Technologies for Learning - Digital Story

Technologies for Learning - Telling Time Website

Technologies for Learning - Rubrics

Rubrics allow educators to communicate expectations, and provide feedback (Flinders University, 2013). However, they can also be used to evaluate digital learning resources (DLR). Having conducted research, design considerations had to be made. These steps were essential in evaluating the Scootle DLR.

Researching DLR rubrics required a systematic approach that was a positive experience. Several rubrics were used as an example to create another rubric. Firstly, a list of search words were made that would yield the results needed. There were no issues encountered in this planning step as prior reading and web-surfing helped. Secondly, information was gathered. According to Shelly, Gunter, & Gunter, (2012), teachers should consider the credibility of sources. It was easy to decipher which were/were not credible. Most sources gathered were not affiliated with professional educational organisations and these were filtered out.   Finally, the information collected was organised. It was important at this stage to think about how the final product was to be structured.  Taking a systematic approach to research by planning, gathering, and organising collected knowledge, and data was a positive experience. Without this approach, it could have been a time consuming and less rewarding experience that may have resulted in a product not suitable.

Designing the rubric was not easy. According to Education Services Australia (2013), rubrics should be used to make performance levels explicit for the teacher. This was a difficult because, in order to create a DLR rubric, it was important to know what to evaluate, and write performance descriptors to identify differences between the levels. The first step in designing a DLR rubric was listing characteristics to evaluate. This was difficult as there were many to consider. It was necessary to consider research on tangential learning and games based learning. Considering these may result in students being engaged or motivated (Breuer & Bente, 2010). However, according to Squire and Jenkins (2003), to motivate students it is necessary to find a balance between entertainment and learning.  The second step in designing the DLR rubric was deciding on the numerical levels. This also proved to be a difficult because a thorough familiarity with the highest quality DLR resource needed to be understood, and the range of capabilities of what is available on the internet. The final step was to ensure that the words used in the rubric were not valueless. According to Moskal (2000), standards in the rubric need to be clearly defined. This may have been achieved in the DLR evaluation rubric as the words used in the rubric allows for detailed analysis of the DLR.

Evaluating the selected DLR resource proved to be an interesting aspect of this entire process. What seemed like an ideal resource, turned out to be a resource that was below average for learning. It has been discovered that criteria such as tangential learning and games based learning in DLR’s are important criteria’s. Having experienced this, an awareness of the importance of such evaluative practices has been instilled.

In conclusion, the process of researching, designing and evaluating a DLR’s has produced a rubric with unique criteria’s such as tangential and games-based learning. It was a rewarding experience yet at the same time a difficult task when designing. Most importantly, it has been discovered that spending time creating a quality rubric to evaluate DLR’s to promote learning.

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).

Mathematics Education - Current Practice in Mathematics Education

Assessment 1: Teaching Plan

Current practice in mathematics education: what it looks like, sounds like and feels like
by Richard Kant

Introduction

The purpose of this paper is to outline current practice in mathematics education. O’Brien (1999) in his article “Parrot Math” discusses constructivism as opposed to behaviourism. Part B of this report presents two lesson plans and discusses the activities conducted.

Part A – Description of current teaching practice in mathematics

There are three different viewpoints in the current teaching practice in mathematics. These different viewpoints are behaviourism, cognitivism and constructivism. Firstly, behaviourism learning is a theory that views learning has occurred when students receive regular, expected responses. Instruction according to behaviourism is repetition and reinforcement (Eggen & Kauchak, 2010).  Secondly, cognitivism views the mind as a storage device. According to cognitivism theory, learning is recalling stored information to demonstrate that learning has occurred. Instruction according to cognitivism is to obtain the learner’s attention and help them make sense of information and store it for later recall (Eggen & Kauchak, 2010). Finally, constructivism is a theory that views the mind as a rhizome. Skills and knowledge are interconnected for it to be recalled as needed. According to constructivism, learning is building knowledge by practical experience and the role of the teacher is to guide problem-solving in a community of learners (Eggen & Kauchak, 2010).

Based on these three views, there are two major types of theories, descriptive theory and prescriptive theory. Firstly, descriptive theory aims to answer what learning is. The outcome of this is attempts to describe learning. Secondly, prescriptive theory attempts to answer how can educators help students to learn and develop? The outcome of this is instructional theory, which provides methods to foster learning (Hogue, 2012).

O’Brien (1999) in his article, “Parrot Math” outlines a constructivist based philosophy to teaching mathematics as opposed to behaviourist approach advocated by a group of well-organised critics. These critics claim that mathematics education should be confined to algorithms of arithmetic. O’Brien (1999) refers to research conducted by Kamii and Dominick (2009) who believe that algorithm of arithmetic, is harmful. Kamii and Dominick’s (2009) research shows that students with “no algorithm” experience performed best on the mental test. However, it could be argued that these students did well in the “no algorithm” test due to considerable experience in mental math methods. On the other hand, students who had algorithm experience failed the test because they were not allowed to use paper and pencil, which is normally the standard procedure (Quirk, 2013). It could be argued that the research carried out may be biased, and it is difficult to work out from the research, as to how algorithm of arithmetic is harmful.

O’Brien (1999) states, critics believe that routine procedures should be transmitted by the teacher with considerable memorization and drill-work. Clements and Battista (1990) like O’Brien (1990), view this as curriculum based on transmission of teaching and learning where students passively “absorb” knowledge created by others. O’Brien (1999) rejects memorisation and practice in favour of maximising “understanding” and developing “powerful thinking skills”. Understanding and thinking skills may be important; however, critics may argue that these skills depend on remembered content (Quirk, 2013). However, in order to remember content, learners should attempt to make sense of and interpret information in a personal way (Eggen & Kauchak, 2010) rather than mindless repetitions and drills to aid memorisation.

According to O’Brien (1999), constructivism is viewed as a fad and new approaches to teaching is criticised by critics. O’Brien (1999) discusses that mathematics teaching should be activity-based, supported by a constructivist philosophy and involving the real basics of classifying, inferring, generalising and hypothesising. He discusses that teachers should harness children’s urge to make sense of things and help them find meaning in maths.  This may be achieved through a constructivist and cognitivist-based philosophies combined with quality mathematics instruction where the professional practitioner is able to improve instruction by being reflective, engaging in professional development, curriculum development and research (Wright, Ellemor-Collins, & Tabor, 2012).

Having considered O’Brien’s (1999) article and scholarly education references, there is no doubt that current practice in mathematics is based on constructivist and cognitivist philosophies as opposed to be behaviourist philosophy. According to Eggen and Kauchak (2010), behaviourism is not a preferred method of instruction however, it can be used to help create a positive environment and control student behaviour. Current practice in mathematics education may look like students of all diversities are actively engaged in making sense of concepts that are presented in a sequential manner appropriate for the developmental level of the students. The classroom may sound low-level noisy where students and teachers are enjoying the learning process through social interaction, instructional games, authentic mathematical tasks, investigations and activities using technological resources (Booker, Bond, Sparrow & Swan, 2010). It seems constructivist learning and teaching is preferable to teacher-centred instruction. According to Booker et al (2010), the role of the teacher is to assist and allow students to construct their own ways of knowing. This can be demonstrated through prescriptive theory. Mathematics classrooms are now environments where students feel supported and help them make sense of mathematics (Reys, Lindquist, Lambdin, & Smith, 2012).

In summary, O’Brien (1999) outlines his viewpoint on how the current teaching practice in mathematics should be. This viewpoint is constructivism as opposed to behaviourism. Based on his view of constructivism, he attempts to provide evidence as to how to help students learn and develop so that it provides methods to faster mathematics learning (prescriptive).

Part B – Lesson Plan & Discussion of Activities

LESSON PLAN 1

Learning Area	Year	Time/Session	Date
Mathematics	Foundation	30 minutes	17.6.13

Topic/Lesson Title: Sequencing Events

PREPARATION

Australian Curriculum, Assessment and Reporting Authority (ACARA) Code

·         “Compare and order duration of events using the everyday language of time (ACMMGOO7)” (ACARA, 2013). ·         Elaboration: “Sequencing familiar events in time order” (ACARA, 2013).

Objectives

At the end of the lesson the students will be able to: 1. Answer questions about everyday family routines. 2. Sequence illustration of events. 3. Sequence the illustrations from Goldilocks and the Three Bears 4. Tell the story with reference to the sequenced pictures.

Preparation / Resources

1. Pictures (SparkleBox, 2006) of everyday events. 2. Make a list of questions to ask for objective one. 3. iPad/ iPad application – Goldilocks and the Three Bears (interactive storybook) by A Tab Tale Production (TabTale, 2013). 4. Goldilocks and the Three Bears illustrations (Goldilocks and the Three Bears, 2011). 5. Props for scenes from story

Summary of Tasks (Non-Differentiated)

1.           Arrange daily routine cards in sequence of events 2.           Listen to interactive storybook on iPad 3.           Students to pretend they are Goldilocks and act out the scenes using props that they must set up. 4.           Students to arrange illustration cards in correct time order according to the story. 5. Students to verbally tell the story with reference to the sequenced pictures

Teaching/Learning Strategies (Non-Differentiated)

Tuning In – determine students’ current knowledge, skills and attitudes through questioning. Think-pair-share – making a list of all activities that students do in a day. Sorting out daily routine cards. Viewing/Open Questioning – interactive storybook on iPad. Students to pay particular attention to the order of events in the story. Mind Map –of the events in the story to help students identify, visualise and record their understanding of the story. Role-play – act out scenes from story in sequence using props. Reflecting (Unfinished sentences) –students are to reflect on their learning by verbally completing incomplete sentences (Appendix E).

LESSON PLAN 2

Learning Area/General Capabilities	Year	Time/Session	Date
Mathematics	Foundation	30 minutes	17.6.13

Topic/Lesson Title: Days of the week

PREPARATION

Australian Curriculum, Assessment and Reporting Authority (ACARA) Code

·         “Connect days of the week to familiar events and actions (ACMMG008)” (ACARA, 2013a). ·         Elaboration: “Choosing events and actions that make connections with students’ everyday family routines” (ACARA, 2013a).

Objectives

At the end of the lesson the students will be able to: 1. Link language such as before, after, yesterday, today and tomorrow, with the days of the week. 2. Answer questions (Appendix C)

Preparation / Resources

1. Days of the week cards. 2. Phrase Cards (Appendix D): 3. Cards with questions from lesson objectives. 4. iPad 5. Days of the week song: http://www.youtube.com/watch?v=OPzIbbvoiMA

Summary of Tasks (Non-Differentiated)

1. Watch song on YouTube http://www.youtube.com/watch?v=OPzIbbvoiMA 2. Arrange days of the week flash cards in order 3. Phrase cards (Appendix D) 4. Questions (Appendix C) Teaching/Learning Strategies (Non-Differentiated)

Tuning In Determine students’ knowledge of days of the week through conversation. Guided Discovery mini-excursion to the school administration office to see the staff dairy and how it is used. Think-pair-share Discuss what the words “before, after, yesterday, today and tomorrow” mean. Viewing –YouTube video and singing along. Pretending Activities Reflecting (Unfinished sentences) – Appendix E

DISCUSSION OF ACTIVITIES

Two mathematics lessons were demonstrated through behaviourist, cognitivist and constructivist philosophies. Based on these philosophies, prescriptive theory was used to allow both students (Appendix A) to maximize understanding, develop thinking skills, make sense and find meaning in the topics. All three philosophies were combined with the dosage of behaviourism being the least in the lessons. Activities were conducted using constructivist techniques and observations of mathematical learning were recorded. Resources used in the lessons maybe in line with research and current practice. The activities conducted, were the beginning stages in the children’s overall mathematics learning plan, and may relate to current, best practices in mathematics education.

Three philosophies were combined in the lessons. Firstly, a behaviourist approach was used to maintain behaviour rather than to use the approach to guide instruction as discussed in the O’Brien article (1999). For example, in the lessons, the classroom environment was positive, emotionally safe, encouraging, and where student’s efforts were praised. According to Eggen and Kauchak, 2010, these behaviourist guidelines can be help create a positive learning environment and maintain student behaviour. Secondly, a cognitivist approach was used to help students find meaning and make sense of the concepts presented. For example, to teach the days of the week, different types of memory knowledge was taken advantage of rather than use of rote learning and repetition, as identified in the O’Brien article (1999). To teach the days of the week, students were asked questions on facts, concepts, procedures and rules (declarative knowledge). This ensured that students were thinking beyond factual understanding. Students were also given clear directions as to how to perform lesson tasks (procedural knowledge), which empowered them to achieve goals. Eggen and Kauchak, 2010 state that these types of knowledge may help students store information in the long-term memory, and therefore, help students find meaning, and make sense of the concepts. Lastly, a constructivist approach was dominantly used in the lessons. The lessons were grounded in social constructivism and students were provided with high-quality representations of content that related to the real world. It also included high levels of interaction and promoted learning with assessment (Eggen and Kauchak, 2010). Social interaction was encouraged throughout the lessons by allowing both students to work as a pair rather than individually on activities. High-quality examples were used throughout the lessons in order for students to understand the topics. For example, students were taken on a mini excursion to the administration office to see a real diary and how it was used. The purpose of this activity was not only to maintain the student’s interests but also to help them construct knowledge about the days of the week by gathering data through their senses (Costa and Kallick, 2000). According to Eggen and Kauchak, 2010, activities of such nature are examples of high-quality representation of content that relates to the real-world. Throughout the lessons, formative assessment was used to informally assess the thinking of the students to ensure that it was congruent with the lesson objectives.

During the lessons, mathematical learning was observed that was underpinned by a constructivist approach. The students demonstrated a sense of achievement by correctly completing games and activities, and it was important to praise them on their achievements. According to Eggen and Kauchak 2010, this form of praise may increase intrinsic motivation. Questioning of students throughout activities revealed that students found meaning in what was being taught. Collaborative activities ensured students were engaged in tasks. All activities were conducted in a manner that was enjoyable, playful, feeling of game-like and taking on a role of information provider and facilitator.  Both O’Brien (1999) and Eggen and Kauchak (2010) view this as a constructivist approach.

A wide range of resources were used during the lessons. For example, iPad, internet, flash cards and real-life props were used to interact with the students. Marsh (2010) states that using a wide range of resources adds dimension to the student’s learning and optimises student learning and therefore promotes learning. An informed choice was made when choosing the resources. For example, real-life props were used to help students memorize the sequence of the story in lesson one. By acting out the scenes in order, students were able to sequence the illustrations in correct order. The iPad was used to tell the story rather than a traditional book. This technological resource was effective as it allowed for various sensory experiences and helped students easily make sense of the story.

The lessons were successful as both students (Appendix A), were able to meet the objectives. The next step would be to refer to the Australian Curriculum documents and prepare lessons that would further advance the students. Alternatively, the lessons could be extended. For example, lesson two could include more challenging questions and phrases. These challenging questions and phrases (Appendix B) may further develop student’s conceptual understanding (McMillian, 2010) of the topic.

The lessons were aimed to engage students in authentic activities that required them to think and understand the topic rather than memorise.  Students were assisted in learning and lessons were based on cooperative work. Both lessons were student centred, requiring social engagement. Resources were selected to develop mathematical thinking and help maximise learning and develop thinking skills.

Conclusion

In summary, it maybe that constructivist approach to teaching and learning is current, best practices in mathematics education, as opposed to a behaviourist approach. Mathematics education begins with effective teaching practices. Elements such as use of assessment, and some use of behaviourist classroom management techniques, are effective practices. An effective mathematics environment is another ingredient required for a constructivist approach. This includes lessons where students are actively engaged, sharing, communicating, using manipulatives, and making connections within mathematics and to the real world.

References

 

Goldilocks and the Three Bears. (2011). Retrieved from http://english4preschool.files.wordpress.com/2011/10/colouring-pages-1.pdf

Australian Curriculum, Assessment and Reporting Authority. (2013). Retrieved from Australian Curriculum, Assessment and Reporting Authority (ACARA): http://www.australiancurriculum.edu.au/Elements/ACMMG007

Australian Curriculum, Assessment and Reporting Authority. (2013a). Retrieved from Australian Curriculum, Assessment and Reporting Authority: http://www.australiancurriculum.edu.au/Elements/ACMMG008

Clements, D.H. & Battisa, M.T. (1990). Constructivist learning and teaching. Arithmetic Teacher, 38(1).

Costa, A., & Kallick, B. (2000). Habits of Mind: A Developmental Series (Books I-IV). Alexandria, VA: Association for Supervision and Curriculum Development.

Dominick, A., & Kamii, C. (2009). The Harmful Effects of "Carrying" and "Borrowing". 10. Retrieved from https://sites.google.com/site/constancekamii/articles-available-for-printing

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

Hogue, R. (2012, March 4th). Theories - descriptive/prescriptive learning theories/instructional design theories. Retrieved from http://rjh.goingeast.ca/2012/03/04/theories-descriptiveprescriptive-learning-theories-instructional-design-theories/

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

McMillan, J. H. (2011). Classroom Assessment: Principles and Practice For Effective Standards-Based Instruction (Fifth ed.). Australia: Pearson.

O'Brien, T. C. (1999). Parrot Math. Retrieved from http://www.professortobbs.com/articles/PDK-Parrot%20Math.htm

Quirk, B. (2013). The Bogus Research in Kamii and Dominick's Harmful Effects of Algorithms Papers. Retrieved from http://wgquirk.com/kamii.html

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

TabTale. (2013). Goldilocks and the Three Bears. A Tab Tale Production.

Wright, R. J., Ellemor-Collins, D., & Tabor, P. D. (2012). Developing Number Knowledge. London: SAGE Publications Ltd.

Appendices

Appendix A

School of Education

GPO Box U1987

Perth Western Australia 6845

Tel: +618 9266 9266

Fax: +618 9266 2547

CRICOS Provider Code 00301J

Dear Parent/Carer:

As part of their development as teachers, teacher education students studying teaching through OUA and Curtin University in the Bachelor of Education (Primary) program enrolled in the unit EDP136 Mathematics Education are required to work with children to learn about children’s understandings of mathematics. This will be achieved by the teacher education students observing the children as they complete some short mathematics activities.

The teacher education students will examine the data collected from their work with the children to create a report about their observations and their learning as a teacher, as a formal assessment requirement of their mathematics education unit. In this process, and in the teacher education students’ formal assignment submissions, the children will not be identified. That is, your child’s name, image, or other features of the work that might identify your child will not be used.

If you are happy for your child to participate in this small study and for his or her work to be used, please sign the form below and return it to the student. If you have questions about the study or activities please contact Audrey Cooke on Audrey.Cooke@curtin.edu.au.

Sincerely,

Audrey Cooke

School of Education

GPO Box U1987

Perth Western Australia 6845

Tel: +618 9266 9266

Fax: +618 9266 2547

CRICOS Provider Code 00301J

Dear Parent/Carer:

As part of their development as teachers, teacher education students studying teaching through OUA and Curtin University in the Bachelor of Education (Primary) program enrolled in the unit EDP136 Mathematics Education are required to work with children to learn about children’s understandings of mathematics. This will be achieved by the teacher education students observing the children as they complete some short mathematics activities.

The teacher education students will examine the data collected from their work with the children to create a report about their observations and their learning as a teacher, as a formal assessment requirement of their mathematics education unit. In this process, and in the teacher education students’ formal assignment submissions, the children will not be identified. That is, your child’s name, image, or other features of the work that might identify your child will not be used.

If you are happy for your child to participate in this small study and for his or her work to be used, please sign the form below and return it to the student. If you have questions about the study or activities please contact Audrey Cooke on Audrey.Cooke@curtin.edu.au.

Sincerely,

Audrey Cooke

Appendix B

Challenging Questions/Phrases

ü  Pretend that tomorrow will be Sunday. What day would today be?

ü  Pretend that tomorrow will be Friday. What day would yesterday be?

ü  Next week, we are going on our excursion on Wednesday. We have to bring the money two days before. What day will that be? We will have the photos at school on the next day. What day will that be?

Appendix C

ü  The last day of the school week is…

ü  The names of the days on the weekend are…

ü  How many days are there in a week?

ü  How many days do we come to school?

Appendix D

                                                                                                        

ü  Yesterday was…

ü  Today is…

ü  Tomorrow will be…

Appendix E

                                                                                                        

Students are to reflect on their learning by verbally completing the following incomplete sentences:

I learnt that…

I think it is important to…

I still want to know…

I felt today was…