Research indicates that teacher effectiveness has a direct impact on learning efficiency. This has triggered the debate over what constitutes effective teaching and learning in mathematics. The increased focus on developing teacher’s abilities to deliver high-quality outcomes suggests that attention must be devoted to the issue of what constitutes effective teaching and learning in our primary schools (Wells, 1999). The current Australian curriculum calls for the adoption of effective teaching and learning practices in primary school mathematics aimed at improving the understanding of the subject.
Recently, there has been an outcry protesting against the current teaching system. There have been complaints by some parents and politicians that demand that we go back to the basics in teaching mathematics in our primary schools. This may suggest that these basics have been abandoned by teachers or are not being taught appropriately. The outcry implies that all we need to teach and emphasize while teaching mathematics in our primary schools today, the central arithmetic skills which were the foundation that many parents went through in the 1970s.
These basic arithmetic skills entailed subtraction, addition, division as well as multiplication (Perso, 2012). However, though the article ‘Back to Basics’ raises important computational concerns that should be addressed, the supposition that these basics are being neglected in our schools is incorrect. Even though some parents might prefer the traditional curriculum, the new curriculum is superior because it ensures that the teachers stick to some interaction models which are helpful in the teaching-learning process. This paper aims at addressing the main concerns raised in the article ‘Back to Basics’, by conducting some activities with students, to demonstrate effective practice that today’s teachers employ.
‘Back to the Basics’
The call for the current teaching and learning methodology to be taken back to the basics may imply that teachers and students are headed in the wrong direction. However, they cannot get back to their traditional ways of doing things as that would mean retrogressing in their system of teaching (Perso, 2012). The reason why the primary school students were not using the calculators some years back was mainly due to the inadequacy of resources. Similarly, machines such as computers were once uncommon, but have recently become more affordable.
The students use the gadgets to make their work easier and thus save time. The learners are also likely to encounter these gadgets in their daily activities in the future as they are now in an era of advanced technology (Bruner, 1961). Moreover, the basic computational skills are still being taught in our schools and have not been neglected as some may argue. However, this is done in moderation and considers the current market demands as it is wrong to over-expose the learners to these basic computational skills which may not be relevant in this era of improved technology.
According to Cole and Wertsch (1996), effective practice in teaching mathematics focuses on the current demands of life and work. The current Australian curriculum as stipulated by The Australian Curriculum Assessment and Reporting Authority (ACARA, 2011), draws insights from the constructivist approach which states that teachers should not merely train pupils to be reckoners as was the case in the ’70s but rather, they should systematically teach the basic skills before embarking on the use of calculators and computers (Springhouse Corporation, 1990). This way, learners do not over-rely on the gadgets as they already have the required skills with them.
Indisputably, the Australian curriculum ensures that all students are equipped with the fundamental skills, facts, and capabilities that are essential to flourish and participate in the world as a global society (ACARA, 2011). The curriculum requires the teachers to appreciate the fact that the learners require to be taught significant learning skills which will aid them in their entire lifetime. As we teach various skills, we carefully plan our lessons which are first introduced by various activities to familiarize the students on a given topic.
This is then followed by other advanced activities that ensure that our students are deeply rooted within the topic in question. Our current curriculum prepares an all-round individual who can take up different careers, which may not necessarily be formal as was the case in the ’70s where the curriculum only focused on formal training. It employs the constructivist approach which considers the view that learners in this century need to go beyond acquiring facts and skills (Bruner, 1961). The constructivist approach is modeled on the works of psychologist Jean Piaget.
Piaget believed that children “understand whatever information fits into his/her established view of the world. When information does not fit, the person must re-examine and adjust his/her thinking to accommodate the new information” (Springhouse Corporation, 1990). Therefore, when teaching a given mathematical concept, a teacher is required to provide vivid and relevant activities which the learners enjoy and can identify with. This way, they can form and retain the concepts taught in their minds. The current Australian curriculum requires the teachers to plan teaching/learning experiences that embrace this theory.
Such experiences should easily connect to the learners’ current knowledge as well as understanding. Learners are encouraged to understand the knowledge they have acquired fully, critically evaluate it and apply it in a variety of real-life situations, and use it to formulate more knowledge. Hence, the teachers in the 21st century present knowledge in several ways to enable the learners to construct the knowledge and make sense out of it and then transform it according to their new demands (Matthews, 1994). Besides, it recognizes the fact that balance is critical in the amount of content or discipline knowledge to be exposed to learners. For instance, learners should only be presented with just the right information that they can hold depending on their level of understanding as required by the Australian curriculum.
Moreover, the Australian curriculum checks that the teachers balance the learners’ processes and skills with mathematical content while teaching the subject (ACARA, 2011). They mainly focus on teaching how to learn and apply the content in real-life situations as well as how to reflect on any given mathematical question. The curriculum focuses on skills such as teacher-student interaction, critical analysis, reflection and justification of mathematical concepts.
Teacher-student interaction can be achieved through the formation of discussion groups where a group of learners is presented with a given mathematical problem to solve as the teacher goes around the groups assisting the learners where necessary. The learners are also taught on how to solve the mathematical equations through reasoning in a logical manner to possess an in-depth understanding. The Australian curriculum also focuses on the learners’ cognitive skills that entail the understanding of the subject, the ability of the learners to make informed decisions, learners’ capability to infer mathematical concepts, their capacity to perform mathematical operations in several ways and their capability to account for the answers (Bruner, 1961).
For instance, while introducing a new topic in a lesson, the teacher has to first find out what his/her learners know and then introduces the topic of the day. This should then be followed by several advanced activities that are aimed at modeling and elaborating on the topic, making it more explicit. While doing this, the teacher should assist and guide the learners in areas they find difficult. The teacher can also provoke them to come up with their convincing mathematical formulas.
When introducing a new mathematical concept, an effective teacher first draws the learners’ attention, highlights the concept and then elaborates on the main ideas. To successfully draw the learners’ attention towards a given aspect in a mathematical concept, the teacher carefully questions, rearticulates or signals to the learners. This helps to put the learners’ sensory experiences into the task (Willis, 2005b). The teacher can then clarify or check the learner’s level of understanding through the use of a question or task.
This helps the teacher to identify the possible areas where individual learners’ need some help and calls for clarification. Also, effective teaching in the current Australian curriculum ensures that the learners are allowed to reflect, recount, recapitulate and review the concepts taught. In this process, the teacher shares other ideas and articulates more procedures as the students record them. This is meant to make the concepts more overt. It is after ensuring that the learners have thoroughly understood the concept in question, that the teacher can then demonstrate other advanced methods of solving mathematical problems.
Subsequently, the teacher may then test the learner’s understanding through the use of open-ended quizzes. Apart from helping to explore the level of children’s understanding, this process also assists in the formulation of more mathematical formulas (ACARA, 2011). In an attempt to demonstrate what constitutes effective practice in our schools today, a few activities were conducted with two students in year four. These activities are also intended to illustrate the outcomes of applying effective teaching and learning mathematics in our primary schools as postulated by the current Australian curriculum.
The first activity which was as stipulated by Willis (2005c) was meant to elaborate on the topic of directions as stipulated by the Australian curriculum, which states that the Year 4 students must be able to use appropriate directions (ACARA, 2011). According to ACARA, the mathematics curriculum for students between years 3 and 6 is meant to stress the significance of students studying meaningful, purposeful and coherent mathematics. ACARA also states that such mathematics also needs to be relevant to the lives of the students in question. In this case, two students were required to identify the different languages of direction that they were familiar with. As such, the two children would be expected to cooperate and work in harmony with one another to achieve the assigned task. This is because one student would be expected to use directional language and help his/her partner to get to where the object in question was placed (ACARA, 2011).
The curriculum for children in these years has been developed in such a way as to enable the students to develop a fundamental understanding of the subjects taught by extending the measurements, numbers, statistical and geometric learning from the early years (ACARA, 2011). As such students in this age group are normally expected to develop a deep understanding of the concepts taught so that they can build reasoning through mental computation skills.
Thus, each of the students was observed tracking the route to take to get to where the object had been placed. Once the first student was through, the second student took over and had to act as the guide. The two students demonstrated the ability to develop a positive attitude towards the task at hand, along with enjoyment and interest. This is in line with the observation made by Booker, Bond, Sparrow, & Swan (2010), who also note that the use of geometry in teaching mathematics enables students to utilize spatial knowledge and ideas in solving various practical problems that they encounter every day.
It was interesting to see the differences between the two students. Student A instructed her partner using the shortest route, while student B gave directions for a less direct route. When the students were questioned about what they had learned in that activity, Student B said that he had learned that learning was fun when you do things practically. Student A also agreed with this comment, and further added that working as a team is fun. To develop this concept, other advanced activities such as local excursions, battleships, and grid pictures should also be conducted. These activities are meant to provide a deeper mental picture for better understanding.
The second activity involved shapes with the key objective being to describe objects according to the different characteristics that they possess. According to ACARA (2011), by the time students complete Year 4, they should be able to describe and compare two-dimensional shapes that come about due to the integration and splitting of common shapes. This activity required the students to first name the various objects on display. After that, the students were then asked to classify the object that they had named with regards to the various shapes, that are boxes and cylinders, among others. The students were then expected to list the different features of the objects that were common to each group. Towards, this end, the students were requested to come up with a name for each of these groups.
As it was proving quite hard for them to do so, they were assisted by the teacher. Once this had been accomplished, a list of all the objects that had been used in different categories was created. The students were then invited to reclassify their collection according to other spatial features. The students classified the boxes in terms of their different shapes with little difficulty, compared with the first time when they had to classify without any guidance. The students managed to distinguish square boxes from the non-square boxes.
The students also managed to distinguish between the symmetrical and the asymmetrical boxes. The students were asked to explain why certain objects were categorized in one group, and not the other. In the future, these students should undertake more advanced activities to further develop their learning skills. This way, the students can sort the objects in terms of their symmetry, or even sort the objects in terms of their dimensions (Willis, 2005c). Such activities may include the classification of different fruits, leaves, and flowers.
The third activity required the students to explain the functions of different shapes. The ACARA has noted that three-dimensional shapes are, by and large, the world of children as it helps them to explore the various geometric senses (ACARA, 2011). Ideally, children should begin by exploring the three-dimensional shapes and objects that they are familiar with such as blocks and balls.
In this particular activity, the children were first asked to list the various shapes that they were familiar with after which they were required to draw a variety of objects and explain why their various parts were in their present shape. Student A drew a school bus but when I asked her why the wheels of the bus were round and not any other shape, she replied ‘just because that’s what they are.’ However, student B added, ‘the wheels are round because this shape is smooth and rolls to move the bus.’ This was a very appropriate answer. This demonstrated how Student A was provided the opportunity to re-examine her way of thinking and adjust her thinking.
Nonetheless, there is still the need to develop this particular concept further by incorporating other related activities. For instance, the students can correct various objects such as bottles, milk packets, cartons among others and elaborate on why they are best suited for their different shapes (Willis, 2005a). However, besides the use of familiar objects, the students should also be exposed to other shapes that they are not familiar with so that they can develop their skills.
The fourth activity was aimed at introducing the students to the concept of measurement. The ACARA curriculum for year 4 students on geometry involves more than the naming of various shapes and objects. It involves helping the children to understand better the spatial world they live in. Because geometry is important in the later years at both primary and secondary schools, there is a need to establish a solid foundation. As such, teachers should endeavor to engage students in exciting and interesting activities, and not tedious or boring worksheets.
In this particular activity, the two students gave a variety of measuring units after which they were presented with various books and boxes of various sizes and shapes. They were then required to estimate what particular box was likely to hold a given amount of books. To achieve this, the students had to explain which side of the book they were looking at to make an appropriate decision. Both of them had no difficulties in doing this, not to mention that they appeared to enjoy the exercise. In the future, there is a need to incorporate more demanding levels of measurements so that the children can exercise their mental capabilities to the fullest.
Undeniably, it is clear that our schools employ effective teaching. From all the activities above, it is evident that the current Australian curriculum addresses the basic mathematical skills thoroughly. Moreover, the learners seem to possess the knowledge and understanding of skills and concepts in the major areas of mathematics which include numbers, direction, shapes, and measurement (Kozol, 1991).
They possess the ability to use mathematical knowledge in various situations and can easily communicate through various mathematical concepts while still enjoying the subject. It is also clear that the Australian curriculum upholds and sustains critical thinking in the teaching-learning process. Moreover, it requires the mathematic teachers to facilitate various classroom activities to enhance the participation of the learners. The various activities enable the learners to expand and intensify their perception of a topic and thus supplement the development of knowledge.
The constructivist approach entails a move from merely focusing on the passing of information by the teacher to the students towards putting more emphasis on promotion of the teacher-student interaction which helps the learners to think openly and visibly (Bereiter & Scardamalia, 1998). This was evident in the activities where the learners interacted with their teacher freely. According to this approach, effective teaching and learning include; detection, depiction, and amplification of the interaction methods that enhance learning in our primary schools.
These activities can be chosen and used suitably for specific motives such as exploring and elaborating the known, testing and broadening the learners’ mathematical judgment capacity. More importantly, the activities are meant to help teachers to make wise resolutions with regards to meeting the learners’ needs in the most suitable manner. These practices are meant to present the teachers with proficient procedures to handle various learners, which is the case in the constructivism theory.
The constructivist approach by Piaget lays more emphasis on the learners’ intellectual growth and postulates that a tutor should not view the learner as a mere vessel to be packed with information, but rather, a learner should be viewed as an associate in active experimentation (Springhouse Corporation, 1990). Moreover, as advocated by the constructivist approach the activities were mainly aimed at developing the learners’ cognitive growth (Wells, 1999). This concept is also supported by the current Australian curriculum. In the previous activities, the focus was on interacting with the learners to develop the concepts systematically and profoundly.
In conclusion, today’s teachers go beyond passing information to their learners. Teachers adhere to the current curriculum which ensures effective approaches in teaching and learning mathematics. The curriculum endorses the application of the constructivist approach in the learning process, which has several benefits to the learners. For instance, it ensures that the learners are actively involved in their education; it guarantees relevancy of the education system to real-life situations and it necessitates correction of previous knowledge.
Indeed, the current curriculum demands that teachers balance the learners’ cognitive processes and skills with mathematical content while teaching the subject. Additionally, they should focus on teaching how to learn and how to apply the content in real-life situations as well as on how to answer any given mathematical question. For this reason, the supposition that the basic computational skills have been disregarded in our schools is erroneous. The current Australian curriculum meets the learners’ requirements at any given time by ensuring effective practice.
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Appendix – Consent Forms