BIG IDEAS:
- Pre-number is early concepts and beginning processes including, determining attributes, matching attributes (matching 2 objects of the sameness and likeness), sorting attributes (Involves 3 or more objects to group according to a defined attribute), comparing (involves 2 objects at a time looking for a mathematically significant attribute like long, longer, heavy/heavier, ordering attributes and patterning (It requires the above mentioned pre number Mathematics concepts together).
- Attribute is the characteristic.
- We encourage the use of all 5 senses when determining attributes. We allow chn to learn by exploring but we have to be careful of sense of taste with smaller children.
- 3 D's of sorting: Decide (concept), Do (skill), Describe (strategy)
- Patterning is an essential prerequisite to later study of Mathematics.
- Counting principles: 1. One to one correspondence – pointing to and counting each object individually. 2. Stable order (1….9) – say the number name in the correct sequence. 3. Cardinal principle – say or tell the number how many items there actually is. 4. Abstraction principle – what can be counted and what cannot be counted. 5. Order irrelevance – does not matter which way the items are counted
- Subitising - skill to rapidly recognize the number of items in a small set without counting usually up to number 9.
- Classifying numbers: Cardinal: numbers that describe how many or how much, Ordinal: numbers that describe position or order, Nominal: numbers as a label.
Pre-number and early number concept, skills and strategies are those that have to be in place before student formally start study of number. If they don't have these concepts, skills and strategies before we start investigating formally number, 1,2,3,4,5 etc. and what it means and the operations, addition, subtraction, multiplication and division, Students will have great difficulty because these pre-number concepts form a basis for their further study of number and form their understanding of number relationships and number patterns. They have to understand those concepts to do with sequencing, matching and order before integrating these concepts to the study of number.
If children can't state the 1 number that tells how many then they don't have cardinality and therefore don't really fully understand the principle of counting to determine how many.
(Jamieson-Proctor,2016)
CONCEPTS, SKILLS AND STRATEGIES
- Concept for sorting: Taking large group of objects and splitting up into small group of number based on attributes.
- Skill for sorting: Moving stuff around to smaller piles to visibly see when the larger number has been sorted.
- Strategies for sorting: Thinking how I can sort it? through the use of sorting chart, cups or containers etc.
(Jamieson-Proctor, 2016)
LANGUAGE MODEL
(Jamieson-Proctor, 2016)
MISCONCEPTION:
Many students who are able to recite the number naming sequence (ie, count orally) to 20 and beyond; recognise, read, and write number words and numerals to 10; and count and model small collections (less than 20), will guess when asked ‘how many’ in a particular collection or to identify which of two single digit numbers presented orally or in written form is the larger/smaller, and/or experience difficulty when counting larger collections (40 or more) accurately.
This could be due to/associated with:
- a failure to understand that counting is a strategy to determine ‘how many’ and/or that the last number counted says how many;
- a mismatch between the oral words and the objects counted (eg, matches objects to syllables, omits certain number names);
- a failure to organise the count to avoid counting objects already counted; and/or
- a superficial understanding of numbers 0 to 10 (ie, limited to simple counts and recognising, reading and writing number names and numerals
(“Common misunderstandings - level 1 trusting the count,” 2015)
ACARA:
(Services, n.d.)
RESOURCES AND IDEAS
(Australia, n.d.)
(Christa Miyoni, 2016)
(Mary Brady, 2013)
- Good number sense is a prerequisite for all later computational development.
- Young chn need to recognize small groups of objects (up to 5 or 6) by sight (subitising) and name them properly.
- Activities involving sight recognition of the numbers of objects in small groups provide opportunities to introduce and use key terms such as less, after, before, 1 more, 1 less.
- To foster a a better number sense, instruction on the numbers through 5 should focus on patterns and develop recognition skills.
- Rational counting is a goal for all young chn using a fixed number-name list, assigning one and only one name to each object counted and realizing that the last object counted represents the number in the group.
- oral counting leads to ways of writing and representing cardinal, ordinal and nominal numbers.
- The relationship between grouping of objects, the number names, the written symbols must be well understood.
(Reys 2012 Chapter 7)
REFERENCES:
Australia, E. S. Home.
Retrieved May 24, 2016, from
(Australia, n.d.)
Australian Curriculum. Retrieved May 25, 2016, from ACARA,
Christa Miyoni (2016, March 1). Prep Subitising and counting on Tutorial Retrieved
(Christa Miyoni, 2016)
Common misunderstandings - level 1
trusting the count. (2015, September 11).
Retrieved May 24, 2016, from http://www.education.vic.gov.au/school/teachers/teachingresources/discipline/maths/assessment/Pages/lvl1trust.aspx
(“Common misunderstandings - level
1 trusting the count,” 2015)
Services, E. Mathematics. Retrieved May 25, 2016, from
http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?layout=1#cdcode=ACMNA003&level=F
(Services,
n.d.)
Jamieson-Proctor, P. R. (2016). Lecture Week 5 Part 1, 2 & 3
Mary Brady (2013, November 2). Counting dot plates Retrieved from
(Mary Brady, 2013)
Reys, L. L. (2012).
Helping Children Learn Mathematics. Wiley.
WEEK 6:
PLACE VALUE refers to the location of a digit in a numeral. The VALUE of the digit is determined by its PLACE. So in 234 the 2 has the value 2 hundreds, the 3 has the value 3 tens and the 4 has the value 4 ones.
FACE VALUE however is consistent for each digit. It doesn't make any difference where the digit is located, it will always have the same face value.
Prime number is a number that is only divisible by itself and 1.
Composite number is divisible by more than two numbers.
1 is only divisible by 1 number and 0 is divisible by every other number. The answer is 0 no matter what you divide 0 by. So 0 has an infinite number of factors. However 0 is not composite.
Google "is 0 composite" to see a range of explanations about why 0 is not a prime nor composite number.
Number Sense is understanding numbers and Number knowledge - informal: number sense. Number sense is an informal understanding of number in everyday use e.g., odd number houses are on the opposite side of the street from even numbers; 1/2 an apple is more than 1/4 of an apple etc.
Number Sense is an INFORMAL understanding of number and numeration is the FORMAL understanding that is taught in schools.
Mental Computation
- Decide what operation to preform (See)
- Select a strategy (Plan)
- Perform the operation (Do)
- Make sense of the answer (Check)
Mental computation strategies are developed rather than taught, number fact strategies are specifically taught.
Mental computation is when children use their own language and develop their own strategies.
Mental computation is when children use their own language and develop their own strategies.
Numeration is the formal understanding of numbers and number notation. Numeration is the content and processes taught in mathematics lessons.
Number sense: Formal: ideas related to numeration and place value/Informal: ideas that we call number sense
LANGUAGE MODEL:
(Jamieson-Proctor 2016)
(Jamieson-Proctor 2016)
MISCONCEPTION:
Many students who are able to identify place-value parts (eg, they can say that there are 4 hundreds 6 tens and 8 ones in 468) and count orally to 100 and beyond, still think about or imagine 2 and/or 3 digit collections additively in terms of ones (ie, 468 is actually understood as the sum of 400 ones, 60 ones and 8 ones).
This could be due to/associated with:
- inadequate part-part-whole knowledge for the numbers 0 to 10 and/or an inability to trust the count;
- an inability to recognise 2, 5 and 10 as composite or countable units (often indicated by an inability to count large collections efficiently);
- little or no sense of numbers beyond 10 (eg, fourteen is 10 and 4 more); and/or
- a failure to recognise the structural basis for recording 2 digit numbers (eg, sees and reads 64 as “sixty-four”, but thinks of this as 60 and 4 without recognising the significance of the 6 as a count of tens, even though they may be able to say how many tens in the tens place)
(“Common misunderstandings - level 2 place-value,” 2015)
ACARA:
(Services, n.d.)
RESOURCES AND TEACHING IDEAS:
(Australia, n.d.)
This is a video that I made to introduce place value of tens and ones to the early foundation years.
(Math &
Learning Videos 4 Kids, 2012)
SYNTHESIS OF THE TEXTBOOK:
- Children must have a clear understanding of our number system if they are going to be mathematically literate.
- They must be able to distinguish the 4 characteristics of our number system (role of zero, additive property of numbers, a base of ten and place value.)
- Place value mats serve as a visual reminder of the quantities involved and provide a bridge toward the symbolic representation of larger numbers.
- The importance of place value is second to none in all later development of number concepts.
- Place value concepts help children compose and decompose numbers and begin to recognize equivalent representations.
- Understanding place value is essential to counting and facilitates operating with larger numbers.
- Place value is not completely developed before operations are introduced.
- Experiences with adding, subtracting, multiplying and dividing whole numbers develops additional competence and understanding of place value.
- Place value is essential for concepts of money and measurement.
(Reys 2012 Chapter 8)
Australia, E.
S. Search. Retrieved May 25, 2016, from
References:
(Australia,
n.d.)
Australian Curriculum. Retrieved May 25, 2016, from ACARA,
BaseTen.
Retrieved May 25, 2016, from
(“BaseTen,”
n.d.)
Common
misunderstandings - level 2 place-value. (2015, September 11).
Retrieved May
25, 2016, from http://www.education.vic.gov.au/school/teachers/teachingresources/di
scipline/maths/assessment/Pages/lvl2place.aspx
(“Common
misunderstandings - level 2 place-value,” 2015)
Jamieson-Proctor, P.
R. (2016). Lecture Week 6 Part 1, 2
& 3
Math &
Learning Videos 4 Kids (2012, December 8). Place value lesson - 1st and
2nd grade
math Retrieved
from
(Math &
Learning Videos 4 Kids, 2012)
Reys, L. L. (2012).
Helping Children Learn Mathematics. Wiley.
Services, E. Mathematics. Retrieved May 25, 2016, from
(Services,
n.d.)
WEEK 7: ALGEBRA
BIG IDEAS:
Algebra
is the relationship
between 2 variables. This relationship can be written and it can
also be spoken. Algebra can be number patterns as well as geometric
patterns (geometry).
Algebra is a different way of thinking it requires problem solving skills, decision making, reasoning and creative thinking, it also helps the brain grow and work out.
Recognise
the pattern
– look at what is happening in the pattern (copy the pattern)
Describe
the pattern
– find out what’s repeating. (Tell me about the pattern?, how can we grow
the pattern?)
Repeating/copying
the pattern
– the elements of the pattern are continuously repeated.
Growing
the pattern
– extending the pattern or making it bigger, adding elements.
Replacing
the pattern
– replacing missing elements in the pattern – 2,4,___, 8
Translating
the pattern
– change the pattern – visual to mathematical. Mathematical to auditory
Children
need to identify the repeating element of a pattern. These
patterns are very important for children to learn.
Algebraic
thinking clarifies and enriches children's thinking about arithmetic.
CONCEPTS, SKILLS AND STRATEGIES:
- Concept: patterns and functions, equivalence and equations, patterns, sequences and generalisations and relationship
- Skill: Being able to find the missing element, to grow and create a pattern and to state the relationship of the pattern
- Thinking: Directly related to with investigation of number
LANGUAGE MODEL:
(Jamieson-Proctor, 2016)
MISCONCEPTION:
The equal
sign is not saying 'here comes the answer'. We need to help children understand what the
equals sign means. When one side is equal to another side there is
balance and this can be shown using a set of scales.
ACARA:
(Acara n.d.)
RESOURCES AND IDEAS:
(Australia n d.)
SYNTHESIS OF THE TEXT:
- Helping chn to think algebraically does not mean adding another topic; it does mean we probably need to teach those areas differently.
- We can help children develop algebraic concepts and habits of algebraic thinking through questioning; helping students model problems, patterns and relations; encouraging them to generalize and expecting them to justify their thinking and statements.
(Reys 2012 Chapter 15)
REFERENCES:
Australia, E.
S. Login. Retrieved May 25, 2016, from
(Australia,
n.d.)
Australian
Curriculum. Retrieved May 25, 2016, from ACARA,
(Acara n.d.)
Jamieson-Proctor, P.
R. (2016). Lecture Week 7 Part 1, 2
& 3
Reys, L. L. (2012).
Helping Children Learn Mathematics. Wiley.
Week 8: MEASUREMENT
BIG IDEAS:
-Measurement involves counting units.
-A unit is a fixed amount that is used to measure
an amount or quantity. -A characteristic is an attribute or property of an item or object.
-links to numbers measuring length, time, volume , mass and area.
1. Identify the attribute (concept)
2. Choose an appropriate unit of measurement for the attribute being measured
3. Measure the object using chosen unit (counting, comparing, ordering, sequencing)
4. Report the number units (How many pencils wide? metres long, square meters in area, cubic meters in volume.
**It is crucial for children to understand the concept of what they are measuring to start with so they can get the understanding that when they are measuring something with counting units, they need to start with arbitrary units like blocks, lego etc. They need to count these units and that's how they measure and then introduce technology.
CONCEPTS, SKILLS AND STRATEGIES:
- Concept of measurement is the size of many attributes.
- Skill is comparing 2 objects on that attribute perceptually, directly and indirectly.
- Thinking strategy is finding the number of units that would represent the object on the attribute that should arise from natural measures before using metric units.
LANGUAGE MODEL:
(Jamieson-Proctor, 2016)
MISCONCEPTION:
Arbitrary units are not regular/not standard. A child might use a much longer pencil to measure the length of a book and another child measures the same book with a much shorter pencil. There will be inaccuracy since they used the same arbitrary unit which is a pencil but has got different size, one is longer and one is shorter.
We then introduce standardized units like tape measure and ruler for the measurement to be accurate in measurement.
ACARA:
(Services n.d)
RESOURCES AND IDEAS:
(Australia n.d.)
(Jacqueline Durant-Harthorne, 2014)
(Sesame Street Games TV, 2014)
(Teaching Without Frills, 2015)
SYNTHESIS OF THE TEXTBOOK:
- Measuring is a process that may be used when determining the size of many attributes.
- Finding equivalent measurements and relating 2 attributes will help chn learn about measuring.
- The language of measurement is crucial to develop a all stages, as is the process of estimation.
- The opportunity to show the practicality of mathematics is by including measuring in the classroom. It can develop other mathematical ideas, to relate mathematics to other topics, and to make mathematics meaningful for children.
(Reys et al Chapter 17)
REFERENCES:
Australia, E. S. Home. Retrieved May 24, 2016, from
(Australia, n.d.)
Australian Curriculum. Retrieved May 25, 2016, from ACARA,
Jacqueline
Durant-Harthorne (2014, December 11). Me and the
measure of things Retrieved from https://www.youtube.com/watch?v=sOEwT3l5fuk
(Jacqueline
Durant-Harthorne, 2014)
Jamieson-Proctor,
P. R. (2016). Lecture Week 8 Part 1, 2 & 3
Reys, L. L.
(2012). Helping Children Learn Mathematics. Wiley
Sesame Street
Games TV (2014, October 6). Sesame street measure
that animal Murray Online game for children Retrieved from
(Sesame
Street Games TV, 2014)
Services, E. Content description. Retrieved May 25, 2016, from
(Services,
n.d.)
Teaching
Without Frills (2015, November 2). Introduction to
standard measurement for
kids: Measuring length in inches with a ruler Retrieved from
(Teaching
Without Frills, 2015)
WEEK 9: GEOMETRY
Geometry is an
organized, logical and coherent structure which looks at the study of shape,
space and measurements, focusing on lines, lanes and solids in a 1D, 2D and 3D
form.
-Tessellation
Art is mathematics and art coming together through different shapes, leaving no
space.
5 basic
skills in geometry
•
Visualising: recognising and describing attributes, similarities and
difference
•
Communication: discussing oral or written, naming, describing concepts
•
Drawing and modelling: sketch and draw 2D and model 3D
•
Thinking and reasoning: classifying, analysing, reasoning, synthesising
similarities and differences
•
Applying geometric concepts and knowledge: linking beween shape and
function e.g. bridge, footballs
**Children
should have experience with the actual shapes and use ‘children’s language’ to
find their own formula.
**Language
is built upon as children develop their understanding through their
experiences. Ensuring that correct terminology is used to avoid confusion.
**Do not teach formulas by rote, have them work and think like a mathematician. Rote learning has no space for proper Geometry. Have experiences to allow students to explore, discover and see for themselves the particular attribute and work formulas out on their own. Also have students to compare formulas to see if different shapes have the same or similar formulas. Having children make their own resources enables them to learn on a deeper level.
CONCEPTS, SKILLS AND STRATEGIES:
- Concept of 1D, 2D and 3D shapes and how can children relate to these? Ex. sphere, children to picture something that looks like a sphere (a ball.)
- Skill is being able to measure the capacity that this particular sphere holds if we fill it up with water. Finding out the formula to measure the outside volume, surface area of the sphere.
- Thinking strategy is finding out how to work out the volume of the sphere? What strategy will I use?
LANGUAGE MODEL:
SYMBOLIC LANGUAGE: Tetrahedron (4 triangles),
Hexahedron (Cube, 6 squares,) Octahedron (8 triangles,)
Dodecahedron (12 Pentagons,) Icosahedron (20 triangles.)
(Jamieson-Proctor, 2016)
MISCONCEPTION:
Paper is not a 2D shape. It has a 3D shape. Why? Paper doesn't have much thickness but it does have thickness. Correct language to be used is rectangular prism or rectangular shape of paper but not a rectangle. A rectangle is simply the outside line. 2D has no thickness like footprint or graphs. 3D has got length, thickness, depth and width.
(Jamieson-Proctor, 2016)
ACARA:
(Acara n.d.)
RESOURCES AND IDEAS:
(Australia n.d.)
SYNTHESIS OF THE TEXTBOOK:
- Geometry is often neglected in primary school.
- Geometry helps us represent the space in which we live and to describe location, movement and the relationships between objects in space.
- Transformation and symmetry can be used to describe our world and to solve problems.
- Building chn's visual skills and reasoning is an important aspect of geometry,
- The ultimate aim of geometry is to be able to solve problems and to appreciate the geometry around us.
(Reys et al Chapter 16)
REFERENCES:
REFERENCES:
Australia, E.
S. Login. Retrieved May 25, 2016, from
(Australia,
n.d.)
Australian
Curriculum. Retrieved May 25, 2016, from ACARA,
(Acara n.d.)
Jamieson-Proctor, P.
R. (2016). Lecture Week 7 Part 1, 2
& 3
Reys, L. L. (2012).
Helping Children Learn Mathematics. Wiley.
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