Numeracy keynote SA

Using key numeracy teaching principles as the basis of leading teaching improvement

Peter Sullivan

Numeracy keynote SA

Abstract

• Supporting improvement in numeracy teaching is both demanding and complex. One way to manage the complexity is to have explicit goals for each step in the improvement process.

• After reviewing other similar lists, I identified six principles that can form the basis, individually and together, of improvement initiatives.

• Using the theme of the teaching of fractions, this session will elaborate each of the principles.

Numeracy keynote SA

What is being recommended about mathematics teaching?

How does this connect to the AC?

Numeracy keynote SA

Describing the proficiencies• 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, …)Numeracy keynote SA

Numeracy keynote SA

The (brand) new UK National Curriculum …all pupils:

become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils have conceptual understanding and are able to recall and apply their knowledge rapidly and accurately to problems reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical languagecan solve problems by applying their mathematics to a variety of routine and nonroutine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.

Numeracy keynote SA

• https://www.education.gov.uk/schools/teachingandlearning/curriculum/nationalcurriculum2014/a00220610/draft-pos-ks4-english-maths-science

Numeracy keynote SA

From Impactful practices• Imagine a world where students, in every

mathematics classroom, are actively engaged with worthwhile tasks that promote mathematical understanding, problem solving and reasoning.

Numeracy keynote SA

• Imagine classrooms where the interactions among students, and with their teacher, are focused on making sense of mathematics, alternative approaches to solving problems, and defending, confirming and verifying possible solutions. These are thinking and reasoning classrooms.

Numeracy keynote SA

An aside

• It is not a problem if we have told students what to do

• It is not reasoning if students are reproducing what we have told them

• It is not understanding unless students can explain in their own words with their own ideas

Numeracy keynote SA

What do you see as the most critical aspect of being a powerful learner of numeracy and literacy?• Powerful learners connect ideas together, they

can compare and contrast concepts, and they can transfer learning from one context to another. They can devise their own solutions to problems, and they can explain their thinking to others.

Numeracy keynote SA

Two task examples that we will use as the basis of the later discussion

Numeracy keynote SA

STRIPED RECTANGLE

  

 

 

If the dotted (blue) rectangle represents what fraction is represented by the striped (red) rectangle?

Work out the answer in two different ways.

3

2

Numeracy keynote SA

REPRESENTING A FRACTION

• If this represents 3 , draw what represents 1

(work this out two different ways)

3

2

2

1

Numeracy keynote SA

What is the point of the six key principles ?

• We can all do these things better (although you will find many of them affirming of your current practice)

• Much advice is complex and hard to prioritise• The principles can provide a focus to

collaborative discussions on improving teaching• The principles can be the focus of observations

if you have the opportunity to be observed teaching

AVAILABLE TO DOWNLOAD FREE FROM

http://research.acer.edu.au/aer/13/ aer

Numeracy keynote SA

Key principle 1:

• Identify important 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

Numeracy keynote SA

Numeracy keynote SA

Feedback - better when they know …

• Where am I going?– “Your task is to …, in this way”

• How am I going?– “the first part is what I was hoping to see,

but the second is not”• Where to next?– “knowing this will help you with …”

Numeracy keynote SA

In terms of learning intentions, we know

• It is difficult to describe the purpose of lessons and teachers benefit from discussions about intentions

• The learning intention should – not restrict – nor tell– nor lower the ceiling – but provide focus for the students– and the teacher

Numeracy keynote SA

What would you say is the learning intention for striped rectangle?

This is taken from the lesson plan

• A fraction is a number. • We can compare, add and multiply fractions

just like we do for numbers, even if the calculation process is a little different.

• You will solve the problem for yourself and explain your thinking

Numeracy keynote SA

Numeracy keynote SA

goals

Key principle 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

Numeracy keynote SA

Numeracy keynote SA

Part 1: Using data

Numeracy keynote SA

• It is as important to know what the students know as it is to know what they do not

• Learning mathematics is not a hierarchy of sequential steps on a ladder, but a network of interconnected ideas

• Students benefit from work on tasks that are beyond what they know– Students at GP 2 can work on GP 4 tasks

Numeracy keynote SA

Numeracy keynote SA

Numeracy keynote SA

Helen has 24 red apples and 12 green apples.

What fraction of the apples are green?

Numeracy keynote SA

Year 5 93%

Numeracy keynote SA

Year 5 24%

Numeracy keynote SA

Helen has 24 red apples and 12 green apples. What fraction of the apples are

green?

• 55% of year 7 students• 67% of year 9 students

Numeracy keynote SA

What does that tell you?

Numeracy keynote SA

Part 2: Connecting with “story”

Numeracy keynote SA

• A chameleon has a tongue that is half as long as its body ...

• … how long would your tongue be if you were a chameleon?

Numeracy keynote SA

Part 3: Creating experience

Numeracy keynote SA

Numeracy keynote SA

Numeracy keynote SA

goals readiness

Key Principle 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

Numeracy keynote SA

Numeracy keynote SA

Why challenge?

• Learning will be more robust if students connect ideas together for themselves, and determine their own strategies for solving problems, rather than following instructions they have been given.

• Both connecting ideas together and formulating their own strategies is more complex than other approaches and is therefore more challenging.

• We need strategies to encourage students to persist

Numeracy keynote SA

Related to those tasks above ..

• To what extent– Are they challenging?– Are they engaging?– Do they allow student decision making– Do they encourage different representations?

Numeracy keynote SA

What is 5 + 5 + 5 + 295 + 295 + 295 ?

Number of students that completed this question

1343

Correct answers 413 (30.8%)

Numeracy keynote SA

I prefer questions we work on in class to be

I prefer

learning

how to do

questions

like this

Much harder

About the same

Much easier

TOTAL

By myself

239 290 69 598Working with other

students78 227 162 467

By listening to the teacher's

explanations first 56 140 100 296

TOTAL

373 657 331 1361

Think about the question What is 5 + 5 + 5 + 295 + 295 + 295 ?

Numeracy keynote SA

I prefer questions we work on in class to be

I prefer

learning

how to do

questions

like this

Much harder

About the same

Much easier

TOTAL

By myself 239118

290103

6921

598242

Working with other students 78

1522764

16213

46792

By listening to the teacher's

explanations first5617

14036

10018

29671

TOTAL373150

657203

33152

1361405

Think about the question What is 5 + 5 + 5 + 295 + 295 + 295 ?

Numeracy keynote SA

Quotes from PISA in Focus 37• When students believe that investing effort in

learning will make a difference, they score significantly higher in mathematics.

• Teachers’ use of cognitive-activation strategies, such as giving students problems that require them to think for an extended time, presenting problems for which there is no immediately obvious way of arriving at a solution, and helping students to learn from their mistakes, is associated with students’ drive.

Numeracy keynote SA

goals readiness

engage

Key Principle 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

Numeracy keynote SA

Numeracy keynote SA

Enabling prompt

Numeracy keynote SA

IF YOU ARE STUCK

• If this represents 7 , draw what represents 2

(work this out two different ways)

3

1

Numeracy keynote SA

IF YOU ARE STUCK

If this represents 11 , draw what represents 5

(work this out two different ways)

Numeracy keynote SA

Extending prompt

Numeracy keynote SA

IF YOU HAVE FINISHED

• If this represents 8 , draw what represents 2.4

(work this out two different ways)

5

4

Numeracy keynote SA

goals readiness

engagedifference

Key Principle 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

Numeracy keynote SA

A revised lesson structure

• In this view, the sequence– Launch (without telling)– Explore (for themselves)– Summarise (drawing on the learning of the students)

• … is cyclical and might happen more than once in a lesson (or learning sequence)

Numeracy keynote SA

Launch

ExploreSummarise

Numeracy keynote SA

CONSOLIDATING THE LEARNING

• If this represents 5 , draw what represents 2

(work this out two different ways)

4

12

1

Numeracy keynote SA

goals

lessonstructure

readiness

engagedifference

Key teaching idea 6

• Fluency is important, and it can be developed in two ways– by short everyday practice of mental calculation or

number manipulation– by practice, reinforcement and prompting transfer

of learnt skills

Numeracy keynote SA

Numeracy keynote SA

One aspect is transfer

• This connects to the consolidating task

Numeracy keynote SA

Another aspect is fluency

7

43+

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?

3423+

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35

?13+

Numeracy keynote SA

The unknowns are different

½

??+

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?

?1.7+

Numeracy keynote SA

? ? ?

? ? ? 0.2+

++

+ +

Numeracy keynote SA

I prefer questions we work on in class to be

I prefer learning how to do questions

like this

Much harder About the same

Much easier TOTAL

By myself 23939.97%

29048.49%

6911.54%

598100%

Working with other students 78

16.7%227

48.61%162

34.69%467

100%

By listening to the teacher's explanations

first 5618.92%

14047.3%

10033.78%

296100%

TOTAL 37327.41%

65748.27%

33124.32%

1361100%

Think about the question What is 5 + 5 + 5 + 295 + 295 + 295 ? (row)

Numeracy keynote SA

I prefer questions we work on in class to be

I prefer learning how to do questions

like this

Much harder About the same

Much easier TOTAL

By myself 23964.08%

29044.14%

6920.85%

59843.94%

Working with other students 78

20.91%227

35.55%162

48.94%467

34.31%

By listening to the teacher's explanations

first56

15.01%140

21.31%100

30.21%296

21.75%

TOTAL 373100%

657100%

331100%

1361100%

Think about the question What is 5 + 5 + 5 + 295 + 295 + 295 ? (column)

Numeracy keynote SA

I prefer questions we work on in class to be

I prefer learning how to do questions

like this

Much harder About the same

Much easier TOTAL

By myself239

17.56%290

21.31%69

5.07%598

43.94%Working with other

students78

5.73%227

16.68%162

11.90%467

34.31%By listening to the

teacher's explanations first 56

4.11%140

10.29%100

7.35%296

21.75%

TOTAL 37327.41%

65714.27%

33124.32%

Think about the question What is 5 + 5 + 5 + 295 + 295 + 295 ? ( % of the overall 1361 responses)

Numeracy keynote SA

I prefer questions we work on in class to be

I prefer learning how to do questions

like this

Much harder About the same

Much easier TOTAL

By myself239 (100%)

118 (49.37%)290 (100%)

103 (35.52%)69 (100%)

21 (30.43%)598 (100%)

242 (40.47%)Working with other

students78 (100%)

15 (19.23%)227 (100%)64 (28.19%)

162 (100%)13 (8.02%)

467 (100%)92 (19.70%)

By listening to the teacher's

explanations first 56 (100%)17 (30.36)

140 (100%)36 (25.71%)

100 (100%)18 (18%)

296 (100%)71 (24%)

TOTAL373 (100%)

150 (40.21%)657 (100%)

203 (30.90%)331(100%)

52 (15.71%)1361 (100%)405 (29.76%)

Think about the question What is 5 + 5 + 5 + 295 + 295 + 295 ?