WRPPA Conference: Gilbert Enoka - Creating and Mantaining a high performance Culture

Issues Surrounding the Integration of Mathematics with Other Learning Areas

Mary-Angela Tombs

The Ministry of Education (MOE) proposes that education should makes links within and across the eight Learning Areas in its key guiding document, The New Zealand Curriculum, or NZC (MOE, 2007). This document suggests that by making such links, educators allow pathways for future learning to be opened up to leaners. However, the gap between theory and practice is evidenced within current literature related to the integration of the learning areas.

The argument for an integrated approach to learning is not new.
In as early as 1916, Dewey promoted learning that was responsive and authentic. Dewey’s thinking about an integrated approach to learning stemmed from his own experience of segregated subject areas:

Almost everyone has had occasion to look back upon his schooldays and wonder what has become of the knowledge he was supposed to have amassed during his years of schooling...but it was so segregated when it was acquired and hence is so disconnected from the rest of experience that it is not available under the actual conditions of life (Dewey 1938, as cited in Beane, 1997, p.6).

The purpose of this essay is to critically explore current debate associated with the integration of mathematics with other learning areas. Literature related to the impact of integration in terms of teacher confidence, students’ academic outcomes and attitudes to mathematics is explored, contrasted and compared. Perspectives from research are evaluated in relation to the writer’s professional experience in order to highlight the strengths and limitations of this literature. Critical perspectives explored are in relation to: technical knowledge needed for integration, pedagogical considerations of integration, and integration to support mathematical content knowledge. Finally this essay investigates models of integration as presented by current literature in order to offer a practical perspective to this debate.

Technical Knowledge Needed for Mathematics Integration

Technical knowledge in this case applies to an understanding of what the technique of integration involves. Many researchers assert that there is ambiguity in education as to what constitutes integration between learning areas (Bosse & Faulconer, 2008; Douville, Pugalee & Wallace, 2003; Fraser, Aitken, Price & Whyte, n.d.; Stinson, Harkness, Meyer & Stallworth, 2009).

When referring to ambiguity related to integrating science and mathematics, Stinson, Harkness, Meyer and Stallworth (2009) promote the need for teacher professional learning to uncover teacher differences in perceptions about integration as a starting point.  From here, a clearer understanding can be developed about what constitutes integration. They also stress the need to provide content-specific support within each discipline so that teachers feel more confident to integrate mathematics and science.

Within an integrated literacy context, Douville, Pugalee and Wallace (2003) state that there is confusion about integrating writing and reading with mathematics. While some teachers perceive that integration involves reading and writing about mathematics, others approach integration from the perspective of reading and writing in mathematics. The place of reading and writing in a mathematics programme is explored in more detail later in this paper. This example further illustrates a level of ambiguity about integration, but as a teacher grows in experience, one may consider they are in a better place to integrate the learning areas more effectively.

Naidoo’s (2010) research highlights the impact of teacher experience and expertise as a determining factor of teacher willingness to integrate different learning areas. This research explores experienced teacher perception that if subjects are integrated, they somehow loose a sense of ‘purity’ in that mixing knowledge waters down the learning. This notion is in contrast to that promoted within the NZC (MOE, 2007), which states that in Years One to Six “teaching and learning programmes are developed through a wide range of experiences across all learning areas, with a focus on literacy and numeracy along with the development of values and key competencies" (p.41).

Therefore, if teachers lack clarity about how to integrate mathematics with other learning areas, or they are well qualified in a particular discipline and reluctant to ‘water down’ that learning area, the success of integration is likely to be undermined. Merely considering the technical aspects of mathematics integration is not enough. It is necessary to make decisions about teaching and learning, based on sound pedagogy.

Pedagogical Considerations of Integration

When considering the pedagogical aspects of integration between mathematics and other learning areas, it is appropriate to consider how the integration of mathematics will benefit the students academically. However, it is important not to reflect solely on academic benefits. In a culturally responsive environment, educators consider social and emotional benefits of individuals and the group as a whole.

According to many writers, there is little research to prove that integrating learning areas improves academic results (Douville, Pugalee & Wallace, 2003; Fraser, Aitken, Price & Whyte, n.d.). However other research can be linked with the benefits of integration through association. For example, Ross and Hogaboam-Gray (1998) claim that integration leads to greater levels of collaboration between students within a group. Lee, McLoughlin and Chan’s (2008) research indicates that collaboration within learning that focuses on facilitating development of learning-centred dialogue between students, helps students to co-construct knowledge, thereby enhancing learning outcomes. According to Laal and Ghodsi (2012), collaborative learning also strengthens attachment and leads to higher levels of social competence and greater self-esteem. Creating a collaborative learning environment, where individuals support one another in order to learn together and benefit the whole learning community is also a culturally sound practice (Bishop & Berryman, 2009).

Further to associated benefits of integration and collaboration, Ross and Hogaboam-Gray’s (1998) research, compares non-integrated contexts with integrated approaches to learning, and shows a clear link between integration and self-efficacy. This research demonstrates that within an integrated context, where students work collaboratively to investigate shared projects, lower levels of reluctance are evident. Whereas in a non-integrative context some students are reluctant to participate due to lower ability in these areas. It could be argued, however, that it is not the integration per se that leads to increased participation, but other factors such as teacher effectiveness, relevant contexts, or the collaborative nature of the learning experiences.

Ross and Hogaboam-Gray (1998) also report that by integrating different learning areas, students are able to draw on their knowledge of one subject area to support them in learning about another subject area. For example, when learning how to construct a graph to present results, the mathematical and statistical knowledge in this area can help a student to compare results within a scientific investigation. This finding is corroborated by more recent research by Douville, Pugalee and Wallace (2003) who also assert that integration helps teachers to learn, as they make connections between disciplines.

Integration to Support Mathematical Content Knowledge

While students who follow an integrated learning approach to solve problems are likely to benefit academically, emotionally and socially, there is significant pressure on teachers to focus on the content knowledge that students need in order to carry out assessment limited to specific learning areas. According to Ross & Hogaboam-Gray (1998), this can result in teachers focusing on these skills rather than providing the broad mathematical education that results from practical investigations.

While focus on particular skills for assessment is typically a secondary school phenomenon, this pressure now extends to primary schools in New Zealand with the introduction of National Standards in Mathematics (MOE, 2012). In my experience, while primary school education provides an ideal context for integration, there is now reluctance among primary school teachers to integrate subject areas, because of the pressures they face to ensure students meet standards. Ironically, the National Standards indicate a need to assess across the curriculum within the context of normal classroom practices in order to establish an overall teacher judgment, rather than assessing merely on the basis of a mathematics test.

Measuring student progress and achievement against the National Standards in Mathematics (MOE, 2012) requires teachers to use realistic examples of mathematical content knowledge being used in everyday situations. For example, one illustration to measure student achievement in mathematics after their first two years at school involves making a decision about how many stamps on a number of letters. Not only are the students required to match one to one, but they also require knowledge about what a stamp is and what it is used for, and what it means to post a letter in order to make sense of the task. Integrating all of these aspects of a mathematical task requires a huge, yet totally realistic integration of content knowledge between learning areas. However, in my experience, teachers are reluctant to draw on all learning areas for assessment against National Standards (MOE, 2012).

Bosse and Faulconer (2008), and Douville, Pugalee and Wallace (2003) refer to the integration between learning areas for helping to build understanding in mathematics. Bosse and Faulconer assert that students benefit from specific instruction in reading and writing in relation to mathematical tasks. Typically, mathematical texts challenge students with a context-specific type of text layout, as well as use of content-specific vocabulary and sentence structures. Students need to develop the ability to quickly move their focus from one part of a page to another and back again, in order to relate the text to the provided illustrations, graphs, etc. They also need to be able to translate numbers into words and vice versa and to interpret symbols quickly.

Bosse and Faulconer (2008) insist that these skills cannot be developed effectively without integrating the content knowledge inherent with understanding written language. This view is corroborated by Ell, Smith, Stensness and Major (in Averill & Harvey, 2010), who descibe the complex process of acquiring mathematical language as a journey:

When children begin to learn mathematics in educational settings they are continuing a cognitive journey from informal, everyday understanding about their world to the formality and abstraction of school mathematics. Part of this journey involves mastering a new vocabulary and symbol system – the special language of mathematics (p.207).

Ell, Smith, Stensness and Major (in Averill & Harvey, 2010) refer to students beginning to learn mathematics, but it can be argued that mastering mathematical vocabulary and symbols remains relevant throughout a student’s learning in mathematics, with the vocabulary becoming more complex and the symbols more specialised.

While the practice of integration between learning areas is not prevalent in most secondary settings where teaching is focused on segregated subjects in order to prepare for assessments, one school in Auckland, New Zealand has experimented and found success with a learner-responsive integrated approach to learning and teaching. Alfriston College curriculum leaders collaborate to blend learning areas within a relevant and authentic context (http://www.alfristoncollege.school.nz/curriculum).    Therefore, as evidenced by Alfriston College, when schools are prepared to innovate, based on sound research and collective experience, integration can be successful and rewarding, even at a secondary school level.

To summarise, the content knowledge within mathematics can only be developed by integrating across learning areas. One could argue that the level of integration detailed here is logical and assumed, yet there is opposition to changing school systems in order to recognise this logic. Once again, perhaps this comes down to teacher perception of what integration actually means.

Models of Integration

According to recent literature by Brough (2008) there are two forms that integration takes: subject-centred integration, and student-centred integration. The first of these takes a thematic approach, making connections across learning areas in order to fully investigate a theme or topic. In this model, the teacher chooses the topics and the investigation is thoroughly teacher planned, well in advance. The student-centred integration model is also thematically based, but investigates an authentic context to solve a real problem or investigate emergent phenomena. Brough advocates the student-centred approach and makes reference to the work of Dewey, referred to earlier within this essay. This model allows for the students to drive the process, rather than teachers pre-determining it.

Burgess (in Averill and Harvey, 2010) also promotes an enquiry-based cyclic approach for teaching statistical thinking, although the activity examples he describes are a combination of subject-centred and student-centred in that the initial topic for investigation, in the form of a problem, is pre-determined by the teacher but becomes student-driven as it develops into an investigation.

Averill, Taiwhati and Te Maro (in Averill and Harvey, 2010) make links between subject-centred integration and cultural responsiveness. The activities they present focus on aspects of Te Ao Māori (a Māori world view) by incorporating Te Reo Māori and other appropriate heritage languages representative of the students in the class into mathematics. They also incorporate aspects of tikanga: Māori tradition and customs. While these activities are rich experiences that integrate literacy and numeracy by including Te Reo, learning languages, english, mathematics and social sciences, Averill, Taiwhati and Te Maro make no mention of the fact that the activities are intended to integrate learning areas – it just happens naturally. This can be typical of a New Zealand primary school setting where the classroom teacher is responsible for curriculum coverage and ensuring that the learning is relevant to the backgrounds and needs of the students. However, my experience is that such integration is limited in practice even within the context of my multicultural school. One could argue that teachers not only feel pressure from standards, but they are reluctant to change their own practice, and do not fully understand the benefits and culturally responsive practice associated with integration across learning areas.

Perhaps, the secondary school dilemma of low teacher skill, confidence and understanding, as reported by Naidoo (2010) , is also a barrier to integration of mathematics at a primary level. Bosse and Faulconer (2008) acknowledge the level of skill and effort invested on the part of the teacher when integrating literacy and mathematics, but state, “Although reading and writing in mathematics may necessitate more skills and practice to master, the mathematical learning derived from reading and writing mathematics far outweighs the burden it places on teachers and students” (p.8).

Conclusion

This essay has unpacked several issues related to the integration of mathematics with other learning areas. Literature related to the technical knowledge, pedagogical knowledge and content knowledge of mathematics has been contrasted and compared in order to identify such issues. The issues explored have included: abiguity about integration techniques, teacher confidence, and teacher reluctance. Several positive outcomes of integration have been detailed including those related to collaborative learning, cultural responsiveness and student confidence.

Two models of integration sourced within literatured have been detailed. A position that teacher confidence is key to the successful integration of mathematics has been presented, with suggestion that professional learning should illuminate teacher perceptions as a strating point for learning about effective integration.

 References

Averill, R., & Harvey, R. Eds.). (2010). Teaching primary school mathematics and statistics: Evidence-based practice. Wellington: NZCER Press.

Beane, J. (1997). Curriculum integration: designing the core of democratic education. New York and London: Teachers College Press, Columbia University.

Bishop, R. & Berryman, M. (2009). The Te kotahitanga effective teaching profile. Set, 2, 27-33.

Bosse, M., & Faulconer, J. (2008). Learning and assessing mathematics through reading and writing. School Science and Mathematics,  108(1), 8-19.

Brough, C. (2008). Student-centred curriculum integration and The New Zealand Curriculum, Set, 2, 16-21.

Douville, P., Pugalee, D., & Wallace, J. (2003). Examining instructional practices of elementary science teachers for mathematics and literacy integration. School Science and Mathematics, 103(8). 388-396.

Fraser, D., Aitken, V., Price, G., & Whyte, B. (n.d.). Connecting curriculum; connecting learning; negotiating and the arts. Retrieved 21 June, 2012, from http://www.tlri.org.nz/tlri-research/research-completed/school-sector/connecting-curriculum-connecting-learning-negotiation

Laal, M., & Ghodsi, S. (2012). Benefits of collaborative learning. Procedia - Social and Behavioural Sciences, 31, 486-490.

Lee, M., McLoughlin, C. & Chan, A. (2008). Talk the talk: learner-generated podcasts as catalysts for knowledge creation. British Journal of Educational Technology, 39(3), 501-521.

Ministry of Education. (2012). National Standards. Retrieved June 26, 2012, from http://nzcurriculum.tki.org.nz/National-Standards

Ministry of Education. (2007). The New Zealand curriculum. Wellington: Learning

Media.

Naidoo, D. (2010). Losing the “purity” of subjects? Understanding teachers' perceptions of integrating subjects into learning areas. Education As Change, 14(2), 137-153.

Ross, J., & Hogaboam-Gray, A. (1998). Integrating methematics, science, and technology: effects on students. International Journal of Science Education. 20(9), 1119-1135.

Stinson, K., Harkness, S., Meyer, H., & Stallworth, J. (2009). Mathematics and Science integration: models and characterizations. School Science and Mathematics, 109(3), 153-161