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Math for America San Diego

Holistic Problems and the Common Core Standards

Susie Amoroso, Yekaterina Milvidskaia, Ovie Soto

Background and Purpose

A Student’s Perspective

A 2nd Year Teacher’s Perspective

A 15th year Teacher/Support Provider Perspective

Q&A – All

- To introduce attendees to holistic problems (HP’s).
- To discuss the implementation of HP’s.
- 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

- ARE YOU NUTS??!!
- Does counting on fingers qualify as “Using appropriate tools strategically”?

BACKGROUND: Synthesis of Common Core’s Mathematical Practices

- Students come to see mathematics is a humanactivity.
- Make sense of problems and persevere in solving them.

- Teachers extend the locus of authority.
- Construct viable arguments and critique the reasoning of others.

- 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: PracticesA Teaching Dilema

- Algebra 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 straight angles?
- What is the sum of the measures of the n interior angles?
- Using your answers above, what is the sum of the measures of the n exterior angles?
- What theorem do the steps above lead to?

- California Geometry (Prentice Hall): #46, p. 162

BACKGROUND: PracticesA 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: PracticesA Teaching Dilemma

Formal Mathematics

Students’ Mathematics

Teacher’s

Knowledge

Enacted

Curriculum

BACKGROUND: Teaching Practices Practices

Problems constrain teachers and teachers constrain problems.

A teaching actionrefers to what teachers in a particular community or culture typically do in the classroom.

A teaching behavioris a typical characteristic of a teaching action.

Teaching Practices

Teaching Behaviors

Teaching Actions

BACKGROUND: PracticesHolistic 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 one-step problem.

BACKGROUND: Purpose Practices

- Questionswe hope to address:
- What are some examples of holistic problems (HP’s)?
- What are some potential benefits and obstacles in using HP’s for teachers?
- How do HP’s connect to the common core mathematical practices?
- What teaching practices do we see as closely connected with the successful implementation of a holistic problem?

A Student’s Perspective Practices

BA in Mathematics (UCSD).

Math 121A and B

Finishing a Masters in Education (UCSD).

Completed 1 year of student teaching.

A Student’s Perspective Practices

Connections between different areas of Mathematics

Algebra, Geometry, and Calculus

Multiple solutions to a single problem

Multiple approaches and ways of thinking

Problems Necessitate Conjectures and Theorems

Find the Center of a given Circle Practices

Step Practices1: Pick any 3 points on a circle and connect them to make a triangle.

A2

A1

A3

Geometric Approach #1Step 2: Construct Perpendicular Bisectors of the Triangle. This can be done using a Compass

Step 3: Practices The perpendicular bisectors intersect in a point, call this point O.

ContinuedClaim: Point O is equidistant from the vertices, and hence is the center of the Circle.

Step 4: The Claim is proved using ThePerpendicular Bisector Theorem:Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment

Algebraic Approach Practices

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 Practices

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.

Step 1: Fold paper to create two semicircles Practices

Step 2: Fold paper again to create another two semicircles

The Tactile ApproachNo hints or cues as to how to approach the Practicesproblem

No “Apply this Theorem, or Use ____ Formula”

What Makes the Problem Holistic?- 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?

Promotes academic debate among Practicesstudents.

CC: Construct viable arguments and critique the reasoning of others.

A Teacher’s PerspectiveReal-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? Practices

- 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 Practices

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° overtake the hour hand?

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 overtake the hour hand?

Step 4: Write and Solve an Equation overtake the hour hand?

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 overtake the hour hand?

#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 overtake the hour hand?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 overtake the hour hand?

Intellectually intriguing

Multiple points of entry

- Some students solved proportionally
- Some solved algebraically
Lend themselves well to differentiation

Connection to Common Core Standards overtake the hour hand?

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 problem-solving strategies based on their strengths

Connection to Common Core Standards (Cont’d) overtake the hour hand?

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 overtake the hour hand?

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 overtake the hour hand?

#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 overtake the hour hand?

#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 overtake the hour hand?

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 overtake the hour hand?

Challenges for Teacher overtake the hour hand?

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 overtake the hour hand?

The value and benefits of using holistic problems

Understanding that change takes time

A Dual Perspective: Teacher/Support Provider overtake the hour hand?

- Personal Background:
- 15 years teaching high school mathematics.
- BA – Math, MA – Math, PhD – Math Ed (May 2010)
- Continue to teach 2 classes at Patrick Henry High School
- MfA Summer Institute Facilitator
- MfA Program Associate Master Teaching Fellows and Field Support
- Graduate of a previous summer institute

Prepare a poster to be presented to your overtake the hour hand?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:

You must find something to measure and say something about how to measure it.

You must use your measurements in some way to defend your answer.

Where do they meet?

Interesting student thinking that surfaced: overtake the hour hand?

Parallel did not mean the same thing to all students even though it had been defined.

Alternative conception: (Jasmine) near and far perceptual intersection.

Where do they meet?

Jasmine’s distinction helped authenticate the task. overtake the hour hand?

Students came up with several productive ways to tell if lines are parallel:

If the “distance” between them ever shrinks, the lines are not parallel. “Distance” can be measured with a ruler.

If you have the Geometer’s sketchpad, you can measure the slopes of the lines. Equal slopes parallelism.

If the lines head in the same direction, then the lines are parallel.

Draw a perpendicular line to one. If it is perpendicular to the other, then the lines are parallel.

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: overtake the hour hand?

The half-plane containing point C.

The half-plane containing point D.

Neither A nor B.

Both A and B.

Where do they meet?

Given the information below, will lines AB and DF meet on: overtake the hour hand?

The half-plane containing point C.

The half-plane containing point D.

Neither A nor B.

Both A and B.

Where do they meet?

Given the information below, will lines AB and DF meet on: overtake the hour hand?

The half-plane containing point C.

The half-plane containing point D.

Neither A nor B.

Both A and B.

Where do they meet?

Given the information below, will lines AB and DF meet on: overtake the hour hand?

The half-plane containing point C.

The half-plane containing point D.

Neither A nor B.

Both A and B.

Where do they meet?

Given the information below, will lines AB and DF meet on: overtake the hour hand?

The half-plane containing point C.

The half-plane containing point D.

Neither A nor B.

Both A and B.

Where do they meet?

Follow-Up: In a previous class discussion, it was established that lines AB and DF would meet in the half-plane containing point D. Assuming this is the case, what would be the angle formed at the point of intersection?

BACKGROUND: Teaching Practices overtake the hour hand?

Problems constrain teachers and teachers constrain problems.

A teaching actionrefers to what teachers in a particular community or culture typically do in the classroom.

A teaching behavioris a typical characteristic of a teaching action.

Teaching Practices

Teaching Behaviors

Teaching Actions

Teaching Practices with Potential to Foster Deductive Reasoning

Goals! Nobody’s Perfect.

Try NOT to invent holistic problems… use the textbooks’ problems whenever possible.

Think about the inception of knowledge. Where does it come from?

1 week model.

Encourage collaboration.

Working with MfA Fellows

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

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, MfAFellow

Fancy Cell Phone Tower Problem

Acknowledgements 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

We would like to thank NSF, The Noyce Foundation, and Math for America for this opportunity.

www.mathforamerica.org/sandiego

BACKGROUND: Language 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

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

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