System Implementation and Monitoring

Regional Session

Winter, 2015

http://sim.abel.yorku.ca

@LNSSIM #WINTERSIM #SIMK12

Tasks in Mathematics Classrooms

Sharing of Implementation Steps and

Monitoring Actions in Like-Role Groups

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Determine where you might be in Ontario given the information in the chart below.

Think on your own for 5 minutes.

City Distance as bird flies (km)

Driving Distance (km)

Windsor 454 763Toronto 398 540Ottawa 549 647Thunder Bay 544 902Niagara Falls 463 653North Bay 245 289

City Distance as bird flies (km)

Driving Distance (km)

Windsor 454 763Toronto 398 540Ottawa 549 647Thunder Bay 544 902Niagara Falls 463 653North Bay 245 289

Determine where you might be in Ontario given the information in the chart below.

Think on your own for 5 minutes.

This is an example of a contextual mathematics task.

Best Evidence Synthesis on Effective Pedagogy in Mathematics

Effective mathematical pedagogy is a coherent system rather than a set of discrete, interchangeable strategies. This pedagogical system encompasses:• A non-threatening classroom environment• Instructional tasks• Tools and representations • Classroom discourse

Effective Pedagogy in Mathematics/Pangarau by Glenda Anthony & Margaret Walshaw, New Zealand (2007)

Quality Instruction in a Math Class

You have a representative sample of your thinking about quality instruction in mathematics grouped according to the four components of an effective pedagogical system in mathematics.

Review and discuss any observations that you have.

The LearnerDiscuss at your tables what the student should be able to do if there is quality instruction.

What is the image of the mathematics learner that emerges if the four components of the pedagogical system are evident?

The learning of mathematics has been defined to include the development of five interrelated proficiencies that, together, constitute mathematical proficiency (NRC 2001):• Conceptual understanding• Procedural fluency• Strategic competence • Adaptive reasoning• Productive disposition

Effective Teaching and Learning

Posing Worthwhile Mathematical Tasks

“It is through tasks that the curriculum and the discipline of mathematics comes alive. Starting with students’ prior knowledge and then creating rich, mathematical tasks, teachers help students to proceed gradually from their informal knowledge of the ideas in the domain to more formal notions.”

(Romberg and Kaput, 1999)

Posing Worthwhile Mathematical Tasks

“Tasks should be created or selected that have the potential to encourage students to wonder why things are, to inquire, to search for solutions and to resolve incongruities”.

(Hiebert et al. 1996)

Rich Tasks

What is also apparent to me is that much of what it takes to make a rich task “rich” is the environment in which it is presented, which includes the support and questioning that is used by the teacher and the roles the learners are encouraged to adopt. That is, an environment in which learners are not passive recipients of knowledge, accepting what is given, but independent assertive constructors of their own understanding, who challenge and reflect. On its own a rich task is not rich – it is only what is made of it that allows it to fulfil its potential.

Jennifer Piggot http://nrich.maths.org/5662

“On its own a rich task is not rich – it is only what is made of it that allows it to fulfil its potential”.

Jennifer Piggott http://nrich.maths.org/5662

Time for Chocolate!

Chocolate Bar Task

• This is an example of a purposeful representative task • These tasks are usually not “contextualized”, however

there was sometimes a hook to engage students• While the mathematics may be explicit, extensive

exposition by the teacher is not necessary as the provision of the model or representation enables the students to generate the mathematical ideas and justification.

• The model, representation, or tool is ideally closely linked to the mathematical concept being developed, in order to be effective.

Purposeful representative tasks

Contextual tasks

Content-specific tasks

Practice and consolidation tasks

In studies of mathematics tasks, four general types have been identified:

Types of Mathematics Tasks

Content-Specific Task

If the perimeter of a rectangle is 64 m what might be the area?

Purposeful representative tasks

Contextual tasks

Content-specific tasks

Practice and consolidation tasks

Content-Specific TaskPurposeful representative tasks

Contextual tasks

Content-specific tasks

Practice and consolidation tasks

• Often an open task

• Although, the student needs specific content knowledge to solve

• Involves investigating, creating, communicating, generalizing and coming to know procedures

Practice and Consolidation Tasks

Factor the expression

x + 5x + 6 = 0

Find the value of

½ + ¾

Purposeful representative tasks

Contextual tasks

Content-specific tasks

Practice and consolidation tasks

2

• Are an important part of a balanced mathematics program

• Provide students with opportunities to solidify mathematical concepts and procedures

• Provide independent practice

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Strategic competence Adaptive reasoning

Purposeful Representative

.

How Tasks Contribute To Students’ Mathematical Proficiency

Mathematical Fluency

Contextual

Content-specific Strategic competence Adaptive reasoning

It is understood that there can be overlap of proficiencies across various tasks. These are the main ones that are developed.

P. Sullivan et al. Teaching with Tasks for Effective Mathematics Learning.

Conceptual Understanding

Type of Task Mathematical Proficiency

• This is one way to monitor the effectiveness of the mathematics teaching and learning in your system

• It is one component of the pedagogical system; however as many researchers have argued it is a critical element

• Even though looking at tasks without student thinking and/or work has limitations, it does give us a window into the mathematics that students experience in our classrooms

Conversation Tool for Reflection on Mathematical Tasks

Conversation Tool for Reflection on Mathematical Tasks

Each grade group will collaboratively discuss a selection of mathematical tasks.

The Conversation tool has been designed as a starting point for your discussion.

Think about the collection of tasks to see if there is a sampling of the four types: contextual, purposeful representative, content-specific and practice/consolidation.

Grade Groups

K-23-5 6-8

9-12

Grade Groups and Rooms

TASK ROOM

6-8 VenetianK-2 Castelli AB3-5 Castelli C9-12 Dolce

12:00 – 12:45

Board Teams

Analyze the tasks that you brought using the conversation tool.

Cross-Board Like-Role Sharing

ROLE ROOMPrincipal/ Vice-Principal VenetianConsultant/Co-ordinator/Teacher

Castelli AB

Superintendent Castelli C

TimeTeam

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See you at the Spring 2015 SIM K-12!

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