# Mathematics Pedagogy

An effective primary school mathematics teacher requires a keen awareness of their competencies. This analysis examines the author's content and pedagogical knowledge and identifies strengths and weaknesses. An action plan will then explain how weaknesses in content and pedagogical knowledge will be addressed.

Content knowledge analysis draws on the "Year 7 Mathematics Curriculum" (ACARA, n.d.) of the Australian Curriculum. Content knowledge was tested with the "Year 7 Eureka Naplan Style Maths Test" (Eureka Multimedia, n.d.), "MP4105 Quizzes" ("Quizzes", n.d.), "thatquiz.org" ("Math", n.d.) and "Khan Academy" (Khan Academy, n.d.) assessment. Pedagogical knowledge is sourced from "Table 1: Domains of Knowledge and supporting questions" Ball, Thames & Phelps (as cited in Hurrell 2013) and the "Australian Professional Standards For Teachers" (AITSL, n.d.).

The first content strand in the mathematics curriculum is "Number and Algebra" (ACARA, n.d.) and contains nineteen content descriptions of which the author is confident to deconstruct and explain seventeen. In the testing phase the author scored 100% on year 7 level questions. A specific area of strength for content and pedagogy is "Compare, order, add and subtract integers (ACMNA280)" (ACARA, n.d.). Using concrete examples such as a thermometer or number line to help students visualise numbers above, including and below zero, ACMNA280 is an area of teaching strength.

An area of weakness is "Round decimals to a specified number of decimal places (ACMNA156)" (ACARA, n.d.). The author is aware of three methods of executing this task. A lack of awareness of the approach recommended by the Australian Curriculum could lead to teaching a non preferred method. Weakness exists in "Apply the associative, commutative and distributive laws to aid mental and written computation (ACMNA151)" (ACARA, n.d.) at the current time the author is unaware of the meaning of the words associative, commutative and distributive in a mathematical context.

Measurement and Geometry has eight content descriptions and the author is confident to deconstruct and explain six. An average score of 97.6% on the Measurement and Geometry tests identified a small area of content weakness. An area of strength can be seen with "Draw different views of prisms and solids formed from combinations of prisms (ACMMG161)" (ACARA, n.d.). As a highly visual person the author has mastered simple strategies for drawing and explaining technical concepts.

"Establish the formulas for areas of rectangles, triangles and parallelograms and use these in problem solving (ACMMG159)" (ACARA, n.d.) Content weakness exists regarding the question of parallelograms and their area and this was evident in the testing. "Describe translations, reflections in an axis, and rotations of multiples of 90Â° on the Cartesian plane using coordinates. Identify line and rotational symmetries (ACMMG181)" (ACARA, n.d.). The author is unaware of best practise in this topic due to a lack of experience with the content. Exposure and a deeper understanding are required for effective teaching.

Statistics and Probability maintains six content areas. The author is confident to deconstruct and explain all six concepts as a score of 100% was achieved in testing. A specific area of strength is "Assign probabilities to the outcomes of events and determine probabilities for events (ACMSP168)" (ACARA, n.d.). This to be a natural area of expertise as the author previously studied a university level statistics unit. The author identifies a weakness and is unaware of the meaning of "single-step" in "Construct sample spaces for single-step experiments with equally likely outcomes(ACMSP167)" (ACARA, n.d.).

Ball, Thames & Phelps identifies six key areas in a teacher's knowledge of mathematical pedagogy:

"Common Content Knowledge (CCK), Specialised Content Knowledge (SCK), Knowledge at the mathematical horizon, Knowledge of Content and Students (KCS), Knowledge of Content and Teaching (KCT), Knowledge of Content and Curriculum". Ball, Thames & Phelps (as cited in Hurrell 2013) Relating to Number and Algebra, Measurement and Geometry and Statistics and Probability the pedagogical principle the author feels is a strength is CCK or "Common Content Knowledge" Ball, Thames & Phelps (as cited in Hurrell 2013) and AITSL Standard Two "Know the content and how to teach it" (AITSL, n.d.). This asks the educator to reflect on their own ability to use the content they are teaching. A question asked in the table is "Are you able to: solve mathematical problems correctly?" Ball, Thames & Phelps (as cited in Hurrell 2013). The testing in this analysis and the authors life experience would indicate yes.

The area of pedagogy the author identifies as a weakness is KCS or "Knowledge of Content and Students" Ball, Thames & Phelps (as cited in Hurrell 2013) and relates to AITSL Standard One "Know students and how they learn" (AITSL, n.d.). Pedagogy here asks the educator to see mathematics from the students' point of view. It asks the educator to "predict what students will find interesting and motivating when choosing an example?" Ball, Thames & Phelps (as cited in Hurrell 2013) and understand how students learn. As a graduate teacher the author has limited experience predicting the interests of students aged four to eleven and is not aware of the types of learning barriers they may present with.

This skills audit has clear implications for the author's teaching future. "Mathematics aims to ensure that students: are confident, creative users and communicators of mathematics, able to investigate, represent and interpret situations in their personal and work lives and as active citizens" (ACARA, n.d.). This aim places a great deal of responsibility on the shoulders of the mathematics teacher. If through a deep understanding of pedagogy the author cannot engage students in learning mathematics they will not be able to fully take their place as "active citizens" (ACARA, n.d.) and the author will have failed in this paramount duty.

The following action plan will identify steps to address the weaknesses identified. The pedagogical weaknesses above relating to "Knowledge of Content and Students" Ball, Thames & Phelps (as cited in Hurrell 2013) will be address thus:

- Join the Mathematical Association of WA (MAWA) and enrol in professional learning opportunities.
- Subscribe to the Australian Association of Mathematics Teachers (AAMT) publication "The Australian Mathematics Teacher" (AAMT, n.d.).
- Develop an understanding of Piaget's Stages of Cognitive Development and read "The Psychology Of The Child" (Piaget & Inhelder, 2000)
- Complete the "AITSL Teacher Toolkit" (AITSL, n.d.).
- Interview my teaching mentor and future colleagues about the strengths and difficulties students have at each year level.
- Interview my friends' children to understand what they enjoy doing and ask about barriers they have to learning mathematics.
- Study the literature e.g. "Pre-Service Mathematic Teachersâ€™ Knowledge of Students about the Algebraic Concepts" (Tanisli & Kose, 2013) to deepen pedagogical understanding.
- Complete the "AAMT Top Draw Exercises" (AAMT, n.d.) that address student and teacher misconceptions related to pedagogy.
- Consider a Graduate Certificate of Education in Learning Difficulties.
- Commit to monthly reviews of pedagogical knowledge and actively seek feedback from mentors.

To address content knowledge weaknesses identified in Number and Algebra, Measurement and Geometry and Statistics and Probability the following action plan will be engaged:

- Identify all concepts within the foundation to year 10 curriculum that are unclear and set these as priority research.
- Enrol in mathematics programs from sources such as Cousera or The Great Courses.
- Complete activities from the Khan Academy or thatquiz.org such as Geometry Warm Up.
- Access professional development opportunities through MAWA, The Australian Mathematics Trust or The Department of Education WA etc.
- Complete the "Top Drawer" (AAMT, n.d.) activities.
- Consider further study such as completing units or an entire qualification e.g. Mathematics and Statistics (BSc).
- Conduct monthly reviews of mathematical content knowledge and address any issues as they arise.

This analysis has identified strengths and weaknesses in the author's mathematical content and pedagogical knowledge and it specifically addresses the three strands of the Australian Mathematics Curriculum. This has clear implications for the author's teaching future. To be an effective mathematics teacher into the future, the content and pedagogical issues would need to be addressed through the action plan introduced.

This annotated bibliography has drawn on the areas of weakness identified in part one of this assignment. Appropriate mathematical journal articles, texts and other resources have been selected that will address these weaknesses.

Hurrell, D. P. (2013). What Teachers Need to Know to Teach Mathematics: An argument for a reconceptualised model. Australian Journal of Teacher Education, 38(11). http://dx.doi.org/10.14221/ajte.2013v38n11.3

When conducting the audit of content and pedagogical knowledge the question was asked "How does one know what one doesn't know?" When content knowledge was concerned the Australian Mathematics Curriculum detailed the required areas of skill and knowledge. Pedagogical knowledge on the other hand was not so clearly defined.

"Figure 2 Mathematical knowledge for teaching" Hill et al., 2008, p. 377 (as cited in Hurrell, 2013) of this article clearly articulates three areas of pedagogical knowledge. KCS Knowledge of Content and Students, KCT Knowledge of Content and Teaching and KCC Knowledge of Content and Curriculum Ball, Thames & Phelps (as cited in Hurrell 2013). "Table 1: Domains of Knowledge and supporting questions" Ball, Thames & Phelps (as cited in Hurrell, 2013) expands on figure 2. It lists reflection questions to assist the educator in evaluating their own position. From "Are you able to: anticipate what students will think?" Ball, Thames & Phelps (as cited in Hurrell, 2013) to "Are you able to: Articulate the strands of the curriculum?" Ball, Thames & Phelps (as cited in Hurrell, 2013) this document arms the educator with a framework to investigate pedagogical practice.

Norris, K. (2014). Raising Teacher Sensitivity to Key Numeracy Competencies in the Early Years. Retrieved from http://ro.ecu.edu.au/theses/1473

This document addresses the weakness identified in pedagogical knowledge identified above. Norris discusses the concept of a mathematical learning trajectory and identifies how early formative assessment can inform the teacher of the trajectory that the mathematics student is on (Norris, K. 2014, pg 7). An awareness of the appropriate trajectory for a young student can help the mathematics teacher to identify early on when a student is at risk of failure and to take appropriate steps to act before said failure occurs (Norris, K. 2014, pg 4).

The early trajectory identifies number sense as a key determinant future mathematics success. This article provides confidence to assess a student's number sense as a beginning indicator of how well they will cope with a rigorous curriculum (Norris, K. 2014, pg 8). By identifying students at risk early and before they fail, proactive measures can be instigated (Norris, K. 2014, pg 19). The thesis shows how targeted interventions then have the potential to address the issues identified.

Department Of Education WA, (2013) First Steps in Mathematics Retrieved from http://det.wa.edu.au/stepsresources/detcms/navigation/first-steps-mathematics/

The First Steps in Mathematics series is broken up into four parts Number, Measurement, Space, Chance and Data (Department Of Education WA, 2013). This evidence based text works to the Western Australian Curriculum Framework but easily aligns to the Australian National Curriculum for mathematics.

An area of pedagogical weakness identified was in Knowledge of Students and Content. The First Steps series can help to address this gap in pedagogical knowledge through the diagnostic map. The diagnostic map allows the educator to see the student on a continuum as they move through five levels from the emergent phase through to the relating phase (Department of Education, 2013). It helps educators to understand a student's responses to given tasks. In addition the diagnostic map explains what a student probably won't be able to do. This diagnostic map clearly informs the educator based on the pedagogical principal of understanding students and what they think. National Research Council. (2009). Mathematics Learning In Early Childhood: Paths Toward Excellence And Equity. Washington, DC. The National Academies Press.

A weakness was discovered relating to "Apply the associative, commutative and distributive laws to aid mental and written computation (ACMNA151)" (ACARA, n.d.) as the author did not know the meaning of the words associative, commutative and distributive laws in a mathematical context.

This text proved to be first class in explaining the concepts. Page 51 shows clearly that these properties relate to the rules of BIMDAS in that the properties apply to the first of the groupings in multiplication-division and addition-subtraction order of operations. The laws identified in the text show how operations can be manipulated without changing the result of the overall operation.

Greenwood, D., Humberstone, B., Robinson, J., Goodman, J., Vaughan, J., Frank, F., (2012). Essential Mathematics For The Australian Curriculum Year 7. Cambridge University Press Melbourne.

The content description "Describe translations, reflections in an axis, and rotations of multiples of 90Â° on the Cartesian plane using coordinates. Identify line and rotational symmetries (ACMMG181)" (ACARA, n.d.) was identified as a weakness. The beauty of this guide is that it not only provides the appropriate information but on page 447 it shows that it aligns to the exact content description identified in the Australian Curriculum.

The text provides informative and visual information to inform background knowledge. It provides a large number of questions and answers to help an educator with skills relating to Cartesian planes, reflections, translations etc. or a student learning the content for the first time. It is presented in an easy to view style with adequate white space and visually pleasing colours. A very helpful technique the text uses is to present 2 dimensional and 3 dimensional line drawings in addition to real photographs such as the bridge shown on page 463 that help students visualise the concepts presented. This helps to address pedagogical issues around relevance as students can now see the real world correlations of the work they are completing.

Australian Curriculum Assessment and Reporting Authority (ACARA) (n.d.) Mathematics Foundation to Year 10 Curriculum Retrieved from http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?y=7&s=NA&s=MG&s=SP&layout=1

An area of weakness identified was "Round decimals to a specified number of decimal places (ACMNA156)" (ACARA, n.d.) The issue arises because there are a number of rounding techniques that can be used and they can all produce slightly different answers. The author required the method recommended by the curriculum so that a standard teaching approach could be taken. After accessing the Australian Curriculum website it was found that this resource was not able to solve the problem. The glossary, elaborations and ScOT dictionary did not offer a solution. The solution was sourced however through a link on the Australian Curriculum website to Scootle (Education Services Australia, n.d.). Scootle provided a video called "How to round off decimals" (Australian Broadcasting Corporation, 2015). This proved to be an excellent solution to addressing this weakness.

Khan Academy, (2011). Area of a Parallelogram. Retrieved from https://www.youtube.com/watch?v=tFhBAeZVTMw

A weakness was identified relating to the area of a parallelogram. "Establish the formulas for areas of rectangles, triangles and parallelograms and use these in problem solving (ACMMG159)" (ACARA, n.d.)

This video resource identified on YouTube clearly and coherently explains the process and formula to calculate the area of a parallelogram. The video uses complex mathematical terms as a way of explaining the process and so would not be suitable to show primary school children in class. However as a content knowledge development tool, three minutes and twenty four seconds in, the video explained the process to calculate the area of a parallelogram for the knowledge development of the author.

Australian Institute for Teaching and School Leadership (AITSL) (n.d.) AITSL Teacher Toolkit Retrieved from http://toolkit.aitsl.edu.au/

The author clearly identified in part one that pedagogical knowledge was lacking and needed to be developed. This online resource is broken in to seven categories Coaching & Mentoring, Performance & Development, Self-Assessment & Reflection, The Standards, Professional Learning, Collaboration and Classroom Practice (AITSL, n.d.).

For the pre-service teacher the Classroom Practice section is highly relevant. The primary guide is the Classroom Practice Continuum and in particular the Graduate column. This outlines what a teacher looks and sounds like and how they organise the learning environment. The web information and PDF documents are strongly enhanced by the use of video content such as those by Patrick Griffin to thoroughly explain the Classroom Practice content from this section of the tool kit.

Math (n.d.) Retrieved from http://www.thatquiz.org/

This website is a comprehensive resource for developing computational abilities. The resource is not linked to the Australian Curriculum for Mathematics however there is the ability to set up exercises and then generate a unique URL for that exercise. This means that exercises could be set for in class practice or added to the school website or learning management system.

This website has a broad range of mathematical challenges. During the skills audit in part one this website was used extensively to test a range of abilities identified in the Australian Curriculum. The author has found this resource to be very effective in drilling and practicing mathematical skills in order to hone proficiency and increase confidence. Australian Association of Mathematics Teachers (AAMT) (n.d.) Big Ideas | Fractions | Topdrawer Retrieved from http://topdrawer.aamt.edu.au/Fractions/Big-ideas

The skills audit identified a weakness in pedagogy for "Knowledge of Content and Students" Ball, Thames & Phelps (as cited in Hurrell 2013) and one of the reflection questions asked "anticipate what students are likely to think?" Ball, Thames & Phelps (as cited in Hurrell, 2013)

The power of this resource is that it brings content knowledge and pedagogy together. The preamble and "Big Ideas" help with a deeper content knowledge. The section "Misunderstandings" in each content area informs pedagogy as it identifies what a student may already think and identifies the common misconceptions students have in their understanding.

Under the Misunderstandings section there are real work samples of a student's misconceptions. This is very helpful as it shows in a visual way how students are likely to show their understanding. The samples emphasise how students may get the right answer for the wrong reason, the sorts of errors that are made and what work samples showing appropriate understanding will look like.

Department Of Education And Training Victoria, (2015) iPads for Education Retrieved from http://www.ipadsforeducation.vic.edu.au/

A primary pedagogical weakness that was identified in part one was "Knowledge of Content and Students" Ball, Thames & Phelps (as cited in Hurrell 2013) and relates to AITSL Standard One "Know students and how they learn" (AITSL, n.d.). In an attempt to take the mountain to Mohammad the author felt it may be possible to bring mathematics instruction to where the students are.

After searching for mathematics apps for iPad and Android it was identified that there are a staggering number of choices. After installing two different apps on the author's Android device the author became aware how dangerous using apps could be for education. Ads promoting other games and in app purchases created strong distractions that reduce the effectiveness of this approach.

The website iPads for Education created by the Victorian Department of Education and Training is an excellent resource. The iPad apps presented here have all been reviewed and deemed appropriate for student use. The site also provides tips and case studies and they publish the results and statistics of their iPad trials such as 85% of primary teachers using iPads reported an increase in learning motivation (Victorian Department of Education and Training, 2015).

In conclusion this annotated bibliography has identified high quality mathematical resources that were related to weaknesses identified in part one of this assignment. These resources have been read or accessed by the author and address the weaknesses in content and pedagogical knowledge identified.

References

Hurrell, D. P. (2013). What Teachers Need to Know to Teach Mathematics: An argument for a reconceptualised model. Australian Journal of Teacher Education, 38(11). http://dx.doi.org/10.14221/ajte.2013v38n11.3

Norris, K. (2014). Raising Teacher Sensitivity to Key Numeracy Competencies in the Early Years. Retrieved from http://ro.ecu.edu.au/theses/1473

Department Of Education WA, (2013) First Steps in Mathematics Retrieved from http://det.wa.edu.au/stepsresources/detcms/navigation/first-steps-mathematics/

National Research Council. (2009). Mathematics Learning In Early Childhood: Paths Toward Excellence And Equity. Washington, DC. The National Academies Press.

Greenwood, D., Humberstone, B., Robinson, J., Goodman, J., Vaughan, J., Frank, F., (2012). Essential Mathematics For The Australian Curriculum Year 7. Cambridge University Press Melbourne.

Australian Curriculum Assessment and Reporting Authority (ACARA) (n.d.) Mathematics Foundation to Year 10 Curriculum Retrieved from http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?y=7&s=NA&s=MG&s=SP&layout=1

Khan Academy, (2011). Area of a Parallelogram. Retrieved from https://www.youtube.com/watch?v=tFhBAeZVTMw

Australian Institute for Teaching and School Leadership (AITSL) (n.d.) AITSL Teacher Toolkit Retrieved from http://toolkit.aitsl.edu.au/

Australian Institute for Teaching and School Leadership (AITSL) (n.d.) AITSL Retrieved from http://aitsl.edu.au/

Math (n.d.) Retrieved from http://www.thatquiz.org/

Australian Association of Mathematics Teachers (AAMT) (n.d.) Big Ideas | Fractions | Topdrawer Retrieved from http://topdrawer.aamt.edu.au/Fractions/Big-ideas

Department Of Education And Training Victoria, (2015) iPads for Education Retrieved from http://www.ipadsforeducation.vic.edu.au/

Australian Broadcasting Corporation, (2015). How To Round Decimals. Retrieved from http://splash.abc.net.au/home#!/media/1477476/ Australian Curriculum Assessment and Reporting Authority (ACARA) (n.d.) Mathematics Foundation to Year 10 Curriculum Retrieved from http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?y=7&s=NA&s=MG&s=SP&layout=1

Education Services Australia, (2015). Scootle Retrieved from https://www.scootle.edu.au/ec/p/home

References

Hurrell, D. P. (2013). What Teachers Need to Know to Teach Mathematics: An argument for a reconceptualised model. Australian Journal of Teacher Education, 38(11). http://dx.doi.org/10.14221/ajte.2013v38n11.3

Australian Curriculum Assessment and Reporting Authority (ACARA) (n.d.) Mathematics Foundation to Year 10 Curriculum Retrieved from http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?y=7&s=NA&s=MG&s=SP&layout=1

Australian Curriculum Assessment and Reporting Authority (ACARA) (n.d.) Mathematics Aims Retrieved from http://www.australiancurriculum.edu.au/mathematics/aims

Australian Institute for Teaching and School Leadership (AITSL) (n.d.) Standards Retrieved from http://www.aitsl.edu.au/australian-professional-standards-for-teachers/standards/list

Australian Institute for Teaching and School Leadership (AITSL) (n.d.) AITSL Teacher Toolkit Retrieved from http://toolkit.aitsl.edu.au/

Eureka Multimedia (n.d.) Eurekas NAPLAN style maths tests year 7 educational software Retrieved from http://www.eurekamultimedia.com.au/naplan-style-maths-tests-year-7-digital-download.html

Quizzes - MPE4105.2015.1.ONCAMPUS Primary Mathematics Education 1 (n.d.) Retrieved from https://blackboard.ecu.edu.au/webapps/blackboard/content/listContent.jsp?course_id=_617063_1&content_id=_3708807_1&mode=reset

Math (n.d.) Retrieved from http://www.thatquiz.org/

Khan Academy (n.d.) Khan Academy Retrieved from https://www.khanacademy.org/

Khan Academy (n.d.) Khan Academy Retrieved from https://www.khanacademy.org/mission/geometry/task/6238433611350016

Piaget, J., & Inhelder, B. (2000) The Psychology Of The Child. Basic Books, New York.

Australian Association of Mathematics Teachers (AAMT) (n.d.) Journals/Home Retrieved from http://aamt.edu.au/Journals

Australian Association of Mathematics Teachers (AAMT) (n.d.) Big Ideas | Fractions | Topdrawer Retrieved from http://topdrawer.aamt.edu.au/Fractions/Big-ideas

Tanisli, D., & Kose, N. Y. (2013). Pre-Service Mathematic Teachersâ€™ Knowledge of Students about the Algebraic Concepts. Australian Journal of Teacher Education, 38(2). http://dx.doi.org/10.14221/ajte.2013v38n2.1

Web Links

http://www.mawainc.org.au/

http://www.aamt.edu.au/

http://www.ecu.edu.au/degrees/courses/graduate-certificate-of-education-learning-difficulties

http://www.murdoch.edu.au/Courses/Mathematics-and-Statistics/

http://www.amt.edu.au/information/for-teachers/

http://www.thatquiz.org/tq-f/math/shapes/

Statistics Intro https://www.youtube.com/watch?v=--r9_R60Jws

https://www.coursera.org/

http://www.thegreatcourses.com.au/

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