Math Education / Teaching & Learning Tips and Strategies

Implementation of Investigative Approaches: 

Challenges, Opportunities and Strategies

Many research have proven that investigative approaches play a very important role in the development of mathematics education. The argument that investigative approaches consume much planning and implementation time, therefore reined teacher from completing the contents was first discussed and evaluated by this article. Small survey has been conducted to assist in the evaluation of the argument. The purposes of this article is (a) to review possible reasons for integrating constructivist and traditional approaches into mathematics class and (b) to review the strategies that have been used to implement the investigative approaches into everyday classroom teachings in order to help teachers plan their teaching and adapt those strategies into their classrooms. Obviously, although investigative approaches are excellent in promoting meaningful mathematics, a bit of traditional touch is useful (in term of its legitimate nature and its firm procedural knowledge and behavioural towards mathematics) to elevate mathematics education in general.  

Introduction

“I don’t have time to try group work or investigative approaches in my classroom.
I’m flat out covering the content so the kids will be properly prepared for the test.”

             It is believed that the National Council of Teachers of Mathematics (NCTM) had published the Principles and Standards for School Mathematics at the right time. It was published while a growing worrisome was emerged about the failure of school mathematics in inducing the meaningful learning and the ability to confidently use mathematics among the school leavers. Bunch of research arose to support and elaborate the investigative, exploratory, open-ended activities which underpin the active learning environment suggested by NCTM and constructivism. However, constructivism theory only provides description on how students learn and understand mathematics but do not configure how teachers should act to induce active learning in their classroom (Draper, 2002). As the result, the statement excerpted above is maybe one of examples of teacher’s gambits when the implementation of the investigative approach is discussed. This statement will be evaluated and discussed critically in this article due to the fact that it becomes among the most popular reasons for not implementing investigative approaches and for the sake of excelling mathematics reforms, it must not simply be ignored (Norton, McRobbie, & Cooper, 2002; Walle, 1999). This article then argues that the investigative approaches shall be integrated with the content-based approach at least in the transition period. Several rationales based on article, journal and book reviews will be provided along with the discussion. Later, several strategies on implementation of investigative learning will be listed and discussed.

            A small survey has been held involving The University of Queensland’s masters students which enrolled in EDUC7303: Navigating Numeracies: Pathways, Pattern and Priorities to study their teaching style and the application of investigative learning in their classroom. Although only a small amount of questionnaires were returned (N=3), the survey shows that there are several items where respondents were unanimous with. All of them have tried to implement the investigative approaches in their classes (varies from 30% to 50% from accumulative time allocated for mathematics per week) and found that investigative approaches elevate their students’ understanding of mathematical concepts. They agreed that investigative approach can be implemented in mathematics classroom but prefer it to be integrated with other approaches rather than employing it alone. No significant stand can be concluded from the survey whether the implementation of investigative approaches consumed too much time or not. Despite that, two of them disagreed that investigative approaches demand little planning time but agreed that time is not the biggest constraint in implementing the investigative approaches. Unfortunately, the number of participants was too small and general conclusion cannot be done according to this survey alone. Therefore, several articles have been reviewed to evaluate the result.

The challenges

            In similar vein to the survey, Walle (1999) stated that problem solving methods (which is an example of investigative approaches) do “eat time” in mathematics periods. He addressed the problem as reformers’ mistakes in the application of reforms mathematics. On the other hand, Norton et al. (2002, p. 47) found that mathematics teachers described investigative approach as “too slow” and does not suit mathematics’ “crowded” syllabus. Although the meaning of “crowded” was not provided, clearly it described the tedious long listed curriculum loads. This statement supports the excerpted argument directly. On the other hand, the term “too slow” describes different thing i.e. the developmental stages of investigative open-ended classroom activities. In USA, the “inappropriate” time taken for the “discovery process” in investigative learning and the reduction of yearly syllabus coverage became the main reasons for the Mathematically Correct anti-reforms group to oppose the reforms. Logically speaking, longer time allocation for the investigative approach means a reduction of mathematics’ core time (Aynes & Sweller, 2000). As the results, an insufficient number of topics were covered in the whole year. Despite that, discovery learning is essential and should not be totally rejected because it is the only way to induce meaningful learning through active participation of the learners in the cognitive construction (Boaler, 1998; Gijbels & Loyens, 2009; Glasersfeld, 1987, 1989; Grimison & Dawe, 2000; Walle, 1999). Actions must be taken to keep in balance both the meaningful discovery and the conceptual and skills attainment of investigative learning so that it may not be misunderstood in general context (Walle, 1999).

             The results that all respondents are prone to adopt variety approaches in their teaching also agree with Norton, et al. (2002) in stating that teachers tend to use variation of different pedagogical approaches in their classroom based on the conviction that investigative approaches are good for the conceptual part whereas the traditional techniques are for the children to pass the examination which was proclaimed as fulfilling teacher’s responsibility to the students and their parents. This is in agreement with the claim that constructivists are prone to ignore the socio-political part of mathematics reforms that there are other socio-political power that established the standard examination and legitimated the academic regulations for entering the higher education (Zevenbergen, 1996). In addition to Norton et al. (2002), Handal & Bobis (2003) found that some teachers tend to combine and adapt different teaching styles into their mathematics classes for different set of reasons. One of the reasons is teachers used to change their teaching styles according to the class context -the condition and level of the students- and the topic to be taught with the belief that different approaches suits different topics (see also Clarke, 1997). Therefore, teachers’ beliefs on students’ academic ability took an important role in choosing teaching approaches in which they believed to be practical for their classes (Bartholomew, 2003). The unfilled gaps about how teachers should act to support investigative learning strengthens this traditional practice (Draper, 2002; Smith III, 1996). Therefore, a good professionalism development program for teachers will improve their beliefs and practices in implementing the constructivist approaches.

             Surprisingly, a lead reformist, Walle (1999) also notified that reformers have forgotten that many basics are still essential to be addressed. This leads to misunderstanding among parents about the nature of reforms mathematics. Clearly, from his presentation, some mathematics skills can be actively constructed and some of them are needed in their basic form. Standard written computational skill, terminologies and basic facts are some examples of those needed in their basic form. Some research or descriptive review should address this issue. Until this point, it seems that in the recent time, some of the content-based mathematics skills such as the written computation are still important so that the children will be able to socially achieve the appropriate legitimate level in the community (Zevenbergen, 1996). In other words, investigative approaches shall not simply replace the entire content-based approaches while the global education system - especially the assessment system - not yet reformed into a constructivism system (Grimison & Dawe, 2000). However, Walle (1999) also said that the most basic in mathematics is that it “makes sense”. Therefore, the combination or integration of both approaches might be successful in targeting both understanding and legitimate purposes of mathematics education with the investigative approach as the main ingredient.

             Should traditional-based being integrated into constructivist classroom? This article will only provide the overview about it. In psychology (generally known as extreme behaviourist camp), the cognitive aspect has been adapted into the psychotherapies for children at-risk by the introduction of Cognitive-Behavioural Therapy which was proven as more effective than behavioral therapy alone (Dryden & Ellis, 2003). Therefore, the same would be happen to mathematics education if some good behaviorist practices being integrated into pure constructivism. In addition, Boaler (2000), in re-evaluating her famous Amber Hill and Phoenix Park experiment, stated that focus should be put on conceptual, procedural and practical aspects of mathematics so that students will learn mathematics cognitively and behaviorally. For instance, Phoenix Park’s student was found to be experienced both methods of learning, the open-ended method as the main approach and the preparation period starting January of the final year which then went intensive in last eigth weeks (Boaler, 1998, p. 49, 2000, p. 117). On the other hand, traditional-based approaches used by Amber Hill only encouraged good mathematics routines and procedurals technique among their students but failed to demonstrate any achievement in the practical aspect of mathematics (Boaler, 2000). This evidence shows that relationship between cognitive and behavior are esssential to produce a perfect mathematics achievers and combination of both approaches might produce better output.

             Moreover, there are several risks of implementing fully pure discovery learning. For instance, teachers have to keep the charcteristic of good problem solving such as good communication and discussion skills, along with the cognitive demands of the task such as high level critical and analytical thinking, throughout the implementation period (Stein, Grover and Henningsen as cited in Erickson, 1999). In constructivism, there is a grey area between assisting the chidren to keep on track and subsiding high level thinking among them. It is not surprising when Boaler (1998) found that fully open-ended approaches may overlook the students who are out of the track just like what has happened to some students in Phoenix Park case. Therefore, teachers must be able to identify the off-tracked students and drag them back onto the track. Unfortunately, in most cases, teachers are lack of expertise especially in handling a complex assessments, understanding students’ thinking in terms of misconception and preconceptions, questioning and explaining skills, and effective pedagogical skills (Carpenter, Fennema, & Franke, 1996; Erickson, 1999; Handal & Bobis, 2003; William, 2005). In the worst cases, teachers might not have the ability to determine the level of cognitive demand of their assessments or projects (Sullivan, Clarke, & Clarke, 2009). By integrating “the guided” part of traditional approach into pure discovery learning approach, tasks can be handled effectively  with proven efficacy (Mayer, 2004).

             Due to the boundless nature of the investigative assignment and teachers’ incompetency in the new curricula, an inappropriate task might be chosen by the teachers, which might divert the student from the intended objectives (Manouchehri, 2004). At other time, teachers might feel unconfident with their ability to implement the new instructional approach and roll back to the traditional approach (Handal & Bobis, 2003; Manouchehri, 2004; Norton, et al., 2002; Smith III, 1996). It is useless to get back to the entire traditional approaches because it has been proven to be successful only in promoting academic achievements but failed to encourage the meaningful mathematics practise in a long-term period (Walle, 1999). Therefore, the disadvantages of constructivism should be addressed by employing the good side of the traditional approaches into constructivists’ classrooms and upgrading teachers’ profesionalism and competency, not by rolling back to traditional approach.

             Finally, Clewell et al.’s (2004) reviews of almost 400 journals, standard documents and books on both traditional-based and standard-based approaches and teachers’ professionalism development programs lead to the extent that both approaches show strong evidences of efficacy in mathematics achievement whereas the inquiry-based approaches are the most significant in science education. Therefore, for the sake of excellent mathematics education, all the evidences discussed before should inspire both camps to open their mind to discuss the possibility of achieving better improvement from an effective combination of both approaches.

The Implementation of Investigative Approaches

            Due to the proven effectiveness of constructivism which have been approved by both constructivist and behaviourist as the most effective teaching strategy (see Dryden & Ellis, 2003; Walle, 1999), the investigative approaches must remain superior. On the other hand, due to the fact that there is little holes in the investigative approaches which water down their efficacy, two strategies should be considered: improve the implementation of investigative learning and employ the good side of traditionalist to cover the uncovered holes. To reach to this extent, both camps must be rational and considerate. Therefore, in later discussion, the implementation of a good constructivist’s investigative activity will take place. It is because, in order to learn mathematics meaningfully, comprehension must become the main purpose in which the knowledge is actively built on existing knowledge and previous experiences. The NCTM has outlined five standard processes for reforms mathematics’ activities i.e. problem solving, reasoning and proof, communication, connection and representation ("Guiding Principles for Mathematics Curriculum and Assessment," 2009, pp. 2-5). In other words, a good constructivist activity establishes opportunities for the students to explore non-routine real mathematical problems, explain and defend their works through discussion with teachers and peers and build senses of meaningful relation between mathematical concepts and into the real-life situation (see also Sullivan, et al., 2009). As the result, students will be competence inside and outside the classroom.

             In addition to NCTM, Walle (1999, webpage) have listed several tips and advice on developing excellent mathematics activity. The first tip is teacher must always look forward for students’ responses by predicting range of questions and answers that might arise during the task. Such preparation will help teachers to keep on track and achieve the intended goals (see also Carpenter et al., 1996). Second, task must be designed to suit the lesson. Some tasks may take a longer time than others therefore unsuitable tasks will results in waste of time. Third, teachers must have clear understanding about the purposes of the task such as either it is intended to introduce the concepts, discover the range of possible procedure or investigate its meaning in wider perspectives. Fourth, focus must be put on the discussion of sense-making, not the answers or how to get the answers. Obviously, discussion on the answers will not promote meaningful learning. Fifth, teacher must know the characteristics of a good problem and should be able to distinguish mathematical problems from mathematical-like creative activities. Such creative activities will consume the valuable time for real mathematics. Finally, teacher must minimise elaboration of conventions and concentrate on the concepts. To do that, teachers must be able to differentiate between mathematical conventions and concepts.

            There are several skills and knowledge that should be employed by a constructivist teacher in order to help in the implementation of the constructivist activities in class. Teacher should have sufficient knowledge about the subject to be taught so that teachers will have the competency and confidence to teach the subject. Besides that, teachers should also know about the range of pedagogical strategies that can be used in order to successfully achieve the intended knowledge and concepts. Last but not least, teacher should have sufficient knowledge of the pedagogical content such as presentation skills, questioning and giving feedbacks so that the strategies will be successively and effectively delivered (Carpenter, et al., 1996; Sullivan, et al., 2009). In order to elevate teachers’ professionalism and competency in understanding the nature of constructivist’s mathematics education, several development programs might be useful. For instance, the Cognitive Guided Instructional (CGI) provides teachers with the skills to understand children’s thinking through critical observation of their acts in completing the task with a conviction that by understanding children’s thinking, teachers will be able to construct an activity that suits the children’s needs and re-organized it, if it is not suitable for the children (Carpenter, et al., 2009). In addition, Goos, Dole and Makar, (2007) have shown that by actively evaluates and reflects on their previous teaching practices and improve their teachings based on the zone-theoretical approach, teacher will be able to improve their teaching approach in order to achieve the intended objectives.

             From a literature review done on about 25 articles (which some of them have been rejected due to inefficiency in demonstrating constructivist activities), it can be concluded that there are five strategies that have been used to implement investigative approaches into everyday mathematics classrooms. The strategies are:-

      1)    Project based assignments- proper organised activities and massive collaborative investigative and discovery works (Civil, 2002; Halverscheid, 2005). Project based normally takes longer time.

      2)    Mathematical Modelling- allows children to actively generate ideas and concepts and test and apply their ideas within unfamiliar realistic situation (English, 2006, 2009)

      3)    Mathematical games- children do hands-on activity and work actively (Bell & Henderson, 2004; Bright, Harvey, & Wheeler, 1985; Civil, 2002; Clarkson, 2008; Flewelling, 2005; Lee, 2008; Simpson, 2005; Wiest, 2006)

      4)    Technology based or technology usage such as calculator, computer softwares and the internet (Frid, 2002; Monaghan, 2004; Thomson, 2008) and

     5)    Thematic approach- by putting mathematics under several appropriate themes (Chronaki, 2000; Handal & Bobis, 2003)

             Clearly, various strategies have been used to implement investigative approaches into the classrooms. Some of them took as short as 10 minutes to be successfully applied but some of them may take a longer period of time. Teachers must creatively and carefully choose and apply strategies that suit their students. Walle’s (1999) tips shall help teachers in choosing, building and implementing an effective investigative, discovery or open-ended assignments.

Conclusions

             The claim that there are tedious list of curricula loads is relatively true but the statement that teachers do not have efficient time to implement the investigative approaches may be questioned. It is true that in mathematics, there are many skills, facts and concepts that must be covered. However, proper planning (by considering tips and techniques to implement constructivism), smart usage of available resources, creative exploitation of existing situation and balanced implementation of open and close mathematics will ensure that both meaningful understanding and social legitimate mathematics status can be achieved. Furthermore, teachers should work harder to improve their teaching skills and professional competency by upgrading their educational knowledge either by joining professional development programs or actively reflects and improves on own teaching practices through action research.

             On the other hand, considering that human is developed by a combination of cognitive and behavioural elements (intelligence and emotional), a combination of both cognitive and behavioural education may produce a better person. Therefore, the value and behaviour in mathematics education such as the discipline of discussion, the critical and analytical thinking, the attitude in decision making and diligence in completing tasks might be useful to educate and improve both sides of human development. To conclude, mathematics education poses high potential to play an important role in improving human behaviour and developing better human. Therefore a bit of traditional touch should be put into constructivist approach so that all previously described as “uncovered holes” could be successfully covered.

             Therefore, for the case of Malaysia, which known by the strong cultural practice within locals and solid relationship between spiritual and academic education, attention and consideration should be put on developing ‘human mathematician’ which excels in all behavioural, emotional, and cognitive aspect through integrated mathematics education.

REFFERENCES

Aynes, P., & Sweller, J. (2000). Why investigation should not be compulsory in the years 9-10 mathematics   courses Report on investigational tasks in mathematics in years 9-10 for advanced and intermediate students (pp. 29-56). New South Wales: Board od Studies NSW

Bartholomew, H. (2003). Ability grouping and the construction of different types of learner in mathematics classroom. Paper presented at the 26th annual conference of the Mathematics Education research Group of Australasia, Geelong, Vic.

Bell, G., & Henderson, C. (2004). 30 Track games for young children. Australian Primary Mathematics Classroom, 9(2), 19-24.

Boaler, J. (1998). Open and closed mathematics: Student experiences and understanding. Journal for Research in Mathematics Education, 29(1), 41-62.

Boaler, J. (2000). Exploring situated insights into research and learning. Journal for Research in Mathematics Education, 31(1), 113-119.

Bright, G. W., Harvey, J. G., & Wheeler, M. M. (1985). Learning and Mathematics Games Journal for Research in Mathematics Education. Monograph., 1, i-189.

Carpenter, T. P., Fennema, E., & Franke, M. L. (1996). Cognitively Guided Instruction: A Knowledge Base for Reform in Primary Mathematics Instruction. The Elementary School Journal, 97(1), 3-20.

Chronaki, A. (2000). Teaching Maths through Theme-Based Resources: Pedagogic Style, 'Theme' and 'Maths' in Lessons. Educational Studies in Mathematics, 42(2), 141-163.

Civil, M. (2002). Chapter 4: Everyday mathematics, mathematicians' mathematics and school mathematics: Can we bring them together? Journal for Research in Mathematics Education. Monograph., 11, 40-62.

Clarke, D. (1997). Informing good practice by teachers, students, and others. Constructive assessment in mathematics: Practical steps for classroom teachers. (pp. 63-74). Berkeley, CA: Key Curriculum Press

Clarkson, P. C. (2008). Exploring the possibilities of using tic-tac-toe to think and communicate about mathematics. Australian Mathematics Teacher, 64(2), 28-33.

Clewell, B. C., de Cohen, C. C., Campbell, P. B., Perlman, L., Deterding, N., Manes, S., et al. (2004). Review of evaluation studies of mathematics and science curricula and professional development models. Unpublished Report. GE Foundation

Draper, R. J. (2002). School mathematics reform, constructivism, and literacy: A case for literacy instruction in the reform-oriented math classroom. Journal of Adolescent & Adult Literacy, 45(6), 520-529.

Dryden, W., & Ellis, A. (2003). Albert Ellis Live! London: SAGE Publications

English, L. D. (2006). Mathematical modelling in the primary school: Children's construction of a customer guide. Educational Studies in Mathematics, 63, 303-323.

English, L. D. (2009). Promoting interdisciplinarity through mathematical modelling. ZDM Matehematical Education, 41, 161-181.

Erickson, D. K. (1999). A problem-based approach to mathematics instructions. Mathematics Teacher, 92(6), 516-521.

Flewelling, G. (2005). Comparing the games we play in the classroom. Australian Mathematics Teacher, 61(1), 20-24.

Frid, S. (2002). Engaging primary students in working mathematically within a virtual enrichment program. Mathematics Education Research Journal, 14(1), 60-79.

Gijbels, D., & Loyens, S. M. M. (2009). Constructivist learning (environments) and how to avoid another tower of Babel: reply to Renkl. Instructional Science, 37(5), 499-502.

Glasersfeld, E. v. (1987). Preliminary to any theory of representation. In C. Janvier (Ed.), Problems of representation in  the teaching and learning of mathematics. Hillsdale, NJ: Lawrence Erlbaum.

Glasersfeld, E. v. (1989). Constructivism in education. In T. Husen & T. N. Postlethwaite (Eds.), The international encyclopedia of education (Vol. Supplement Vol. 1, pp. 162-163). Oxford/ New York: Pergamon Press.

Goos, M., Dole, S., & Makar, K. (2007). Supporting an Investigative Approach to Teaching Secondary School Mathematics: A Professional Development Model. Paper presented at 30th annual conference of the Mathematics Education Research Group of Australasia. MERGA.

Grimison, L., & Dawe, L. (2000). Report supporting investigations for the advanced and intermediate courses of the NSW mathematics years 9-10 syllabus. Report on investigational task im mathematics in year 9-10 for advanced and immediate students. (pp. 4-28). New South Wales: Board of Studies NSW

Guiding Principles for Mathematics Curriculum and Assessment (2009).  Retrieved October 13, 2009, from http://www.nctm.org/standards/content.aspx?id=23273

Halverscheid, S. (2005). Features of mathematical activities in interdisciplinary, project-based learning. ZDM 37(3), 200-207.

Handal, B., & Bobis, J. (2003). Instructional styles in the teaching of mathematics thematically. International Journal for Mathematics Teaching and Learning (Online Journal)  3 October 2003. Retrieved October 25, 2009, from http://www.cimt.plymouth.ac.uk/journal/handalbobis.pdf

Lee, Y. L. (2008). Educational game themes of a fraction brick game. Paper presented at the ASCILITE 2008 Melbourne: hello! where are you in the landscape of educational technology? : proceedings : 25th annual ASCILITE conference : November 30 - December 3, 2008. from http://www.ascilite.org.au/conferences/melbourne08/procs/lee-poster.pdf

Manouchehri, A. (2004). Implementing mathematics reform in urban schools: A study of the effect of teachers' motivation. Urban Education, 39(5), 472-508.

Mayer, R. E. (2004). Should there be a three-strikes rule againts pure discovery learning?: Case for guided methods of instruction. American Psychologist, 59(1), 14-19.

Monaghan, J. (2004). Teachers' activities in technology-based mathematics lessons. International Journal of Computer for Mathematical Learning, 9, 327-357.

Norton, S., McRobbie, C. J., & Cooper, T. J. (2002). Teachers' responses to an investigative mathematics syllabus: Their goals and practices. Mathematics Education Reseacrh Journal, 14(1), 37-59.

Simpson, D. (2005). Place value trading games. Prime Number, 20(2), 6-8.

Smith III, J. P. (1996). Efficacy and Teaching Mathematics by Telling: A Challenge for Reform. Journal for Research in Mathematics Education, 27(4), 387-402.

Sullivan, P., Clarke, D., & Clarke, B. (2009). Converting mathematics tasks to learning opportunities: An important aspect of knowledge for mathematics teaching. Mathematics Education Reseacrh Journal, 21(1), 85-105.

Thomson, I. (2008). Integrating technology into the mathematics classroom. Teaching Mathematics, 33(3), 28-34.

Walle, J. A. V. d. (1999). Reform mathematics vs the basics: Understanding the conflict and dealing with it. Retrieved October 25, 2009, from http://www.mathematicallysane.com/analysis/reformvsbasics.asp

Wiest, L. R. (2006). Fostering number sense through digits and dice. Australian Primary Mathematics Classroom, 11(4), 10-14.

William, D. (2005). Keeping learning on track: Formative assessment and the regulation of learning. Paper presented at the 20th biennal conference of the Australian Association of Mathematics Teachers, Sydney.

Zevenbergen, R. (1996). Constructivism as a liberal bourgeois discourse. Educational Studies in Mathematics, 31(1/2), 95-113.