Teacher led Inquiry into Fostering Reasoning and

Motivation

October 28,2011North Carolina Council of Teachers of Mathematics Conference Greensboro, NC

Tyrone Washington Tyler Brannan Landon JordanAyanna Franklin Kirsten Childs Chris LimerEmily Thrasher Latoya Clay Elizabeth PawelkaDr. Hollylynne Lee Will Hall Amber Searfoss

North Carolina State University

Overview

• Scholars Engage in Modified Action Research Cycle

• Habits that Promote Motivation• Habits that Promote Reasoning and

Sense-making • Finding/Creating Activities• Selection of Exemplar Activities

o FERMI Problems Involving Distance and Rateo Prime Numbers Activity

• Discussion

Modified Action Research Cycle

• Choosing a Theme o Informed by sessions at NCTM Conference in April

2010• Structured Readings Assigned

o Articles about student motivationo Articles about reasoning and sense-makingo Research-based stories, suggestions, etc.o Variety of contexts

• Discussion and Framework creation• Designing /Choosing Tasks • Implementation• Reflections and Making Changes

Four Habits That Promote Motivation

• Don’t beg, buy, or force students to participate. Be inviting.

• Foster students’ sense of ownership

• Praise students in ways that reward effort, not ability

• Encourage student cooperation instead of competition

National Council of Teachers of Mathematics (n.d.). You have the power to motivate! Last retrieved October 28, 2010 from http://www.nctm.org/resources/content.aspx?id=16481

Four Habits That Promote Reasoning And Sense-making

• Analyzing a problem

• Implementing a strategy

• Reflecting on a solution

• Seeking and using connections across different domains and representations

National Council of Teachers of Mathematics (2009). Focus in high school mathematics: Reasoning and sense making. Reston, Va.: Author.

Common Core - Practice Standards

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.

http://www.corestandards.org/the-standards/mathematics/introduction/standards-for-mathematical-practice/

Selection of Activities

• Scholars Selected Activitieso Each scholar was responsible for creating or

finding a task that incorporated reasoning and sense-making and the four habits with a justification of their choice.

• Discussion in Groups about the Activitieso In groups, we discussed our activities and

decided which activity was most fitting to go to final vote.

• Final Votingo 4 activities were voted on and 2 will be

presented.

Two Example Tasks

• Focused Fermi Problems• Prime Numbers Task

FERMI Framework

Central Points of a FERMI Problem (N.Pylypiw, 2010)

oFocus on smaller problemsoEstimate when necessaryoRemember to roundoMake realistic assumptions and state themo Include units

Why Use FERMI Problems?

To encourage • discussion and reasoning about

mathematical concepts and terminology• student ownership of the question and

process• students to think “outside their normal

capabilities”• motivation in learning and build foundations

on reasoning and sense-making

FERMI - Implementation

• Create groups of 3 or 4o Assign each group a different FERMI problem

OR o Assign groups the same FERMI problem.

• Give ~10 minutes to work on the problem• Give ~5 minute to present their process and

solution. o The presentation should focus on defending the

estimations and assumptions made.o Encourage student-student questioning

FERMI Problem and Solution

When should Mrs. Childs leave her house in order to arrive at the White House at 7 pm?

276 miles at an average speed of 70 mph (assumption) 3.94 hours about 4 hours

That puts us at 7pm – 4 hrs = 3:00 pm Major cities between the White House and Raleigh.

Using a map, for a 4-hour trip, 2 hours is halfway: Richmond, VA

Traffic should add about 45 minutes (assumption) From 2:15 pm to 7:00 pm is 4 hours and 45

minutes.

FERMI Problems- Car Maintenance

What is the yearly cost of gas for a driver who lives 20 miles away from school/work?

What is the yearly vehicle mileage total for an average household?

How much is the yearly cost of car insurance for a student?

How many gallons of gasoline does one car use each year in the United States? And what is the total cost of this amount of gasoline?

Motivating Students With FERMI Problems

• Don’t beg, buy, or force. Invite.o Problems are written using topics that students can

relate too These topics will initiate an internal motivation within

students because they can relate to these

• Foster a sense of ownership in your studentso Each assumption that is made comes directly from the

studentso The groups of students are solely responsible for

coming up with their unique solutions

Motivating Students With FERMI Problems• Always listen and invite students to improve

o Students assumptions may not always be logicalo Instructors may be called on to guide students to more

reasonable assumptions about a problem

• Encourage cooperation instead of competitiono Group work requires cooperationo When working with these problems, no solution is ever

“wrong"

Using FERMI Problems to Promote Reasoning and Sense-Making• Analyzing a problem

• Implementing a strategy

• Seeking and using connections across different domains and representations

• Reflecting on a solution

Two Example Tasks

• Fermi Problems• Prime Numbers Task

Prime Number Task

Eratoshenes’ Sieve Method• Students systematically go through a list

of numbers and cross out those that have factors

• The remaining numbers are primeDevelops a deeper understanding of prime

numbers though students’ own investigation

Possible areas of use• Introducing prime numbers• Reinforcing students’ understanding of

prime numbers• Creating a better understanding of

factorization

Prime Number Task1. Pick a number between 10 and 100.2. List all the numbers, in order, up to the number you have picked,

starting with 2.3. Circle the 2; this is your first Eratos number!4. Now, count by 2s up to the number you picked, crossing out all the

numbers you land on.5. Next, move from the 2 you circled before, and circle the next number

that isn’t crossed out (it should be 3). This is your next Eratos number!6. Starting at the 3, count up by 3s and cross out any number you land on.7. Continue this procedure (circling the next number that isn’t crossed

out and then count up using that number) until you have either crossed out or circled all the numbers.

8. Once you have finished step #6, collect all the circled numbers and these are all the Eratos numbers up to the number you picked.

Prime Number Task

Extend the task using:• Worksheet:

• Downloadable programs for the students’ TI-83/84 Calculators:  o http://www.ticalc.org/pub/83plus/basic/math/arithmetic/facto

ring/date.html

Spreadsheet: calculates whether numbers are divisible by each “Eratos” number.  A “1” means it has not been crossed out.  

Prime Number Task

Promoting Motivation Through the Prime Number Task • Historical exploration  o What do these Eratos numbers have in common?

• Self-guided task• Making the activity as challenging as they want.

• Attainable task • Interactive task• Collaboration not competition• Students can help each other decide whether to cross

out the numbers or not.

Using the Prime Number Task to Promote Reasoning and Sense-makingOpportunities for• Generalization and reflection

• Students will examine larger numbers• Students will reflect on the common properties of

the numbers left or crossed out.• Implementation of strategies

• Students may implement other strategies to decide if a number is an Eratos number or not.

• Recognition of connections• Students will see the relationships of the Eratos

numbers

Implementation and Reflection

• Several Noyce Scholars have used FERMI problems in their classeso Generally positive feedbacko Students drew on previous knowledge and

experience from other disciplines to make sense of the problem.

• We have not had a chance to implement the Prime Number Task yet.o But we hope to in the coming semester!

References

Boyer, K. (2002). Using active learning strategies to motivate students. Mathematics Teaching in the Middle School, 8(1), 48 – 51.

Chazan, D. (1999). On teachers’ mathematical knowledge and student exploration: A personal story about teaching a technologically supported approach to school algebra. International Journal of Computers for Mathematical Learning, 4, 121 – 149.

Chazan, D. (2000). Beyond formulas in mathematics and teaching: Dynamics of the high school algebra classroom. Teachers College Press: New York.

Cuoco, A., Goldenberg, E., & Mark, J. (2010). Organizing a curriculum around mathematical habits of mind. MathematicsTeacher, 103(9), 682 – 689.

Mark, J., Cuoco, A., Goldenberg, E., & Sword, S. (2010). Developing mathematical habits of mind. Mathematics Teaching in the Middle School, 15(9), 505 – 509.

National Council of Teachers of Mathematics (2009). Focus in high school mathematics: Reasoning and sense making. Reston, Va.: Author.

National Council of Teachers of Mathematics (n.d.). You have the power to motivate! Last retrieved October 28, 2010 from http://www.nctm.org/resources/content.aspx?id=16481

References

Websites for FERMI Problemso http://www.physics.uwo.ca/science_olympics/events/puzzles/f

ermi_questions.html

o http://www.jlab.org/~cecire/garden/fermiprob.html

o http://www.physics.umd.edu/perg/fermi/fermi.htm

o http://en.wikipedia.org/wiki/Fermi_problem

Website for Prime Number Activityo  

http://www.ticalc.org/pub/83plus/basic/math/arithmetic/factoring/date.html

Population Educationo http://www.populationeducation.org/

Extra FERMI Problems

• How tall would a stack of a million sheets of paper be? • What would be the length of a billion people holding hands?• How big a field would you need to contain one million

people?• Estimate the number of blades of grass a typical suburban

house's lawn has in the summer. • How many frames are in a Walt Disney animated movie

such as Tarzan?• What is the length in miles of the US Interstate Highway

system?• How many square inches of pizza are consumed by all the

students at the NC State University during one semester?

Extra FERMI Problems

• How many pencils would it take to draw a straight line along the entire Equator of the earth?

• If all the string was removed from all of the tennis rackets in the US and laid out end-to-end, how many round trips from Detroit to Orlando could be made with the string?

• How many drops of water are there in all of the Great Lakes?• What distance will your fingernail grow today?• How many golf balls can be fit in a typical suitcase? • How much milk is produced in the US each year? • If the entire population of Canada lined up to do the wave,

how long would it take?• How many flat tires are there in the US at any 1 time?• How many math problems does a math teacher grade in a

year?

Questions, Comments, Concerns… To learn more about the Noyce

Mathematics Education Teaching Scholars program at NC State please visit us at http://ced.ncsu.edu/2/noyce

Email Ayanna Franklin (adfrankl@ncsu.edu) for copies of activities