# Mathematics and Pedagogy

Mathematics and Pedagogy;Please read all the three attachment in details. Assignment has two sections as elaborated in detail. First section has two areas learning trajectory approach & choose between play based, constructivism or differentiation. The second part is the lesson plans for year 4/5 and the topic is NUMBER only and can have more than one descriptor from Australian curriculum. The lesson plan creation designed by first learning students strengths, prior knowledge and capabilities – an inclusive approach is important. You can use various online links for tasks, worksheets or games to scaffold the learning goal Please use the template which is a step to step guide to complete the assignment. Use the resources and my notes to further understand as this will lead to designing this learning resource for my second assessment hence a good level of detail is important.

First assignment has two main components: first, Critical Review of two Mathematics pedagogies and second, designing lesson plans in the form of a learning trajectory from

https://online.cdu.edu.au/ultra/courses/_52730_1/cl/outline\

**Critical Review of two Mathematics pedagogies**

First you can choose two mathematical pedagogies and critically analyse them. I suggest that you take learning trajectory as one approach and choose second one as of your choice. .

See the document below to analyse the pedagogies:

https://nzmaths.co.nz/sites/default/files/images/EdPractices_19.pdf

https://research.acer.edu.au/cgi/viewcontent.cgi?article=1022&context=aer

Second you review either play based learning or constructivism or differentiation approach

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**Designing lesson plans in the form of a learning trajectory**

For second part of the assignment 1 you are required to choose a topic from ‘**Number Only NOT ALGEBRA’**. In order to have different assignments, each year I assign different strand to students to work with. In one way, this is also to avoid plagiarism**. You can choose more than one descriptor.**

Here are few tips for designing lesson plans**:**

Go through the research papers given at learnline. Read Journal articles and develop a good understanding of learning trajectories. The key elements or learning trajectories are a learning goal and a good learning sequence based on prior knowledge.

First of all you need to choose a descriptor in number for year 4/5 of the curriculum:

http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?layout=1

For instance, if you choose the following year 6 descriptor:

Investigate equivalent fractions used in contexts (ACMNA077)

Then you need to decide what you are doing in each of 3 to 8 lessons on the chosen descriptor so that learning gets progressively more difficult, and is based on prior knowledge from the previous years (see the curriculum).

You may use any model, e.g. Burner’s model breaks representation of mathematical ideas down into 3 phases: enactive, Iconic, symbolic. Perhaps follow these three stages to teach equivalent fractions. Start with concrete materials (enactive stage), move on to diagrams and visuals (iconic phase), and finally move towards symbolic representations.

Perhaps use one lesson for revision of previous work on Number, one on each of Bruner’s stages and one lesson focusing on application of these ideas. Note that whatever you do needs to be presented on a website for the children to use in assignment 2 so you need to bear that in mind, choosing good web based materials and links to help to teach the content. Include some such links in ass 1.

I suggest you present the content with a description of what you plan to do, then a discussion of prior knowledge (see the curriculum for measurement), then a table with rows for lesson 1, 2, … and columns for the aim of the lesson and examples of activities including some weblinks. Then a short paragraph to end stating whether you are going to use a Wix, Webquest or any other tool to present ass 2, and a reference list. Eg **PLEASE REMEMBER THE CONCEPTS WILL BE EXTENDED AS A DIGITAL RESOURCE SO ENSURE TO DEMONSTRATE A CLEAR PLAN FOR THIS ASSIGNMENT 1 eg of assignment 2 link – > ****https://alisonkungel.wixsite.com/areaperimeterwebques**

**Possible layout: (Note there are many possible ways of presenting this)**

**Critical Review of two Mathematical Pedagogies:**

You can take the six key principles for effective teaching of Mathematics described by Peter Sullivan in Teaching Mathematics: Using research -informed strategies.

**Learning Trajectories****………….**

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**Designing Lesson plans in the form of a learning trajectory**

__Title__ e.g. Number in year 5.

__Introduction of learning:__ Curriculum descriptor (Part of Number in about year 4, 5, 6) and level and brief description of what you plan to do

__Prior knowledge linked to the topic__ (see the curriculum for Number)

__Proficiencies or key ideas __ Discuss how you will address the proficiency strands in your topic

__Learning trajectory: __ (Shows learning getting progressively more difficult. Suitable Learning Tasks/ Use of varied technology/Catering for diversity: ie some scaffolding or extension activities.)

Aim | Description | Examples of activities and weblinks | |

Lesson 1 | |||

Lesson 2 | |||

Lesson 3 | |||

Lesson 4 | |||

Lesson 5 | |||

… | |||

__Conclusion:__ A short paragraph to end stating whether you are going to use a Wix, Webquest or any other tool to present ass 2 and how you plan to do this.

__References__

Inclusion of a reference list with reference to the **curriculum** and other readings on teaching your topic, such as the relevant chapters in the textbook (Siemon et al.).

**Preferably can include figures, diagrams and the word count is 3000 words.**

Assignment details

Analyse two mathematical pedagogies

Your Topic will be same from year level either 4/5/6

Select descriptor from Australian curriculum

Which descriptor you are going to develop your learning trajectory

This will have two parts – in first you will critically review two mathematical pedagogies so it is not a description or an essay but a critical analysis.

Critical review will be discussed how to address this.

Second part of this assignment to develop framework of a learning trajectory, a layout will be provided to get an idea of how to work with the two parts.

__Week 1__

What will the student learn and what are the different strategies or ways of engaging our students. What details are we going to focus and achieve that, different learning models have different approach. Banking model is one learning method by Freire, Second is Piaget’s developmental theories.

Ernest’s ideas are:

Instrumentalist approach: An accumulation of utilitarian facts, rules and skills

Platonist approach: A body of knowledge which is discovered not created.

Problem Solving: A dynamic and continually expanding field of human creation, invention a cultural product.

Constructivism theory by Piaget is that learning is an active process rather than a passive process. It consists of Assimilation and Accomodation. Assimilation refers to the process of taking new information and fitting it into the existing schema – means whatever existing cognitive structure we have, we are taking that new information into that existing cognitive structure or pre-existing schema. Accomodation refers newly acquired information to revise and re-develop exusting schema. Sometimes gain of new knowledge helps to improve your existing knowledge and improve your existing cognitive structure.

Social constructivism by Vygotsky is that the communities and social interaction contributes towards learning. As we grow up and within our social interactions we learn outside of the formal environment or classroom. In some parts they need some help.

Communities of practice is when you join some association if you want to lean about maths, what is happening in terms of projects, methods and what are the outcomes and what has worked.

Brain based research, there are certain studies that the human brain has different parts of the brain. When you are involve in different tasks certain parts of the brain is active.

__Week 2__

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How the students progress in their learning in a specific mathematical domain. The description of the children’s thinking and learning and how we proceed with that. The focus can be computational skills, reasoning or develop their analytical skills.

What cognitive structure will emphasise and how to ensure they are learning that topic. Topic could be fractions and developing mental processes and developing some relational thinking and level of activities in detail. This is to ensure that they are achieving the goals of learning. Focus on year 4/5/6. Use one year level for the assignment. You can have one or more descriptors.

It is important to be aware of the prime knowledge they have the cultural background learning style and abilities. This will help to develop lesson plans that are inclusive to all.

Another is the language which is the mathematical language ie mathematical language, symbols, operators when we working with problem solving activities another is the language to explain those concepts.

The second trajectory is a pathway ie mental processes. How you’ll make sure that the different pathways. You may use concrete materials, different manipulatives or group activities or other ways you can think of different pathways. Also how are you bringing involvement and engagement for those pathways.

It is also important to know what tasks will help that developmental progression for your students. You will build the student’s knowledge from basic concept and skills and whether they need different activities, scaffolding for a hierarchical knowledge in a sequenced way.

Designing lesson plans is one task however it is very important about how to provide set of instructions and has to be in a sequenced way to develop mathematical concepts. The student may adopt different methods to reach the goal, hence it is important to know how and whether they are relevant.

Teachers will develop a learning trajectory with their pathway from their level of understanding. This may turn out differently and hence it is important to design and analyse the tasks according to the suitability of the tasks for your students. The learning trajectory has to be from a child’s perspective and they should understand how to apply mathematical concepts, symbols and operators and how to present their work and this will add up to assessing their learning.

Concepts that can be used for assignment

For example, confronted with the division misconception just referred to, a teacher could ask students to investigate the difference between 10 :– 2, 2 :– 10, and 10 :– 0.2 using diagrams, pictures, or number stories.

Open-ended tasks are ideal for fostering the creative thinking and experimentation that characterize mathematical “play”. For example, if asked to explore different ways of showing 2/3, students must engage in such fundamental mathematical practices as investigating, creating, reasoning, and communicating.

Skill development can often be incorporated into “doing” mathematics; for example, learning about perimeter and area offers opportunities for students to practice multiplication and fractions. Games can also be a means of developing fluency and automaticity. Instead of using them as time fillers, effective teachers choose and use games because they meet specific mathematical purposes and because they provide appropriate feedback and challenge for all participants.

Students need opportunities to practice what they are learning, whether it be to improve their computational fluency, problem solving skills, or conceptual understanding. Skill development can often be incorporated into “doing” mathematics; for example, learning about perimeter and area offers opportunities for students to practice multiplication and fractions. Games can also be a means of developing fluency and automaticity. Instead of using them as time fillers, effective teachers choose and use games because they meet specific mathematical purposes and because they provide appropriate feedback and challenge for all participants.

Essential steps for Learning Trajectory below:

Example of learning trajectories

Some readings and references that might help you in assignment 1 & 2,

Australian Curriculum Assessment and Reporting Authority (ACARA). (2011b). The Australian curriculum: Mathematics. Retrieved from https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/rationale/

Australian Curriculum Assessment and Reporting Authority (ACARA). (2015). Mathematics: Sequence of content. Retrieved from

https://www.australiancurriculum.edu.au/media/3680/mathematics_-_sequence_of_content.pdf

Board of studies Teaching and Educational Standards NSW (Bostes) (2012). NSW syllabus for the Australian curriculum: Mathematics K-10 Syllabus. Retrieved from

http://syllabus.bostes.nsw.edu.au/mathematics/mathematics-k10/

Chapters on Number in your textbook:

Siemon, D. (2015). Teaching mathematics. South Melbourne, Vic.: Oxford University Press.

Readings on Number:

Numeracy professional development projects. (2008) Teaching fractions, decimals and percentages. Retrieved from

http://nzmaths.co.nz/sites/default/files/Numeracy/2008numPDFs/NumBk7.pdf

Stacey, K , Moloney, K and Steinle, V. (1996). Students’ Understanding of Decimals: An Overview . Retrieved from

https://extranet.education.unimelb.edu.au/SME/TNMY/Decimals/Decimals/backinfo/refs/molostac.htm

Misconceptions and Error Patterns. Retrieved from

http://ptgmedia.pearsoncmg.com/images/9780135009109/downloads/Ashlock_Ch1_MisconceptionsandErrorPatterns.pdf

Unit readings and lecture materials on Number e.g., topics such as fractions, decimals, percentages etc.

See templates emailed to you, such as templates for teaching fractions and percentages.

Reading on language development

Rubenstein, R. N. (2007). Focused strategies for middle-grades mathematics vocabulary development. Mathematics Teaching in the Middle School, 13(4), 200-207.

http://www.collectedny.org/wp-content/uploads/2017/10/strategies-in-vocabulary-development.pdf

Dictionaries such as the online Jenny Eather dictionary (link below):

Eather, J. (2011). A maths dictionary for kids. Retrieved 2011, July 20, from http://www.amathsdictionaryforkids.com/

Or your Origo handbook:

Anderson, J., Briner, A., Irons, C., Shield, M., Sparrow, L., & Steinle, V. (2008). The Origo handbook of mathematics education. Australia: Origo Education.

Possible websites

Introduction: Resources for the classroom

Fraction song:

Fractions, Decimals and Percentages

Search for other similar material

Other useful websites for maths teaching

Good websites for maths

http://nctm.org Maths website

http://www.ixl.com/ Maths and other subjects

https://www.mathsisfun.com Maths website

https://illuminations.nctm.org Maths website

http://nrich.maths.org Maths website

http://www.amathsdictionaryforkids.com/ Dictionary for maths

http://www.tesaustralia.com/mathematics-primary-teaching-resources/ Maths resources

http://www.sparklebox.co.uk/maths/#.VsQZpuZ8vqU Maths resources

http://nlvm.usu.edu/ Maths resources

http://coolmath4kids.com Maths resources

Different models and activities

Fraction games and activities:

Numberline for fractions, decimals, percentages:

Number line

Fraction wall:

Decimals:

Showing tenths:

Showing hundredths (and percentages):

Place value chart:

Number Expanders:

Common fractions, decimals and percentages