Lead Teacher Workshop 3

Divisibility Rules Warm Up

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Purpose of this session is…

Share and discuss examples of mid-year reporting to parents.

Continue to explore the mathematics behind the National Standards with a focus on measurement.

Effective Mathematics Pedagogy - engaging learners in mathematics.

Overview (9.10 – 12.00)

•Any current issues?•Share report examples•Formative Assessment– Dylan Wiliam•Effective pedagogy and Engaging Learners in Mathematics – rich tasks•Unpacking Measurement in the Standards•What’s new – keep updated

Mid Year Reports

•Successes •Feedback from parents•Possible modifications for the next report

•Critique the report examples

Reviewing written reports

Reporting achievement in relation to National Standards

Experiencing difficulties

Working towards the

standard

Working at the standard

Working above the standard

Working well above

Well below Below At Above Well above

Working below

Working at Working above

“Harry is working_____ the National Standard for his age”

Effective Mathematics Pedagogy and

Engaging Learners in Mathematics

Effective Teaching Cycle

Assess

Analyse data

Plan

Teach

Practice/Apply

Assessment in the NZC

“The primary purpose of assessment is to improve students’ learning and teachers’ teaching as both student and teacher respond to the information that it provides……”

The New Zealand Curriculum, p.39

Assessment of learning

Assessment for learning

Formative Assessment – Dylan WiliamProfessor of Educational Assessment at the University of London. Also works with

Paul Black – co-authors of Inside the Black Boxhttp://www.ltscotland.org.uk/learningaboutlearning/aboutlal/biogs/biogdylanwiliam.asp

.

Discuss…

• Main points that you found of interest• How you do / might implement these

ideas into your school.

Should formative assessment be recorded?If so – how?• Modelling book• Teachers feedback comments in student books• On planning units• Other anecdotal notebook• Self/peer assessment in maths diaries

The expectations defined by the standards include how a student solves a given problem, not only the student’s ability to solve it so….

•Provide tasks with multiple possible solution strategies

Using Different Problem Types

Different Problem Types

1. Martin opened his book and noticed that the sum of the two pages was 157.

What page numbers were showing?

2. 67 + 59 =

Open ended problems are something they need to think about, not simply a disguised way of practising already demonstrated algorithms

Can one number be divisible by 8 different digits, 7 digits, 6 digits?

Measurement Problem Types

Task 2.Describe the difference in length between your own hand and the giant’s hand. (Provide a range of tools such as cubes, rulers etc. for children to choose and use at their leisure)

Task 1.Use cubes to measure the length of your hand and the giants hand.

The Giants Hand

procedural

open ended

Task 2.Describe the difference in length between you’re your own hand and the giant’s hand.

Consider possible responses from the children and place them on the Standards to identify next learning steps.

• It must be accessible to everyone at the start.• It needs to allow further challenges and be extendable.• It should invite learners to make decisions.• It should involve learners in speculating, hypothesis

making and testing, proving and explaining, reflecting, interpreting.

• It should not restrict learners from searching in other directions.

• It should promote discussion and communication.• It should encourage originality/invention.• It should encourage 'what if' and 'what if not' questions.• It should have an element of surprise.• It should be enjoyable.Ahmed (1987), page 20

How rich was this task?

• It must be accessible to everyone at the start.• It needs to allow further challenges and be extendable.• It should invite learners to make decisions.• It should involve learners in speculating, hypothesis

making and testing, proving and explaining, reflecting, interpreting.

• It should not restrict learners from searching in other directions.

• It should promote discussion and communication.• It should encourage originality/invention.• It should encourage 'what if' and 'what if not' questions.• It should have an element of surprise.• It should be enjoyable.Ahmed (1987), page 20

How rich was this task?

While there is a place for practice and consolidation, “tasks that require students to engage in complex and non-algorithmic thinking promote exploration of connections across mathematical concepts” p.97

Using rich tasks to engage learners in mathematics

Without a problem, there is no mathematics.Holton et al. (1999)

‘A richer problem’

If this is a handprint of the giant - how tall is the giant?

Turning a terrific task into a terrific lesson

In your groups, consider…

1. Mathematical opportunities

2. Possible responses

3. ‘Enablers’ and ‘Extenders’

If this is the giant’s handprint, how tall is the giant?

1. Mathematical opportunities

• Use of self-chosen standard or non-standard units.

• Proportional / Multiplicative thinking, i.e. A person is ? times their hand length?

• Statistical thinking

If this is the giant’s handprint, how tall is the giant?

2. Possible responses?

• No idea.• Guess only.• Measure in hand lengths,• Measure using whole number

standard units e.g. cm• Measure using standard units to

nearest tenth.• Use of statistical relationship

graphs.

‘Enablers’ – using hands as non-standard units

• Draw and cut around your own hand. • Use that to measure your height.• How could you use this idea to with the

giants hand?.Mathematical opportunities in enabling task:•Estimate first•Place hands with no overlaps or gaps (‘tiling’)•Measure from the same baseline.•Hand units can be partitioned (e.g. halves) to be more accurate. •Understand the limitations of non-standard units.

“Extenders”

• Does it make any difference if the giant handprint is a woman or a man?

• What if the giant was only a child?

• How does this compare to the World’s tallest man in the Guinness book of

records?

• It must be accessible to everyone at the start.• It needs to allow further challenges and be extendable.• It should invite learners to make decisions.• It should involve learners in speculating, hypothesis

making and testing, proving and explaining, reflecting, interpreting.

• It should not restrict learners from searching in other directions.

• It should promote discussion and communication.• It should encourage originality/invention.• It should encourage 'what if' and 'what if not' questions.• It should have an element of surprise.• It should be enjoyable.Ahmed (1987), page 20

Which of the rich task criteria did this problem meet?

However, keeping mathematics interesting and fun should not be at the expense of content.

Turning a terrific task into a terrific lesson1. Select an engaging problem that is both

achievable and challenging.

2. Know the mathematical opportunities provided in the problem.

3. Consider possible responses from the children.

4. Consider ‘enablers’ and ‘extenders’ to make the task both achievable and challenging.

5. Formatively assess what a child can do and..

6. Identify next learning steps.

Jack and the Beanstalk

learning centre

Jack and the Beanstalk learning centre ideasAdapted from NZAMT maths week 1999, google

1. How heavy is the golden egg? (mass)2. How much are the beans worth? (money)3. How many beans in a handful? (volume/statistics)4. How long did it take to chop down the beanstalk?

(time)5. How long will it take to grow a beanstalk?

(time/length)6. How tall was the giant (length)7. How do I get to the Giant’s castle? (position)8. Design a giant’s castle using…. (shape)9. Design a patterned glove to fit either your own or

the giant’s hand (using symmetry)

What’s New – Keeping up to date

Thought for the day

Remember that frequently…

The student knows more than the teacher about what he has learned even though he knows less about what he was taught.

 Just because you’ve taught it doesn’t mean

they’ve learned it!