Acknowledgment of Country. We acknowledge the elders and people, past and present, of the Aboriginal people, as the traditional owners of the land on

Acknowledgment of Country

We acknowledge the elders and people, past and present, of the Aboriginal people, as the traditional owners of the land on which

we meet today; and recognise their strength, resilience and capacity.

Open-ended Enquiry in MathsDay 1

• Working Mathematically and Authentic Enquiry

• Questioning models

• Problem solving

• Teacher sharing session

• Internet sites

Open-ended Enquiry in MathsDay 2

• Extending Open Investigations• Articulating Mathematical

Thinking and Learning - Maths Blogs and Reflection tools

• The Open-Ended Approach and Lesson Study (Japan et al)

• Learning styles (M I) and programming

• Technology in Mathematics (research and shared experience)

• Quality Teaching and Assessment

• Final Reflection

A Game to Start our Thinking

Double, Halve or StayAn activity for two to four playersEquipment: two different coloured diceDecide on one coloured dice to represent the tens and the

other one to represent the ones. Choose a target number between 5 and 122. Players take turns to roll the dice. Once the dice are rolled a number is formed, then may be doubled, halved or kept unchanged to create a score. The players repeat the process and continue to add or subtract their scores, until the target number is reached exactly, or after a number of rounds the player closest to the target number wins.

Numeracy

• Numeracy is the ability to effectively use the mathematics required to meet the general demands of life at home and at work, and for participation in community and civic life.

• As a field of study mathematics is developed and/or applied in situations that extend beyond the general demands of everyday life.

Why Mathematics ?

• Why Mathematics?

• Making sense of the world

Why Mathematics ?

• Think Pair Square

• Make a list of contexts or occupations in the real-world where mathematics needs to be understood and used well

Working Mathematically

• Affinity Diagram

• Write one aspect of what it means to be working mathematically on each post-it note

• Group sort in silence (individually)

• Move post-it notes in silence (individually)

• Refine the group sort with collaborative discussion and decision-making

• Assign headings to each group


Syllabus definition

• questioning• applying strategies• communicating• reasoning• reflecting

• Make connections to the double, halve or stay game and to the world relevance of mathematics list

• (HO) - Working Mathematically Outcomes overview


• Teaching Mathematics

• stages and outcomes

• content

• process

Table group reflectionImplications for teaching from the

affinity diagram

Authentic Inquiry

• Jigsaw Reading - three groups

• Table group reflection If …. Then ?

• Reforming Education:

• The Pursuit of Learning Through Authentic Inquiry in Mathematics, Science and Technology

• Watters and Diezmann, QUT, Brisbane

Questioning Models

• Examples of open, closed and extended investigations

• (HO) - Working Mathematically

• Activity In pairs select several closed questions and write open questions and extended investigations for these (topics on cards)

• Journal Writing - Five Whys?• Why is it important to provide

students with open questions and extended investigations?

• Closed and open questions

Questioning Models

• overview of models• divergent questions• Bloom’sTaxonomy - question stems• Thinker’s Keys (Tony Ryan)• De Bono - 6 Thinking Hats• Socratic Dialogue• Weiderhold’s Matrix

• Group Activity - each group studies one questioning model and identifies areas of mathematics teaching and learning where this model could be used

• Whole Group Sharing

• Understanding questioning models

Problem Solving

• Collaborative problem solving

• Group structures and roles

Problem Solving

• The Problem• Working in groups of 4 find a solution to

the following problem:

• A farmer takes pigs and cows to the local stock sale. Each pen at the saleyards is the same size holding either 2 cows or 5 pigs. If he delivers a total of 81 animals to be sold and fills 30 pens, how many cows did he have to sell?

• Discuss a variety of ways in which you could solve this problem

• Recorder / Reporters and Observers to report to the whole group

• (HO) - Cooperative Learning Structures


• Group structures

• Group roles

• - manager

• - recorder / reporter

• - clarifier / encourager

• - observer

Problem Solving

• Table group brainstorm• What are the elements of

problem solving?

• Structured Brainstorm • Whole group thinking and

ideas captured on an A1 chart


• Cooperative Learning Strategies (HO)

Problem Solving

• make a table• act it out• simplify the problem• trial and improve• work backwards• look for a pattern• draw a picture of graph• try all possibilities• write an equation• guess and check • make a model• process of elimination• …. whatever works for me!

(HO) - Problem Solving Strategies

• Strategies for problem solving

Problem Solving

• TAPE DIAGRAMS• A tape diagram offers students a thinking tool to visually represent a

mathematical problem and transform the words into an appropriate numerical operation

• I had 3 apples, how many more do I need to buy to make 10?

• 22 sweets are divided among three children in a family. The twins have the same number each, while their younger sister has 6. How many sweets does each twin have?

Problem Solving

• TAPE DIAGRAMS

• Activity

• Solve problems using tape diagrams• Create a tape diagram, then draft three different

problems that could be represented by that tape diagram

Problem Solving

• Newman’s Error Analysis

• Transformation Skills

• (HO) - overview

• Newman’s Error Analysis

• Professor Anne Newman’ research into children’s problem solving barriers

Reflection Journals

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Teacher Sharing Session

.

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Internet Sites

Websites that have a focus on interactive activities and problem solving in mathematics

• CAP website - Maths on the Net - Research Modules• TaLe• http://www• Diane’s Favourites (HO)• Others to share …..

http://www.mathgames.com.au/

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Reflection

From the black and white photographs provided, select one photograph that says something to you aboutyour learning and participation in the workshopsessions today.

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Extending Open Investigations

Multo

Fay’s NinesNumber tilesIs it fair?Possible prisms and

wrappingThe traffic surveyHappy and sad numbers

RICH TASKS

.

Articulating Mathematical Thinking and Learning

Articulating mathematical thinking and learning through writing

Maths Blogs

MATHS BLOGS

A middle years project forPSP school communities

AmandaSchofield - QT Numeracy Consultant- North Coast Region

.

Articulating Mathematical Thinking and Learning

Some Examples:

Bloom’s verbs and stems

Apollo Parkways Primary School reflection stems (on cards)

Quality Improvement tools (Tool Time book)

Visual imagery eg photographs Free writing

REFLECTIONTOOLS

“The Open-ended Approach”

• “…provides students with experience in finding something new in the process …”

(Shimada - 1970s)• “ In the open-ended approach

students are often asked to not only show their work, but also to explain how they got their answers or why they chose the method they did.”

(Schoenfeld 1997)

“The Open-ended Approach”

PROBLEM

solution

solution

solutionsolution

solution

IDEAS / QUESTIONS / PROBLEMS

Advantages of “The Open-ended Approach”

1. Students participate more actively in lessons and express their ideas more frequently2. Students have more opportunities to make comprehensive use of their mathematical knowledge and skills3. Every student can respond to the problem in some significant ways on his/her own4. The lesson can provide students with a reasoning experience5. There are rich experiences for students to have the pleasure of discovery and to receive the approval from fellow students.

The Open-ended Approach and Lesson Study

Lesson Study is an ongoing, collaborative, professional development process that was developed in Japan.Many teachers in other countries are interested in this process, particularly in light of TIMSS (Third International Mathematics and Science Study) results, in the 1990s, which highlighted the advanced performance and deeper thinking in mathematics by Japanese students.

The Lesson Study Process

• Groups of teachers identify an area of need in student learning and progress in their classes that is in need of improvement.

• They then enquire into developments in teaching that are likely to have an impact on this aspect of student learning

• The group spends between one and three years working together:

• - planning interventions in lessons that may improve student learning

• - teaching and collaboratively closely observing these ‘research lessons’

• - carefully discussing the outcomes

• - writing up what happens - ‘failures’ as well as ‘successes’


• Choose a research theme

• Focus the research

• Create the lesson

• Teach and observe the lesson

• Discuss the lesson

• Revise the lesson

• Document the findings


• Select a broad goal, such as increasing your students’ ability to reason mathematically, or increasing their confidence in their mathematical abilities

• Select a unit to focus on and analyse the current abilities and needs of your students

• Select a lesson to develop together, being sure to look at how the skills for that lesson fit in the continuum of skills across the grades. Also think about how evidence of student thinking can be observed during the lesson.

• Teach the lesson and observe it• Get together and discuss and analyse the lesson• After discussing your observations, work together to revise the

lesson, and then have another teacher teach the lesson, then repeat the observation and discussion.

The Open-ended Approach and Lesson Study Process

• Example

• Focus area - multiplying and dividing decimal numbers in Grade 5 (Japan)

• The problem:

A 2 metre length of wire weighs 24.8 grams. If you have 6 metres of the same wire, how much will it weigh?


• Video segment 1 - Introduction

• Mr Masahiro Seki sets up the problem by showing students a wire. After helping students think about things that change as the length of wire changes, he poses today’s problem.

QuickTime™ and a decompressor

are needed to see this picture.


• Video segment 2 - Individual problem solving

• Mr Seki circulates around the classroom, asking questions and making suggestions. He also records students’ responses on the seating chart he carries with him. After the individual problem solving time, Mr Seki calls on some students to write down their solutions on the blackboard.




• Video segment 3 - Whole class discussion

• The class discusses the solutions written by their classmates. They note that some strategies involve multiplication by 3 while others do not. Mr Seki orchestrates the discussion as the class tries to articulate the reasons behind both approaches.




• Video segment 4 - Summarising and consolidating students’ understanding

• Mr Seki summarises the whole class discussion. He then asks the students to write a journal entry




• Video segment 5 - Final comment

• Final comment at the end of the post-lesson discussion by Professor Toshiakira Fujii, Tokyo Gakugei University




PERSONAL REFLECTION

Consider the implications and application of these processesto your own teaching

Reflection Journal writing

How does this learning impact on my reflection on my currentteaching practice in teaching mathematics?

Using an Enquiry-Based Approach

National Teacher Research Panel - Conference Summary:

Teaching and Learning Mathematics using an Enquiry-BasedApproach

Mark Richards, Lancaster Girls’ Grammar School, Lancaster UK

Activity: Individual reading of the report

Learning Styles

Overview of Learning Styles:- Mumford (1996) - theoretical, pragmatic, reflective, activist - Rose (1985) - visual, auditory, kinaesthetic- Gagne (1985) - nine phases of learning- Kroehnert (1990) - nine guidelines for teaching situations- Myers-Briggs - identification of a person’s type (and the impact of this on learning)- Grasha-Riechmann - categorisation of social indicators and the classroom preferences of these categories - competitive, collaborative, avoidant, participant- Herrmann Brain Dominance Index (HBDI) - four categories of thinkers/ learners - analyser, organiser, sensor, explorer- Multiple Intelligences - originally 7, now 8 and more being investigated (eg emotional, financial ….)- Dunn and Dunn (1994)- five major stimuli to which students respond in learning situations - environmental, emotional, sociological, physical, psychological

- Global Vs Analytical

Learning Styles

MULTIPLE INTELLIGENCESIt is not how smart are you but how are you smart?

Howard Gardner 1983 onwards

View video clip from: Multiple Intelligences: Discovering the Giftedness in ALL

Thomas Armstrong (USA)

(HO) - Smart pizza sheet

Learning Styles

MULTIPLE INTELLIGENCES

Programming using multiple intelligences mapped againstHigher order thinking (Bloom’s Taxonomy) - an example

Sample units:- fractions- money- area and volume- numeration

(HO) - copies of sample units

Learning Styles

MULTIPLE INTELLIGENCES

Activity:Using the appropriate mathematics syllabus, select a topicthat may lend itself to using a 48/56 grid and enter some activities in the appropriate cells

Know Understand Apply Analyse Evaluate Create

Verb/Lin

Log/Mat

Vis/Spa

Mus/Rhy

Bod/Kin

interper

Intraper

Environ

Technology in Mathematics

• Research on the effects of technology use on the teaching and learning of mathematics - ppt

• CAP Maths in Technology workshops

• Table group activity -• Share with others the

technology used in your mathematics teaching and learning activities

Technology in Mathematics

• Mathematics learning opportunities in virtual environments or virtual worlds

• Game environments• Software packages eg Kahootz

• Maths Education in Second Life - ppt

Quality Teaching and Assessment

Table group activity:Discuss and identify the elements ofthe NSW Quality Teaching frameworkthat will be supported directly by theuse of open-ended enquiry in mathematicsteaching and learning.

Quality Teaching and Assessment

Fishbone DiagramWhat is the effect of using open-ended questions, extended investigations and rich tasks as assessment strategies?

Fishbone headings:- teachers- teaching- students- thinking- record keeping- reporting

Final Reflection

A Social View of Language

FIELD

WHAT is being communicated

Field is built up by:•naming•describing•defining•classifying•sequencing•cause / affect•part / whole•class / subclass

To develop field knowledge students need:•familiarity with language•opportunity to experience field knowledge

EVERYDAY (familiar) ABSTRACT (technical)

Field = Subject Matter• has its own way of constructing language•has characteristic language patterns

TENOR

Tenor reflects:• roles• relationships

Tenor indicates:•power•status•authority•expertise•degree of familiarity•feelings

WHO is involved in the communication

INFORMAL (personal) FORMAL (impersonal)

MODE

Mode reflects appropriate choice of * spoken text or * written text

HOW is the communicationconducted

SPOKEN WRITTEN(spontaneous oral interaction) (crafted edited organised)

READING

The four sources of information:

• Semantic

• Grammatical

• Graphological

• Phonolological

READING

The four roles of the reader:

• Text decoder - I can say the words

• Text participant - I understand what I am reading

• Text user - I can use the text

• Text analyst - I can analyse, reflect on, respond to the text (bias, point of view…)

VOCABULARY

“The English language is a vast sea and people often communicate using only what they know of the shallows on the shore.”

“Reading 2 Learn” - David Rose

The lesson sequence applied to a mathematics problem:

*

VOCABULARYacnodebimodalcatenarydeltoideccentricFermat’s last theoremgnomonhemicycleimaginary numberjoulekinematicslocusmultifoil

normalogivePascal’s trianglequadratic equationradianscalenetowers of Hanoiunconditional inequalityvicious circlewalkx-axisy-axiszone

Asking Questions to Find the Answer

• Three Level Guide

I. The answer is specified in the text

II. The answer is in the text but I need to read between the lines

III. The answer will be found if I read the text and think beyond / outside of it

• 3H StrategyWhere do I find the answer to a

question?

I. Here - Is it here in one sentence in the text?

II. Hidden - Is it found by joining together information from two or more places in the text, OR from information in the text and what I already know?

III. Head - Is it in my background knowledge: what I already know and does it require inferences and evaluations beyond the text?

Find x

Expand

Literacy

• Literacy is the ability to communicate purposefully and appropriately, in and through a wide variety of contexts, modes and mediums. Whilst English has a particular role in developing literacy, all curriculum areas, including mathematics, have a responsibility for the general literacy requirements of students, as well as for the literacy demands of their particular discipline

Technology

• Information and Communication technology (ICT) has been developed with the significant utilisation of mathematics, and a range of opportunities exists within the teaching and learning of mathematics to utilise ICT.

New terrorist threat!

• A public school teacher was arrested today at John F Kennedy airport as he attempted to board a plane while in possession of a ruler, a protractor, a set square, a slide rule and a calculator.

• At a morning press conference, Attorney General John Ashcroft said he believes the man is a member of the notorious Al-gebra movement.

• He did not identify the man who has been charged by the FBI with carrying weapons of maths instruction. “ Al-gebra is a problem for us,” Ashcroft said. “They desire solutions by means and extremes, and sometimes go off on tangents in a search for absolute value. They use secret code names like ‘x’ and ‘y’ and refer to themselves as ‘unknowns’ , but we have determined that they belong to a common denominator of the axis of medieval with coordinates in every country. As the Greek philanderer Isosceles used to say ‘There are 3 sides to every triangle’.”

• When asked to comment on the arrest, President Bush said, “If God had wanted us to have better weapons of maths instruction, He would have given us more fingers and toes.”

• White House aides told reporters they could not remember a more intelligent or profound statement by the president.

Documents

Acknowledgment of Country. We acknowledge the elders and people, past and present, of the Aboriginal people, as the traditional owners of the land on