Play + Connections

According to Askew (1999), having a "connectionist orientation" (p. 98) towards teaching numeracy was what distinguished a group of highligh effective teachers of numeracy from other less effective teachers of numeracy.

Here is my list of important principles of a connectionist orientation towards numeracy teaching based on ideas discussed by Askew (1999).

Teachers ought to...

1. Have a consistent and coherent set of beliefs with regards to their mathematics teaching
2. Understand and teach children to understand the connections or relationship between different components of mathematics such the inverse relationship between addition and subtraction, and between multiplication and division, as well as between different strands of mathematics, e.g. number and measurement
3. Use differnt ways of representing mathematical concepts, but in a way which shows how each representation connections to one another - e.g. explore differnt ways of representing the concept of 1, differnt strategies of calculating 44 +89
4. Observe, value and find interest in understanding children's thinking including the processes they go through before arriving a final answer
5. Have a deep understanding of numeracy - by paying attention to efficiency and effectivenss of strategise being applied to a variety of mathematical contexts

References for the Play + ? posts

Here are my references:

Ailwood, J. (2003). Governing early childhood education through play. Contemporary Issues in Early Childhood, 4(3), pp. 286 299

Askew, M. (1999). Issues in teaching numeracy in primary schools. Buckingham: Open University Press.

Bragg, L. A. (2006). “Hey, I’m learning this.” Australian Primary Mathematics Classroom, 11(4), 4-9

Cutler, K., Gilkerson, D., Parrott, S., and Browne, M. (2003). Developing math games. Young Children, 58, 22-27

Dockett, S., and Fleer, M. (1999) Play and pedagogy in early childhood: bending the rules, Sydney: Harcourt

Brace Haynes, M. (2000). Mathematics education for early childhood: a partnership of two curriculums. Mathematics Teacher Education and Development, 2, 93-104

Perry, B. & Dockett, S. (2002). Ch 5: Young children’s access to powerful mathematical ideas. In L. D. English (ed). Handbook of international research in mathematical education. Mahwah, N.J.: Lawrence Erlbaum Associates

Queensland Studies Authority (QSA). (2006). Early Years Curriculum Guidelines. Brisbane: Queensland Studies Authority.

Waite-Stupiansky, S. and Stupiansky, N. G. (1999). Games that teach: spice up winter days by reinforcing math skills with challenging games. Instructor, 108(5), 16-18

Appropriate resources + play

As mentioned above, games are a useful resource to engage children’s interests in maths learning. According to Bragg (2006), in spite of games being seen as warm-up activities that take place before actual learning, if used properly, games can in fact be used to constitute a central part of a mathematics lesson as they build positive environments for learning, enhance students’ motivation and self esteem towards mathematics, promote mathematical learning, stimulates mathematical discussion and interactions.

However, I also believe that it is important to select the right tool for the job. Therefore in my observations of children's play I believe that I should take special care to understand their thinking and reasoning in order to make appropriate and careful assessments of children's current level of mathematical understanding. I should note what their interests are, what they are able to do, in addition to the misconceptions they might hold.

My principles for choosing resources includes avoiding choosing resources or using them in a way that:
• There is no clear or explicit mathematical concept being practiced or taught
• Are mathematically incorrect, and confuse children and lead them to develop misconceptions
• Are developmentally inappropriate for children’s learning
• Are quite frankly, very boring
• Can only done one way, with one right answer, found using one strategy

Rather I should choose good resources and use them in a way that:
• Promotes interactions and mathematical discussions using appropriate mathematical language
• Encourages mathematical thinking and different strategies to be applied to reach conclusions – and that children are asked to justify their positions and transfer mathematical knowledge from one context to another
• Is of interest and meaningful to the children
• Makes explicit links to specific mathematical concepts and aide children’s knowledge development and understanding in these areas

For this reason I’ve taken some games that I have seen children enjoy and turned them into math games. My favourite among these are number guess who and 100-chart battleships because it necessitates turn-taking which promotes interactions, can be changed to increase the level of difficulty, have clear mathematical purposes and foci and because well...they’re fun (See my maths resources post for photographs and further explanation of these resources).

Challenge + children’s interests + play

Cutler, Gilkerson, Parrott and Browne (2003) highlight the importance of establishing rich and meaningful environments that allow children to explore mathematical concepts through play. I believe that meaningful environments for children must take into account their interests as it encourages them to engage in learning activities and persist through difficulties they may encounter, and persisting through difficulties is usually how anyone learns anything. Furthermore, Perry & Dockett (2002) lists “not knowing” and “wanting to know” (p.98) as two conditions that tends to exist simultaneously, which motivate humans to learn. I believe the idea of a challenge connotes children not knowing something, while the notion of children’s interests motivates children to develop a wanting to know.

So how can teachers facilitate challenge? Firstly by believing that children are capable learners and can rise up to challenges (Perry & Dockett, 2002) and secondly, by knowing the developmental sequence of mathematical concepts (Cutler, Gilkerson, Parrott & Browne, 2003). For example, if a teacher knows that counting all as an addition strategy tends to occur and generally needs to be understood before moving onto the more difficult strategy of counting-on (NSW Department of Education and Training, 2005), we know to challenge children who understand how to perform addition using counting-all strategy to try the counting-on strategy in order to improve efficiency and deepen mathematical understanding.

What about facilitating children’s interests? In a way, by providing play situations, children’s interests emerge. In early childhood meaningful environments for learning are created through teachers first paying attention to the subject of children’s curiosity through observing children’s thinking and questioning, then drawing out mathematical understandings relevant to those interests (Perry & Dockett, 2002). I believe this can become more difficult as the mathematical concepts become more complicated, but nevertheless it is not a reason for teachers to resort to boring worksheets. According to Waite-Stupiansky and Stupiansky (1999), playing maths-games can “take the drudgery out of practicing [and learning] math and add the challenge of thinking more efficiently”, improve students’ mathematical abilities and increase their excitement with regards to learning maths.

Warm and responsive relationships + dialogue + play

According to Perry and Dockett (2002), warm and responsive relationships are important to facilitate play and results in learning. Through the establishment of such relationships, teachers assume “the role of provocateur” (p. 98) by posing questions, including elements of surprise, asking children to mathematically justify reasons for their positions, encouraging collaboration, and reasoning with children by making explicit “the logical consequences of the positions [children] adopt” (p.98). Mathematical concepts need to be taught explicitly (Haynes, 2000). However, that does not necessarily mean taking a transmissive approach, which conceptualise children as being passive learners – empty vessels that receive knowledge which teachers pour into them. Instead, by taking a social-constructivist approach, we view children as active participants in their own learning and constructors of meaning and social interactions as being integral for optimising such meaning-making (Perry & Dockett, 2002). Through warm and responsive relationships we can provoke children to pose and investigate mathematical problems as well as discuss mathematical ideas, hypotheses, strategies and understandings using mathematical reasoning, in a way which makes mathematical learning experiences fun and enjoyable.

Teachers’ sound mathematical content knowledge + play

Hayes (2000) states that "to optimise the potential for the development of concepts emerging through play, teachers also need sound content knowledge of [mathematical] concepts... in order to address the 'what' in teaching" (p.99). In other words, simply allowing children to play and trusting that that mathematical concepts will arise as a result of children engaging in play is not enough. This is because it does not guarantee that children will develop coherent and systematic understandings and appreciation of these mathematical concepts. Teachers need to know mathematical concepts well in order to first, recognise mathematics learning opportunities created through children's play and second, scaffold children's mathematical knowledge construction. So, what kind of mathematical thinking can be created in play situations? Hayes (2000) suggests a few examples:

Geometric thinking – e.g. when children contemplate ideas about space, or space, and how children themselves fit into a space
Algebraic thinking – e.g. when children recognise or make patterns and discuss relationships between objects
Statistical thinking – e.g. when children sort objects into categories and discuss how many Numerical thinking – e.g. when children count objects such as the number of candles on their birthday cake
Measurement thinking – e.g. when children compare their height or size of objects

Furthermore, as teachers we should be constantly improving our own understanding of mathematical content knowledge in order to know how to further children's mathematical learning.

Play + ? = Maths Learning

The following series of reflections result from the maths games we played in workshops in weeks 5 and 6. I have been thinking about the concept of play and how it can be used to enhance children's mathematics learning.

Play is a concept which has being considered as being “the essence of early childhood practice” (Dockett & Fleer, 1999 p.2) and critical for learning and development. In spite of this, Dockett and Fleer (1999) argue that many early childhood educators are unable to articulate, nor defend the value and benefits of play because they lack frameworks for understanding what play is and how it actually benefits children’s learning. Furthermore, Ailwood (2003) highlights the complex nature of developing an understanding of play, as the term itself is socially constructed, being conceptualised according to discourses, such as the Frobelian romantic or nostalgic discourse and the Piagetian developmental discourse. These discourses influence educators’ understandings of the purpose of play, which in turn influences how they plan and create environments for play in when constructing their curriculum.

For this reason, I believe it is important as an early childhood pre-service teacher to reflect on, critique and improve upon my own understandings of what it means for play to be viewed as a "context for learning" (QSA, 2006, p. 41) in order to better facilitate, stimulate and provide environments that best provoke the kind of play that actually leads to mathematical learning.

I will be posting a series of reflections on principles or components that make play an effective teaching and learning tool. Stay tuned!

The beginning processes

*Click on these tables to enlarge*

References:

Irons, Rosemary Reuille. (1999). Numeracy in early childhood Educating Young Children: Learning and Teaching in the Early Childhood Years, 5 (3), 26-32.

Irons, R. R. (1999). Numeracy in Early Childhood. Educating Young Children: Learning and Teaching in Early Childhood. 5(3), 26-32

Metaphors for Mathematics

In the week 2 workshop we explored different conceptualisations of what learning maths is like.

The following metaphors are useful images to articulate how I picture mathematics learning and teaching...



From our discussions, the useful analogy of a brick wall was mentioned. I find this a useful way of conceptualising mathematics learning as learners must build up their "wall of mathematics knowledge" sequentially, that is, basic or foundational concepts need to be well understood before more difficult concepts can be learnt or taught.

The second metaphor is that of a language. If children truly understand mathematical concepts, they should be able to transfer their knowledge from one context to another, so that their mathematics learning should become a way of thinking. Therefore mathematical learning experiences ought assist children develop their ability to use mathematics as Irons (1999) suggests, to reason and problem solve with, rather than to use the prescribed strategy to obtain the correct answer in classroom mundane activities which often consists of filling in worksheets.

After some further research, I discovered that I was not the first to think of maths in this way. Sterenberg (2008) suggests that the metaphor of mathematics as a language "encouraged a consideration of the humanistic dimensions of mathematics" (p.89. Sterenberg (2008) stated that "language can be described as a systematic method of communicating through oral and written words" (p.96) and just like language, "the purpose of mathematics is to communicate" (p.97). In fact, mathematics is often been described by mathematicians and philosophers such as Galileo, as the universal language. Sterenberg (2008) argues that it is more accurate to consider the ideas of mathematics, rather than the language of mathematics, as being universal because like human languages, mathematics is human creation, which is niehter static nor unchanging.

Other metaphors for mathematics that emerged from the participants' discussion in the study include images of "a battle", "a mountain" and "a bridge".

For the pre-service teacher participants in the study, mathematics is a battle because it was linked to experiences of fear, struggle and even survival. It is also a mountain because it is looming and sometimes menacing, and although they wanted to form a relationship with it, like they want to climb Mount Everest, they find even the thought of it challenging and unnerving. Furthermore, mathematics is a bridge, because it is difficult to build, it must be constructed over a somewhat long period of time, and its construction means overcoming obstacles such as mechanical failure of machinery, and unavailability of raw materials, just like at times the tools and strategies used by teachers to teach mathematics is not appropriate and inhibits learners from constructing mathematical knowledge and understanding.


Sterenberg (2008) argues that investigations of metaphors of mathematics helps to create a "shared communicative space" (p.89) and can be helpful for aiding already-held perceptions about maths learning of teachers and pre-service teachesr alike. I also believe that reflecting on their personal images of mathematics is very important because our beliefs about what we teach not only affects how we teach but how students learn. Additionally, discussing metaphors of mathematics with students is important as it can give teachers valuable insight into how students feel towards mathematics learning. Students' dispositions towards mathematics will no doubt be affected by how they conceptualise mathematics, which will in turn also affect their learning in mathematics.


References:

Irons, R. R. (1999). Numeracy in Early Childhood. Educating Young Children: Learning and Teaching in Early Childhood. 5(3), 26-32

Sterenberg, G. (2008). Investigating teahcers' images of mathematics. Journal of mathematics Teacher Education, 11(2), 89-105