away from upper cord equals 12.
4 lbs times 3 units away from upper cord also equals 12, so the sides of the
upper bar balance.
Common Core State Standards for Mathematics
Standards for Mathematical Practice
1.
Make sense of problems and persevere in solving them.
2.
Reason abstractly and quantitatively.
3.
Construct viable arguments and critique the reasoning of others.
4.
Model with mathematics.
5.
Use appropriate tools strategically.
6.
Attend to precision.
7.
Look for and make use of structure.
8.
Look for express regularity in repeated reasoning.
Standards for Mathematical Practice addressed by weight balancing
problem
1.
Make sense of problems and persevere in solving them.
2.
Reason abstractly and quantitatively.
3.
Construct viable arguments and critique the reasoning of others.
7.
Look for and make use of structure.
8.
Look for express regularity in repeated reasoning.
Were there others?
Some Content Standards addressed by weight balancing problem
Are there others?
Connecting the Standards for Mathematical Practice to the Standards
for Mathematical Content
The Standards for Mathematical Practice describe ways in which developing student practitioners of the
discipline of mathematics increasingly ought to engage with the subject matter as they grow in
mathematical maturity and expertise throughout the elementary, middle and high school years.
Designers of curricula, assessments, and professional development should all attend to the need to
connect the mathematical practices to mathematical content in mathematics instruction.
SO,….
How can you design lessons that target different Practice Standards
and also connect to Content Standards?
The Standards for Mathematical Content are a balanced combination of procedure and understanding.
Expectations that begin with the word “understand” are often especially good opportunities to connect
the practices to the content. Students who lack understanding of a topic may rely on procedures too
heavily. Without a flexible base from which to work, they may be less likely to consider analogous
problems, represent problems coherently, justify conclusions, apply the mathematics to practical
situations, use technology mindfully to work with the mathematics, explain the mathematics accurately
to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In
short, a lack of understanding effectively prevents a student from engaging in the mathematical
practices.
In this respect, those content standards which set an expectation of understanding are potential “points
of intersection” between the Standards for Mathematical Content and the Standards for Mathematical
Practice. These points of intersection are intended to be weighted toward central and generative
concepts in the school mathematics curriculum that most merit the time, resources, innovative energies,
and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional
development, and student achievement in mathematics.
Common Core State Standards for Mathematics, pg 8.
Standards for Mathematical Practice #5:
Use appropriate tools strategically.
Solve the quadratic equation x 2  x  2 x  4 .
versus
Describe two (or more) different ways you could solve the equation x 2  x  2 x  4 . Explain how your
solution methods are related to each other. Then choose one of your methods and use it to solve the
equation, being sure to explain the steps you are taking to solve the equation.
List four different properties of parallelograms.
versus
What are some ways you can construct a parallelogram? Describe and create two (or more) different
constructions for a parallelogram and indicate what property or properties of parallelograms you used in
your construction. (Here, parallelogram can be replaced with some other “special" quadrilateral, such
as a kite, rhombus, rectangle, square, general trapezoid, right trapezoid, isosceles trapezoid, cyclic
quadrilateral or our own favorite "special" quadrilateral.).
CAS and Equation Solving
When you use the Solve command on a CAS calculator like the TI-89 or TI-Nspire CAS to solve an
equation, it jumps straight to the correct answer without showing any work!! That’s great if all we want
is the answer, but what if we also want to work on developing or assessing a student’s ability to solve
equations. For example, suppose you want to see if students not only can obtain a solution to an
equation like 3x – 1 = 4 – 7x, but also understand the process of solving such an equation. This
equation can be entered on the TI-Nspire CAS and then solved in the “standard” manner of “isolating”
the variable and then simplifying. Once a solution is obtained, it can be checked by substituting back
into the equation using the | key.
One strength of this method is that, when students do an incorrect step, such as subtracting 10 from each
side of the equation 10x = 5 rather than dividing each side by 10, the TI-Nspire CAS makes the error
readily apparent, as shown above at the right. This is in stark contrast what happens in real life when a
student will swear that subtracting 10 from each side of 10x = 5 produces the equation x = –5!!
Another equation to try that illustrates the
"teaching" power of CAS is
2 y  3  2 y  3  1 . To solve this
equation, commands under the Algebra
menu, such as Expand, Common
Denominator, and Proper Fraction
will need to be used. As you solve the
equation, a wonderful surprise, AND
learning opportunity will present itself at the
bottom of the calculator screen.
SAMPLE COMMON CORE MATH ASSESSMENT ITEM
DEVELOPED BY ILLINOIS TEACHERS
Key Point in all this:
Don't always let the content
drive your lessons.
Instead, think about the
practices first, then the content!
Or at the very least, think about
the practices and content at the
same time.
You'll be pleasantly surprised as
to how well the content will
take care of itself when the
practices are the focus and are
done well.
Use of Multiple Representations
Giving students opportunities to demonstrate their understanding of the connections between multiple
representations of mathematical ideas and concepts will help in their development into be flexible
thinkers and problem solvers. In Principles and Standards for School Mathematics (NCTM, 2000), the
National Council of Teachers of Mathematics proposed that "Instructional programs from
prekindergarten through grade 12 should enable all students to - … Select, apply, and translate among
mathematical representations to solve problems" (pg. 360). One highly advocated theoretical view on
mathematical learning is based on the premise that multiple representations can be utilized to help
students develop deeper, more flexible understandings of concepts and processes. If we consider
multiple representations of concepts to be important, then that means we will have to look for ways to
assess students understandings of these representations, and technology can be invaluable in doing this!
What follows are my views on the use of multiple representations in the teaching and learning of
mathematics along with a rubric that I have been working on (for many, many years) for assessing work
on problems involving use of technology and different representations. You may find this information
and rubric useful in your own teaching (and I would appreciate any comments and suggestions you
might have concerning the rubric)
WHY USE GRAPHICAL, NUMERICAL, AND SYMBOLIC REPRESENTATIONS
IN THE TEACHING AND LEARNING OF MATHEMATICS? BECAUSE THEY:






are inherent in mathematics;
provide multiple concretizations of a concept;
may be used to mitigate difficulties students have in accomplishing certain tasks;
provide students with more mathematical tools for solving problems (assuming they view the
different representations as tools);
may make mathematics more interesting to students;
can help students develop deeper, more flexible understandings of a concept by having them:
 generate the different representations of the concept and then view and reflect upon common
properties and differences between the different representations,
 “see” the concept or idea from several points of view (i.e., different representations) and then
build a web of connections between the different viewpoints (representations), and
 utilize the different representations to model and solve a problem.
ASSESSING WORK THAT INVOLVES USE OF TECHNOLOGY AND
MULTIPLE REPRESENTATIONS; FOCUS ON STUDENT’S ABILITY:






To use appropriate solution methods (not just their ability to get the correct answer);
To recognize and use different representations (symbolic, numerical, graphical) of a given problem
situations and of concepts related to that problem situation;
To “construct” different representations (symbolic, numerical, graphical) using appropriate tools
(such as graphing calculators, CAS calculators, and various types of computer software);
To recognize how the same procedure or concept can be applied to different representations of the
problem situation;
To recognize connections between solution methods that use different types of representations;
To justify their use of various procedures, concepts, processes, representations, and connections, to
solve the problem.
NOVICE LEVEL





Use (at least attempted) of only one type of representation
Use of inappropriate procedures, concepts, or methods
Inaccurate, incomplete, or missing descriptions of solutions
No justification of processes or representations used in problem solution
No evidence of recognition of connections between solutions involving different representations
 Serious mathematical errors present
APPRENTICE LEVEL





Use (at least attempted) of two different types of representations
Use of some appropriate procedures, concepts, and methods
Poorly-developed descriptions of solutions
Vague or unclear justifications of processes and representations used in problem solution
Evidence of vague, unclear recognition of connections between solutions involving different
representations
 Minor mathematical errors present
COMPETENT LEVEL





Use of two (possible more) different types of representations
Use of mostly appropriate procedures, concepts, and methods
Reasonably well-developed descriptions of solutions
Fairly clear justifications of processes and representations used in problem solution
Evidence of at least partial recognition of connections between solutions involving different
representations
 Few, if any, minor mathematical errors present
PROFICIENT LEVEL





Use of at least two different types of representations
Use of appropriate procedures, concepts, and methods
Well-developed, accurate descriptions of solutions
Justifications/explanations of processes and representations used in problem solution
Evidence of recognition of connections between solutions involving different representations
 No mathematical errors present
DISTINGUISHED LEVEL






Use of three or more different types of representations
Use of appropriate, possibly unique, procedures, concepts, and methods
Very well-developed, accurate, concise descriptions of solutions
Precise, detailed justifications of processes and representations used in problem solution
Discussion of the non-use of certain methods or representations
Clear evidence of understanding of connections between solutions involving different
representations
DISTINGUISHED LEVEL
Leaps Tall Buildings in a Single Bound
PROFICIENT LEVEL
Leaps Short Buildings in a Single Bound
COMPETENT LEVEL
Leaps Short Buildings with a Running
Start and Maybe a Boost
APPRENTICE LEVEL
Bumps into Buildings
NOVICE LEVEL
Does Not Recognize Buildings
5
5
8
4
3
1
2
1
2
6
9
8
4
1
3
9
15
7
5
4
10
2
6
10
13
3
12
7
14
11
Geometric Play
The figure below consists of 20 identical
squares. Divide the figure into 5 pieces,
each consisting of 4 squares, so that no 2
pieces have the same shape. Note that each
square in each piece must share at least one
side with another square in that piece.
Geometric Play
The figure below consists of 20 identical
squares. Divide the figure into 5 pieces,
each consisting of 4 squares, so that no 2
pieces have the same shape. Note that each
square in each piece must share at least one
side with another square in that piece.
1
34 46
11
22 44
69
12
24
48
70
9
18
42 49
73
23
32 56 74
How Far to Zequop?
The Situation:
The seven towns of Tidville, Ubania, Vogton,
Wimpster, Xendic, Yunjar, and Zequop are
indicated on the map on the next page by the letters
A to G, not necessarily in order. The roads
connecting the towns run strictly north-south and
east-west. The distance between adjacent dots on
the map is exactly one mile. Thus, the town labeled
C is 5 miles (by road) from the town E. Each town
contains a sign that gives the length of the shortest
route to two other towns. These seven signs, one
from each town, are shown on the next page. The
last sign, though, has been left incomplete.
The Problem:
Discover which town is where, put each sign in its
correct town, and provide the correct distance to
Zequop that is missing on the last sign.
A
F
D
E
B
C
G
1
Tidville
Ubania
3
4
4
Wimpster 6
Xendic
3
2
Ubania
Vogton
7
3
5
Xendic
Yunjar
6
3
3
Vogton
5
Wimpster 6
6
Yunjar
Zequop
4
4
7
Tidville
Zequop
1
?
Wimpster
Sign 2 F
Tidville
Sign 6 E
A
Zequop
Sign 4 A
D
Yunjar
Sign 3 B
F
E
Xendic
Sign 7 D
B
C
G
Ubania
Sign 5 C
Vogton
Sign 1 G
7
Tidville
Zequop
1
3