Early mathematical thinking

Louise Hodgson 2014

Outline of presentation

• Background• Mathematical processes• Role of the Early Childhood Educator• Mathematising• Mathematising in a play based curriculum

Pre school children’s knowledge of mathematics predicts their later school success…it predicts later reading achievement…and predicts College science achievement across subjects.

The quantitative , spatial, and logical reasoning competencies of mathematics may form a cognitive foundation for thinking and learning across subjects.

Clements and Sarama 2011.

All children need robust knowledge of mathematics in their earliest years, however not all children have opportunities …

Some 6 yr olds have not acquired the mathematical knowledge that other children acquire at 3 years of age

Children from low income families less able to explain mathematical ideas and processes.

Clements and Sarama 2011

• Mathematical processes - problem solving, reasoning and proof, communication and connections and habits of mind such as curiosity, imagination and persistence, collectively are as important as mathematical content in a high quality early childhood mathematics program.

Clements and Sarama 2009

EYLF Outcome 4: Children are confident and involved learners.

Children develop dispositions for learning such as curiosity, cooperation, confidence, creativity…, enthusiasm, persistence, ... Children develop a range of skills and processes such as problem solving, inquiry,… hypothesising, researching and investigating”. (EYLF, 2009)

EYLF Outcome 5: Children areeffective communicators.

Spatial sense, structure and pattern, number, measurement, data, argumentation, connectionsand exploring the world mathematically are thepowerful mathematical ideas children need to become numerate. (EYLF, 2009 p38)

Early childhood teachers often believe they are “doing mathematics” when they provide puzzles, blocks and counting songs. Even when they teach mathematics, the content is not the main focus but usually embedded in a fine motor or reading activity…Such an approach is ineffective, owing to lack of explicit attention to mathematical concepts.

Clements and Sarama 2011; Chen and McCray 2014

Early Childhood Teachers need:

To understand

1. What to teach (content)

2. Whom to teach (developmental pathways)

3. How to teach (strategies for supporting thinking about the content)

Teachers need integrated knowledge of all three components. Clements and Sarama 2011; Chen and McCray 2014

The notion of play

• Important to be clear about the notion of play, the relationship between the playing and learning and position of adults towards play.

• What is your understanding of play in a play based curriculum?

Play based curriculums

• Play is not some Laissez-faire event• The teacher constructs the curriculum in

close interaction with the children, informed by children’s interests and the teachers mandatory goals.

Intentional pedagogy in mathematics permeates all areas in a classroom throughout the day. The term “mathematising” describes this approach.

“Mathematising” refers to the process of taking an everyday situation or problem in daily life and framing it in mathematical terms.

“Mathematising involves children in the construction and re-construction processes of mathematical reasoning, problem solving, … and communication”Chen and McCray, 2014

Mathematising

• “Mathematising is the basis of mathematical thinking that underlies both operational-procedural thinking and mathematical problem solving. Helping young children to appropriate strategies of mathematising is a core element of the stimulation of their mathematical literacy”

Van Oers 2013

Mathematising

• Children must learn to mathematise their informal experiences by abstracting, representing and elaborating them mathematically. If they do not, they miss the opportunity to learn the language of mathematics in all its multifaceted forms.

Clements and Sarama 2011

In other words…

As educators, we support young children’s mathematical thinking and development when we purposefully and intentionally organise children’s playful experiences for mathematical purposes.

This process is called“mathematising.”

Mathematical thinking

“Playful activity for young children is the interactional niche for the development of mathematical thinking”• “Mathematical thinking should start out as

mathematics in play (rather than direct instruction on elementary maths operations) and then be fostered into mathematics as play”

Van Oer, 2013

What does this mean for our practice?

Mathematising in a play based curriculum

In a genuine play based curriculum; • mathematising is provoked and

encouraged as a way of dealing (collaboratively) with the quantitative and spatial dimensions of reality

• Play is the format for all student activities• Children work out solutions to problems

collaboratively under the guidance of a teacher or more knowledgable peer

• Van Oers, 2013

Mathematising in a play based curriculum

Effective teachers:• Ask questions that encourage children to

reflect on their actions• Ask for arguments that explain their ideas

“Are you sure?”

“Can you convince us”

…contributes to the sense of the questions and enhances children’s ability to participate more successfully with others.

Van Oers 2013

Mathematising in a play based curriculum

• To support children develop mathematical reasoning and a mathematical space for focussed communication, the following mediums are helpful

• Picture books• Gestures• Schematic representations

Schematic representations

• Children produce more sophisticated schematic representations of quantities and their relationships as long as it is meaningful to the children and in the context of their play.

• Van Oers, 2013

Picture books in mathematics learning

Picture books in mathematics teaching enable children to encounter problematic situations, ask questions, search for answers, consider different points of view, exchange views with others and guide them toward higher levels of proficiency.

Van den Heuvel-Panhuizen, 2013

Picture books can have a dual function…

1.An informal and spontaneous activity that children engage in

2.An intentional activity organised and directed by the teacher

Mathematising story books 

Draw the people in your family from the shortest to the tallest.

What if some teddies fell out? Draw a picture to show how many teddies were in the bed and how many fell out. Can you find all the ways?

What will happen after 7 more sleeps?

“What would it be like if there weren’t any numbers in our world?”

A four year olds response

How can we foster a culture of mathematising in our early

childhood classrooms?

Role of the early childhood educator

• Model mathematical language.• Ask probing questions.• Build on children’s interests and

natural curiosity.• Provide meaningful experiences.• Scaffold opportunities for learning

& model strategies.• Monitor children’s progress and

plan for learning.

Role of the early childhood educator

• “It is important in a play based curriculum that the teacher has the curriculum requirements for mathematics learning always in mind, when collaborating in children’s projects. With this content in mind he/she should decide in advance which of these should be explored and practiced with children in the context of their play” FIJMA 2012

Mathematise teachable moments as they occur

“I wonder if we’ll have enough cupcakes for everyone to share”

Bobby Bear NCTM Illuminations

“Can you find all the different ways to dress the teddies Charlotte”?

Mathematics in play“How many cups would you need to make a triangle building with four

levels?”

Play spacesRole play (home, shoe shop, bank, post office, doctor, restaurant, supermarket)

Construction (blocks, tracks, linking materials)

Display area (peg line, pinboards, magnet board)

Play trays (sand, water, multiple objects e.g. buttons, pasta, shells, leaves)

Mini-worlds (story/drama,cloth or sand tray environments, toy animals, people, vehicles

Play spacesModelling & painting

Graphics (drawing,writing, recording,shapes)

Reading and listening

areas (story-telling,picture books, rhymes,songs, CDs, music &percussion

“How many milk cartons do we have to build our reading igloo?”

(ACMNA002)

Challenges

• The elaboration of play as a context for meaningful mathematics learning

• The tension between instruction and construction

Mathematics as PlayMy friend Louise has an apple stall at the market. She sells her apples in bags of ten. The other day she had a box of thirty four apples. How many bags of apples could she put on her stall? (ACMNA014)

Somebody told me the calculator can count for us. Lets investigate what happens when we press 1+1===

Mathematics as Play

“Your job at the factory is to put chocolates into bags and send them to the shops. Each bag has exactly ten chocolates in it. At the end of the day you have to draw a picture to show how many chocolates you have packed”. (ACMNA014)

“The future of mathematical thinking in young children strongly depends on the quality of early years teachers to recognise mathematical actions in children, to see the potential of play activities and play objects and to guide children into the future where they can participate creatively and autonomously in mathematical communications”

Van Oers, 2013

References• Chen, J. Q., & McCray, J. (2014). Intentional Teaching: Integrating the

Processes of Instruction and Construction to Promote Quality Early Mathematics Education. In Early Mathematics Learning (pp. 257-274). Springer New York.

• Clements, D. H., & Sarama, J. (2011). Early childhood mathematics intervention. Science, 333(6045), 968-970.

• Fijma, N. (2012). Learning to communicate about number. In Developmental Education for Young Children (pp. 253-269). Springer Netherlands.

• Rudd, L. C., Lambert, M. C., Satterwhite, M., & Zaier, A. (2008). Mathematical language in early childhood settings: What really counts? Early Childhood Education Journal, 36(1), 75-80

• Sarama, J., & Clements, D. H. (2009). Early childhood mathematics education research: Learning trajectories for young children. Routledge.

• Van den Heuvel-Panhuizen, M., & Elia, I. (2013)The role of picture books in young children’s mathematics learning. In Reconceptualizing Early Mathematics Learning (pp. 183-203). Springer Netherlands.

• Van Oers, B. (2013). Communicating about number: Fostering young children’s mathematical orientation in the world. In Reconceptualizing Early Mathematics Learning (pp. 183-203). Springer Netherlands.

• Van Oers, B. (2013). Challenges in the innovation of mathematics education for young children. Educational Studies in Mathematics, 84(2), 267-272.