Engaging with the research presented here will deepen your understanding of the difference and benefits of teaching and learning mathematics instrumentally and relationally, and what the evidence says actually works in mathematics teaching using research-informed strategies.
COUNTING AND GROUPING
Adults assume that learning to count is easy, because we cannot remember the time before we could count. This research explains the principles, the sequence, and the challenges.
Mathematics in the Early Grades: Counting & Cardinality Victorian State Government’s Education Development Centre explains that even if young children can recite the number sequence (i.e. say one, two, three, etc) we cannot assume that they can count small sets of objects. The important principles and challenges involved in learning to count are identified.
The Link Between Place Value and Operations This book emphasises the vital link between place value and operations, finding that a good understanding of addition, subtraction, and place value is crucial for strategies in multiplication, division, fractions, algebra, and statistics. It explains that students initially use counting to solve addition and subtraction problems, then start to think strategically, first with smaller whole numbers, then with larger ones, and later with fractions, decimals, and integers.
Teachers’ Understanding of Addition and Subtraction with Whole Numbers George J. Roy’s study examined teachers’ understanding of number concepts and operations, adding and subtracting in base 8 (rather than base 10) to allow prospective teachers to reason about addition and subtraction with whole numbers in similar ways that students’ reason in base 10.
Place value is so much more than naming the column a digit is in. This research explains the deeper understandings about place value, their complexity and importance.
Children’s Understanding of Two-Digit Place Value University of Virginia’s research finding that the development of young children’s number sense and understanding of the base 10 system is essential for the acquisition of more complex number skills in later years
Place Value (page 7) University of Notre Dame Australia’s research on page 7 explores the complexity of standard place value.
Rethinking Concrete Manipulatives Clements and McMillen’s NCTM-published research finding the effect of using materials in understanding mathematical concepts depends on the teaching.
Multiplication and division goes much further than ‘groups’ and ‘tables’. This research explains the importance of multiplicative thinking, and of understanding, intentionally using, generalising, and explaining the distributive property.
The Key Role of Multiplicative Thinking Di Siemon’s research finds that access to multiplicative thinking has been identified as the single, most important reason for the eight-year range in mathematics achievement in Years 5 to 9, and explains that while elements of multiplicative thinking are represented in the Australian Curriculum, the connections between these and how they contribute to the development of multiplicative thinking over time is not entirely clear. (3rd lecture from top of page)
From Additive to Multiplicative Thinking Di Siemon’s research explains the characteristics of multiplicative thinking as a capacity to work flexibly and efficiently with an extended range of numbers, and an ability to recognise and solve a range of problems involving multiplication or division including direct and indirect proportion, and – the means to communicate this effectively in a variety of ways. The research demonstrates the difference between multiplicative and additive thinking, identifies lack of multiplicative thinking as the major barrier to learning mathematics in the middle years. (Select the blue download button at the top right of the page.)
The Distributive Property Nelis Vermeulen’s research explains that children have a intuitive grasp of the distributive property, and the importance of developing their capacity to intentionally use it, to generalise it, and to explain it.
Fractions are so much more than ‘equal parts’. This research explains the 5 ways to understand fractions, and how to teach fractions multiplicatively.
The Inter-relatedness of the 5 Sub-constructs of Fractions Charalambous’ research explains that fractions comprise a multifaceted notion encompassing five interrelated subconstructs (part-whole, ratio, operator, quotient and measure), and that when teaching fractions, teachers need to scaffold students to develop a profound understanding of the different interpretations of fractions.
3 Concepts Underpinning the 5 Sub-constructs Annie Mitchell’s paper demonstrates how the 5 sub-constructs of Fractions are underpinned by three concepts: partitioning, equivalence, and unit-forming, explaining that these concepts provide better categorisation of the underlying fraction concepts.
Unifying the 5 Sub-constructs into 3 Elements Doug Clarke’s research has used student interviews to unify the 5 sub-constructs into 3 big ideas – identification of the unit, partitioning and the notion of quantity (how “big” a fraction is).
Partitioning to Create Fractions Di Siemon’s research explains that focusing fractions understanding on partitioning underpins the fraction sub-construct of fractions as division.
Units in Length Measurement Yanik, Helding, and Flores’ research into the sub-construct of measure explains how the number line is a practical model as the base unit is identified and reiterated from the zero point.
Procedural and Conceptual Knowledge Forrester and Chinnappan’s research distinguished between the procedural and conceptional knowledge of teachers in operating with fractions, finding that procedural knowledge was dominant, leading to a dominance of procedural teaching.
Teaching Fractions for Conceptual Understanding Fazio and Siegler’s research identified that, although understanding fractions is essential for learning algebra, geometry and other aspects of higher mathematics, students’ difficulties with fractions stem from a lack of conceptual understanding. They recommends ideas for introducing fraction concepts, ideas to help older students understand fraction magnitudes, ways of helping students use fractions to solve rate, ratio and proportion problems, and methods to increase teacher’s conceptual knowledge of fractions.
Decimals are so much more than the decimal point. This research explains the reason some common misunderstandings arise about decimals.
Decimal Misconceptions Steinle’s paper identifies that common misconceptions students have about decimals have been caused by procedural knowledge, and suggests remedies involving developing conceptual understanding including the relationship to the denominator in fractions, the importance of ‘one’, and multiplicative place value.
Geometry is more than naming shapes and objects, and Measurement is more than naming the size of shapes and objects. This research explains the relationships between Geometry, Measurement, and Number.
Geometry, Spatial Thinking, and Measurement This chapter explains the steps in spatial reasoning that children go through from recognising, naming features and properties, and analysing shapes and objects. It identifies that measurement connects number and space, and begins with understanding length, area and volume involving spatial structure (squares and cubes), and volume involving measurement (liquid). (Chapter 6)
Australian Curriculum Measure and Geometry Tom Lowrie, Tracy Logan, and Brooke Scriven’s chapter examine the connectivity between measurement and geometry and the inter-connectivity with number, identifying the potential for teachers to develop rich, conceptually-connected learning opportunities.
Mathematical Content Knowledge for Teaching Elementary Mathematics Evidence indicates that cognitive development, along with spatial visualisation skills, plays a greater role in learning geometry than memory skills. The focus of this paper is on mathematical content knowledge for teaching elementary mathematics with particular attention to geometry and measurement.