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…