What's Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success (28 page)

So each way of replacing 2 of the 12 beans with an
x
corresponds to a way to put the remaining 10 beans in bowls. So if we can figure out how many ways there are to choose 2 beans and replace them with an
x,
then we will know how many arrangements of beans in bowls there are. How many ways are there to pick the first bean? Well, there are 12, because you can pick any bean. Then there are 11 ways to pick the second bean, because you’ve already picked 1. So there are 12 × 11 = 132 ways to pick 1 bean and then pick another bean. But, there is another thing we need to be careful of. There are 2 ways to pick each pair of
x
’s, because you can switch the order in which you picked them. So 132 counts every pair of
x
’s twice, and we need to divide by 2. So the number of ways of replacing 2 beans by x’s, which is the same as the number of ways to put 10 beans in 3 bowls, is (12 × 11)/2 = 66.

Can you see how this method would work if you changed the number of beans? What about if you changed the number of bowls?

Partitions

You could use Cuisenaire rods to help with this problem.

The number 3 can be broken up into positive numbers in four different ways:

Or maybe you think that 1 + 2 and 2 + 1 are the same, so there are really only three ways to break up the number.

Decide which you like better and investigate partitions for different numbers using your rules.

Solution

For this one, you’re on your own! Coming up with a general pattern for how many partitions a number has is an unsolved problem. Welcome to cutting-edge
mathematics.

Recommended Curriculum

K–5

• TERC: Investigations (Grades K–5). Publisher: Pearson Scott Foresman. https://investigations.terc.edu.

• ST Math. Publisher: MIND Research. www.mindresearch.org.

• Bridges (Grades K–5). Publisher: The Math Learning Center. www.mathlearningcenter.org/bridges.

• Contexts for Learning.

6–8

• College Preparatory Mathematics (CPM) Core Connections Series. Publisher: CPM Educational Program. www.cpm.org.

• Connected Mathematics Project. Publisher: Michigan State University. https://connectedmath.msu.edu.

Intervention/Supplemental

• ST Math. Supplemental curriculum. Publisher: MIND Research. www.mindresearch.org.

• Math 180. Publisher: Scholastic. http://teacher.scholastic.com/products/math180.

9–12

• Core-Plus Mathematics, CPMP. Publisher: McGraw-Hill. http://wmich.edu/cpmp.

• CPM Core Connections Series. Publisher: CPM Educational Program. www.cpm.org.

• Interactive Mathematics Program, IMP. Publisher: It’s About Time Interactive. http://mathimp.org.

Further Resources

Books

Boaler, Jo.
Mathematical Mindsets: Unleashing Students’ Potential Through Creative Math, Inspiring Messages and Innovative Teaching
. Hoboken, NJ: Wiley, 2015.

Boaler, Jo, and Cathy Humphreys.
Connecting Mathematical Ideas: Middle School Video Cases to Support Teaching and Learning
. Portsmouth, NH: Heinemann, 2005.

Cohen, E. G., and R. A. Lotan.
Designing Groupwork: Strategies for the Heterogeneous Classroom.
New York: Teachers College Press, 2014.

Driscoll, M.
Fostering Algebraic Thinking: A Guide for Teachers, Grades 6–10.
Portsmouth, NH: Heinemann, 1999.

Driscoll, M., R. W. DiMatteo, J. Nikula, and M. Egan.
Fostering Geometry Thinking: A Guide for Teachers, Grades 5–10
. Portsmouth, NH: Heinemann, 2007.

Harris, P. W.
Building Powerful Numeracy for Middle and High School Students
. Portsmouth, NH: Heinemann, 2011.

Hiebert, J., T. P. Carpenter, E. Fennema et al.
Making Sense: Teaching and Learning Mathematics with Understanding.
Portsmouth, NH: Heinemann, 1997.

Humphreys, Cathy, and R. Parker.
Making Number Talks Matter: Developing Mathematical Practices and Deepening Understanding, Grades 4–10.
Portland, ME: Stenhouse, 2015.

Lamon, S. J.
Teaching Fractions and Ratios for Understanding: Essential Content Knowledge and Instructional Strategies for Teachers
. New York: Routledge, 2011.

Mason, J.
Learning and Doing Mathematics
. Hertfordshire, England: Tarquin, 2000.

Mason, J., L. Burton, and K. Stacey. (2010).
Thinking Mathematically
. Upper Saddle River, N.J.: Pearson, 2010.

Moses, R. P. and C. E. Cobb Jr.
Radical Equations: Math Literacy
and Civil Rights
. Boston: Beacon Press 2001.


Parrish, S.
Number Talks: Helping Children Build Mental Math
and Computation Strategies, Grades K–5, Updated with Common
Core Connections.
Sausalito, CA: Math Solutions, 2014.


Petersen, J.
Math Games for Independent Practice: Games to
Support Math Workshops and More
. Sausalito, CA: Math Solutions, 2013.

Pólya, G.
How to Solve It: A New Aspect of Mathematical Method
. Princeton, N.J.: Princeton University Press, 2014.


R4 Educated Solutions.
Making Math Accessible to English Language
Learner: Practical Tips and Suggestions
. Bloomington, IN: Solution Tree, 2009.

Shumway, J. F.
Number Sense Routines: Building Numerical Literacy Every Day in Grades K–3
. Portland, ME: Stenhouse., 2011


Small, M.
Good Questions: Great Ways to Differentiate Mathematics Instruction
. New York: Teachers College Press, 2012.


Small, M., and A. Lin. More Good Questions: Great Ways to Differentiate Secondary Mathematics Instruction. New York: Teachers College Press, 2010.


Stein, M. K., M. S. Smith, M. A. Henningsen, and E. A. Silver.
Implementing Standards-Based Mathematics Instruction: A Casebook for Professional Development, Second Edition
by Mary Kay Stein et al. New York: Teachers College Press, 2009.

Swetz, F. J.
Mathematical Expeditions: Exploring World Problems
Across the Ages
. Baltimore: Johns Hopkins University Press, 2012.

Tate, M. Worksheets Don’t Grow Dendrites: 20 Instructional Strategies That Engage The Brain. Thousand Oaks, CA: Corwin, 2010.

Wiggins, G., and J. McTighe.
Understanding by Design
. Upper resources). Saddle River, NJ: Pearson, 2005.

Wiliam, D.
Embedded Formative Assessment
. Bloomington, IN: Solution Tree, 2011.

Web Sites

Balanced Assessment: http://balancedassessment.concord.org

Conceptua Math (grades 3–6): Visual and conceptual learning, www .conceptuamath.com

Dan Meyer’s resources: http://blog.mrmeyer.com

Estimation 180: www.estimation180.com

GeoGebra: http://geogebra.org/cms

Hour of Code: http://hourofcode.com/us

Mathalicious (grades 6–12): Real-world lessons for middle and high school, www.mathalicious.com

The Math Forum: www.mathforum.org.

The Mathematics Assessment Project, Shell Center: http://map.mathshell.org/materials/index.php


National Council of Teachers of Mathematics (NCTM): www.nctm.org (You need to be a member to access some of the resources.)

NCTM Illuminations: http://illuminations.ntcm.org

NRICH: http://nrich.maths.org

Number Strings: http://numberstrings.com


Shell Centre for Mathematical Education Publications: www.mathshell.com

Teach the Hour of Code: http://code.org

Video Mosaic Collaborative: http://videomosaic.org

Visual Patterns (grades K–12): www.visualpatterns.org

YouCubed: www.youcubed.org

Apps

ABCya.com: Virtual Manipulatives

Bedtime Math Foundation: Bedtime Math

BrainQuake Inc.: Wuzzit Trouble.

DragonBox: Algebra 5+, Algebra 12+, Elements

Educreations

Explain Everything

MicroBlink: PhotoMath

MIND Research Institute: BigSeed and KickBox

Motion Math: Zoom, Hungry Fish, and Fractions!

Notability

The Math Learning Center: Number Rack

Scholastic: Sushi Monster

TapTapBlocks

Wolfram Group LLC: Wolfram|Alpha

Games

Calculation Nation: http://calculationnation.nctm.org

Mathbreakers: www.mathbreakers.com

Minecraft: https://minecraft.net

Puzzle Books

Berlekamp, E., and T. Rodgers.
The Mathemagician and Pied
Puzzler: A Collection in Tribute to Martin Gardner.
Natick, MA: AK Peters, 1999.


Berlekamp, E. R., J. H. Conway, and R. K. Guy.
Winning Ways for
Your Mathematical Plays,
2nd ed. Wellesley, MA: AK Peters, 2001.

Bolt, B.
The Amazing Mathematical Amusement Arcade.
Cambridge, England: Cambridge University Press, 1984.


———.
Mathematical Cavalcade.
Cambridge, England: Cambridge University Press, 1992.

———.
A Mathematical Pandora’s Box.
Cambridge, England: Cambridge University Press, 1993.


———.
A Mathematical Jamboree.
Cambridge, England: Cambridge University Press, 1995.


Cornelius, M., and A. Parr.
What’s Your Game?
Cambridge, England: Cambridge University Press, 1991.


Gardner, M.
Riddles of the Sphinx: And Other Mathematical Puzzle Tales.
Washington, DC: Mathematical Association of America, 1987.


———.
Mathematical Puzzle Tales.
Washington, DC: Mathematical Association of America, 2000.


———.
The Colossal Book of Mathematics.
New York: W.W. Norton, 2001.


Moscovich, I.
Knotty Number Problems & Other Puzzles.
New York: Sterling, 2005.

Tanton, J.
Solve This: Math Activities for Students and Clubs.
Washington, DC: Mathematical Association of America, 2001.

Preface to the New Edition

1
. Keith Devlin,
What Is Mathematics? Introduction to Mathematical Thinking
(Spring 2014), https://www.coursera.org/course/maththink.

2
. Jo Boaler, “Research Suggests That Timed Tests Cause Math Anxiety,”
Teaching Children Mathematics
20, no. 8 (April 2014): 469–74. Available for download at www.youcubed.stanford.edu.

3
. Jo Boaler, “Ability and Mathematics: The Mindset Revolution That Is Reshaping Education,”
Forum
55, no. 1 (2013): 143–52. Available for download at www.youcubed.stanford.edu.

4
. Jo Boaler, “Fluency Without Fear: Research Evidence on the Best Ways to Learn Math Facts” (January 2014). Available for download at www.youcubed.stanford.edu.

5
. “How to Learn Math: For Students,” Stanford University, http://online.stanford.edu/course/how-to-learn-math-for-students-s14.

6
. Odula O. Abiola and Hakirat S. Dhindsa, “Improving Classroom Practices Using Our Knowledge of How the Brain Works,”
International Journal of Environmental & Science Education
7, no. 1 (January 2012): 71–81.

7
. “PISA 2012 Results,” OECD, http://www.oecd.org/pisa/keyfindings/pisa-2012-results.htm.

8
. Katherine Woollett and Eleanor A. Maguire, “Acquiring ‘the Knowledge’ of London’s Layout Drives Structural Brain Changes,”
Current Biology
21,
no. 24 (December 2011): 2109–14, http://www.cell.com/current-biology/abstract/S0960-9822(11)01267-X.

9
. Boaler, “Ability and Mathematics.”

10
. Jason S. Moser, Hans S. Schroder, Carrie Heeter, T. P. Moran, and Yu-Hao Le, “Mind Your Errors: Evidence for a Neural Mechanism Linking Growth Mind-set to Adaptive Posterror Adjustments,”
Psychological Science
22, (2011): 1484–89.

11
. http://online.stanford.edu/course/how-to-learn-math-for-students-s14.

12
. Elena Silva and Taylor White,
Pathways to Improvement: Using Psychological Strategies to Help College Students Master Developmental Math
. The Carnegie Foundation for the Advancement of Teaching, 2013.

13
. Lawyers’ Committee for Civil Rights of the San Francisco Bay Area (LCCR),
Held Back: Addressing Misplacement of 9th Grade Students in Bay Area School Math Classes
, January 2013, http://www.lccr.com/wp-content/uploads/Held_Back_9th_Grade_Math_Misplacement.pdf.

Introduction: Understanding the Urgency

1
. Programme for International Student Assessment (PISA),
Learning from Tomorrow’s World: First Results from PISA 2003
(Italy: Organisation for Economic Co-operation and Development [OECD], 2003).

2
. “in
chapter 2
. Higher Education in Science and Engineering,” National Science Foundation: Science and Engineering Indicators 2014, http://www.nsf.gov/statistics/seind14/index.cfm/chapter-2.

3
. “Science, Technology, Engineering, and Math: Education for Global Leadership,” US Department of Education, http://www.ed.gov/stem.

4
. MathAlive!
Math Relevance to U.S. Middle School Students: A Survey Commissioned by Raytheon Company
,
2012, http://www.mathmovesu.com/sites/default/files/Math-Relevance_rtn12_studentsmth_results_2012.pdf.

5
. Silva and White,
Pathways to Improvement
.

6
. Susan L. Forman and Lynn A. Steen,
Beyond Eighth Grade: Functional Mathematics for Life and Work
(Berkeley: National Center for Research in Vocational Education, University of California, 1999), 14–15.


7
. “Conrad Wolfram: Teaching Kids Real Math with Computers,” TED, http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers?language=en.

8
. Robert P. Moses and Charles E. Cobb Jr.,
Radical Equations: Math Literacy and Civil Rights
(Boston: Beacon Press, 2001).

9
. Lynn A. Steen, ed.,
Why Numbers Count: Quantitative Literacy for Tomorrow’s America
(New York: College Entrance Examination Board, 1997).

10
. Julie Gainsburg,
The Mathematical Behavior of Structural Engineers
(Stanford University, 2003),
Dissertation Abstracts International
A 64, no. 5: 34.

11
. Celia Hoyles, Richard Noss, and Stefano Pozzi, “Proportional Reasoning in Nursing Practice,”
Journal for Research in Mathematics Education
32, no. 1 (2001): 4–27.

12
. Richard Noss, Celia Hoyles, and Stefano Pozzi, “Abstraction in Expertise: A Study of Nurses’ Conceptions of Concentration,”
Journal for Research in Mathematics Education
33
,
no. 3 (2002): 204–29.

13
. Gainsburg,
Mathematical Behavior
, 36.

14
. “How to Learn Math: For Teachers and Parents,” Stanford Graduate School of Education, http://scpd.stanford.edu/instanford/how-to-learn-math.jsp.

15
. J. Lave, M. Murtaugh, and O. de la Rocha, “The Dialectical Construction of Arithmetic Practice,” in Barbara Rogoff and Jean Lave, eds.,
Everyday Cognition: Its Development in Social Context
(Cambridge, MA: Harvard University Press, 1984).

16
. Jean Lave,
Cognition in Practice: Mind, Mathematics and Culture in Everyday Life
(Cambridge, England: Cambridge University Press, 1988).

17
. Jo Masingila, “Learning from Mathematics Practice in Out-of-School Situations,”
For the Learning of Mathematics
13
,
no. 2 (1993): 18–22.

18
. Terezinha Nunes, Analucia D. Schliemann, and David W. Carraher,
Street Mathematics and School Mathematics
(New York: Cambridge University Press, 1993).

19
. Lave,
Cognition in Practice.

20
. Jo Boaler,
Experiencing School Mathematics: Traditional and


Reform Approaches to Teaching and Their Impact on Student Learning,

revised and expanded edition (Mahwah, NJ: Lawrence Erlbaum, 2002).


21
. Carolyn Maher, “Is Dealing with Mathematics as a Thoughtful Subject Compatible with Maintaining Satisfactory Test Scores? A Nine-Year Study,”
Journal of Mathematical Behavior
10, no. 3 (1991): 225–48.

22
. Boaler,
Experiencing School Mathematics.

23
. All the names of schools, teachers, and students in this book are pseudonyms. Schools that are involved in research studies are always promised complete anonymity, a requirement of University Institutional Review Boards.

1/What Is Math? And Why Do We
All
Need It?

1
. Dan Brown,
The Da Vinci Code
(New York: Doubleday, 2003).

2
. Margaret Wertheim,
Pythagoras’s Trousers: God, Physics, and the Gender Wars
(New York: W. W. Norton, 1997), 3–4.

3
. Boaler,
Experiencing School Mathematics.

4
. Keith Devlin,
The Math Gene: How Mathematical Thinking Evolved


and Why Numbers Are Like Gossip
(New York: Basic Books, 2000), 7.

5
. Patricia C. Kenschaft,
Math Power: How to Help Your Child Love Math, Even If You Don’t,
revised edition (Upper Saddle River, NJ: Pi Press, 2005).

6
. W. W. Sawyer,
Prelude to Mathematics
(New York: Dover, 1955), 12.

7
. Nicholas Fiori, “In Search of Meaningful Mathematics: The Role of Aesthetic Choice,” Stanford PhD dissertation, http://searchworks.stanford.edu/view/6969509.

8
. Simon Singh,
Fermat’s Enigma: The Epic Quest to Solve the World’s


Greatest Mathematical Problem
(New York: Anchor Books, 1997).

9
. Ibid.

10
. Ibid., xiii.

11
. Quoted in ibid., 6.

12
. Fran Schumer, “Jersey; In Princeton, 
Taking On Harvard’s Fuss About Women,”
New York Times,
June 19, 
2005.

13
. Imre Lakatos,
Proofs and Refutations: The Logic of Mathematical Discovery
(Cambridge, England: Cambridge University Press, 1976).

14
. W. H. Cockcroft,
Mathematics Counts: Report of Inquiry into the


Teaching of Mathematics in Schools
(London: Her Majesty’s Stationery 
Office, 1982).

15
. Robin Wilson, in Donald J. Albers, Gerald L. Alexanderson, and Constance Reid,
More


Mathematical People: Contemporary Conversations
(Boston: Harcourt 
Brace Jovanovich, 1990), 30.

16
. Devlin,
Math Gene,
76.

17
. George Pólya,
How to Solve It: A New Aspect of Mathematical Method
(New York: Doubleday Anchor, 1971), v.

18
. Leone Burton, “The Practices of Mathematicians: What Do They Tell Us About Coming to Know Mathematics?”
Educational Studies in Mathematics
37 (1999): 36.

19
. Peter Hilton, popular quote.

20
. Reuben Hersh,
What Is Mathematics, Really?
(New York: Oxford University Press, 1997), 18.

21
. Devlin,
Math Gene,
9.

22
. Jo Boaler, “Britain’s Maths Policy Simply Doesn’t Add Up,”
Telegraph
, August 14, 2014, http://www.telegraph.co.uk/education/educationnews/11031288/Britains-maths-policy-simply-doesnt-add-up.html.

23
. Fiori,
Practices of Mathematicians.

24
. Boaler,
Experiencing School Mathematics.

25
. David A. Reid,
Conjectures and Refutations in Grade 5 Mathematics
(doctoral dissertation, Indiana University, 2002).

26
. Gloria E. Keil,
Writing and Solving Original Problems as a Means of Improving Verbal Arithmetic Problem Solving Ability
(unpublished doctoral dissertation, Indiana University, 1965).


2/What’s Going Wrong in Classrooms? Identifying the Problems

1
. Alan H. Schoenfeld, “The Math Wars,”
Educational Policy
18, no. 1 (2004): 253–86.

2
. William L. Sanders and June C. Rivers,
Cumulative and Residual Effects of Teachers on Future Student Academic Achievement
(Knoxville: University of Tennessee, Value-Added Research and Assessment Center, 1996).

3
. S. Paul Wright, Sandra P. Horn, and William L. Sanders, “Teacher and Classroom Context Effects on Student Achievement: Implications for Teacher Evaluation,”
Journal of Personnel Evaluation in Education
11
,
no. 1 (1997):
57–67.

4
. Heather R. Jordan, Robert L. Mendro, and Dash Weersinghe,
Teacher Effects on Longitudinal Student Achievement: A Preliminary Report on Research on Teacher Effectiveness
(paper presented at the National Evaluation Institute, Indianapolis; Kalamazoo, MI: Western Michigan University, 1997).

5
. Linda Darling-Hammond, “Teacher Quality and Student Achievement: A Review of State Policy Evidence,”
Educational Policy Analysis Archives
8, no. 1 (2000).

6
. L. Rosen, “Calculating Concerns: The Politics of Representation in California’s ‘Math Wars,’” University of California, San Diego.

7
. Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, and William L. Cole,
Algebra: Structure and Method
(Evanston, IL: Houghton Mifflin, McDougal Littell, 2000), 1.

8
. Ibid., 3.

9
. Daniel Fendel, Diane Resek, Lynne Alper, and Sherry Fraser,
Interactive Mathematics Program
(Emeryville, CA: Key Curriculum Press, 1997), 189.

10
. Ibid., 194.

11
. Ibid., 203–4.

12
. Wayne Bishop, “Another Terrorist Threat,”
Math Forum,
March 11, 2003.

13
. “Professor Jo Boaler,” Stanford University, http://stanford.edu/~joboaler/.

14
. Scott Jaschik “Casualty of the Math Wars,” Inside Higher Ed, October 15, 2012, https://www.insidehighered.com/news/2012/10/15/stanford-professor-goes-public-attacks-over-her-math-education-research.

15
. “Programme for International Student Assessment (PISA),” OECD, http://www.oecd.org/pisa/.

16
. E. G. Johnson and N. L. Allen,
The NAEP 1990 Technical Report

(No. 21-TR-20), Washington, DC: National Center for Educational 
Statistics, 1992.

17
. Jo Boaler,
Experiencing School Mathematics: Teaching Styles,


Sex and Setting
(Buckingham, England: Open University, 1997).

18
. Thomas P. Carpenter, Megan L. Franke, and L. Levi,
Thinking Mathematically: Integrating Arithmetic & Algebra in the Elementary School
(
Portsmouth, NH: Heinemann, 2003).

19
. Sarah Flannery,
In Code: A Mathematical Journey
(Chapel Hill, NC: Algonquin Books, 2002), 38.

20
. Hersh,
What Is Mathematics, Really?
,
27.

21
. Jo Boaler and Jim Greeno, “Identity, Agency and Knowing in Mathematics Worlds,” in Jo Boaler, ed.,
Multiple Perspectives on Mathematics


Teaching and Learning
(Westport, CT: Ablex, 2000), 171–200.

22
. A. Murata, personal communication, 2006.

23
. Alan H. Schoenfeld, “Confessions of an Accidental Theorist,”
For the


Learning of Mathematics
7, no. 1 (1987): 37.

24
. Hilary Rose, “Reflections on PUS, PUM and the Weakening of Panglossian Cultural Tendencies,”
The Production of a Public Understanding


of Mathematics
(Birmingham, England: University of Birmingham, 1998), 4.

25
. Ibid., 5. 


3/A Vision for a Better Future: Effective Classroom Approaches

1
. “Program for Complex Instruction,” Stanford University, http://cgi.stanford.edu/group/pci/cgi-bin/site.cgi.