ArticlePDF Available

Abstract

The mathematical content knowledge (MCK) and pedagogical content knowledge (PCK) of primary and elementary teachers at all levels of experience is under scrutiny. This article suggests that content knowledge and the way in which it is linked to effective pedagogies would be greatly enhanced by viewing mathematical content from the perspective of the ‘big ideas’ of mathematics, especially of number. This would enable teachers to make use of the many connections and links within and between such ‘big ideas’ and to make them explicit to children. Many teachers view the content they have to teach in terms of what curriculum documents define as being applicable to the particular year level being taught. This article suggests that a broader view of content is needed as well as a greater awareness of how concepts are built in preceding and succeeding year levels. A ‘big ideas’ focus would also better enable teachers to deal with the demands of what are perceived to be crowded mathematics curricula. The article investigates four ‘big ideas’ of number – trusting the count, place value, multiplicative thinking, and multiplicative partitioning – and examines the ‘microcontent’ that contributes to their development.
Developing the Big Ideas of Number
Introduction
The notion of ‘big ideas’ of mathematics is not a new one, but little has been specifically written
about it. It has been alluded to in the seminal works of Schulman (1986) who spoke of the ‘syntactic
structures within mathematics’ and Ma (1999) who discussed the idea of ‘knowledge packages’. Charles
(2005) described 21 ‘big ideas’ of mathematics and noted, as did Clarke, Clarke & Sullivan (2012) that it
would be unlikely to obtain universal agreement from teachers and teacher educators about precisely what
the ‘big ideas’ should be.
Siemon, Bleckly & Neal (2012) took a particular stance in discussing the ‘big ideas’ of number in
terms of how they were presented in the Australian Curriculum: Mathematics (Australian Curriculum,
Assessment and Reporting Authority (ACARA), 2012) and described six ‘big ideas of number’ which
form the basis of the graphic illustration that follows (Figure 1). None of the six ideas presented by
Siemon et al. are the same as any of those presented by Charles (2005) apart from Proportional
Reasoning which Charles termed Proportionality. However, the ideas presented are embedded in some of
Charles’ ‘big ideas’.
Charles’ (2005) first ‘big idea’ is termed Numbers and he discusses ‘counting numbers’ which
effectively parallels Siemon et als. (2012) Trusting the Count. Charles’ second ‘big idea’ is The Base Ten
Numeration System in which corresponds with Siemon et al’s. Place Value. However, as part of his
second ‘big idea’, Charles also discusses the idea that “each place value to the left of another is ten times
greater than the one to the right” (2005, p. 13) which is an essential element of the idea of Multiplicative
Thinking as described by Siemon et al. Similarly, Charles has embedded elements of Siemon et al.’s
Multiplicative Partitioning in his first ‘big idea’ of Numbers where he discusses fractions and rational
numbers and in his fourth ‘big idea’(Comparison) where he discusses fractions and percentage.
One of the strengths of Siemon et al’s. (2012) work is that it highlights that there is a hierarchical
aspect to the development of the six ideas and they present this in a table showing approximate age levels
at which it is reasonably expected children would have an understanding of each ‘big idea’ (Appendix 1).
This has been adapted to form the graphic (Figure 1) which also serves to illustrate how elements of each
‘big idea’ necessarily develop alongside other ideas. For example, Siemon et al. note that Multiplicative
Partitioning should be well developed by the end of Year Six, yet it is clear that many aspects or pre-
conditions for its full development are present when children learn about Trusting the Count, Place Value
and Multiplicative Thinking.
Figure 1: Development of the ‘Big Ideas’ of Number
1
Chris Hurst
Curtin University
c.hurst@curtin.edu.au
Derek Hurrell
University of Notre Dame Australia
Derek.Hurrell@nd.edu.au
The relationship between the ‘big ideas’ as depicted in Figure 1, should be considered alongside the
set of criteria for determining the extent of development of children’s understanding of each idea. This is
shown in a variety of tables which accompany each ‘big idea’ and highlights ‘landmark’ or critical points
of development within each of the ‘big ideas’. These tables were constructed using the article by Siemon,
Bleckley and Neal (2012) as a reference point and is also informed by diagnostic maps from First Steps in
Mathematics (Department of Education, Western Australia, 2013b) (FSiM) and by the work of Siemon,
Beswick, Brady, Clark, Faragher and Warren (2011), Reys et al. (2012), and Van de Walle, Karp and Bay-
Williams (2013).
Of the six ‘big ideas’, four of them; trusting the count, place value, multiplicative thinking and
(multiplicative) partitioning, are firmly rooted in the primary school setting. The final two, proportional
reasoning and generalising algebraic reasoning are developmentally more suitable in secondary school
(Siemon et al, 2012). The remainder of this paper will examine the ‘big ideas’ situated in the primary
school years, with a particular emphasis on multiplicative thinking. It will briefly examine each of the big
ideas, how they are inter-related and how they become apparent and are enacted in the primary classroom.
A ‘lens’ has been applied to teachers’ subject matter knowledge to see if there is a need for it to be
enhanced to position them better to assist students in moving through the trusting the count ‘phase’ to
place value, to multiplicative thinking and to (multiplicative) partitioning. It needs to be recognised at the
outset that this progression through the ideas is not a linear process, but a developmental one.
Trusting the count
The first of the big ideas is trusting the count. Originally the term trusting the count was coined by
Willis (2002) to highlight how students may not understand that the number said at the end of the
counting act represented the total, and was invariant, in that if counted again the same number would be
reached. In more recent times the definition of trusting the count has broadened from just being the
invariant result, to also mean “…a child’s capacity to access flexible mental objects for the numbers 0
-10” (Seimon, Beswick, Brady, Clark, Faragher & Warren, 2011, p.197).
Although a detailed account of trusting the count will not be pursued here it is not to underestimate
the importance of trusting the count or the difficulty in the teaching and learning of it. It is however an
acknowledgement that there is much research and literature available to guide, particularly the early
childhood teacher, through good pedagogical practices to position the students to be able to achieve this
particular big idea ( e.g. Department of Education, Western Australia, 2013b; Gelman & Gallistel, 1978).
It is perhaps timely to acknowledge that it is almost axiomatic to suggest that all big ideas are constructed
from many ‘little ideas’, and so it is with trusting the count. Siemon, Bleckly and Neal (2012) identify 11
such ideas which support trusting the count, a list which can be further enhanced through the work of
First Steps in Mathematics (Department of Education, Western Australia, 2013b) and others. This
amalgamated list has been developed and is presented as Table 1.
Table 1
Trusting the Count – Key Criteria
Early number experiences Classifying, grouping, ordering, patterns underpin the
development of this idea
Subitising or instant recognition of small groups can be a means of quantifying
Purpose of counting or subitising is to quantify
Each object is counted once – one to one correspondence
Collections can be compared on a one-one basis
In a count, the last number signifies quantity
Arrangement of objects in a count does not change the quantity
Basic addition facts always give the same result irrespective of arrangement
Counting numbers (the number string) are always said in the same order
There are multiple ways of seeing grouping of objects
Small numbers can be seen as the combination of other numbers
The part-part-whole relationship can be used as the basis for operating
2
Addition and subtraction situations can be considered in terms of a whole and two parts, one of
which is unknown or missing
Additive thinking is employed to solve problems with small numbers
Counting on and back can be used to solve simple problems
Skip counting to find the total will give the same result as one-one counting
Share portions from a quantity and know that there more portions there are, the smaller will be
the portions
Most teachers, particularly those in the early childhood setting would recognise the elements in Table
1, appreciate their place in the development of trusting the count and have a clear understanding of
appropriate pedagogy. They would also acknowledge the understandings developed during the trusting
the count phase have implicit links, and overlaps, even if not immediately developed, with the second ‘big
idea’ place value. Particularly, the notion of the part-part-whole relationship, and the patterns which have
been established with the counting numbers. At the time of moving the students into numbers beyond ten
it is highly likely that these understandings will be emerging and in need of attention in the teaching and
learning.
Place Value
What some teachers may find less obvious is the importance of making the connections between
trusting the count and place value more explicit. A view that students will intuitively develop an
understanding of place value perhaps deserves further scrutiny. Place value is a complex process which is
“…subject to considerable inter-individual variability” (Moeller, Pixner, Zuber, Kaufmann & Nuerk,
2011, p. 1839), and Table 2 is an indication of this complexity showing the variety of criteria which are
needed to be understood. Major (2012) wrote about how this complexity is quite often masked by
condensing all of these key criteria into one seemingly simple construct, that of defining place value as a
way to say, read and write numbers. Further Major alludes to the fact that because students can achieve
the act of saying, reading and writing numbers this can often mask the fact that they are unable to
generalise the multiplicative relationships within the place value system, an issue also recognised by other
researchers (Irwin, 1996; Kamii, 1986; Thomas, 2004). Table 2 is a composite of the research from
sources such as: Department of Education, Western Australia (2013b); Siemon, Beswick, Brady, Clark,
Faragher and Warren (2011); Reys et al. (2012); and Van de Walle, Karp and Bay-Williams (2013).
Table 2
Place Value – Key Criteria
Order of digits makes a difference
There are patterns in the way we read, write and say numbers
Place value columns have names
Zero can hold a place
Ten group is seen as a special entity (use of ten frames)
Number value = face x place
Ten group can be applied to ‘ten tens’ or ‘ten hundreds’ and so on
We can skip count by ten, hundred both forwards and backwards (in place value parts)
Numbers can be partitioned in flexible ways using standard and non-standard partitions
Number partitioning can be shown as indicative of digit value and place value. For example, 26 = 20 + 6
or (2 x 10) + (6 x 1)
Not only does a developing understanding of place value have an impact on the immediate success of
students when moving from single to multi-digit numbers, it also has impact on future mathematical
attainment. Place value is fundamental to the eventual development of algebraic reasoning (Ketterlin-
Geller & Chard, 2011, Wu, 2001). This is another illustration of the overlap and parallel development
between the six ‘big ideas’.
3
Teacher knowledge of mathematics is an essential component of effective teaching (Ball, Hill &
Bass, 2005; Young-Loveridge & Mills, 2009) and effective teaching of place value requires an
understanding of the learning progression. When looking at place value we would argue that its
development runs through three phases. The first phase is unitary value; unitary value being the
placement of the number in the number string i.e. 37 is after the number 36. This is a concept which is
perhaps not as easy as it might seem, as Moeller et al. (2011) insisted that children must automatically
apply place value rules to place the tens and ones in the correct ‘bins’; something which according to
Gervasoni and Sullivan (2007), 27% of Year 2 students find problematic. Being able to place the numbers
into ‘bins’ is important, as students who are better in determining which of two symbolic numbers is the
larger, enjoy higher achievement in mathematics (De Smedt, Noël, Gilmore & Ansari, 2013).
The second phase is quantity value, that is, 36 is 30 + 6, this phase is built on additive thinking and
where standard partitioning along place value lines is employed. This understanding of place value is
particularly important in employing mental computation strategies (Thompson, 2009). In his pithy two-
page 2009 article, Thompson (2009) gives the following example:
                

 !"
#$$%$ &&%$&&'&()
#$%% '&()
# $ ' ' *'&(
+
 ,
,"-),,." /
He concluded that for all of the four operations, the digits in the tens (and hundreds) column are seen as
quantities in their own rights, 40 is not seen as four in the tens column or even 4 x 10, but forty. Further,
he concludes that this is highly desirable until formal written algorithms are required (Thompson, 2009).
The third and final phase is a column value understanding of place value. That is that 36 is 3 x 10
and 6 x 1, the kind of understanding that is vital in being fluent with many standard written algorithms.
Many lower and middle primary school teachers are well versed in the use of trading games and
structured and unstructured materials to promote the first and second phases of place value but can at
times find the third phase a challenge. This third understanding of place value is an important pre-
requisite for multiplicative thinking (Thomas, 2004). As stated previously, it should be understood that
there is a certain amount of multiplicative thinking which is developing simultaneously with trusting the
count, and an increased amount with working towards an understanding of place value (See Figure 2).
The column value understanding of place value relies on a developing understanding of
multiplication. There is an argument (Graveiimeijer & van Galen, 2003) to suggest that a combination of
procedural (memorisation of basic multiplication and division facts) and a conceptual understanding of
multiplication are both required. To move the students through quantity value place value, which is
mostly additive in nature, an alternative approach emphasising the significance of the size of the unit and
the number of those units in determining quantity is required (Confrey & Maloney, 2010). Larsson (2013)
cautioned that if students who use additive thinking are left to practise multiplication facts, algorithms
and other procedures, this may not provide them with the opportunity to develop the understanding of
multiplication as something more than repeated addition of equal groups. Traditionally, teaching
multiplication and division begins with the relationship between repeated addition and multiplication
(Confrey & Smith, 1995) an approach which reflects a ‘repeated addition’ understanding of
multiplication. This ‘repeated addition’ understanding does not necessarily provide the required broader
view and the qualitative change in students thinking which is ultimately required (Barmby, Harries,
Higgins & Suggate, 2009). This broader view is characterised as requiring: replication (rather than joining
as in addition/subtraction); the binary rather than unary nature of multiplication, and the notion of two
distinct and different inputs; commutativity for multiplication but not division; and distributivity
(Barmby, Harries, Higgins & Suggate, 2009).
4
One method for trying to build a conceptual understanding of multiplication is the multiplicative
array. Whilst for the purposes of this article a focus will be made of multiplicative arrays, it should be
noted that other representations, such as the number-line also need to be employed to develop a rich
understanding of column value place value. Literature (e.g. Moseley, 2005) calling on the use of multiple
representations in mathematics education suggested that students who can call upon a broader range of
representations have an increased understanding of concepts.
Multiplicative arrays refer to representations of rectangular arrays in which the multiplier and the
multiplicand are exchangeable. These arrays are seen by some as powerful ways in which to represent
multiplication (Barmby, Harries, Higgins & Suggate, 2009;Young-Loveridge & Mills, 2009). Young-
Loveridge (2005) asserted that they have the potential to allow students to visualise commutativity,
associativity and distributivity, and added that array representation of multiplication should be employed
alongside other representations, to “allow students to develop a deeper and more flexible understanding of
multiplication/division and to fully appreciate the two-dimensionality of the multiplicative process” (p.
38-39). Nunes and Bryant’s (1995) research supported the strength of arrays in relation to developing a
conceptual understanding of commutativity. Wright (2011) states that multiplicative arrays embody the
binary nature of multiplication, and contended that as a representation they have value in that they also
connect to other mathematical ideas of measurement of area and volume and Cartesian products. All this
research lends credibility to the multiplicative array being able to satisfy Barmby, Harries, Higgins and
Suggate’s (2009) ‘broader view’. Certainly at least eight of the 13 criteria listed in Table 3 (big ideas for
multiplicative thinking) can be addressed through the use of multiplicative arrays.
Table 3
Multiplicative Thinking – Key Criteria
Can double count by representing one group (e.g., hold up four fingers) and counting repetitions of that
group, simultaneously keeping track of the number of groups and the number in each group.
Cyclical pattern of 100-10-1 is repeated from ones to thousands
Cyclical pattern of 100-10-1 is repeated beyond 1000s to millions
Ten times multiplicative relationship exists between places
Multiplicative relationship extends to numbers less than one, that is to the right of the decimal point
There is symmetry in the place value number system based around the ones place so that the pattern in
naming wholes is reflected in naming decimals
The multiplicative relationship between quantities is expressed as ‘times as many’ and ‘how many times
larger or smaller’ a number is than another number
Multiplicative arrays are used to visualise and represent multiplicative situations
Multiplicative situations can be represented as equal-groups problems, comparison problems,
combinations (Cartesian) problems and area/array problems.
Basic number facts to 10 X 10 are recalled and patterns in number facts are investigated
Commutative property of multiplication is understood and can be shown to be linked to arrays
The multiplicative situation is understood Factor X Factor = Multiple with the meanings of the terms
clearly understood
Prime and composite numbers are understood and linked to multiplicative arrays
Distributive property of multiplication over addition is applied
Division and multiplication are known as the inverse of one another
Partition division involves finding the size of each group and quotation division involves finding the
number of groups
Multiplicative arrays are linked to the concept of area
Number facts can be extended by powers of ten
Numbers move a place each time they are multiplied or divided by 10
Measurement units have the same multiplicative relationship as the Base Ten Number System
Cartesian products can be represented symbolically and in tree diagrams
Both research (Ma, 1999) and anecdotal evidence would suggest that the complexity of the
understandings of place value required to assist the students towards further mathematical understandings
5
is not well understood by many teachers. This rich understanding of the specialised content knowledge
(Hill, Ball and Schilling, 2008) of place value seems to elude some teachers. It would be of interest to
examine how many teachers are able to articulate both, the complexity of place value, and how the
different forms of place value fit into the curriculum.
Multiplicative Thinking
According to Siemon, Bleckly and Neal (2012), the third big idea is multiplicative thinking. In their
research Clark & Kamii (1996) found that 52% of fifth grade students were not sound multiplicative
thinkers, and the work of Siemon, Breed, Dole, Izard, and Virgona (2006) showed that up to 40% of Year
7 and 8 students performed below curriculum expectations in multiplicative thinking and at least 25%
were well below expected level. Further, Siemon, Breed, Dole, Izard, and Virgona declared that the
students who are not well established with multiplicative thinking do not have the foundational
knowledge and skills needed to participate effectively in further school mathematics, or to access some
post-compulsory training opportunities. If we accept, that in order to understand multiplication we need
the flexibility which place value affords in dealing with larger numbers, then the progression from
trusting the count, through place value, and into multiplicative thinking is a reasonable one.
Multiplicative thinking is fundamental to the development of important mathematical concepts and
understandings such as algebraic reasoning (Brown & Quinn, 2006), place value (this as argued earlier is
cyclical, a developing understanding of multiplicative thinking improves place value understanding,
which in turn improves multiplicative thinking, which in turn…), proportional reasoning, rates and ratios,
measurement, and statistical sampling (Mulligan & Watson, 1998; Siemon, Izard, Breed & Virgona,
2006). Siegler et al. (2012) advocate that knowledge of division and of fractions (another part of
mathematics very much reliant on multiplicative thinking) are unique predictors of later mathematical
achievement.
Multiplicative thinking is not easy to teach or to learn. Whereas most students enter school with
informal knowledge that supports counting and early additive thinking (Sophian & Madrid, 2003)
students need to re-conceptualise their understanding about number to understand multiplicative
relationships (Wright, 2011). Multiplicative thinking is distinctly different from additive thinking even
though it is constructed by children from their additive thinking processes (Clark & Kamii, 1996).
Multiplicative thinking is more than the capacity to remember and utilise multiplication facts. What
is required is the development of the ability to apply these facts to a variety of situations which are
founded on multiplication. Jacob and Willis (2003) proposed five broad stages for the development of
multiplicative thinking: One-to-One Counting; Additive Composition; Many-to-One Counting;
Multiplicative Relations; and Operating on the Operator.
In the one-to-one counting phase the students are grappling with the basics of counting and do not
see the relevance of the many-to-one count, that is, they may know what it means to hand out a given
quantity but this is viewed additively and not multiplicatively (Jacob & Willis, 2003). At this point
students are not able to use a row by column structure (an array) to work out a number of squares, and
they resort to additive strategies (Batista, 1999). Stage 2, additive composition, is when the students
understand the principle of trusting the count, that is, that the last number said indicates the quantity. At
this stage, through skip-counting, the students can use groups to count more efficiently (Jacob & Willis,
2003). It is important at this stage that the children manipulate materials to facilitate the move to
recognising the multiplicative situation, as the materials will help them to: recognise and then count the
number in each group, the number of groups and the total; describe multiplicative situations without
necessarily finding a total; and transfer these understandings to the division situations (Jacob & Willis,
2003).
The third stage is the development of many-to-one counting. Many-to-one counting is when the
students can hold two numbers in their head simultaneously, the number of groups and the total in each
group. At this stage they do not necessarily understand the relationship between multiplication and
division in that they may not transfer all of the understandings gained with multiplication to the division
situation, and they may not consistently employ the inverse relationship between the two operations or the
6
commutative property of multiplication. At the fourth stage, Multiplicative Relations, the students are
able to employ the commutative, distributive and inverse properties of multiplication and division (Jacob
& Willis, 2003; Mulligan & Watson, 1998). They are also aware that the three aspects of multiplication;
the multiplicand, the multiplier and the product, are involved in the multiplicative situation (Jacob &
Willis, 2003). It is at this stage that the need for manipulative materials is decreasing, as students need to
describe when the operations of multiplication and division became objects of thought rather than actions
(Sophian & Madrid, 2003; Wright, 2011). This is the stage which is described by Jacob and Willis (2003)
as one in which students treat the numbers in a problem situation as variables, a concept which is quite
abstract in nature.
Traditionally the approach has been to facilitate students’ multiplicative thinking through a process
of making links with repeated addition (Confrey & Smith, 1995). This is an approach which may stand to
reinforce additive rather than multiplicative thinking and may be detrimental to the variety of situations to
which multiplication needs to be applied (Wright, 2011). This concern has led some researchers to look
for alternative constructs to create this bridge (Confrey & Smith, 1995; Sophian & Madrid, 2003). Rather
than building from an additive construct, some researchers have advocated the use of a primitive
multiplicative operation”, a splitting construct (Confrey and Smith, 1995, p. 66). A splitting construct is
where multiple versions of an original are made such as is seen in a tree-diagram or in doubling and
halving (Confrey and Smith, 1995). By adopting the splitting construct teachers may be able alleviate
some of the issues where students will wrongly apply additive thinking to multiplicative situations, and in
the case of older students, multiplicative thinking (particularly proportional strategies) in additive
situations (Van Dooren, De Bock &Verschaffel, 2010). According to the research of Harel, Behr, Post and
Lesh (1994) students often have the inappropriate additive model presented to them by teachers who
themselves use this intuitive additive model. It may be useful for teachers to be given the opportunity to
‘see’ multiplication as standard and non-standard multiplicative partitioning (splitting) and to create
situations where they are mindful of the way in which questions and tasks are phrased and are aware of
the power of particular representations. Doing so would be beneficial in that there is then a direct and
discernible link between the third big idea of multiplicative thinking and the fourth, (multiplicative)
Partitioning and this sits firmly within the notion of splitting. In 1988 Confrey, wrote:
“There is a concept of multiplication which is not adequately described by the idea of
repeated addition nor the ratio concept. This concept of multiplication is connected to the
actions of splitting and of magnifying (or shrinking) and starts with the unit (one) rather
than the arithmetic origin (zero). A child’s experience of the actions of splitting and
magnifying develops relatively independently of his/her experience with counting or
enumerating, although a mapping does occur from one to the other.” (p. 4)
The fundamental differences between the counting and splitting constructs are succinctly compared by
Corley, Confrey, Maloney, Lee and Panorkou (2012) (Figure 2).Drew Corley
Counting World Splitting World
Fundamental Action: Count as iterative enumeration Split as repeated action
Basic operation: Addition and subtraction Division and multiplication
Origin: 0 (zero) 1
Unit: One n-split
Comparison: Subtraction Ratio
Sequence, function,
repeated operation:
Arithmetic Geometric
Figure 2: Comparison between counting and splitting structures (adapted from Confrey, 1994)
Multiplicative Partitioning
Multiplicative Partitioning is referred to by Confrey (2009) as equipartitioning, and is desribed as
being the set of cognitive behaviours needed to produce equal-sized groups, parts, or combinations of
wholes and parts. Therefore, equipartitioning is essential when starting to work in the difficult to teach
and learn area of rational numbers and their various representation (Anthony & Ding, 2011; Capraro,
2005; Nunes & Bryant, 2009; Usiskin, 2007), and is the foundation of division and multiplication, ratio
7
and rate (Siemon et al., 2011). Some of the complexity in this ‘big idea’ may be illustrated through the
fact that in constructing a learning trajectory for equipartitioning, Confrey (2012) outlines 16 levels of
cognitive proficiency (Appendix 3). These are illustrated in Appendix 1 in terms of statements of
understanding. These statements are in no particular order.
Charalambous’s (2010) proposed that “...strong mathematical knowledge for teaching supports
teachers in using representations to attach meaning to mathematical procedures...” (p. 273). He further
asserted that “...strong MKT supports teachers in giving and co-constructing explanations that illuminate
the meaning of mathematical procedures.” (p. 274). If these propositions are correct then it is not
unreasonable to suggest that the reverse may also be true. Weak mathematical knowledge for teaching
would impede teachers in using representations to attach meaning to mathematical procedures, and
impede teachers in co-constructing explanations that illuminate the mathematics. Given that research
points to teachers having difficulty with the topic of rational numbers (Moseley, Okamoto & Ishida, 2007;
Tirosh, 2000; Zhou, Peverly & Xin, 2006) this is problematic. It indicates that the big idea of
(multiplicative) Partitioning may not be being taught and learned as effectively as it should.
Conclusion
One of the aims of the Australian Curriculum was to make the curriculum ‘deep’ rather than ‘wide’
(National Curriculum Board, 2009). Even so, if each of the content descriptors is taken individually the
capacity of any teacher to cover all of the content would be severely strained. What may be of benefit is
to think at a more macro-level, think in big ideas and then attach the content descriptors to those big ideas
rather than teach to the content descriptors with the notion that the big ideas will emerge. To do this,
teachers need to be given the professional courtesy of being helped towards an understanding of the big
ideas and their importance. In this paper we have attempted to give some insight to what the big ideas
may be, what they mean to the classroom practitioner and how they develop through and within each
other. As was indicated earlier, the final two big ideas, proportional reasoning and generalising algebraic
reasoning are developmentally more suitable in secondary school (Siemon et al, 2012) and consequently
were not be addressed in this paper.
References
Australian Curriculum, Assessment and Reporting Authority (ACARA) (2012). Australian curriculum
mathematics. Retrieved from http://www.australiancurriculum.edu.au/Mathematics/Curriculum/F-
10.
Anthony, G., & L. Ding (2011). Teaching and learning fractions: Lessons from alternative example
spaces. Curriculum Matters, 7, 159-174.
Ball, D., Hill, H., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well
enough to teach third grade, and how can we decide? American Educator, 14-22, 43-46. Retrieved
from http://deepblue.lib.umich.edu/bitstream/handle/2027.42/65072/Ball_F05.pdf?sequence=4
Battista, M. C. (1999). Spatial structuring in geometric reasoning. Teaching Children Mathematics, 6,
171-177.
Bramby, P., Harries, T., Higgins, S., & Suggate, J. (2009). The array representation and primary children’s
understanding and reasoning in multiplication, Educational Studies in Mathematics, 70, 217–241.
Capraro, R. M. (2005). The mathematics content knowledge role in developing preservice teachers'
pedagogical content knowledge. Journal of Research in Childhood Education, 20(2), 102-118.
Charalambous, C. Y. (2010). Mathematical knowledge for teaching and task unfolding: An exploratory
study*. The Elementary School Journal, 110(3), 247-278.
Charles, R.I. (2005). Big ideas and understandings as the foundation for early and middle school
mathematics. NCSM Journal of Educational Leadership, 8(1), 9–24.
Clarke, D.M., Clarke, D.J. and Sullivan, P. (2012). Important ideas in mathematics: What are they and
where do you get them? Australian Primary Mathematics Classroom, 17(3), 13-18.
Clark, F. B., & Kamii, C. (1996). Identification of multiplicative thinking in children in grades 1-5.
Journal for Research in Mathematics Education, 27, 41-51.
8
Confrey,J . (1988). Multiplication and splitting: Their role in understanding exponential functions. In M.
Behr, C. La Compagne, & M. Wheeler( Eds.), Proceedings of the tenth annual meeting of the North
American Chapter of the International Group for Psychology of Mathematics Education (pp. 250-259).
DeKalb, IL.
Confrey, J. (1994). Splitting, similarity, and the rate of change: New approaches to multiplication and
exponential functions. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning
in the learning of mathematics (pp. 293-332). Albany, NY: State University of New York Press.
Confrey, J. (2012). Better measurement of higher-cognitive processes through learning trajectories and
diagnostic assessments in mathematics: The challenge in adolescence. In V. Reyna, M. Dougherty, S.
B. Chapman, & J. Confrey (Eds.), The adolescent brain: Learning reasoning, and decision making.
Washington, DC: American Psychology Association.
Confrey, J., & Maloney, A. (2010). The construction, refinement, and early validation of the
equipartitioning learning trajectory. In K. Gomez, L. Lyons & J. Radinsky (Eds.), Proceedings of the
9th International Conference of the Learning Sciences (Vol.1, pp.968-975). Chicago, Ill: International
Society of the Learning Sciences.
Confrey, J., & Smith, E. (1995). Splitting, covariation, and their role in the development of exponential
functions. Journal for Research in Mathematics Education, 26(1), 66-86.
Corley, D., Confrey, J., Maloney, A., Lee, K., and Panorkou, N. (2012). Retrieved 10/07/2014 from
http://gismo.fi.ncsu.edu/wp-content/uploads/2012/11/2012NCCTM-Unpacking-Equipartitioning-
Standards-in-the-CCSS-M-October.pdf.
Department of Education, Western Australia. (2013a). First steps in mathematics: Overview. Retrieved
from: http://det.wa.edu.au/stepsresources/detcms/portal/
Department of Education, Western Australia. (2013b). First steps in mathematics: Number. Retrieved
from: http://det.wa.edu.au/stepsresources/detcms/portal/
De Smedt, B., Noël, M., Gilmore, C. & Ansari, D. (2013). How do symbolic and non-symbolic numerical
magnitude processing skills relate to individual differences in children's mathematical skills? A review
of evidence from brain and behavior, Trends in Neuroscience and Education, 2(2), 48-55.
Gelman, R., & Gallistel, C. R. (1978). The child's understanding of number. Cambridge, MA: Harvard
University Press.
Gervasoni, A., & Sullivan, P. (2007). Assessing and teaching children who have difficulty learning
arithmetic. Educational & Child Psychology, 24(2), 40-53.
Harel, G., Behr, M., Post, T., & Lesh, R. (1994). The impact of number type on the solution of
multiplication and division problems: Further considerations. In G. Harel & J. Confrey (Eds.), The
development of multiplicative reasoning in the learning of mathematics. New York: State University of
New York Press.
00123445.$$6/+"*"
*1"7""8*
Journal for Research in Mathematics Education39. /9 $$
Irwin, K. (1996). Making sense of decimals. In J. T. Mulligan and M. C. Mitchelmore (Eds.), Children’s
number learning (pp. 243-257). Adelaide: Australian Association of Mathematics Teachers.
Jacob, L. & Willis, S. (2003). The development of multiplicative thinking in young children. In: 26th
Annual Conference of the Mathematics Education Research Group, 6 - 10 July 2003, Deakin
University, Geelong.
Kamii, C. (1986). Place value: an explanation of its difficulties and education implications for the primary
grades. Journal for Early Childhood Education, 1(2), 75-86.
Ketterlin-Geller, R., & Chard, D. J. (2011) Algebra readiness for students with learning difficulties in
grades 4–8: Support through the study of number. Australian Journal of Learning Difficulties, 16(1),
65-78.
Larsson, K. (2013). Mulitplicative thinking in relation to commutativity and forms of representation. In:
Jarmila Novotná and Hana Moraová (Ed.), Tasks and tools in elementary mathematics: . Paper
presented at International Symposium Elementary Maths Teaching, SEMT '13, Prague 2013 (pp. 179-
9
187). Prague: Charles University, Faculty of Education.
Ma, L. (1999). Knowing and teaching elementary mathematics. Mahwah, NJ: Lawrence Erlbaum
Associates.
Major, K. (2012). The Development of an Assessment Tool: Student Knowledge of the Concept of Place
Value. In J. Dindyal, L. P. Cheng & S. F. Ng (Eds.), Mathematics education: Expanding horizons
(Proceedings of the 35th annual conference of the Mathematics Education Research Group of
Australasia). Singapore: MERGA.
Moeller, K., Pixner, S., Zuber, J., Kaufmann, L., & Nuerk, H.-C. (2011). Early place-value understanding
as a precursor for later arithmetic performance–A longitudinal study on numerical development.
Research in Developmental Disabilities, 32, 18371851. doi:10.1016/j.ridd.2011.03.012
Moseley, B. (2005). Students’ early mathematical representation knowledge: the effects of emphasising
single or multiple perspectives of the rational number domain in problem solving. Educational Studies
in Mathematics, 60, 37–69. doi:10.1007/s10649-005-5031-2.
Moseley, B., Okamoto, Y., & Ishida, J. (2007). Comparing US and Japanese elementary school teachers'
facility for linking rational number representations. International Journal of Science and Mathematics
Education 5, 165-185.
Mulligan, J. & Watson, J. (1998). A developmental multimodal model for multiplication and division.
Mathematics Education Research Journal, 10(2), 61–86.
National Curriculum Board. (2009). Shape of the Australian curriculum: Mathematics. Retrieved from
http://www.acara.edu.au/verve/_resources/australian_curriculum_-_maths.pdf
Norris, K. & Swan, P. (2013). Make it Count: Intentional Teaching Activities to Develop Counting Skills
K 2. Perth: A-Z Type.
Nunes, T., & Bryant, P. (2009). Paper 3: Understanding rational numbers and intensive quantities. Key
Understandings in Mathematics Learning. Retrieved 15/12/10, 2010, from
http://www.nuffieldfoundation.org/sites/default/files/P3_amended_FB2.pdf.
Reys, R.E., Lindquist, M.M., Lambdin, D.V., Smith, N.L., Rogers, A., Falle, J., Frid, S., & Bennett, S.
(2012). Helping children learn mathematics. (1st Australian Ed.). Milton, Qld: John Wiley & Sons,
Australia.
Schulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15,
4-14.
Siegler R.S., Duncan G.J., Davis-Kean P.E., Duckworth K., Claessens A., Engel M., Susperreguy M.I., &
Chen M. (2012). Early Predictors of High School Mathematics Achievement. Psychological Science,
23 (7), 691-697.
Siemon, D., Bleckly, J. and Neal, D. (2012). Working with the Big Ideas in Number and the Australian
Curriculum: Mathematics. In B. Atweh, M. Goos, R. Jorgensen & D. Siemon, (Eds.). (2012).
Engaging the Australian National Curriculum: Mathematics – Perspectives from the Field. Online
Publication: Mathematics Education Research Group of Australasia pp. 19:45.
Siemon, D., Breed, M., Dole, S., Izard, J., & Virgona, J. (2006). Scaffolding Numeracy in the Middle
Years – Project Findings, Materials, and Resources, Final Report submitted to Victorian Department
of Education and Training and the Tasmanian Department of Education, Retrieved from
http://www.eduweb.vic.gov.au/edulibrary/public/teachlearn/student/snmy.ppt
Siemon, D., Izard, J., Breed, M., & Virgona, J. (2006). The derivation of a learning assessment
framework for multiplicative thinking. Paper presented at the 30th Conference of the International
Group for the Psychology of Mathematics Education – Mathematics in the Centre. Prague.
Siemon, D., Virgona, J. & Corneille, K. (2001). The Final Report of the Middle Years Numeracy Research
Project: 59. Retrieved from
http://www.education.vic.gov.au/Documents/school/teachers/teachingresources/discipline/maths/mynu
mfreport.pdf
Siemon, D., Beswick, K., Brady, K., Clark, J., Faragher, R., & Warren, E. (2011). Teaching Mathematics:
Foundations to the middle years. Melbourne: Oxford University Press.
Sophian, C. (2004). Mathematics for the future: Developing a Head Start curriculum to support
10
mathematics learning. Early Childhood Research Quarterly, 19, 59-91.
Sophian, C., & Madrid, S. (2003). Young childrens' reasoning about many-to-one correspondences. Child
Development, 74(5), 1418-1432.
Thomas, N. (2004). The development of structure in the number system. Paper presented at the 28th
Conference of the International Group for the Psychology of Mathematics Education, Bergen, Norway.
Thompson, I. (2009). Place Value?. Mathematics Teaching, (215), 4-5.
Tirosh, D. (2000). Enhancing prospective teachers' knowledge of children's conceptions: The case of
division of fractions. Journal for Research in Mathematics Education 31, 5-26.
Usiskin, Z. P. (2007). The future of fractions. Arithmetic Teacher 12(7), 366-369.
Van de Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2013). Elementary and middle school
mathematics: Teaching developmentally. (8th Ed.). Boston: Pearson.
Van Dooren, W., De Bock, D., & Verschaffel, L. (2010). From addition to multiplication… and back. The
development of students additive and multiplicative reasoning skills. Cognition and Instruction,
28(3), 360-381.
Willis, S. (2002). Crossing Borders: Learning to count. Australian Educational Researcher, 29(2), 115:
130.
Wright, V. J. (2011). The development of multiplicative thinking and proportional reasoning: Models of
conceptual learning and transfer. (Doctoral dissertation). University of Waikato, Waikato. Retrieved
from http://researchcommons.waikato.ac.nz/.
Wu, H. (2001). How to prepare students for algebra. American Educator, 25(2), 10–17.
Young-Loveridge, J. (2005). Fostering multiplicative thinking using array-based materials. Australian
Mathematics Teacher, 61(3), 34-40.
Young-Loveridge, J., & Mills, J. (2009). Teaching multi-digit multiplication using array-based materials.
In R. Hunter, B. Bicknell, & T.Burgess (Eds.), Crossing divides (Proceedings of the 32nd annual
conference of the Mathematics Education Research Group of Australasia pp. 635-643). Palmerston
North, NZ: MERGA
Zhou, Z., Peverly, S.Y., & Xin, T. (2006). Knowing and teaching fractions: A cross-cultural study of
American and Chinese mathematics teachers. Contemporary Educational Psychology 31(4), 438-457.
11
Appendix 1 Siemon, Bleckly & Neal (2012) p.25
By the end
of:
Big Idea Indicated by:
Foundation
Year
Trusting the Count Access to flexible mental objects for the numbers to ten based on part-part-
whole knowledge derived from subitising and counting (e.g., know that 7 is
1 more than 6, 1 less than 8, 5 and 2, 2 and 5, 3 and 4 without having to
make or count a collection of 7)
Year 2 Place-value Capacity to recognise and work with place-value units and view larger
numbers as counts of these units rather than collections of ones (e.g., able to
count forwards and backwards in place-value units)
Year 4 Multiplicative
Thinking
Capacity to work flexibly with both the number in each group and the
number of groups (e.g., can view 6 eights as 5 eights and 1 more eight).
Recognises and works with multiple representations of multiplication and
division (e.g., arrays, regions and ‘times as many’ or ‘for each’ idea).
Year 6 (Multiplicative)
Partitioning
Ability to partition quantities and representations equally using
multiplicative reasoning (e.g., a fifth is smaller than a quarter, estimate 1
fifth on this basis then halve and halve remaining part again to represent
fifths), recognise that partitioning distributes over previous acts of
partitioning and that numbers can be divided to create new numbers
Year 8 Proportional
Reasoning
Ability to recognise and work with an extended range of concepts for
multiplication and division including rate, ratio, percent, and the ‘for each’
idea, and work with relationships between relationships
Year 10 Generalising Capacity to recognise and represent patterns and relationships in multiple
ways including symbolic expressions, devise and apply general rules
Appendix 2
Big Idea – Multiplicative Partitioning – Key Criteria
Quantities and collections can be shared to create equal parts
Can repeatedly halve and double numbers and quantities – e.g., use successive splits to show that one half is
equivalent to 2 parts in 4, 4 parts in 8 etc.
There is a relationship between the number of parts and the size and name of the parts and the number of
parts increases as the size or share decreases
Percentages, fractions and decimals express the relationship between two quantities.
A quantity can be partitioned into a number of equal portions to show unit fractions so that say one third is
more than one fourth etc.
The relative magnitude of a fraction is dependent on the relationship between the numerator (how many
parts) and denominator (total parts)
Construct of fraction as division can be used to produce equal parts (equipartitioning)
Fractions are used to describe quotients and operators
Fractions are used to describe part-whole relations
Fractions are used to describe simple ratios
Percentages are special part: whole ratios based on 100.
Any given percentage can be used as a ratio to generate an infinite number of equivalent fractions (e.g., 50%
= ½, 2/4, 3/6 etc.)
Common fractions and decimal fractions can be compared, ordered and renamed in conceptual ways
Multiplicative strategies are used to represent fractions, decimals and percentages in diagrams
Fractions are renamed as equivalents where the total number of parts (denominator) and required number of
parts (numerator) are increased by the same factor
Fractions with unlike denominators can be compared
Benchmark fractions, decimals and percentages, which are the equivalents of one another, are understood and
used to solve problems
12
Appendix 3 Equipartitioning Learning Trajectory Matrix (grades K-8) (Confrey, 2012)
Proficiency Levels
1. Equipartition collections
2. Equipartition single wholes
3. Justify the results of equipartitioning
4. Name a share with reference to the referent unit
5. Re-assemble: n times as much
6. Qualitative Compensation
7. Composition of splits (singular wholes)
8. Quantative compensation
9. Reallocation
10. Property of equality
11. Assert continuity principle
12. Equipartition multiple wholes
13. Composition of splits (multiple wholes)
14. Co-splitting
15. Distributive property (multiple wholes)
16. Generalise: a among b = a/b
13
... Creation of numbers is established in a unique way (Fritz et al., 2013;Hurst and Hurrell, 2014;Posamentier, 2015;Penn, 2021). Permutation and combination of numbers in different ways to generate different sets of numbers which when study critically; the set of numbers produce have marvelous and miraculous in pattern, structure, series and arrangement (Liljedahl, 2004;Itaketo, 2010;Mulligan et al., 2010;Yesildere and Akkoc, 2010;Ernest, 2015;Hessman, 2020). ...
Conference Paper
Full-text available
This study used the application of Geographic Information System (GIS) to analyse students’ staff index in Public Secondary Schools (PSSs) of Osun West Senatorial District (OWSD), Nigeria with a view to providing information that could guide the ineffective distribution of human resources (teaching staff) to enhancing secondary school education planning. The specific objectives are to determine the number of staff and students of the public secondary schools in the district and examine the staff index. Data for this study were obtained through questionnaire administration. Areas with secondary schools were stratified into three categories based on population density: suburb (less than 10,000 people), semi-urban (between 10,000-19,999 people), and urban (20,000 people and above). Eleven urban with a total of 59 secondary schools, twelve semi-urban with a total of 16 secondary schools, and twenty-eight suburb areas were identified with a total of 41 secondary schools. All the schools were sampled while students’ enrolment and staff data were collected from the principal of each school. Findings revealed, that the largest proportion of settlements with inadequate staff strength is the suburb with eleven (11) public secondary schools, a mean value of 26 and standard deviation of 2.87, urban settlement category have seven (7), with a mean value of 23 and standard deviation of 6.007 and semi-urban with just one (1) school, denoting 26 mean value and 0.0 standard deviation. Summary of the findings across OWSD revealed inadequate staffing in urban (36.8%), semi-urban (5.3%), and suburb (57.9%) as the mean score of the student to staff is 44 and the standard deviation is 13.771. It is therefore recommended that government and other school administrators should come together to employ adequate teaching staff to secondary schools at all categories of settlements.
... Creation of numbers is established in a unique way (Fritz et al., 2013;Hurst and Hurrell, 2014;Posamentier, 2015;Penn, 2021). Permutation and combination of numbers in different ways to generate different sets of numbers which when study critically; the set of numbers produce have marvelous and miraculous in pattern, structure, series and arrangement (Liljedahl, 2004;Itaketo, 2010;Mulligan et al., 2010;Yesildere and Akkoc, 2010;Ernest, 2015;Hessman, 2020). ...
Conference Paper
Full-text available
Kifilideen trinomial theorem of negative power of is theorem which is used to generate the series and terms of a trinomial expression of negative power of in an orderly and periodicity manner that is based on standardized and matrix methods. Negative power of Newton binomial theorem had been used to produce series of partial fractions of a compound fraction. The establishment of the negative power of of trinomial theorem would extend the number of compound fraction in which series (expansion) can be produced. This study applied Kifilideen expansion of negative power of of Kifilideen trinomial theorem for the transformation of compound fraction into series of partial fractions with other developments. Kifilideen theorem of matrix transformation of negative power of of trinomial expression in which three variables are found in parts of the trinomial expression was developed. The development would ease the process of evaluating such trinomial expression of negative power of . This standardized and matrix method used in arranging the terms of the Kifilideen expansion of negative power of of trinomial expression yield an interesting results in which it is utilized in transforming compound fraction into series of partial fractions in a unique way.
... Creation of numbers is established in a unique way (Fritz et al., 2013;Hurst and Hurrell, 2014;Posamentier, 2015;Penn, 2021). Permutation and combination of numbers in different ways to generate different sets of numbers which when study critically; the set of numbers produce have marvelous and miraculous in pattern, structure, series and arrangement (Liljedahl, 2004;Itaketo, 2010;Mulligan et al., 2010;Yesildere and Akkoc, 2010;Ernest, 2015;Hessman, 2020). ...
Conference Paper
Full-text available
The Kifilideen trinomial theorem of positive power of n built on matrix and standardized procedures had been developed and implemented in company with the Kifilideen general power combination formula which helps to determine the terms in the kif expansion of trinomial expression of positive power of n. This research work inaugurated Kifilideen trinomial theorem of negative power of – n by employing matrix and standardized techniques. Matrix was used in this study to arrange the terms of the series of the negative power of the Kifilideen trinomial theorem. The Kifilideen general power combination formula of any term in the series of the Kifilideen trinomial theorem of negative power of – n was invented. The Kifilideen general term formula to determine the term of a given power combination was also originated. It has been proving that the theorem and formulas generated are accurate, reliable, easy and interesting. The theorem helps in generating the terms of Kifilideen trinomial theorem of negative power of – n in an orderly form and makes it easy in obtaining the power combination that produce any given term and vice versa.
... Creation of numbers is established in a unique way (Fritz et al., 2013;Hurst and Hurrell, 2014;Posamentier, 2015;Penn, 2021). Permutation and combination of numbers in different ways to generate different sets of numbers which when study critically; the set of numbers produce have marvelous and miraculous in pattern, structure, series and arrangement (Liljedahl, 2004;Itaketo, 2010;Mulligan et al., 2010;Yesildere and Akkoc, 2010;Ernest, 2015;Hessman, 2020). ...
Conference Paper
Full-text available
This study reviews recent literature on the engineering properties of Warm Mix Asphalt (WMA) containing Waste Plastic Bottle (WPB) and Waste Sachet Water (WSW) as a modifier and as a partial replacement for conventional bitumen. The study summarizes various contributions elucidating the various WPB and WSW utilized, their growing production and usage, WPB and WSW material preparation and treatment, physical composition and different engineering test methods adopted by previous studies such as Penetration, Softening point, Ductility, Viscosity, Flash and fire point, Loss on heating, Specific gravity, Stability and Flow. This study showed a significant improvement in the engineering properties of WPB modified WMA compared to the unmodified sample. The review also showed that WPB and WSW improve the engineering properties of warm mix asphalt when they were used separately. Furthermore, the sasobit manufacturer recommended a 3% addition of sasobit to bitumen when aiming at maximum temperature reduction and to achieve optimum performance. Previous researchers adopted the sasobit manufactures recommendation which reduces the production temperature of asphalt concrete by 30 oC. Therefore, it is recommended that further studies should use WSW and WPB in a combined form for use as a modifier in improving WMA. Also, further study should vary the percentage of sasobit to be blended with bitumen in other ascertain the modification that is most suitable for Nigeria’s condition using the locally available materials in asphalt production.
... Creation of numbers is established in a unique way (Fritz et al., 2013;Hurst and Hurrell, 2014;Posamentier, 2015;Penn, 2021). Permutation and combination of numbers in different ways to generate different sets of numbers which when study critically; the set of numbers produce have marvelous and miraculous in pattern, structure, series and arrangement (Liljedahl, 2004;Itaketo, 2010;Mulligan et al., 2010;Yesildere and Akkoc, 2010;Ernest, 2015;Hessman, 2020). ...
Article
Full-text available
The positive power of Kifilideen trinomial theorem based on matrix and standardized approach had been developed and implemented alongside the general power combination formula which helps to determine the terms in the kifilideen expansion of positive power of n of trinomial expression. This study inaugurated negative power of − n of kifilideen trinomial theorem using standardized and matrix methods. Matrix was used in this research work to arrange the terms of the series of the negative power of the Kifilideen trinomial theorem. The general formula of the power combination of any term in the series was invented. The general formula to determine the term of a given power combination was also originated. It has been proving that the theorem and formulas generated are accurate, reliable, easy and interesting. The theorem helps in generating the terms of Kifilideen trinomial theorem of negative power of-n in an orderly form and makes it easy in obtaining the power combination that produce any given term and vice versa.
... Bei der Addition und Subtraktion lassen sich Zahlen, auch in der Zehner-Bündelungsschreibweise des Stellenwertsystems, als Vielfache von 1 betrachten. Bei der Multiplikation erhält der Multiplikator jedoch eine andere Bedeutung (Hurst & Hurrell, 2014). Beispielsweise gilt dies bei der Aufgabe 3 · 4 wahlweise für die 3 (drei Vierer-Mengen) oder für die 4 (die 3, viermal). ...
Chapter
Full-text available
Dieser einführende didaktische Kommentar erläutert die in Abbildung 1 dargestellten Schwerpunkte des Kompetenzerwerbs zur Leitidee Zahlen und Operationen. Ein wesentliches Anliegen des Textes ist es, an ausgewählten Beispielen die Bedeutung eines verstehensorientierten Mathematikunterrichts für langfristig erfolgreiches Lernen zu veranschaulichen und Zusammenhänge zwischen zu erwerbenden Kompetenzen zu verdeutlichen. Damit macht dieser Einführungstext auch Überlegungen transparent und nachvollziehbar, die zu der vorliegenden Auswahl der Schwerpunkte des Kompetenzerwerbs und der entwickelten Diagnose- und Fördermaterialien geführt haben. Diese Informationen, so die Hoffnung, tragen zu einer zielgerichteten Verwendung der Materialien im Unterricht bei. Die vorliegenden Diagnose- und Fördermaterialien des LISUMs verfolgen zusammenfassend wenigstens drei eng miteinander verbundene Ziele: (a) Sie geben in den Erläuterungen Anregungen und Orientierung für eine Weiterentwicklung des Mathematikunterrichts, (b)sie konkretisieren diese in Form von Aufgaben und (c) sie möchten auf diesem Wege einen Beitrag zu einer gelingenden tagtäglichen Förderung der Schülerinnen und Schüler leisten. https://bildungsserver.berlin-brandenburg.de/rlp-online/c-faecher/mathematik/materialien/materialien-zur-diagnose-und-foerderung-im-mathematikunterricht-leitidee-zahlen-und-operationen
... Siemon et al. (2012, p.24) comment that these ideas are 'very big ideas in Number without which students' progress in mathematics will be severely restricted'. As Hurst and Hurrell (2014) point out, the work on 'big ideas' by Siemon et al. (2012) has a particular strength as it highlights the hierarchical and connected nature of these ideas. Hurst (2015) further argues that a model for a curriculum which uses 'big idea' thinking as a way of organising content could support teachers in planning for connecting content and ensuring development of those ideas over time, thus supporting progress and depth of learning. ...
Research
Full-text available
‘Learning about Progression’ is a suite of research-based resources designed to provide evidence to support the building of learning progression frameworks in Wales. ‘Learning about Progression’ seeks to deepen our understanding of current thinking about progression and to explore different purposes that progression frameworks can serve to improve children and young people’s learning. These resources include consideration of how this evidence relates to current developments in Wales and derives a series of principles to serve as touchstones to make sure that, as practices begin to develop, they stay true to the original aspirations of A Curriculum for Wales – A Curriculum for Life. It also derives, from the review of evidence, a number of fundamental questions for all those involved in the development of progression frameworks to engage.
... ,Hurst & Hurrell (2014),Siemon et al., (2011), andWright (2011) remind us that repeated addition and subtraction become inefficient with increased sophistication of problems. For example, if 21 ...
Article
Full-text available
The importance of multiplicative thinking in supporting students' learning of key topics and success in further mathematics is widely and clearly stated in mathematics education literature. However, multiplicative thinking is not clearly stated in curriculum documents of many countries including Australia. In the Australian Curriculum: Mathematics, it is implied throughout as multiplication and division with emphasis on recall of multiplication facts, factors, and multiples of a number, grouping into equal sets, and the use of mental solution strategies. In this theoretical paper, we discuss how this lack of an explicit framework for students' development of multiplicative thinking might be remedied by: (i) making curriculum and teaching to be explicit about multiplicative thinking from early primary school (F-3), and (ii) implementing three key teaching stages for developing multiplicative thinking in students throughout the middle primary (4-6) and junior secondary years (7-8).
... ,Hurst & Hurrell (2014),Siemon et al., (2011), andWright (2011) remind us that repeated addition and subtraction become inefficient with increased sophistication of problems. For example, if 21 ...
Article
The importance of multiplicative thinking in supporting students' learning of key topics and success in further mathematics is widely and clearly stated in mathematics education literature. However, multiplicative thinking is not clearly stated in curriculum documents of many countries including Australia. In the Australian Curriculum: Mathematics, it is implied throughout as multiplication and division with emphasis on recall of multiplication facts, factors, and multiples of a number, grouping into equal sets, and the use of mental solution strategies. In this theoretical paper, we discuss how this lack of an explicit framework for students' development of multiplicative thinking might be remedied by: (i) making curriculum and teaching to be explicit about multiplicative thinking from early primary school (F-3), and (ii) implementing three key teaching stages for developing multiplicative thinking in students throughout the middle primary (4-6) and junior secondary years (7-8).
ResearchGate has not been able to resolve any references for this publication.