Mathematics Education - Effective Mathematics Teaching and Learning

Assessment 2 - Report on Effective Mathematics Teaching and Learning
by Richard Kant (2013)

Effective mathematics and teaching and learning, is a cogwheel of many components and characteristics. These may include knowledge of how children learn mathematics, knowledge of teaching strategies, the impact of a teacher’s personal attitude toward mathematics, teaching strategies used by the teacher, teacher’s content knowledge, and knowledge of resources (including curriculum documents). Research may suggest that these various characteristics may contribute to positive student outcomes in mathematics education.

Children learn mathematics if it makes sense and is meaningful to them. This may be achieved if children are actively involved in the learning process. According to Reys, Lindquist, Lambdin, & Smith (2012), active involvement refers to children interacting with a wide range of resources, exploring and making sense, and reflecting on what they have done. For example, to see how shapes rearranged make other shapes, children may use pattern blocks and other concrete materials to help them visualise the geometric shapes and attributes. This exploring and sense making activity provides children with the opportunity to be actively involved through use of resources. They also have the opportunity to explicate or reflect by talking, elaborating, and discussing with their peers and the teacher on their observations and discoveries.

Communicating about mathematics may also help children make sense and meaning when learning mathematics. This is important in the learning process as this provides the teacher valuable insight about students thinking and understanding  (Reys, Lindquist, Lambdin, & Smith, 2012). For example, in the Shapes from Squares video (WGBH, 1997), the teacher allowed students to communicate their understanding of constructing shapes from squares by asking students to record their shapes by drawing. This allowed students not only to keep a track of their thinking, but also to verbalise and communicate what they were thinking as they completed the activity. This learning process also provided the teacher with information about student learning, understanding and misconceptions.

Finally, children may make sense and meaning of mathematics, when learning proceeds from concrete to abstract concepts. Piaget discusses this in his Four Stages of Cognitive Development (Eggen & Kauchak, 2010) stating that children need to start with concrete thinking first and then move to abstract reasoning. For instance, moving from Geoboards to GeoGebra (International GeoGebra Institute, 2013) may be an example of moving from concrete to abstract. GeoGebra is a free-software that combines geometry, algebra, tables, graphing, statistics and calculus in one package. From a geometric perspective, Geoboard activities are hands-on concrete activities and it provides algebraic foundations for more abstract thinking, which GeoGebra software provides. Understanding how children learn mathematics goes hand-in-hand with the teacher having knowledge of teaching strategies.

It is essential that teachers have knowledge bank of teaching strategies. Using the correct strategy for the right situation may allow students to “see mathematics as a sensible, natural and enjoyable part of the environment” (Booker, Bond, Sparrow, & Swan, 2010, p. 5). Allowing students to build on what they already know, mathematically and experientially, may be one such strategy. According to Sullivan (2011), it is important that students connect learning with experience. For example, year two students’ may already have knowledge of halves, quarters and eighths of shapes [ACMNA033], (ACARA, 2013a) from a previous unit. The classroom teacher may connect this prior learning and experience when teaching time to the quarter-hour [ACMMGO39] (ACARA, 2013b) when preparing the lesson. According to Booker, Bond, Sparrow, & Swan, (2010), it is particularly important that knowledge is connected in explicit ways. In particular, spatial ideas and number understanding underpins measurement whereas geometry evolves from everyday experiences (Booker, Bond, Sparrow, & Swan, 2010). It may be that these types of connections may result in effective learning. 

Another strategy is, ensuring that students are engaged in rich and challenging mathematical tasks (Sullivan, 2011). This means that mathematics learning must be interesting for the students where the teacher presents a variety of relevant tasks that are meaningful and relevant. For example, to foster engagement in a geometry lesson on nets, a teacher may ask students to use Toblerone packaging to see how many different nets they can make. This activity may be fun, interesting and potentially engaging for the students (as opposed to watching a teacher demonstrate the various nets) because they are using an interesting model (Toblerone packaging) to discover solutions to a practical situation.

Planning to support students who need help and challenging other students who are ready is another strategy (differentiation) in teaching mathematics. Sullivan (2011), states teacher interaction with students is important and students should be encouraged to interact with each other including asking and answering questions. Most importantly teachers should differentiate student support according to the needs of the students (Sullivan, 2011). This was demonstrated in the video, Shapes from Squares video (WGBH Educational Foundation, 1995), where the teacher had students working in groups. The teacher moved from group to group interacting with the students and asking them questions to help them understand concepts and solve problems without giving them an answer. Differentiation was evident in one scene (WGBH Educational Foundation, 1995), where the teacher helped a student understand the meaning of sides through a series of well-planned questions. The teacher in the video (WGBH Educational Foundation, 1995) also challenged the students by asking them to name the shapes created. These practical strategies (direct and indirect differentiation) may help support students who need help and challenge other students. The strategies used to teach mathematics may be influenced by teacher’s attitudes and beliefs towards the subject and that these, may have an effect on student attitudes (Relich, Way, & Martin, 1994).

Teachers, who have a positive attitude toward mathematics as a subject and towards the teaching of mathematics, may develop positive attitudes towards mathematics in their pupils. According to Relich, Way, & Martin (1994), many teachers view mathematics as a subject that was not a pleasurable experience at school and this may have an impact on the students they teach. For example, some teachers may dislike mathematics because they do not see the relevance to the real-world. It may be that for these teachers, a deeper appreciation of the importance of mathematics is lacking. In the case of a teacher who cannot see how mathematics relates to the real world, these teachers may be shown how it does relate to the real-world. For example, understanding how mathematics is nature’s language and how it communicates directly with individuals may help them see how it relates to the real-world. This deeper appreciation and engagement may be encouraged and developed at pre-teacher service level. Once this is achieved, these teachers may pass on this positive attitude to the students they teach. 

Many teachers may possess negative attitudes towards mathematics. According to Relich, Way, & Martin (1994), it may be that negative teacher attitudes towards mathematics can be changed at teacher training level so that it correlates with student achievement, enjoyment and perception of mathematics. An extract from A National Statement on Mathematics for Australian Schools (Australian Council, 1991, p.31):

     An important aim of mathematics education is to develop in students                                       positive attitudes towards mathematics and their involvement in it…The notion of having a positive attitude toward mathematics encompasses both liking mathematics and feeling good about one’s own capacity to deal with situations in which mathematics is involve.

This statement shows the significance of student’s positive attitudes towards mathematics and stresses the importance of developing and retaining positive attitudes towards mathematics. Research and studies have shown that teacher attitudes towards the teaching of mathematics are “important determinants of student attitudes and performance in mathematics” (Relich, Way, & Martin, 1994, p. 59). For example, a teacher who has a negative attitude about teaching geometry may “have a powerful impact on the atmosphere and ethos of the mathematics classroom” (Relich, Way, & Martin, 1994, p. 59). Students may be perceptive to this negativity which may influence the atmosphere and ethos of the classroom. Therefore, a positive attitude towards mathematic and the teaching of it are important characteristics. Equally important, in the cogwheel of effective mathematics learning and teaching is the teacher’s knowledge for the teaching of mathematics. 

Subject matter knowledge and pedagogical content knowledge are the two important types of knowledge that may be required to teach mathematics. Subject matter knowledge is a central requirement for teaching, which entails helping children learn, and understanding what is to be taught (Sullivan, 2011). Teachers may refer to the Australian Curriculum, Assessment and Reporting Authority (ACARA) mathematics curriculum documents to develop knowledge about what is to be taught. Buchman (1984), states that it is unrealistic to expect teachers to plan lessons if they are ignorant about subject matter and therefore a component that teachers need to know. Specific mathematics subject content knowledge may be developed through regular ongoing professional development. It could be argued that relying on content knowledge is necessary, but may not be sufficient. Sullivan (2011), states that knowing how to solve a mathematical problem may be completely different to knowing how to help children solve mathematical problems. It may be that pedagogical knowledge (Sullivan, 2011) should be combined with subject matter knowledge in order to be more sufficient in helping children solve mathematical problems. Pedagogical knowledge of content and teaching, and content and students (Sullivan, 2011) could be combined with knowledge of mathematics subject content. For example, in teaching geometry the teacher may need to understand how to sequence particular curriculum content for instruction and evaluate the advantages and disadvantages of resources and representations. Geometry teachers may also pay attention to visual and intuitive thinking and be aware of that in order to help children solve geometric problems. It may be important that teachers do not attribute errors to procedural matters, rather take into consideration conceptual issues that may have contributed to that error. The prescriptions they provide the students must match the diagnosis. Knowing how to use subject matter knowledge is what teachers may have to focus on. It may be important that teachers have strategies to foster the development and sustaining of subject matter knowledge and pedagogical knowledge (Sullivan, 2011). These strategies may include ongoing, collaborative, school-based teacher professional learning that may include pedagogical studies and a detailed knowledge of resources to support learning. 

Teacher’s knowledge of resources may be a critical component in the characteristics of effective mathematics classroom teaching, and learning. The variety of resources to teach and support student learning in mathematics may include the use of information, real/virtual manipulatives, technology and curriculum documents (Marsh, 2010, p. 242). Physical materials such as paper models, pattern blocks, and geoboards are some materials that may be used to help students develop geometric representations.  Research has proven that these types of physical teaching aids are helpful in developing visualisation and spatial reasoning skills (Reys, Lindquist, Lambdin, & Smith, 2012). Geoboards may also be used to help students explore a variety of mathematical topics. For example, students may stretch bands around pegs to form line segments and polygons, and make discoveries about perimeter, area, angles, congruence, and fractions. Virtual versions of the manipulative may be used as an open-ended educational tool in the classroom as well as a variety of electronic geometry manipulatives, which can be accessed from the internet. It is valuable for teachers to use real and virtual manipulatives, especially if the activities help children learn the subject matter (Marsh, 2010) and it focusses on the learning needs of the students.  Information technology resources make it possible for classroom environments to be multi-sensory (Reys, Lindquist, Lambdin, & Smith, 2012). For example, shape-making computer programs allow students to create highly sophisticated shapes. Geometry and measurement iPad applications combine visual image, text, animation, and sound to help students not only develop at different rates but also help students become computer-literate. However, it may be that this be kept in perspective as non-computer based teaching can sometimes be just as effective (Reys, Lindquist, Lambdin, & Smith, 2012). Other digital resources such as Scootle (2012) may support teachers teaching the Australian Curriculum mathematics learning area. For example, teachers of all levels are able to search for a wide variety of digital resources on the Scootle (2012) website on specific learning areas and topics. Teachers preparing learning experiences on geometry have access to 430 digital classroom resources and 85 teacher reference materials (Scootle, 2012) that directly provide students and teachers with support materials that are aligned to the Australian, state and territory curriculums.

Curriculum documents are another resource available to teachers to teach measurement and geometry. These documents may be used to support both teacher and student learning. Mathematics Scope and Sequence document (ACARA, 2012) published by the Australian Curriculum, Assessment and Reporting Authority (ACARA) is one such document that may support teachers in planning measurement and geometry learning experiences. Teachers may use this document to prepare learning experiences that are logical in scope and sequence. To help support teacher learning, a glossary (ACARA, 2013c) published by ACARA contains common key terms used in the content descriptions. This document may be used to help teachers revise common key terms or use it as a reference when preparing learning experiences. 

Children may make sense and meaning of mathematics by being actively involved, communicating about mathematics, and proceeding from concrete to abstract concepts. In order to help student make sense and meaning of mathematics, teacher’s need knowledge of teaching strategies such as building on what children already know, engaging them in rich and challenging mathematical tasks and supporting students who may need help and challenge others who are ready. These strategies may be influenced by a teacher’s attitudes towards the subject and teaching. Subject matter knowledge, pedagogical content knowledge, and sustaining teacher improvement by ongoing collaboration and professional development, may be important. Knowledge of resources to support learning may be critical in the element of effective mathematics teaching. Technological, real/virtual manipulatives and curriculum documents are some resources that may help support student learning.

References

WGBH (Producer). (1995). Teaching Math: A Video Library, K-4 - Shapes from Squares [Motion Picture]. Boston. Retrieved from http://www.learner.org/vod/vod_window.html?pid=888

Scootle. (2012). Retrieved from http://www.scootle.edu.au/ec/p/home

ACARA. (2012). Mathematics Scope and Sequence: Foundation to Year 6. NSW: ACARA. Retrieved from http://www.australiancurriculum.edu.au/Australian%20Curriculum.pdf?Type=0&a=M&e=ScopeAndSequence

ACARA. (2013a). Mathematics / Year 2 / Number and Algebra / Fractions and decimals. Retrieved from http://www.australiancurriculum.edu.au/Elements/ACMNA033

ACARA. (2013b). Mathematics / Year 2 / Measurement and Geometry / Using units of measurement. Retrieved from http://www.australiancurriculum.edu.au/Elements/ACMMG039

ACARA. (2013c). The Australian Curriculum Mathematics (Glossary). NSW: ACARA. Retrieved from http://www.australiancurriculum.edu.au/Australian%20Curriculum.pdf?Type=0&a=M&e=Glossary

Australia, E. S. (2013). Educational value standards for digital resources. Educational Value Standard.

Booker, G., Bond, D., Sparrow, L., & Swan, P. (2010). Teaching Primary Mathematics. NSW: Pearson.

Buchmann, M. (1984). The priority of knowledge and understanding in teaching. Norwood NJ: Ablex.

Eggen, P., & Kauchak, D. (2010). Educational Psychology: Windows on Classrooms (8th ed.). New Jersey: Pearson.

Institute, I. G. (2013). GeoGebra. Retrieved from http://www.geogebra.org/cms/en/

Marsh, C. (2010). Becoming a teacher: knowledge skills and issues (5th ed.). Australia: Pearson Education.

Relich, J., Way, J., & Martin, A. (1994). Attitudes to Teaching Mathematics: Further Development of a Measurement Instrument. Mathematics Education Research Journal, 6(1), 56-69. Retrieved from http://www.merga.net.au/documents/MERJ_6_1_RelichWay%26Martin.pdf

Reys, Lindquist, Lambdin, & Smith. (2012). Helping Children Learn Mathematics (10th ed.). John Wiley and Sons Inc.

Sullivan, P. (2011). Australian Education Review - Teaching Mathematics: Using research-informed strategies. VIC: Australian Council for Educational Research (ACER).