# Picture Writing

## Using Students' Pictorial Representations to Promote Mathematical Thinking

Your feedback is important to us!

After viewing our curriculum units, please take a few minutes to help us understand how the units, which were created by public school teachers, may be useful to others.

## Resources

Bibliography for Teachers

Bagnall, Bernard. "Children's Mathematical Writing."
*
NRICH
*
,
*
University of Cambridge,
*
Accessed March 31, 2014, http://nrich.maths.org/6296. An article outlining many reasons to record, as well examples that highlight need to listen to students' explanations.

"Bansho (Board Writing)."
*
Ontario Ministry of Education:
*
Capacity Building Series, Secretariat Special Edition #17, February 2011. Accessed June 8, 2014. http://www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/CBS_bansho.pdf. Resource explaining how to use bansho as an instructional strategy to allow students to build understanding of concepts and solve problems in ways that make sense to them.

Carruthers, Elizabeth, and Maulfry Worthington.
*
Children's Mathematics: Making Marks, Making Meaning.
*
London: Sage Publications, 2006. Explores development of young children's mathematical graphics, the connection between these marks and abstract symbols, and how they reveal deep levels of thinking.

Carpenter, Thomas P., Elizabeth Fennema, Megan Loef Franke, Linda Levi, and Susan B. Empson.
*
Children's Mathematics: Cognitively Guided Instruction
*
. Portsmouth, NH: Heinemann, 1999. Excellent resource to help to further understand children's mathematical thinking, as well as different types of story problems.

"Concrete-to-Representational-to-Abstract (C-R-A) Instruction."
*
University of Kansas Special Connections.
*
Accessed June 8, 2014. http://www.specialconnections.ku.edu/?=instruction/mathematics/teacher_tools/concrete_to_represenational_to_abstract_instruction. Overview of C-R-A Instruction.

Fong Ho Kheong, Chelvi Ramakrishnan, and Bernice Lau Pui Wah.
*
Math in Focus: Singapore Math Teacher's Edition, Book A, Grade 1.
*
Singapore: Marshall Cavendish Education. 2011. Teacher's guide for implementing the
*
MIF
*
curriculum.

"Mathematics Standards."
*
Common Core State Standards Initiative
*
. Accessed July 1, 2014. http://www.corestandards.org/Math/. This resource provides access to K-12 mathematical practice standards(
*
how
*
to teach content) and math content standards officially adopted by all states except AK, IN, MN, NE, OK, TX, & VA.

"Maximizing Student Mathematical Learning in the Early Years."
*
Ontario Ministry of Education:
*
Capacity Building Series, Special Edition #22, September 2011, 1-8. Accessed March 28, 2014. http://www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/cbs_maximize_math_learning.pdf. Resource for early mathematical learning that includes starting points for improving student learning and achievement, how to encourage "math talk" and practical tips for creating a mathematics-rich environment.

Saundry, Carole, and Cynthia Nicol. "Drawing as Problem-Solving: Young Children's Mathematical Reasoning Through Pictures." In Novotná, J., Moraová, M. and Stehliková, N. (eds.).
*
Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education,
*
5 (2006): 57-63. Prague: PME. Accessed May 25, 2014. http://www.emis.de/proceedings/PME30/5/57. Investigation of how young children think through mathematical problem solving.

Sheridan, Susan Rich. "A Theory of Marks and Mind: The Effect of Notational Systems on Hominid Brain Evolution and Child Development with an Emphasis on Exchanges Between Mothers and Children." In press,
*
Medical Hypotheses Journal
*
. http://www.drawingwriting.com/MMNota.pdf. Integrated research suggesting that human brain was spatial before it was linguistic.

————. "Scribbles." Submitted to
*
Medical Hypotheses Journal,
*
Leeds, UK., Accessed June 7, 2014. http://drawingwriting.com/Scribbs.pdf. Integrated research suggesting that literacy drove the elaboration of speech.

Smith, Stephen P. "Representation in School Mathematics: Children's Representations of Problems."
*
A Research Companion to Principles and Standards for School Mathematics,
*
2003, 263-274. Accessed May 25, 2014. http://393methods1.wikispaces.com/file/view/Smith.pdf. Chapter offers significant insight into children's idiosyncratic representations.

Stephens, Mitchell.
*
The Rise of the Image and the Fall of the Word
*
. New York: Oxford University Press, 1988. Explores birth of spoken language, markings, and writing.

Woleck, Kristine. "Listen to Their Pictures: An Investigation of Children's Mathematical Drawings."
*
The Roles of Representation in School Mathematics, 63rd Yearbook (2001)
*
, Chapter 17.
*
*
Accessed May 25, 2014. http://elem-math.wiki.educ.msu.edu/file/view/Woleck_ChildDrawings.pdf. One teacher's experience and research with students' mathematical drawings and the importance of listening and attention to what they represent.

"Words."
*
Radiolab
*
, Season 8, Episode 2. Aired August 10, 2010. Accessed May 20, 2014. http://www.radiolab.org/story/91725-words/. Includes interviews, research and stories pertaining to a life/world without/before words.

Worthington, Maulfry, and Elizabeth Carruthers. "Children's Mathematical Graphics in Early Education."
*
Children's Mathematics Network
*
. Updated July 21, 2014. Accessed July 23, 2014. http://www.childrens-mathematics.net/. Contains articles and resources that further explore bi-numeracy, the development of children's own invented symbolism, the difference between
*
modeling
*
and
*
examples
*
, galleries with children's mathematical graphics, etc.

Worthington, Maulfry, and Elizabeth Carruthers. "The Art of Children's Mathematics: The Power of Visual Representation." July 2005. Accessed April 1, 2014. http://www.childrens-mathematics.net/art_of_childrens_mathematics.pdf. Paper exploring the relationship between children's early drawing and their "mathematical graphics".

List of Materials for Classroom Use

- "Bansho": chart paper/butcher paper/blackboard/(interactive) whiteboard

- Paper: blank and graph paper of varying sizes, shapes and colors, notepads, etc.

- Writing utensils: pencils, crayons, colored pencils, markers, pens, clipboards, etc.

- Manipulatives: base 10 blocks, two-sided counters, 10 frames (blank and filled showing different quantities), snap cubes, unifix cubes, pattern blocks, color 1" tiles, beans (or other found items that can be used as counters), Cuisenaire rods, fraction bars, tangrams, etc.

- Other math tools: hundred chart, number line, calculators, clocks, etc.

- Materials for games: dice, decks of cards, etc.

- Technology to help record students' strategies: cameras, iPads, video cameras

- Anything else that you think could be used to promote mathematical thinking