Key Principles of Effective Math Instruction: Outline and Examples

Dr. Mohamed Matar, Palestine Ministry of Education Aarnout Brombacher, Brombacher and Associates and RTI

4 December 2013

Overview of presentation

• What is mathematics & how is it used?

• What does it mean to know/do mathematics?

• What mathematics to teach and how to teach mathematics in the early years?

• Implications for countries:

– Teachers and teacher training

– Curriculum

– Other issues

• What is mathematics & how is it used?

• What does it mean to know/do mathematics?

• What mathematics to teach and how to teach mathematics in the early years?

• Implications for countries:

– Teachers and teacher training

– Curriculum

– Other issues

What is mathematics and how is it used?

Problem “real world”

Solution “real world”

Mathematical problem

Mathematical solution

Formulation “translation”

Interpretation “translation”

“Calculation”

Mathematics as:

The memorisation of facts, rules, formulas and procedures needed to determine the answers to questions.

Mathematics as:

A meaningful, sense-making, problem solving activity.

• What is mathematics & how is it used?

• What does it mean to know/do mathematics?

• What mathematics to teach and how to teach mathematics in the early years?

• Implications for countries:

– Teachers and teacher training

– Curriculum

– Other issues

… to adopt a composite, comprehensive view of successful mathematics learning… we have chosen mathematical proficiency to capture what we think it means for anyone to learn mathematics successfully.

Mathematical proficiency, as we see it, has five strands

Adaptive Reasoning

Strategic Competence

Conceptual Understanding

Productive Disposition

Procedural Fluency

ability to formulate, represent, and solve

mathematical problems

comprehension of mathematical concepts, operations, and relations

inclination to see mathematics as sensible, useful, and worthwhile

capacity for logical thought, reflection, explanation, and

justification

carrying out procedures flexibly, accurately,

efficiently, and appropriately

… the mathematics assessment is organized around two dimensions: a content dimension specifying the subject matter or content domains to be assessed in mathematics, and a cognitive dimension specifying the thinking processes that students are likely to use as they engage with the content.

• Three cognitive domains: knowing, applying, and reasoning were used.

• Knowing refers to the student’s knowledge base of mathematics facts, concepts, tools, and procedures.

• Applying focuses on the student’s ability to apply knowledge and conceptual understanding in a problem situation.

• Reasoning goes beyond the solution of routine problems to encompass unfamiliar situations, complex contexts, and multi-step problems.

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Grade 4 TIMSS achievement in Mathematics Cognitive Domains

Knowing Applying Reasoning

• What is mathematics & how is it used?

• What does it mean to know/do mathematics?

• What mathematics to teach and how to teach mathematics in the early years?

• Implications for countries:

– Teachers and teacher training

– Curriculum

– Other issues

Focal outcomes for Early Childhood Level:

• Number sense and operations

• Spatial sense and geometry

• Patterns and classification

• Measurement and comparison

Focal outcomes for Primary Level:

• Number concepts and operations

• Geometry and patterns

• Mathematics application

• Number Sense

– Being able to work fluently and flexibly with numbers and number concepts.

• 8 + 7 = ___ & 7 + 5 = ___

• 43 – 18 = ___

• 36 + 45 = ___ & 36 + 49 = ___

– Having a rich understanding of the meaning of number.

– Having a wide range of effective calculation strategies for solving a large variety of problems involving number.

• Operations with numbers

– When learning operations, the common progression begins with addition, then subtraction, then multiplication and then division; however, research does not support such a linear path.

– Instead, children may begin with problems, including all operation, with smaller numbers that are contextualized and refer to concrete objects.

– With practice and exposure, as well as the encouragement to use multiple strategies, children will become fluent with the basic facts while at the same time using these basic facts to solve more complex problems.

A case study: The case of Jordan

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A case study:

The case of Jordan

منطقة تعليمية 12

مشرفا 20

مدرسة 43

معلمة 300

طالب 12000

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Counting activities

• Rote counting activities:

– Counting in ones

– Counting rhymes and songs

– Counting is steps

• Rational counting activities:

– Counting small sets of counters in ones

– Counting out small groups of counters

– Estimating and counting larger sets of counters in ones

– Counting in groups

– Counting large sets of counters in groups

Manipulating number activities

• Single digit arithmetic

• Arithmetic with multiples of ten, hundreds and thousands

• Completing tens (hundreds and thousands) including adding and subtracting to and from multiples of ten

• Bridging tens (hundreds and thousand)

• Doubling and halving – to develop efficient division and multiplication strategies

• Interrelated multiplication facts

Solving problems

• Problems that support the development of addition and subtraction including:

– Change, Combine, and Compare problems

• Problems that support the development of the division concept:

– Sharing, and Grouping

• Problems that support the development of the multiplication concept:

– Repeated addition, and Situations with a grid or array type structure

• Problems that support the development of the following concepts:

– Fractions, Ratio, rate and proportion including sharing in a ratio.

• What is mathematics & how is it used?

• What does it mean to know/do mathematics?

• What to teach and how to teach mathematics in the early years?

• Implications for countries:

– Teachers and teacher training

– Curriculum

– Other issues

Implications-Challenges

• Changing teaching and learning from what children learn to how children learn:

• The role of curriculum: designing, development, implementation and evaluation

• Assessment :From assessment of learning toward assessment for learning

• Policy change…, and

• Teacher training

Implications – national vision and leadership

• Articulating the national imperative for a change in teaching approach.

• Providing support to the different communities both through encouragement and through holding the various role players accountable and responsible .

• Establishing a roadmap. The roadmap may prioritize the early grades.

• Taking a long-term approach.

Implications – MoE

Curriculum, materials development, teacher training

• Creating and articulating a clear vision of an alternate teaching approach.

• Making the necessary adaptations to: curricula,

• teaching and learning materials,

• national assessments etc.

Implications – Pre-service teacher training community

• Training pre-service teachers in alternate pedagogies.

• How do we expose this community to these alternate pedagogies?

• What support does this community need to make the transition?

Implications – In-service teacher training community

• Supporting teachers in developing alternate mental models with regard to what it means to teach

• Workshops vs. teacher coaching

• Managing transitions

• How do we expose this community to these alternate pedagogies?

• What support does this community need to make the transition

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About Reform, but in Education.. • “No experience is better than bad experience for school

teachers…”

• Teaching to the test is not bad if the test is good.

• “The main purpose of evaluation is not to prove but to improve..”

• In Finland, people know that “if you’ve trained as a teacher you must really be some thing special,,

• “Doing what matter is good, but doing only what really matter is the best…”

• “Successful policy involves 20% design and 80% implementation” . Hong Kong Education reform plan.

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Is it a difficult/ easy task…

• With the situation in the region…

• With the absence of the visions and priorities…

• With these gaps between the general education sector and the higher education one.. and,,,

• With the dominant culture of evaluation as a tool of prove not improve,,

• The task will be a difficult one,, god may help..

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Thanks again

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Aarnout and Matar

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