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Six principles of effective mathematics teaching. Peter Sullivan Monash University. NT 2012. First, let us consider this task sequence. The following sequence seeks to introduce students to the nature of volume of prisms and cylinders (area of the end × length)
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Six principles of effective mathematics teaching • Peter Sullivan • Monash University NT 2012
First, let us consider this task sequence • The following sequence seeks to introduce students to • the nature of volume of prisms and cylinders (area of the end × length) • surface area of prisms (total area of the faces) • efficient methods of calculating volume and surface area of rectangular prisms • ways in which surface area and volume are different NT 2012
This is a rectangular prism made from cubes. What is the volume of this prism? What is the surface area? NT 2012
A set of 36 cubes is arranged to form a rectangular prism. • What might the rectangular prism look like? • What is the surface area of your prisms? NT 2012
A rectangular prism is made from cubes. It has a surface area of 22 square units. What might the rectangular prism look like? NT 2012
Key teaching idea 1: • Identify big ideas that underpin the concepts you are seeking to teach, and communicate to students that these are the goals of your teaching, including explaining how you hope they will learn NT 2012
How might the AC help? NT 2012
Year 7 Year 6 Year 8 NT 2012
Three content strands (nouns) • Number and algebra • Measurement and geometry • Statistics and probability NT 2012
Using the content descriptions • Get clear in your mind what you want the students to learn • Make your own decisions about how to help them learn that content • Overall • what do these suggest are the overall goals, the big ideas, the important focus, etc NT 2012
AC content descriptions: Using Units of measurement • Year 6 • Connect volume and capacity and their units of measurement • Year 7 • Calculate volumes of rectangular prisms • Year 8 • Choose appropriate units of measurement for area and volume and convert from one unit to another • Develop the formulas for volumes of rectangular and triangular prisms and prisms in general. Use formulas to solve problems involving volume • Year 9 • Calculate the surface area and volume of cylinders and solve related problems • Solve problems involving surface area and volume of right prisms NT 2012
So far there is not much difference from what you are doing • It is the proficiencies that are different NT 2012
In the Australian curriculum • Understanding • (connecting, representing, identifying, describing, interpreting, sorting, …) • Fluency • (calculating, recognising, choosing, recalling, manipulating, …) • Problem solving • (applying, designing, planning, checking, imagining, …) • Reasoning • (explaining, justifying, comparing and contrasting, inferring, deducing, proving, …) NT 2012
The proficiencies – why do we change from “working mathematically”? • These actions are part of the curriculum, not add ons • Mathematics learning and assessment is more than fluency • Problem solving and reasoning are in, on and for mathematics • All four proficiencies are about learning NT 2012
Choosing tasks will be a key decisions • If we are seeking fluency, then clear explanations followed by practice will work • If we are seeking understanding, then very clear and interactive communication between teacher and students and between students will be necessary • If we want to foster problem solving and reasoning, then we need to use tasks with which students can engage, which require them to make decisions and explain their thinking NT 2012
What would you say to the students were the goals the “36 cubes” task? • Would you write that on the board? • What would you say to the students about how you hope they would learn? NT 2012
What proficiencies are associated with the 36 cubes task? NT 2012
goals NT 2012
Key teaching idea 2: • Build on what the students know, both mathematically and experientially, including creating and connecting students with stories that both contextualise and establish a rationale for the learning NT 2012
What % of Year 5 Victorians can do this? • A rectangular paddock has a perimeter of 50 metres. Each long side has a length of 15 metres. • What is the length of each short side? metres NT 2012
56% of students could do this • A rectangular paddock has a perimeter of 50 metres. Each long side has a length of 15 metres. • What is the length of each short side? metres NT 2012
What % of Year 7 Victorians (no calculator) can do this? NT 2012
12 % of Year 7 Victorians (no calculator) can do this NT 2012
This tells us • Around half of the year 5 students are ready for challenging tasks about perimeter • Very few year 7 students have a sense of volume NT 2012
1mm of rain on 1 sq m of roof is 1 L of • water. • Design a tank for this building that • captures all of the rain that usually falls • this month. NT 2012
goals readiness NT 2012
Key teaching idea 3 • Engage students by utilising a variety of rich and challenging tasks, that allow students opportunities to make decisions, and which use a variety of forms of representation NT 2012
How might those activities TOGETHER contribute to learning? NT 2012
goals readiness engage NT 2012
Key teaching idea 4: • Interact with students while they engage in the experiences, and specifically planning to support students who need it, and challenge those who are ready NT 2012
Focusing on the SA = 22cm2 activity • How might we engage students who could experience difficulty with it? NT 2012
goals readiness difference engage NT 2012
Key teaching idea 5: • Adopt pedagogies that foster communication, mutual responsibilities, and encourage students to work in small groups, and using reporting to the class by students as a learning opportunity NT 2012
First the teacher told a story about tatami mats that emphasised the notion of area as covering NT 2012
