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How Students Should be Taught Mathematics

01 Sep

After completing the online ‘How to Learn Math‘ class give by Prof. Boaler from Stanford University, I wanted to note down many valuable references, resources and 1-page summary provided, so that I can use them in the future. (You can read another blog entry related to this: Impressions after completing the ‘How to Learn Math’ class.)

"Please submit a word that, in your opinion, describes the most important aspect of a student's ideal relationship with mathematics."

“Please submit a word that, in your opinion, describes the most important aspect of a student’s ideal relationship with mathematics.”

1-page summary:

Mathematics classrooms should be places where students:

Develop an inquiry relationship with mathematics, approaching math with curiosity, courage, confidence & intuition.

Talk to each other and the teacher about ideas – Why did I choose this method? Does it work in other cases? How is the method similar or different to methods other people used?

Work on mathematics tasks that can be solved in different ways  and/or with different solutions.

Work on mathematics tasks with a low entry point but a very high ceiling – so that students are constantly challenged and working at  the highest and most appropriate level for them.

Work onmathematics tasks  that are complex, involve more than one method or area of mathematics, and that often, but not always, represent real world problems and applications.

Are given growth mindset messages at all times, through the ways they are grouped together, the tasks they work on, the messages they hear, and the assessment and grading.

Are assessed formatively – to inform learning -­‐ not summatively to give a rank with their peers. Students should regularly receive diagnostic feedback on their work, instead of grades or scores. Summative assessments are best used at the end of courses.

Mathematics classrooms should be places  where students believe:

Everyone can do well in math.

Mathematics problems can be solved with many different insights and methods.

Mistakes are valuable, they encourage brain growth and learning.

Mathematics will help  them in their lives, not because they will see the same types of problems in the real world but because they are learning to think quantitatively and abstractly and developing an inquiry relationship with math.

References

Session 1 References

Community College Performance

The Carnegie Foundation for the Advancement of Teaching. “Strengthening Pre-collegiate Education in Community Colleges: Project Summary and Recommendations.” A Report from Strengthening Pre-collegiate Education in Community Colleges (SPECC). Stanford, Calif.: The Carnegie Foundation for the Advancement of Teaching, 2008.

Stereotype Threat

Steele, Claude M. (2011).  Whistling Vivaldi:  How stereotypes affect us and what we can do (issues of our time). New York, NY:  W. W. Norton and Company.

Steele, C. M. (1997). A threat in the air. How stereotypes shape intellectual identity and performance. The American psychologist, 52(6), 613–629.

Spencer, S. J., Steele, C. M., & Quinn, D. M. (1999). Stereotype Threat and Women’s Math Performance. Journal of Experimental Social Psychology, 35(1), 4–28. doi:10.1006/jesp.1998.1373

Shih, M., Pittinsky, T. L., & Ambady, N. (1999). Stereotype susceptibility: Identity salience and shifts in quantitative performance. Psychological Science, 10(1), 80–83. doi:10.1111/1467-9280.00111

Ambady, N., Shih, M., Kim, A., & Pittinsky, T. L. (2001). Stereotype susceptibility in children: Effects of identity activation on quantitative performance. Psychological Science, 12(5), 385–390. doi:10.1111/1467-9280.00371

Perry, T., Steele, C., & Hilliard, A. (2003).  Young, gifted, and black:  Promoting high achievement among african-americans.  Boston, MA: Beacon Press.

Messages

Yeager, D. S., Purdie-Vaughns, V., Garcia, J., Apfel, N., Brzustoski, P., Master, A., et al. (2013). Breaking the Cycle of Mistrust: Wise Interventions to Provide Critical Feedback Across the Racial Divide. Journal of Experimental Psychology: General.

Cohen, G. L., Garcia, J., Apfel, N., & Master, A. (2006). Reducing the Racial Achievement Gap: A Social-Psychological Intervention. Science, New Series, 313(5791), 1307–1310.

Hulleman, C. S., & Harackiewicz, J. M. (2009). Promoting Interest and Performance in High School Science Classes. Science, 326(5958), 1410–1412. doi:10.1126/science.1177067

Geoffrey L. Cohen, Julio Garcia, Nancy Apfel, and Allison Master. Reducing the Racial Achievement Gap: A Social-Psychological Intervention. Science 1 September 2006: 313 (5791), 1307-1310. [DOI:10.1126/science.1128317]

Walton, G. M., & Cohen, G. L. (2007). A question of belonging: Race, social fit, and achievement. Journal of Personality and Social Psychology, 92(1), 82. doi:10.1037/0022-3514.92.1.82

A Mathematician’s Lament.

Lockhart, P. (2009). A mathematician’s lament. New York, NY : Bellevue Literary Press.

Session 2 References

Brain Plasticity

Abiola, O. O., & Dhindsa, H. S. (2011). Improving classroom practices using our knowledge of how the brain works. International Journal of Environmental & Science Education, 7(1), 71–81.

Woollett, K., & Maguire, E. A. (2011). Acquiring “the Knowledge” of London’s Layout Drives Structural Brain Changes. Current Biology, 21(24), 2109–2114. doi:10.1016/j.cub.2011.11.018

Maguire EA, Woollett K, Spiers HJ. (2006) London taxi drivers and bus drivers: a structural MRI and neuropsychological analysis. Hippocampus. 16(12):1091-101.

http://www.today.com/id/36032653/ns/today-today_health/t/meet-girl-half-brain/#.UeGbixbfvCE

Karni, A., Meyer, G., Rey-Hipolito, P., Adams, M., Turner, R., Ungerleider, L. (1998). The acquisition of skilled motor performance: Fast and slow experience-driven changes in primary motor cortex. Proc. Natl. Acad. Sci. USA Vol. 95, pp. 861–868, February 1998. Colloquium Paper

Mindset

Dweck, Carol S. (2007).  Mindset:  The new psychology of success.  New York, NY:  Random House.

Blackwell, L. S., Trzesniewski, K. H., & Dweck, C. S., (2007). Implicit theories of intelligence predict achievement across an adolescent transition: A longitudinal study and an intervention. Child Development, 78 (1). 246-263.

Dweck, C. (2009). Mindsets and equitable education. Educational Leadership.

Other Dweck references are available for download here:

http://www.stanford.edu/dept/psychology/cgi-bin/drupalm/cdweck

Session 3 References

Mistakes

Tugend, A. (2011). Better By Mistake. The Unexpected Benefits of Being Wrong. Riverhead Books: New York.

Moser, J.S., Schroder, H.S., Heeter, C., Moran, T.P., Lee, Y.-H., 2011. Mind your errors: evidence for a neural mechanism linking growth mindset to adaptive post-error adjustments. Psychological Science 22, 1484 – 1489.

Boaler Study of English Schools and Follow Up Study of Adults

Boaler, J. (2002). Experiencing School Mathematics: Traditional and Reform Approaches to Teaching and Their Impact on Student Learning. (Revised and Expanded Edition ed.). Mahwah, NJ: Lawrence Erlbaum Association.

Boaler, J. (1998). Open and Closed Mathematics: Student Experiences and Understandings. Journal for Research in Mathematics Education, 29(1), 41-62.

Boaler, J. (2012, 8-15 July). From Psychological Imprisonment to Intellectual Freedom – The Different Roles that School Mathematics Can Take in Students’ Lives. Paper presented at the 12th International Congress on Mathematical Education, Seoul, Korea.

Ray Peacock

In:

Boaler, J. (2009) What’s Math Got To Do With It?How Parents and Teachers Can Help Children Learn to Love Their Least Favorite Subject. Penguin: New York.

&

Boaler, J. (2010). The Elephant in the Classroom. Helping Children Learn & Love Maths. Souvenir Press: London.

&

Boaler, J. (2011). Elefanten i klassrummet: Att hjälpa elever till ett lustfyllt lärande i matematik [The elephant in the classroom: Helping children to a joyful learning in mathematics] (E. Trägårdh, Trans). Stockholm, Sweden: Liber.

Didactic Contract

Brousseau, G. (1997). Theory of didactical situations in mathematics (N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield, Eds.). Dordrecht, Netherlands: Kluwer Academic Publishers.

Timed Tests

Ashcraft, M. (2002). Math Anxiety: Personal, Educational and Cognitive Consequences. Current Directions in Psychological Science.  11, 5, 181-185.

Beilock, S. (2011). Choke: What the Secrets of the Brain Reveal about Getting it Right When you have To. Simon & Schuster, Free Press: New York.

Beilock, S., Gunderson E., Ramirez, G., & Levine, S. (2009). Female Teachers’ Math Anxiety Affects Girls Math Achievement. Proceedings of the National Academy of Sciences February 2, 2010, Vol 107, 5, 1860-1863.19-23.

Boaler, J. (July 3, 2012). Timed tests and the development of math anxiety. Education Week.

Faust, M.W. (1992). Analysis of Physiological Reactivity in Mathematics Anxiety. Unpublished doctoral dissertation. Bowling Green State University, Bowling Green, Ohio.

Hembree, R. (1990). The Nature, Effects, and Relief of Mathematics Anxiety. Journal for Research in Mathematics Education, 21, 33-46.

Lyons, I. and Beilock, S. (2011)  Mathematics anxiety: separating the math from the anxiety.  Cereb Cortex 22: 2102 – 2110.

Ramirez, G., Gunderson, E. A., Levine, S. C., & Beilock, S. L. (in press). Math anxiety, working Memory and Math Achievement in Early Elementary School. Journal of Cognition and Development.

Richardson, F. C, & Woolfolk, R.L. (1980). Mathematics Anxiety. In I.G. Sarason (Ed), Test anxiety: Theory, research and application (pp271-288). Hillsdale NJ:Erlbaum.

Young, C.B., Wu, S.S. & Menon, V. (2012).  The Neurodevelopmental Basis of Math Anxiety. Psychological Science Online First. March 20, 2012. doi:10.1177/0956797611429134

Number Sense

Boaler, J. (2009) What’s Math Got To Do With It? How Parents and Teachers Can Help Children Learn to Love Their Least Favorite Subject. Penguin: New York.

Gray, E. and D. Tall. (1994). Duality, Ambiguity, and Flexibility: A “Proceptual” View of Simple Arithmetic. Journal for Research in Mathematics Education, 1994. 25 (2), 116-140.

Seeley, Cathy (2009). Faster isn’t Smarter  (2009). Sausalito, CA:Math Solutions Publications

Some Good wesbites and apps to help with number Sense (recommened by Dan Meyer):

Motion Math

Conceptua Math

Buzz Math

ST Math

Dragon Box

Refraction

Laurent Schwartz:

Un mathématicien aux prises avec le siècle. Paris 1997, Autobiography, English translation: A mathematician grappling with his century. Birkhäuser, 2001

Session 4 References

Cathy Humphrey’s Teaching:

Boaler, J., & Humphreys, C. (2005).  Connecting mathematical ideas: middle school video cases to support teaching and learning. Portsmouth, NH: Heinemann.

Mason, J., Burton, L. & Stacey, K. (1982). Thinking mathematically. London: Addison-Wesley.

Assessment

Black, P., & Wiliam, D. (1998). Inside the Black Box: Raising Standards through Classroom Assessment. Phi Delta Kappan, October, 139-148.

Butler, R. (1988). Enhancing and Undermining Intrinsic Motivation: The Effects of Task-Involving and Ego-Involving Evaluation on Interest and Performance. British Journal of Educational Psychology, 58, 1-14.

White, B., & Frederiksen, J. (1998). Inquiry, modeling and metacognition: making science accessible to all students. Cognition and Instruction, 16(1), 3-118.

Ability Grouping

Burris, C.C., J.P. Heubert and H.M. Levin (2006), “Accelerating Mathematics Achievement Using Heterogeneous Grouping”, American Educational Research Journal, Vol. 43, No. 1, pp. 105-136.

Boaler, J. (2009) What’s Math Got To Do With It?How Parents and Teachers Can Help Children Learn to Love Their Least Favorite Subject. Penguin: New York.

Boaler, J. (2010). The Elephant in the Classroom. Helping Children Learn & Love Maths. Souvenir  Press: London

Boaler, J. (2012, 8-15 July). From Psychological Imprisonment to Intellectual Freedom – The Different Roles that School Mathematics Can Take in Students’ Lives. Paper presented at the 12th International Congress on Mathematical Education, Seoul, Korea.

Nunes, T., Bryant, P., Sylva, K., & Barros, R. (2009).  Development of maths capabilities and confidence in primary school.  London: Department for Children, Schools and Families.

NRC.  (2004).  Engaging Schools: Fostering High School Students’ Motivation to Learn. Rosenthal, R.; Jacobson, L. (1968). Pygmalion in the classroom. New York: Holt, Rinehart & Winston.

Dixon, A. (2002) Editorial, FORUM, 44(1), pp. 1.

Kutnick, P., Sebba, J., Blatchford, P., Galton, M. (2005) The Effects of Pupul Grouping. A Literature Review. Department for Education and Skills: London. Research Report, number 688.

 Maggie Lampert:

Lampert, M. (2001). Teaching problems and the problems of teaching. New Haven: Yale University Press.

Michelle Obama:

Japanese Math Video referenced, from the TIMSS study:

http://timssvideo.com/67

Mindset Websites:

http://mindsetonline.com

http://www.mindsetworks.com

Task Makeovers from Dan Meyer:

http://blog.mrmeyer.com/?p=17162

Session 5 References

Gray, E. M., & Tall, D. D. (1994). Duality, ambiguity, and flexibility: A “proceptual” view of simple arithmetic.  Journal for Research in Mathematics Education, 25, 116–146.

Slavit, D. (1998). The role of operation sense in transitions from arithmetic to algebraic thought.

Educational Studies in Mathematics, 37, 251–274.

Thurston, W.  (1990).  Mathematical Education.  Notices of the AMS 37, 844–850.

Kieran, C. (2013). The false dichotomy in mathematics education between

conceptual understanding and procedural skills: An example from algebra.

In K.R. Leatham (Ed.), Vital directions for mathematics education research

(pp. 153-171). New York: Springer.

Kiernan, C.  (1992).  ‘The learning and teaching of school algebra,’ in D. Grouws (ed.), The Handbook of Research on Mathematics Teaching and Learning. Macmillan, New York, 390-419.

Sawyer, W. W. (1964). Mathematician’s delight. New York: Penguin Book.

Number Talks:

Boaler, J. (2009). What’s Math Got To Do With It?: How Parents and Teachers Can Help Children Learn to Love their Least Favorite Subject.  New York: Penguin.

Harris, P.W. (2001). Building (Powerful) Numeracy for Middle and High school students. Portsmouth NH: Heinemann:

Parker, R. (1993). Mathematical Power: Lessons from a Classroom. Portsmouth NH: Heinemann:

Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies, grades K-5. Sausalito, CA: Math Solutions Publications.

Parker, R. (1993). Mathematical power: Lessons from a classroom. Portsmouth, NH: Heinemann.

Mentioned websites/apps

Wolfram Alpha.

http://www.wolframalpha.com

http://doodlecastpro.com

http://www.airsquirrels.com/reflector/

Session 6 References

Pólya, G. (1957). How to solve it: A new aspect of mathematical method (2d ed.). Garden City, N.Y.: Doubleday.

Fullilove, R. E., & Treisman, P. U. (1990). Mathematics achievement among African American undergraduates at the University of California, Berkeley: An evaluation of the Mathematics Workshop Program. Journal of Negro Education, 59 (30), 463-478.

Burton, L.: 1999, ‘The Practices of mathematicians: What do they tell us about coming to know mathematics?’, Educational Studies in Mathematics 37, 121–143.

Hersh, R.  (1999).  What is Mathematics, Really?  New York: Oxford University Press.

Wilder, R.  (1984).  ‘The Role of Intuition,’ in Campbell, D. & Higgins, John (eds).  Mathematics.  People.  Problems.  Results: Volume ii.  Belmont, CA: Wadsworth.

Davis, P., R. Hersh, L. Steen, & William C. Waterhouse.  (1983) The Mathematical Experience and Mathematics Tomorrow.  Physics Today 36.11: 76.

Davis, P., R. Hersh, R. (1999). The Mathematical Experience. Mariner Books.

Session 7 References

Algebra:

Confrey, J. (1998, September). What do we know about K-14 students’ learning of algebra. In The nature and role of Algebra in the K-14 curriculum: Proceedings of a national symposium (pp. 37-40). National Academies Press.

Hacker, A. (2012). Is Algebra Necessary? New York Times.

Moses, R., & Cobb, C. E. (2002). Radical equations: Civil rights from Mississippi to the Algebra Project. Beacon Press.

Kieran, C. (1992). The learning and teaching of school algebra. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning. New York, NY: Macmillan.

Kieran, C. (2013). The false dichotomy in mathematics education between conceptual understanding and procedural skills: An example from algebra. In K.R. Leatham (Ed.), Vital directions for mathematics education research (pp. 153-171). New York: Springer.

http://blogs.oregonstate.edu/smedcohort2010/files/2011/02/Smith-Hillen-Catania_2007.pdf

Smith, M. S., Hillen, A. F., & Catania, C. L. (2007). Using Pattern Tasks to Develop Mathematical Understandings and Set Classroom Norms. Mathematics Teaching in the Middle School, 13(1), 38–44.

http://blogs.oregonstate.edu/smedcohort2010/files/2011/02/Smith-Hillen-Catania_2007.pdf

Küchemann, D. (1978). Children’s Understanding of Numerical Variables. Mathematics in School, 7(4), 23–26.

Kuchemann, D.: 1981, ‘Algebra’, in K. Hart (ed.), Children’s Understanding of Mathematics: 11-16, Murray, London, pp. 102–119.

Mason, J., Graham, A., & Johnston-Wilder, S. (2012). Developing Thinking in Algebra. Sage.

Confrey, J. (1998, September). What do we know about K-14 students’ learning of algebra. In The nature and role of Algebra in the K-14 curriculum: Proceedings of a national symposium (pp. 37-40). National Academies Press.

Moses, R., & Cobb, C. E. (2002). Radical equations: Civil rights from Mississippi to the Algebra Project. Beacon Press.

Kieran, C. (1981). Concepts associated with the equality symbol. Educational studies in mathematics, 12(3), 317-326.

Kozol, J. (2012). Savage inequalities: Children in America’s schools. Random House Digital, Inc.

Harvard Calculus:

http://www.math.harvard.edu/~knill/pedagogy/harvardcalculus/

Bell, A. W., Malone, J. A., & Taylor, P. C. (1989). Algebra-an exploratory teaching experiment. Shell Centre for Mathematical Education, University of Nottingham.

Driscoll, M. (1999). Fostering Algebraic Thinking: A Guide for Teachers, Grades 6-10. Heinemann, 361 Hanover Street, Portsmouth, NH 03801-3912.

RAND, M. S. P. (2002, October). Mathematical proficiency for all students: Toward a strategic research and development program in mathematics education (DRU-2773-OERI). Arlington, VA: RAND Education & Science and Technology Policy Institute.

Jo’s summer school teaching in:

Boaler, J. (2009). What’s Math Got To Do With It?: How Parents and Teachers Can Help Children Learn to Love their Least Favorite Subject.  New York: Penguin.

Mathematics Assessment Resource Service

Sources for Algebra Tasks:

Driscoll, M. (1999). Fostering Algebraic Thinking: A Guide for Teachers, Grades 6-10. Heinemann, 361 Hanover Street, Portsmouth, NH 03801-3912.

The DVD and book, “Thinking mathematically: Integrating arithmetic and algebra in elementary school” by T.P. Carpenter, M.L. Franke, and L. Levi (2003); Portsmouth, NH: Heinemann.

Kieran, C. (2013). The false dichotomy in mathematics education between conceptual understanding and procedural skills: An example from algebra. In K.R. Leatham (Ed.), Vital directions for mathematics education research (pp. 153-171). New York: Springer.

http://www.utdanacenter.org/k12mathbenchmarks/tasks/tasks.php

Session 8 References

Rose, Hilary (1998). Reflections on PUS, PUM and Weakening of Panglossian Cultural Tendencies. The Production of a Public Understanding of Mathematics. Birmingham, England: University of Birmingham, p4

Boaler, J. (1993). The role of contexts in the mathematics classroom: do they make mathematics more “real”? For the Learning of Mathematics, 13(2), 12-17.

Boaler, J. (2009). What’s Math Got To Do With It?: How Parents and Teachers Can Help Children Learn to Love their Least Favorite Subject.  New York: Penguin.

Dan Meyer on contexts:

(blog.mrmeyer.com/?p=8002), (http://blog.mrmeyer.com/?p=17162), (http://threeacts.mrmeyer.com/).

Grouws, D., Tarr, J., Chávez, O., Sears, R., Taylan, D., & Soria, V.  (2013).  Curriculum and Implementation Effects on High-School Students’ Mathematics Learning from Curricula Representing Subject-Specific and Integrated Content Organizations.  Journal for Research in Mathematics Education, 44(2), 416-463.

Tarr, J. E., Grouws, D. A., Chávez, Ó., & Soria, V. M. (2013). The Effects of Content Organization and Curriculum Implementation on Students’ Mathematics Learning in Second-Year High School Courses. Journal for Research in Mathematics Education, 44(4), 683-729.

Jo’s research:

Boaler, J. (2009). What’s Math Got To Do With It?: How Parents and Teachers Can Help Children Learn to Love their Least Favorite Subject.  New York: Penguin.

Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings. Journal for research in mathematics education, 41-62.

Boaler, J. (2002). Experiencing School Mathematics: Traditional and Reform Approaches To Teaching and Their Impact on Student Learning, Revised and (Hardback)(Series: Studies in Mathematical Thinking and Learning Series).

Boaler, J. (2005, September). The ‘Psychological Prisons’ from which they never escaped: The role of ability grouping in reproducing social class inequalities. In Forum (Vol. 47, No. 2, pp. 135-144). Symposium Journals.

Papers available for download from:

www.joboaler.com

Flannery, S. (2000). In code: A mathematical journey. Algonquin Books.

Advice for parents:

http://yano.co.uk/2012/05/dont-let-maths-muddle-you-2/

http://www.racetonowhere.com

A selection of Keith Devlin’s books:

Devlin, K. J. (2000). The math gene: How mathematical thinking evolved and why numbers are like gossip. New York: Basic Books.

Devlin, K. (1996). Mathematics: the science of patterns: the search for order in life, mind and the universe. Macmillan.

Devlin, K. (2000). The language of mathematics: making the invisible visible. Macmillan.

Jo’s response to the attacks on her work:

http://stanford.edu/~joboaler/

Resources for mathematics puzzles.

Flannery, S.  (2002), In Code: A Mathematical Journey. Algonquin Books.

Boaler, J. (2009) What’s Math Got to Do With It? Penguin: New York. Appendix A

Puzzle books, in order of difficulty.

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

Bolt, B. (1992) Mathematical Cavalcade.  Cambridge, England: Cambridge University Press.

Bolt, B. (1993) A Pandora’s Mathematical Box. Cambridge, England: Cambridge University Press.

Bolt, B. (1995) A Mathematical Jamboree. Cambridge, England: Cambridge University Press.

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

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

Berlekamp, E. & Rodgers, T. (1999) The Mathematician and Pied Puzzler: A collection in tribute to Martin Gardner. Natick, Mass: A K Peters.

Gardner, M (1987). Riddles of the Sphinx: And other mathematical puzzle tales. Washington DC: Mathematical Association of America

Gardner, M (2000). Mathematical puzzle tales. Washington DC: Mathematical Association of America

Gardner, M (2001). The Colossal Book of Mathematics. New York: W.W Norton

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

Bergekamp, E. R., Conway, J.H & Guy, R.K. (2001). Winning Ways for your Mathematical Plays. 2nd ed. Wellesley, Mass: AK Peters

 

 
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Posted by on September 1, 2013 in CogSci, Math, psychology

 

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