- Number knowledge consists of:
- Formal ideas related to numeration and place value
- Informal ideas that we call number sense (Jamieson-Proctor, 2016a) - Mental computation is a numeration skill that allows people to calculate mentally (Jamieson-Proctor, 2016a)
- Mental computation develops from:
- Number sense
- Exploring a range of useful mental strategies
- Knowledge of multiple facts (Jamieson-Proctor, 2016a) - To compute mentally students must:
- Decide what operation to perform
- Select a strategy for carrying out the operation
- Perform the operation(s)
- Make sense of the answer (Jamieson-Proctor, 2016a) - The decimal point separates the whole from the fraction part of the number (Jamieson-Proctor, 2016a)
- There are three types of numbers in the world:
- Cardinal numbers
- Ordinal numbers
- Nominal numbers (Jamieson-Proctor, 2016a) - There are seven types of number subsets:
- Prime and composite numbers
- Square and cubic numbers
- Triangular numbers
- Odd and even numbers
- Pascal's triangle
- Exponents, integers and real numbers
- Directed number (Jamieson-Proctor, 2016a) - Place value helps to:
- Form a picture
- Help to calculate
- Help to estimate
- Help to learn new numbers (Dietz, 2016) - We use the Hindu-Arabic numeration system. This system has four important characteristics:
1. Place value
2. Base of ten
3. Use of zero
4. Additive property (Reys, Lindquist, Lambdin, Smith, Rogers, Falle, Frid & Bennett, 2012) - Two types of materials help young children develop place value:
1. Ungrouped materials
2. Pre-grouped materials (Reys et al., 2012) - Australian curriculum documents recommend the use of calculators and computers in school (Reys et al., 2012)
- Number theory provides an avenue to extend and practise mathematical skills (Reys et al., 2012)
- Prior to this I had very little knowledge of mental computation because it has never exactly been my strong suit
- This week also changed my understanding about place value and the use of calculators. Prior to this week I didn't know that the curriculum placed importance on children using calculators in maths. Whenever I was taught in school calculators were frowned upon
Demonstrate your understanding of the mathematical concept and related skill and strategies children need to assimilate and be able to use, that are related to the topic of number and place value
- There were many different concepts covered this week, including number, numeration, number sense and place value.
- The concept of place value base 10 "...means that any number can be represented using only 10 digits (0-9)" (Reys et al., 2012, p. 169). In place value "the position of a digit represents its value" (Reys et al., 2012, p. 168)
- The skill of place value base 10 include:
- trading
- regrouping
- renaming (Reys et al., 2012) - The thinking strategies of place value base 10 is to make place value mats and use concrete materials to understand
Language model for concept
 |
| Figure 1.21: Jamieson-Proctor, R. (2016). Language model for place value. Retrieved from http://leo.acu.edu.au/mod/forum/discuss.php?d=403262 |
- A teaching strategy for place value is to use different concrete materials in teaching place value. Concrete materials can include beans, straws, counters, MAB and many others.
- Other resources include:
- Place value concept
- Place value song
Describe/demonstrate a specific misconception children might have in relation to number and place value
- A common misconception about number and place value is that children have an understanding of the size of numbers greater than 100, even if they can count that high (Reys et al., 2012)
- A way to re-mediate this misconception is to have children have hands on practice with concrete materials, which display the size of numbers greater than 100
- This can be done by using resources such as beans, counters, MAB blocks. Beans and counters can be glued on to cardboard to show the size of numbers
- Place value can first be seen in Year 1, ACMNA014, Number and Algebra Strand, Number and Place Value substrand
- The content description for ACMNA014 is "count collections to 100 by partitioning numbers using place value" (ACARA, 2016).
- The elaborations for ACMNA014 are:
- "understanding partitioning of numbers and the importance of grouping in tens;
- understanding two-digit numbers as comprised of tens and ones/units" (ACARA, 2016) - Attached are three Scootle resources which aid in the teaching of ACMNA014:
 |
| Figure 1.22: Scootle Resource One Retrieved from http://www.scootle.edu.au/ec/search?accContentId=ACMNA014&userlevel=(1) |
 |
| Figure 1.23: Scootle Resource Two Retrieved from http://www.scootle.edu.au/ec/search?accContentId=ACMNA014&userlevel=(1) |
 |
| Figure 1.24: Scootle Resource Three Retrieved from http://www.scootle.edu.au/ec/search?accContentId=ACMNA014&userlevel=(1) |
Resources for students to understand the concept of place value:
Resources for students to understand the skills related to the concept of place value:
Resources for students to understand the teaching strategies related to the concept of place value:
- Number expander - can also be made in paper by students
 |
| Figure 1.25: Hands On. (2015). Place value. Retrieved from http://www.handson.co.uk/primary-resources/maths-numeracy/place-value.html |
- MAB blocks
 |
| Figure 1.26: Alexander, A. (2014). Math: Place value with base ten blocks. Retrieved from http://wes1stgradeparentblog.blogspot.com.au/2014/09/math-place-value-with-base-ten-blocks.html |
- Base 10 sticks - can be made by students and can be made by many different materials
 |
| Figure 1.27: The Crafty Classroom.com. (2012). Base ten sticks. Retrieved from https://au.pinterest.com/pin/78672324710128926/?from_navigate=true |
Provide a concise synthesis of the textbook chapter/s related to number and place value
Chapter 8 - Extending number sense: Place value
- We use the Hindu-Arabic numeration system. This system has four important characteristics:
1. Place value
2. Base of ten
3. Use of zero
4. Additive property (Reys et al., 2012) - Development of place value promotes number sense and rests on 2 key ideas:
1. Explicit grouping or trading rules are defined and consistently followed
2. The position of a digit determines the number being represented (Reys et al., 2012) - Place value (in the Hindu-Arabic number system) means that any number can be represented using only 10 digits, 0-9 (Reys et al., 2012)
- Two types of materials help young children develop place value:
1. Ungrouped materials
2. Pre-grouped materials (Reys et al., 2012) - Place value models may be either proportional or not proportional (Reys et al., 2012)
- Later errors in computation can often be traced back to a lack of understanding of place value
- In developing place value and establishing number names it is far better to skip beyond the teens and start with the larger numbers (Reys et al., 2012)
- Counting suggests many patterns (Reys et al., 2012)
- The calculator can be successfully used to illustrate regrouping with very large numbers (Reys et al., 2012)
- Practice in skip counting helps decrease bumps in place value learning (Reys et al., 2012)
Chapter 10 - Solving mathematical problems with mental and written strategies, calculators and estimation
- Australian curriculum documents recommend the use of calculators and computers in school (Reys et al., 2012)
- Finding the balance between mental, written and calculator (or computer) methods is an essential aspect of the Australian curriculum (Reys et al., 2012)
- Children should be encouraged to always try mental computation before using paper and pencil or a calculator (Reys et al., 2012)
- Estimation is a valuable process that produces answers that are close enough to allow for good decisions without performing elaborate or exact computations (Reys et al., 2012)
- There are three different points at which to perform the estimation process:
1. Before solving a problem
2. During the problem
3. After solving the problem - Using compatible numbers is often helpful in estimation
Chapter 14 - Extending students with number theory
- Number theory allows the opportunity to extend and connect mathematical ideas met in other chapters (Reys et al., 2012)
- Number theory provides an avenue to extend and practise mathematical skills (Reys et al., 2012)
- Classifying numbers as odd or even is one of the first number theory topics that children encounter (Reys et al., 2012)
- Children begin to learn about factors and multiples when learning about multiplication and division (Reys et al., 2012)
- Prime numbers are numbers which are greater than 1 that has only 2 factors, 1 and itself (Reys et al., 2012)
- A number with more than 2 factors is known as a composite number
- Every composite number can be expressed through prime factorisation (Reys et al., 2012) - A number is divisible if there is no remainder (Reys et al., 2012)
- Polygonal numbers include square and triangle numbers (Reys et al., 2012)
Alexander, A. (2014). Math: Place value with base ten
blocks. Retrieved from http://wes1stgradeparentblog.blogspot.com.au/2014/09/math-place-value-with-base-ten-blocks.html
Australian
Curriculum Assessment and Reporting Authority. (2016). Mathematics. Retrieved from http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?layout=1#level1
Dietz,
J. (2016). EDMA202/262 Tutorial Week 6.
Retrieved from http://leo.acu.edu.au/mod/book/view.php?id=1237883&chapterid=32751
Education
Services Australia. (2016). Scootle:
Mathematics. Retrieved from http://www.scootle.edu.au/ec/search?accContentId=ACMNA014&userlevel=(1)
Hands On. (2015). Place value. Retrieved from http://www.handson.co.uk/primary-resources/maths-numeracy/place-value.html
Jamieson-Proctor,
R. (2016a). EDMA202/262 mathematics
learning and teaching 1: Week 6 Part 1. Retrieved from http://leo.acu.edu.au/mod/book/view.php?id=1237883&chapterid=32746
Jamieson-Proctor,
R. (2016b). Language model for place
value. Retrieved from http://leo.acu.edu.au/mod/forum/discuss.php?d=403262
Math
& Learning Videos 4 Kids. (2015). Place
value first grade: Tens and ones. Retrieved from https://www.youtube.com/watch?v=1F3AycEDksY
Mr.
Peters’ Classroom. (2015). Place value
song: Rude by MAGIC! Parody. Retrieved from https://www.youtube.com/watch?v=gsvrhKka1nc
Mr.
R’s Songs for Teaching. (2011). Place
value math song: Ones, tens, hundreds. Retrieved from https://www.youtube.com/watch?v=5W47G-h7myY
Reys,
R., Lindquist, M., Lambdin, D., Smith, N., Rogers, A., Falle, J., Frid, S.,
& Bennett, S. (2012). Helping
children learn mathematics. Queensland, Australia: John Wiley & Sons
Australia, Ltd.
Teacher
Tipster. (2010). Teacher tipster: Place
value song. Retrieved from https://www.youtube.com/watch?v=ATgnG0M3S3Q
The Crafty
Classroom.com. (2012). Base ten sticks. Retrieved from https://au.pinterest.com/pin/78672324710128926/?from_navigate=true
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