Math for America San Diego
Holistic Problems and the Common
Core Standards
Susie Amoroso, Yekaterina Milvidskaia, Ovie Soto
Outline
1. Background and Purpose
2. A Student’s Perspective
3. A 2nd Year Teacher’s Perspective
4. A 15th year Teacher/Support Provider
Perspective
5. Q&A – All
Purpose
1. To introduce attendees to holistic problems
(HP’s).
2. To discuss the implementation of HP’s.
3. To demonstrate the ability of HP’s to support
the Common Core Standards’ eight
mathematical practices.
BACKGROUND: Common Core’s 8 Mathematical Practices
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 and express regularity in repeated reasoning.
BACKGROUND: Connecting Common Core
Math Practices with Proof Schemes Taxonomy
Common Core:
• Reason abstractly and quantitatively
• Use appropriate tools strategically
• Look for and make use of structure
• Look for and express regularity in repeated reasoning
Deductive (Proof Scheme): Harel and Sowder (1998)
• Ability to pause and probe into the meaning of the symbols
• Make changes to expressions/equations/geometric objects,
anticipating their effects and compensating for them in a goal-oriented
fashion
• Attend to the generality aspect of conjectures
• Attend to regularity in a process using it as a reason results generalize
BACKGROUND: Synthesis of Common Core’s
8 Mathematical Practices
1. ARE YOU NUTS??!!
2. Does counting on fingers qualify as
“Using appropriate tools strategically”?
BACKGROUND: Synthesis of Common Core’s Mathematical Practices
1. Students come to see mathematics is a human activity.
• Make sense of problems and persevere in solving
them.
2. Teachers extend the locus of authority.
• Construct viable arguments and critique the reasoning
of others.
3. Teachers help students learn to reason deductively.
• Reason abstractly and quantitatively.
• Look for and make use of structure.
• Look for and express regularity in repeated reasoning.
• Attention to precision.
BACKGROUND: A Teaching Dilema
Algebra A polygon has n sides. An interior angle of the polygon and
an exterior angle form a straight angle.
a. What is the sum of the measures of the n straight angles?
b. What is the sum of the measures of the n interior angles?
c. Using your answers above, what is the sum of the measures
of the n exterior angles?
d. What theorem do the steps above lead to?
California Geometry (Prentice Hall): #46, p. 162
BACKGROUND: A Teaching Dilemma
If the problem were changed (see below), what potential
benefits and obstacles might a high school Geometry teacher
encounter? What might students try?
A polygon has n sides. An interior angle of the polygon and an
exterior angle form a straight angle. What is the sum of the
measures of the n exterior angles?
Adapted from California Geometry (Prentice Hall): #46, p. 162
BACKGROUND: A Teaching Dilemma
Students’
Mathematics
Formal
Mathematics
Teacher’s
Knowledge
Enacted
Curriculum
BACKGROUND: Teaching Practices
Problems constrain teachers and teachers constrain problems.
A teaching action refers to what teachers in a particular
community or culture typically do in the classroom.
A teaching behavior is a typical characteristic of a teaching
action.
Teaching
Practices
Teaching
Actions
Teaching
Behaviors
BACKGROUND: Holistic vs. Non-Holistic Problems
A holistic problem refers to a problem where one must figure
out from the problem statement the elements needed for its
solution—it does not include hints or cues as to what is
needed to solve it.
A non-holistic problem, on the other hand, is one which is
broken down into small parts, each of which attends to one
or two isolated elements. Often each of such parts is a onestep problem.
BACKGROUND: Purpose
Questions we hope to address:
1. What are some examples of holistic problems (HP’s)?
2. What are some potential benefits and obstacles in using
HP’s for teachers?
3. How do HP’s connect to the common core mathematical
practices?
4. What teaching practices do we see as closely connected
with the successful implementation of a holistic problem?
A Student’s Perspective
• BA in Mathematics (UCSD).
• Math 121A and B
• Finishing a Masters in Education (UCSD).
• Completed 1 year of student teaching.
A Student’s Perspective
• Connections between different areas of Mathematics
o Algebra, Geometry, and Calculus
• Multiple solutions to a single problem
o Multiple approaches and ways of thinking
• Problems Necessitate Conjectures and Theorems
Find the Center of a given Circle
Geometric Approach #1
Step 1: Pick any 3 points on a circle and connect them to make
A2
a triangle.
A1
A3
Step 2: Construct Perpendicular Bisectors of the Triangle. This
can be done using a Compass
Continued
Step 3: The perpendicular bisectors intersect in a point, call this
point O.
Claim: Point O is equidistant from the vertices, and hence is the
center of the Circle.
Step 4: The Claim is proved using The Perpendicular Bisector
Theorem: Any point on the perpendicular bisector of a segment is equidistant from
the endpoints of the segment
Algebraic Approach
Step 4: Set the two expressions equal to each other to find the
point of intersection. This point will be the center of the Circle.
Geometric Approach # 2
Step 1: Construct a chord that is a diameter (proved by Thale’s
Theorem).
Step 2: Repeat.
Claim: The point of intersection of two diameters is the Center. This
can be proved by definition of a diameter and intersection of lines.
The Tactile Approach
• Step 1: Fold paper to create
two semicircles
• Step 2: Fold paper again to
create another two
semicircles
What Makes the Problem Holistic?
No hints or cues as to how to approach the problem
No “Apply this Theorem, or Use ____ Formula”
Multiple ways of solving the problem
Geometric, Algebraic, & Logical
Multiple ways of thinking
Perceptual, Empirical, & Deductive
Necessitates a Particular Mathematical Concept
Why must the bisectors of the chords intersect at the center of a
circle?
A Teacher’s Perspective
Promotes academic debate among students.
CC: Construct viable arguments and critique the
reasoning of others.
Real-life thinking - Students need to figure out an
approach before applying tools they have.
CC: Use appropriate tools strategically.
Holistic Problems afford different learning modalities.
This give students better opportunity to find an approach
that works for them.
Who am I?
• Susan Amoroso
• In 2008, earned B.S. in Mathematics and was selected
as a Math for America Fellow
• Just completed 2nd year of teaching
• Low-income, high-minority high school in San Diego
County, CA
• Attended first MfA San Diego Summer Institute in July
2009
My Intro to Holistic Problems
At MfA San Diego Summer Institute in July
2009.
Knew right away this was different:
• No lecture
• No explicit discussion of teaching strategies
• Given holistic problems to solve
• Worked in small groups
• Every group shared its solution
#1: At what time after 4:00 will the
minute hand of a clock overtake the
hour hand?
Step 1: Started
with a picture
of such a clock.
Step 2: Divided the clock face into
360°
Thus, the starting
locations are 0° for the
minute hand and 120°
for the hour hand.
In one minute, the minute
hand moves 6° and the
hour hand moves 0.5°.
Step 3: Create a Table
Time (in Location of Minute
minutes)
Hand
(in degrees)
0
0°
1
6°
2
12°
3
18°
t
6t
Location of
Hour Hand (in
degrees)
120°
120.5°
121°
121.5°
120 + 0.5t
Step 4: Write and Solve an
Equation
6t = 120 + 0.5t
5.5t = 120
t ≈ 21.82 ≈ 21 minutes 48 seconds
The minute hand will overtake the hour
hand at approximately 21.82 minutes, or
21 minutes and 48 seconds.
Follow Up Problems
#2 – At what time after 7:30 will the hands of a
clock be perpendicular?
#3 – Between 3:00 and 4:00 Noreen looked at
her watch and noticed that the minute hand
was between 5 and 6. Later, Noreen looked
again and noticed that the hour hand and the
minute hand had exchanged places. What
time was it in the second case?
What I Learned About Learning
Mathematics
Power of images (and alternatives)
• e.g., Degrees vs. proportions
Okay for students to see that math is messy
Mathematical procedures are tools;
quantitative reasoning is the goal
What I Learned About Teaching
Mathematics
Intellectually intriguing
Multiple points of entry
• Some students solved proportionally
• Some solved algebraically
Lend themselves well to differentiation
Connection to Common Core
Standards
Reflect real-life nature of problem-solving
Require students to make sense of the
problem
Often provide multiple points of entry –
students develop their own problemsolving strategies based on their strengths
Connection to Common Core Standards
(Cont’d)
Require students to reason quantitatively
(in context) and attend to the meaning of
quantities
Students must communicate effectively
by making clear mathematical arguments,
explaining their reasoning, justifying
conclusions and critiquing others’
solutions
Fast Forward Two Years
Knew this was the type of learning environment I
wanted to create in my classroom
Currently, I teach very traditionally
• Homework review
• Toolbox Notes
• Independent Practice
Effective use of holistic problems is a complex process
Systems of Equations
#1 – Tree A is 2 feet tall and growing at a
rate of 1 foot each year. Tree B is 6 feet
tall and growing at a rate of 0.5 feet each
year. In how many years will the trees be
the same height and what will that height
be?
Students: Represented their work in a table…
My Goal: Solving by Graphing
Systems of Equations
#2 – Tree C is 4 feet tall and grows 0.5 feet
each year. Tree D is 7.3 feet tall and
grows 0.25 feet each year. In how many
years will the trees be the same height
and what will that height be?
Students: Represented their work in a table…
Goal: Solving by Substitution or Elimination
Challenges for Students
Not being able to get started
Poor organization
Addition errors, particularly on #2
Lack of perseverance
Calculating growth for fractions of a year
Partial Table of Values for #2
Years of
Growth
Height of Tree C
Height of Tree D
0
4
7.3
1
4.5
7.55
2
5
7.8
3
5.5
8.05
12
10
10.3
13
10.5
10.55
14
11
10.8
15
11.5
11.05
16
12
11.3
17
12.5
11.55
Challenges for Teacher
How to help students get started
Getting students to share solutions with the class
Time issues
Moving from contextualized to de-contextualized
The Big Picture
The value and benefits of using holistic
problems
Understanding that change takes time
A Dual Perspective: Teacher/Support Provider
Personal Background:
1. 15 years teaching high school mathematics.
2. BA – Math, MA – Math, PhD – Math Ed (May 2010)
3. Continue to teach 2 classes at Patrick Henry High School
4. MfA Summer Institute Facilitator
5. MfA Program Associate Master Teaching Fellows and Field
Support
6. Graduate of a previous summer institute
Where do they meet?
Prepare a poster to be presented to your classmates
answering the following question. Will these two lines
meet on the left side of the page, the right side of the
page, both sides of the page, or never?
Explain your reasoning with the following rules:
1. You must find something to measure and say
something about how to measure it.
2. You must use your measurements in some way to
defend your answer.
Where do they meet?
Interesting student thinking that surfaced:
1. Parallel did not mean the same thing to all students
even though it had been defined.
2. Alternative conception: (Jasmine) near and far
perceptual intersection.
Where do they meet?
1. Jasmine’s distinction helped authenticate the task.
2. Students came up with several productive ways to tell
if lines are parallel:
a. If the “distance” between them ever shrinks, the lines are not
parallel. “Distance” can be measured with a ruler.
b. If you have the Geometer’s sketchpad, you can measure the
slopes of the lines. Equal slopes  parallelism.
c. If the lines head in the same direction, then the lines are
parallel.
d. Draw a perpendicular line to one. If it is perpendicular to the
other, then the lines are parallel.
e. Draw a transversal. Check for congruent angles in the “same
position of both groups of angles formed”.
Where do they meet?
Given the information below, will lines AB and DF meet on:
A. The half-plane containing point C.
B. The half-plane containing point D.
C. Neither A nor B.
D. Both A and B.
Where do they meet?
Given the information below, will lines AB and DF meet on:
A. The half-plane containing point C.
B. The half-plane containing point D.
C. Neither A nor B.
D. Both A and B.
Where do they meet?
Given the information below, will lines AB and DF meet on:
A. The half-plane containing point C.
B. The half-plane containing point D.
C. Neither A nor B.
D. Both A and B.
Where do they meet?
Given the information below, will lines AB and DF meet on:
A. The half-plane containing point C.
B. The half-plane containing point D.
C. Neither A nor B.
D. Both A and B.
Where do they meet?
Given the information below, will lines AB and DF meet on:
A. The half-plane containing point C.
B. The half-plane containing point D.
C. Neither A nor B.
D. Both A and B.
Follow-Up: In a previous class discussion, it was
established that lines AB and DF would meet in the halfplane containing point D. Assuming this is the case, what
would be the angle formed at the point of intersection?
BACKGROUND: Teaching Practices
Problems constrain teachers and teachers constrain problems.
A teaching action refers to what teachers in a particular
community or culture typically do in the classroom.
A teaching behavior is a typical characteristic of a teaching
action.
Teaching
Practices
Teaching
Actions
Teaching
Behaviors
Teaching Practices with Potential to Foster Deductive Reasoning
Teaching Actions
Teaching Behaviors
Dealing with student thinking Ensure that debate/discussion is truly public
Eliciting student responses
Ask students to communicate her solution process
Direct the class to similarities/differences between
solution processes and underlying mental imagery.
Ask for alternatives in the presence of correct
solutions
Devolve responsibility to
students
Encouraging them to make and prove conjectures
Allow errors to persist
Introduce new
tools/concepts
Through attention to students’ natural capacity to
be puzzled (intellectual need)
Goals! Nobody’s Perfect.
Working with MfA Fellows
1. Try NOT to invent holistic problems… use the
textbooks’ problems whenever possible.
2. Think about the inception of knowledge. Where
does it come from?
3. 1 week model.
4. Encourage collaboration.
Cell Phone Tower Problem
A cell phone tower sends out a signal that reaches a
fixed distance in every direction. Some can carry calls
up to 5 miles from the tower while less powerful towers
can only carry calls within a mile of the tower. If there is
a cell tower at Point A that reaches 5 miles, how can we
determine what space is in the calling area? If there is
a cell tower at Point B that reaches 3 miles, how can we
determine what space is in the calling area? Where
does the calling area of the towers overlap?
Vanessa Davis, MfA Fellow
Fancy Cell Phone Tower Problem
Verizon has come out with a new tower plan that will
provide faster internet service and clearer calls.
However, this new service requires two cell towers.
The cell phone user must be within 10 miles of both
cell towers combined. (So if a person is 7 miles from
one tower, she can only be 3 miles away from the
other.) Verizon plans to put the towers 4 miles apart.
If A and B are the locations of the two towers, what is
the calling area for this new service?
Vanessa Davis, MfA Fellow
Acknowledgements
We would like to thank NSF, The Noyce Foundation,
and Math for America for this opportunity.
www.mathforamerica.org/sandiego
BACKGROUND: Language
Conjecture, Proving & Proof Schemes
A conjecture is an observation made by a person who has doubts about its
truth. A person’s observation ceases to be a conjecture and becomes a fact in
her or his view once the person becomes certain of its truth.
Proving is the process employed by an individual to remove or create doubts
about the truth of an observation.
A person’s proof scheme consists of what constitutes ascertaining and
persuading for that person.
- Harel & Sowder, 1998, p. 244