Concluding the EAB023 Journey: Some final thoughts

This concept map maps out the blog-entries which represent much of my reflections throughout the EAB023 learning journey. (Click on the image to enlarge)

The explorations of mathematical concepts which I have undertaken through this unit have been very rewarding for me as a future maths teacher. I have enjoyed the first-hand experience of being challenged to think mathematically, and use everyday language to explain and justify the reasoning I use to reach a solution. I also found this extremely helpful for me as a teacher as my aim is to stimulate children to engage their natural curiosity and use it to explore and develop systematic understandings of mathematical concepts (Australian Association of Mathematics Teachers and Early Childhood Australia, 2006).

In the future, I wish to engage in more professional development opportunities so that my own love of mathematics learning will not come to a close. Furthermore according to Cockburn (2008), on-going professional development is also important and valuable as it informs teaching practitioners to develop and improve teaching practices that are informed by empirical research. Many of the readings which I have accessed in order to inform my blog-posts have really taught me a lot of valuable things about mathematics teaching. I am more thankful than ever for the researchers who dedicate their careers to conducting good quality research to inform and improve teaching practices. Sadly, according to Cockburn (2008) the amount of educational research being applied to professional practice is still very limited. Perhaps it is because much of the teaching style recommended by research requires more time and effort than traditional approaches. For example, teaching mathematics without resorting to text-books and worksheets takes a lot more time and effort, both in the preparation stages and reflection stages of teaching and learning. However, it is so much more worthwhile.

Throughout this unit my ideas about mathematics learning have grown and I have realised that although quality resources do play a role in facilitating mathematical understandings, there are other factors which are even more critical when it comes to developing children’s understanding of mathematics. These include: how the resource is used, the sequence of mathematical concepts being taught/learnt, the teaching style adopted by the teacher (which determines the type of relationships that exist between the teacher and child, and between children and their peers), the identification of the connections that exist between different mathematical concepts, and the teachers’ ability to mathematise, recognise mathematising and identify common misconceptions. As early mathematical development and its benefit for future learning is so well documented through research (Godfrey, 2006), I believe it is of critical importance for teachers to adopt approaches that encourage children to see themselves as mathematicians and to develop a love for learning mathematics that stretches beyond getting the right answer. This has become an important part of mathematics learning for me as I have grown to change my perceptions about mathematics as being straight forward and unambiguous. This is evident as I grew to enjoy mathematical problems where there is more than one solution and more than one strategy can be used to solve it.

Lastly, the value of reflection as a tool for evaluating one’s own learning is an important component of mathematics learning which I have come to appreciate more throughout this unit (Griffiths, 2000). I have found that creating this blog required me to analyse my own learning at a much deeper level than I would have, had I simply participated in the workshops alone. Similarly, the workshops provided me with the stimulus to think more deeply about mathematical concepts. I see now that the two go hand-in-hand. Writing is a way of documenting thoughts tangibly, and I see how having a tangible copy of the process of my learning journey will assist me as a future teacher. In my future as a teacher of mathematics, I hope I will be just as motivated to keep reflecting on my learning.

References:

Australian Association of Mathematics Teachers & Early Childhood Australia. (2006). Position paper on early childhood mathematics. Retrieved 18 Oct, from http://www.aamt.edu.au/content/download/722/19512/file/earlymaths.pdf

Cockburn, A. D. (2008). How can research be used to inform and improve mathematics teaching practice? Journal of Mathematics Teacher Education, 11, 343-347.

Godfrey, R. (2006). Early mathematics development and later achievement: further evidence. Mathematics Education Research Journal, 18(1), 27-46.

Griffiths, V. (2000). The reflective dimension in teacher education. International Journal of Educational Research. 33, 539-555.

Function machines and hands on learning

In my readings I found this brilliant teaching idea for teaching functions to kindergartens by Willoughby (1997). I feel that this may even be a valuable learning experience for children in the older grades too. I would love to try this idea with children in an actual lesson. Please note that the function machine my graphic refers to is a large box big enough for a child to fit inside comfortably, decorated so that it looks like a machine, used for teaching functions. It must also contain a slot for the input and a slot for the output.




(Please click to enlarge this graphic)

As I reflected on this lesson plan by Willoughby (1997), I contrasted the potential learning described with Warren and Cooper's (2005) research findings, which stated that "the use of child volunteers to act as IN and OUT with respect to the function machine was a strong distractor in classroom 3." According to Willoughby (1997) the above-mentioned learning experience was one which he has carried out many times. Furthermore a photograph of a child remaining engaged while participating in this activity was included in the article. This led me to wonder, how could this be? Perhaps it was because of the fact that in the lesson conducted by Willoughby (1997), the child inside the box remained hidden to the rest of the class and therefore was less of a distraction. Or perhaps it could have been the fact that the child inside the box was in control of the transformation from recieving the input to the providing of the output. This was unlike the learning experience described by Warren and Cooper (2005) where several volunteers were responsible for only one aspect of the transformation, i.e. "Frank gives the green stick to Ned; Ned puts green stick in box; Researcher changes the green stick to red stick and gives this to Bonnie; Bonnie gives the red to Frank and the teacher records the change on the IN/OUT table." (p.157)

This reflection has reminded me that in designing learning experiences and assessments for children, simply having concrete examples, hands-on experiences and well-designed resources is not enough - aspects such as what the children are paying attention to must also be taken into consideration. As teachers we must ask ourselves, "What are the children learning?" and "How do I know he or she is learning it?"

References:

Willoughby, S. (1997). Functions from kindergarten through sixth grade. Teaching Children Mathematics, 3(6), 314-318.

Warren, E. & Cooper, T. (2005). Introducting functional thinking in year 2: a case study of early algebra teaching. Contemporary Issues in Early Childhood, 6(2), 150-160.

Authentic contexts for learning chance and data

Week 11’s workshop was all about the strand of chance and data. The main activity surrounded the use of Smarties to create an authentic context to learn chance and data concepts.
I work at an OSHC. In my experience working with children in primary school, chance and data is often a dreaded strand of maths, notoriously well-known for useless worksheets such as the one I witnessed this afternoon when discussing with the children what they have been learning in maths:

After reading through this worksheet, I discussed its contents with the girl, a grade 4 student, who had completed it for homework just a few days ago. When I asked if she enjoyed doing this worksheet, she answered, “No, not particularly, it was easy, but it was boring. But then again maths is boring, so yeah. ” Then I asked her is there anything about it that she liked, and she optimistically pointed out that at least she got to make a choice about whether to colour the bars in colour or not, and she chose not to.

It was not just the lack of authentic context that annoyed me about the worksheet, but that fact that its design did not really highlight the usefulness of graphs. If you look at the worksheet in detail, it asks the student to graph the table of information provided at the top of the page, then use the graph to answer the following questions: Which was the city with the highest recorded temperature? Which was the city recorded the same temperature? How much hotter was Darwin than Hobart? Much of this information would have been just as easy to answer without constructing the graph, let alone the last question which would have been easier to calculate without the graph. I was astonished and annoyed by how chance and data, a topic which lends itself so easily to authentic investigations based on real-life contexts or other hands-on learning experiences, could be treated in this way. In the same afternoon at work, while playing connect four with another child in grade 1, an authentic context arose for her to record some real data, and she did so on her own accord.

This grade 1 girl started to tally the number of wins each of us achieved as a result of playing a series of games of Connect Four. In actual fact, we did not play as many games as was shown on this piece of paper. The score started off as:

She accurately recorded the results and made statements about them, “You’ve won four games and I’ve only won one”. “If you win one more game then I’ll put an across mark and you would have won five games.”

After playing four more games the score table read:

She said, “If I win one more game we’ll be tied.” We talked about where she got the idea of doing up a table from. She said that she picked it up from watching other girls keep score when they play Connect Four.

Then the call for afternoon tea was announced and she quickly scribbled in the remaining tally-marks. When I asked her what she was doing she said, “I’m just mucking around, look we both won lots of games”.

If we had more time, we could have explored different ways of graphically representing the information she collected or made up with post-its, bundle sticks, stickers and so forth. I am confident that she would have enjoyed that experience and gained a lot more from it than the girl in grade 4 seemed to have from completing her homework. This led me to think deeper. I wonder, what does actually constitute an authentic context for learning chance and data? Furthermore, is having an authentic context the crucial deciding factor in determining if this strand of maths is taught well or not?

Nisbet, Langrall and Mooney (2007) conducted a study which raised the research question "How do students knowledge of real-life contexts affected their ability to analyse some data provided to them?" The result of the study suggest that when primary-aged students were given sets of data related to a topic area which they had special knowledge and interest in, they used their understandings of the real-life context to "rationalise their data or their interpretations, in taking a critical stance towards the data, and in ways that were not necessarily productive or pertinent in addressing the task at hand." (p.16) According to Nisbet, Langrall and Mooney (2007), providing opportunities for learners to integrate contexual knowledge and statistical information through investigating real-life data is important. However, teachers need to be aware that learners can be just as easily distracted by their contextual knowledge, which can lead them to disengage with the mathematical problem or task. This hypothesis seems to be supported by English and Watters (2005) who found in their study that children who were participants in their study used their informal knowledge to relate to and identify important problem information, but at times became "absorbed in applying their informal knowledge" (p.72).

Therefore it is clear that providing data which is taken from real-life contexts or may be of interest to children is not enough. As teachers we need to crtically reflect on our lesson designs so as to assist students to use their informal knowledge to critically evaluate data where it is appropritate, as well as develop statictical literacy in considering the data itself and how it applies to the problem (English & Watters, 2005). Perhaps activities where students take part in collecting the data, as well as manipulating it to solve problems which they pose for themselves would be appropriate (see fishing game blog-post).

References:

English, L. D. & Watters, J. J. (2005). Mathematical modelling in the early school years. Mathematical Education Research Journal. 16(3), 58-79.

Nisbet, S. Langrall, C., & Mooney, E. (2007). The role of context in students' analysis of data. Australian Primary Mathematics Classroom, 12(1), 16-22.

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!