Part A notes

Question 1

What would your response be to someone who told you that numeracy was all about number?

Answer 1

• Numeracy definition
  
  • Being numerate involves having fluency in mathematical skills and knowledge; a clear understanding of the contextual requirements; the disposition to choose and use mathematics confidently; and a critical appreciation of how mathematics can be used
  • Numeracy involves the ability to apply number and computational skills to everyday purposes, together with the ability to interpret the quantitative information that pervades everyday life (Westwood, 2008)
  • Maths used in circumstances other than mathematics class
• Why is numeracy important?
  
  • Numeracy prepares students to manage the requirements of everyday life efficiently, in a range of situations that involve the use of mathematics e.g. Shopping, cooking etc. and school leavers in finance
  • Numerate individuals are also able to participate more effectively and successfully in wider community and social spheres
  • The growth of a socially just society depends upon individuals gaining appropriate levels of numeracy
• How are maths and numeracy different?
  
  • Mathematics conveys the power of conceptualising and abstracting; numeracy conveys the power of context and practicality
  • Mathematics does not need to consider the real world as it can focus purely on abstract constructs and ideas regardless of their potential applications; numeracy is the application of mathematics in authentic contexts
  • Mathematics is organised using categories from the past; numeracy focuses on the way quantitative knowledge is used in the information age
  • Mathematics is most encountered in educational institutions; numeracy is mostly in the world around us.
  • Numeracy comprises not only mathematics skills, but also the way in which they are used. This involves marrying the mathematical meaning of the symbols and operations to their contextual application, and thinking simultaneously about both.
• Being numerate
  
  • Numeracy technician (begins understanding)
  • Numeracy participant (begins understanding)
  • Numeracy user
  • Numeracy analyst.
  • Being numerate within a context involves a blend of three types of know-how: mathematical, contextual and strategic. Contextual know-how requires an understanding of how the mathematics is shaped by the context.
  • Being numerate involves having:
  • A fluency in mathematical skills and requisite knowledge
  • A clear understanding of the contextual requirements
  • The disposition to choose and use the mathematics appropriate to the context with confidence (involves the use of representational, physical or digital tools)
  • A critical appreciation of how mathematics can be used, and misused, in order to question or judge the appropriateness of its use.

Question 2

Discuss the contention that numeracy is contextual.

Answer 2

• Numeracy definition
  
  • Being numerate involves having fluency in mathematical skills and knowledge; a clear understanding of the contextual requirements; the disposition to choose and use mathematics confidently; and a critical appreciation of how mathematics can be used
  • Numeracy involves the ability to apply number and computational skills to everyday purposes, together with the ability to interpret the quantitative information that pervades everyday life (Westwood, 2008)
  • Maths used in circumstances other than mathematics class
• Why is numeracy important?
  
  • Numeracy prepares students to manage the requirements of everyday life efficiently, in a range of situations that involve the use of mathematics e.g. Shopping, cooking etc. and school leavers in finance
  • Numerate individuals are also able to participate more effectively and successfully in wider community and social spheres
  • The growth of a socially just society depends upon individuals gaining appropriate levels of numeracy
• Being numerate
  
  • Numeracy technician (begins understanding)
  • Numeracy participant (begins understanding)
  • Numeracy user
  • Numeracy analyst.
  • Being numerate within a context involves a blend of three types of know-how: mathematical, contextual and strategic. Contextual know-how requires an understanding of how the mathematics is shaped by the context.
  • Being numerate involves having:
  • A fluency in mathematical skills and requisite knowledge
  • A clear understanding of the contextual requirements
  • The disposition to choose and use the mathematics appropriate to the context with confidence (involves the use of representational, physical or digital tools)
  • A critical appreciation of how mathematics can be used, and misused, in order to question or judge the appropriateness of its use.
• Critical numeracy
  
  • Critical numeracy is the ability to make discerning decisions about everyday issues that involve the use of mathematics
  • According to Stoessiger (2002), there are four main aspects to being critically numerate. These are;
    
    • Being able to critique mathematical information
    • Being able to interpret or decode mathematical situations
    • Using mathematics in a self-reflective way
    • Using mathematics to operate more powerfully in the world

Question 3

Discuss the different domains from Shulman’s PCK model with particular focus on the intersections.

Answer 3

PCK= Pedagogical Content Knowledge

• Knowledge types
  
  • Knowledge of the curriculum
    
    • Knowing about the structure of the Australian Curriculum, the specific content to be taught at the relevant year, and how the mathematical proficiency strands are incorporated throughout that content.
  • Knowledge of students as mathematics learners (KSL)
    
    • Knowing about the typical development of student understanding and their likely responses to mathematical tasks, e.g. knowing that many students struggle to develop understanding of fractions as numbers, to compare and order them, and locate them on the number line.
  • Knowledge of mathematical content (KMC)
    
    • Knowing the key mathematical ideas to be taught, e.g. that fractions are rational numbers that can also be represented as decimals and percentages, and knowing how to use these various forms of rational numbers in solving problems.
  • General pedagogical knowledge (PK)
    
    • Knowing generally applicable strategies for managing and organising the classroom, e.g. how to get the attention of the class and organise transitions from one activity to the next.
  • Pedagogical content knowledge (PCK)
    
    • Knowing a variety of ways to present mathematical content and to assist students to develop their understanding of it, e.g. knowing that 10 × 10 grids can be useful models for helping students to understand decimals, and being aware of the limitations and affordances of this and other decimal representations.
  • Knowledge of the purposes and values of education
    
    • Knowing the overarching aims of the curriculum and the values that underpin it.
  • Knowledge of educational contexts
    
    • Knowing about the workings of a school and/or system and the wider educational community, their governance structures and cultures.

Question 4

Define the terms learning and development. With your understanding of learning and development how would you respond to someone who said “I was never any good at maths so I’m not expecting my daughter/son to be any good!”

Answer 4

• Learning is about lasting change that is not simply a consequence of maturation or development, but rather of experience
  
  • Development relates to changes that occur as a result of maturation. Just as children develop physically, they also develop socially, emotionally and cognitively. Children’s stages of development have an impact on the ways in which they learn, and on the kinds of skills, knowledge and understandings that they are able to acquire.
  • Children learn from their environment/context. IF their parents dislike maths and have a negative view of it and present that view to their children, then their children are more likely to dislike maths
  • Its about when you show them maths, if you begin to introduce it when they are younger their number sense will be better by the time they get to school

Question 5

What would your reaction be to seeing a teacher’s daily work pad which had as a description for one of the lessons as, “Today I will be teaching the proficiency strand of reasoning”.

Answer 5

• The proficiency strands can’t be focused on in one lesson. They need to be achieved for each and every topic. They come at different times for different children depending on the level they are at.
  
  • If we are seeking fluency, then clear explanations followed by practice will work.
  • If we are seeking understanding, then very clear and interactive communications between teacher and students will be necessary
  • If we want to foster problem solving and reasoning, then we need to use tasks with which students can engage, which require them to make decisions and explain their thinking.
  • There is no content in their description. The proficiency strand is the how but they have not included the what or the why.

Question 6

Describe what the four proficiency strands are and where they fit in the Australian Curriculum.

Answer 6

• Understanding= conceptual understanding rather than procedural understanding (understanding why something works not just how to do it)
  
  • Fluency- Being able to get the correct answers- it does not rely on speed but accuracy
  • Problem solving- Can’t be problem solving if you know how to do it. It’s about solving something you don’t know how to do.
  • Reasoning- relies on all other strands
    Strands are intertwined- they rely on each other but should be viewed as individual strands
  • If we are seeking fluency, then clear explanations followed by practice will work.
  • If we are seeking understanding, then very clear and interactive communications between teacher and students will be necessary
  • If we want to foster problem solving and reasoning, then we need to use tasks with which students can engage, which require them to make decisions and explain their thinking.
  • Content strands=nouns and proficiency strands= verbs
    
    • Content is what needs to be taught and the proficiency strands is how they will be taught.
  • Overarching
  • Can be found in the year level descriptors

Question 7

List and explain the 6 principles of effective teaching as defined by Muir (2008a).

Answer 7

• Make connections
  
  • Using real life examples and questions
  • Connect to what they already know
• Challenge all students
  
  • Stretching students of all levels
  • Create activities that allow for all levels and abilities
• Teach for conceptual understanding
  
  • Ensure they understand the concept and why it works not just that it does work
• Purposeful discussion
  
  • Have a reason for chatting rather than just talking
• Focus on mathematics
  
  • Make sure you are trying to get mathematical development out of the lesson
• Positive attitudes
  
  • Teacher needs to have positive attitudes to influence the student’s feelings towards mathematics

Question 8

At the top of page 54, Siemon et al. (2015) state that “Planning is intimately related to assessment.” What does this mean?

Answer 8

• You must plan what you are going to assess to ensure;
  
  • You have taught it
  • You have met the curriculum points
• The planning cycle
  
  • Planning- knowing your students and where they are at
  • Teaching- Are multiple intelligences being addressed? Does the lesson provide for differentiation?
  • Assessment
    
    • Diagnostic assessment (where at)
    • Formative assessment (ongoing and providing feedback),
    • Summative (End of topic)
• Evaluation- determines the effectiveness of teaching and learning through analysis of evidence

Question 9

Define the terms: subitising,perceptual subitising and conceptual subitising,and how we can support the development of these.

Answer 9

• Subitising- An instant recognition of numbers (usually in small groups)
• Perceptual subitising- recognising numbers up to 4 (without maths)
  
  • Do games with dice
• Conceptual subitising- recognising numbers up to 10 and larger instantly by adding 2 things together, e.g. 8 is 5+3 (using maths)
• Support- using the numbers in everyday life, allowing them to recognise these numbers in a variety of situations, giving them many opportunities to use the numbers 1 to 10 as they are most commonly used
• E.g. Using playing cards, dice, arrays

Question 10

A student often is one out in the solution to simple addition and subtraction questions, for example 6 + 3 = 8 or 8 – 3 = 6. What might this be indicating and why might they be arriving at these solutions? What are you going to do about it?

Answer 10

Can’t count on. Board games

• Ask them how they are getting their answers to check if they have a misunderstanding
• Use concrete, representational and then abstract ideas to teach it again
  
  • Allow them to practice the new way
  • They may have a misunderstanding when counting (can’t count on). E.g. To add 6+3 they start from 6 then say 7 and then 8.
  • Struggle to subitise
• Use counters, use dots, then go back to sums

Question 11

1. One of the most common errors in bridging the 100 is when a student recites the sequence of “…98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 200, 201, 202…”.
   What do you think has contributed to this?
   What would be an appropriate teaching response?

Answer 11

• Counting sequence error
• Misunderstanding with place value
  
  • Showing place value e.g. 109=109 100+9. 110=100+10.
  • Place it up around the room
• Don’t stop at 100 when teaching- stop at 130.

Question 12

What do we mean if we use the terms renaming, regrouping or trading? What activities could you use to support this learning?

Answer 12

• Renaming= Process that involves renaming large numbers by their parts. E.g. The number 335 could be renamed 33 tens and 5 ones
• Regrouping= The process of making groups of ten when adding or subtracting two digit numbers or more e.g. Carrying and borrowing.
• Trading= Swapping 1 ten for 10 ones
• Introduce trading or regrouping before standard algorithms so that children have developed a deep understanding of tens as abstract composite units
• Bundling using straws- Activity with Rosie