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Are doing mathematics and being mathematical separate ideas? The answer lies in our understanding of mathematics itself. Devlin (2005) explains that mathematics is 'recognizing and manipulating patterns' while Barton (2009, p.5) describes being mathematical as pursuing an enquiry. When combined, the two ideas represent the exploration of mathematical concepts through the use of problem-solving and reasoning (Baroody, Coslick, & Wilkins, 1998, p.1-13). Pratt (2006, p.52) supports this theory by claiming that in order 'to 'do' maths ... we must have a problem to solve'. He continues by suggesting that the word 'problem' is replaced with 'enquiry', creating lessons that promote the learning of mathematics through the application and development of knowledge and skills. This, sequentially, should prevent the skills from being considered as isolated pieces of information.
Investigative and enquiry based learning can empower children (Wassermann, 2000, p.14) by allowing them to be actively involved in, and have control over, their learning. The use of open-ended investigation has 'the potential to increase the math talk in the classroom' (National Research Council, 2009, p.246), and with language playing a critical role in cognitive development (Vygotsky & Bruner, cited in Stierer & Maybin, 1993, p.xi), it is logical that the use of mathematical language is considered when judging the quality of teaching and learning (OfSTED, 2010).
During a recent lesson observation, I witnessed pupils discussing their responses to the question, 'The answer is 42. What is the question?' The children were captured by the openness of the task and enjoyed communicating their ideas and the reasoning behind them. The National Council of Teachers of Mathematics (2009, p.3) discusses the value of exchanging ideas when learning mathematics and suggests that it can, 'help learners sharpen their ability to reason, conjecture, and make connections'. Teachers do, however, need to be able to 'scaffold the discussion by [using] careful questioning' (Bottle, 2005, pp.122-123) in order to guarantee that the discussion is valuable, develops understanding and remains open.
The use of open-ended questioning does, however, require that children accept that there may not be an ultimate goal to work towards (Yeo, 2007, p.7). This poses several challenges, including the possibility that unexpected learning may occur (Yeo, 2007, p.9). Good teaching, however, means being able to transform unexpected discoveries into opportunities for further learning (Idris, 2006, p.53). I observed an example of this during a lesson [Appendix A] where pupils were invited to use a map to investigate the distances of possible routes to given destinations. Pupils began their investigation by specialising; selecting a destination and then calculating the distances using a scale. The majority of pupils focused, as anticipated, on the roads, however one group chose to compare pedestrianised routes with those of vehicles, resulting in unexpected discussions that linked measuring distances to time and speed. Their reasoning was that they conjectured that some destinations might be reached more easily by travelling on foot and they tested this during the investigation. This example reveals that the children, when presented with an open-ended enquiry, were thinking creatively, and demonstrating their ability to inter-link mathematical concepts and pose further questions when presented with a real life context.
This example also confirmed that pupils were using and applying in mathematics by practicing the appropriate skills identified in guidance by the TheÂ Department for Education and Skills (DfES., 2006a, p.4). With these skills also being attributed to investigative work (Yeo & Yeap, 2010, p.4), it is reasonable to assume that mathematical investigations will support the achievement of the National Curriculum's using and applying objectives. Assessing the level or achievement of the objectives has the potential, however, to be problematic (Klavir & Hershkovitz, 2008, p.2) although this can be addressed by teachers working collaboratively with pupils to evaluate the effectiveness of their investigations (TDA., 2008, p.8, Q28).
I observed an example of pupils using and evaluating their investigative skills during a lesson where they, when presented with a number puzzle [Appendix B], began by specialising using a given example, and then formed conjectures about patterns that might appear. This provided them with a focus for their enquiry, and the confidence to test their ideas which resulted in the majority of pupils forming generalisations about the patterns created by the numbers. Each group then explained the reasoning behind their chosen methods and conclusions with the rest of the class communicating their thoughts on the effectiveness of the chosen strategies. The pupils work [Appendices C & D] clearly shows that they were able to form conjectures at various points in the investigation, suggesting that they were building on their existing knowledge, a process identified by Piaget as essential for cognitive development (Slavin, 1994, p.32) and also a vital component to constructivist learning (Boghossian, 2006, p.714). The children who were able to generalise, did so as a result of effective communication and adopting a systematic approach to their investigation. Conversely, some pupils struggled to identify any numerical relationships as a result of deficiencies in their ability to calculate efficiently. This type of struggle can, however, be beneficial to learning.
John Stewart Mill (n.d.) once said, 'The pupil, who is never required to do what he cannot do, never does what he can do'. This philosophy of education is supported by Vygotsky's claim (Slavin, 1994, p49) that children need to move out of their comfort zone if they are to achieve their potential and theÂ DfES (2006a, p.8) upholds this idea by explaining that challenging tasks are crucial when developing problem solving strategies. Nevertheless, although expectations need to be high (TDA., 2008, p.8, Q1) they also need to be realistic (Malone, 2003, p.239) and therefore it is essential that all teachers are aware of children's current levels of understanding and, as a result, plan appropriately differentiated activities (TDA., 2008, p.8, Q10) that enabled all children to succeed (Kendall-Seatter, 2005, p.3).
In summary, by combining directed teaching of mathematical techniques with teaching the processes of investigative mathematics (DfES, 2006b, p65), children can learn to use investigative skills collectively to solve problems and to explore the world around them. Using these skills creates successful learners who can use mistakes to help them to progress and who enjoy learning (Rose, 2009, p.34). By developing questioning skills, children can learn to form insightful conjectures that they will be motivated to test and prove. Communicating will allow them to extend their ideas (Cockcroft, 1982, p.73) and open, challenging and meaningful problems will inspire intrinsic motivation (Pratt, 2006, p.51) and allow them to 'do' mathematics and be mathematical.