08/09/2022

Mathematics Education to Develop Students Agency
Part III: Summary
Mathematical Thinking and Activities

Focused onPrimary
School Math.
Part I.
Number&Operations
Part II.
Fractions＆Proportionalit
yISODA Masami, Prof/PhD

Part III.
Other Strands

Instead of just saying Higher Order Thinking Skills in
General and Authentic Mathematics, we preferred touse Terminology to design mathematics’ lessons andtask sequence. It is terms to explain mathematicalthinking (invariant), at first and conceptual differences(variant), at second.
Can you explain what is higher order thinking in mathematics?
Higher order thinking is explained by Mathematical Thinking
For example:
Instead of just saying Critical Thinking, we discussed dialectic approaches as a part ofproblem-solving approaches which explains how to design arguments in classroom byusing students’ misconception on the manner ‘if your saying is true, what will happen?’See Part II, Lesson 1 & 3, and Part III, Lesson 4
Discuss possibility of generality by using Counter example, by changing
Representations, by posing the questions with other cases.

Director of CRICED
University of Tsukuba, Japan

12On SEA-BES: CCRLS in Mathematics
Appreciation

Instead of just saying Higher Order Thinking Skills in
General and Authentic Mathematics, we preferred touse Terminology to design mathematics’ lessons andtask sequence. It is terms to explain mathematicalthinking (invariant), at first and conceptual differences(variant), at second.

Reflection

Can you explain what is Authentic Mathematics in mathematics?
Authentic Mathematics is explained by Problem Solving Approaches on
the appropriate task sequence which uses what students already learned.
Example 2:
Instead of just saying authentic mathematics,

See Part III, Lesson 1 & 4• We discussed the problematic appeared from children on Problem Solving
Approach on the task sequence which are explained by second terminology.

3Acquisition• We discussed the nature of mathematics curriculum, not the nature of puremathematics, which includes huge numbers of inconsistency/contradictions.

Context 1: Real World Problem Solving
Context 2: Problem Solving on theappropriate task sequence41

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Continue…

What is Authentic Math on SEA-BES: SSRLS?

Ontology
Idealism
Plato: idea• Mathematics isunique common
language in theworld

DiscoverReasonable, harmony& beautifulness on thesystem.
Materialism
Aristotle
Invention• Mathematics is
unique subject forunderstanding
others throughRe-presenting.

Modeling, instruments
and representation.
Epistemology
Idealism:
Hegel’s Dialectic• Mathematicalactivity is basedon human activity
withcompetitivenessandunderstanding
othersLakatos’s Proof andRefutation
Plato’s Dialectic
Polya’s problem solving
Freudental’smathematization

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Continue…

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Summary: Basic Principles
Constructivism
Piaget, Dewey& Freudenthal
Learning
through
Reflection
SocialConstructivism
VigotskyThinking isdevelopedthroughinternalizationofcommunication

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1. Principle for Teaching Approach
Teaching is the activity by using what students learned before andpreparation for Future It is based on the appropriate task sequence.
 Task is given by teacher and problematic emergent from students.
 Problem solving approach or teaching through problem solving are not aimed to find answers for giventask but aimed to realize students’ arguments to solve problematic for students.
 On problem solving approach, the objective of lesson is not to solve given task but to solve problematicfor students by themselves.
 Problematic is not easily to solve by using what students already learned however if they utilize, extend
and arrange various representations of learned ideas, they are able to solve. It is new creation bystudents and points of arguments in the lesson.
 Misconceptions are the essential product of appropriate learning through mathematics curriculum and apart of task sequence. We should not deny them based on what teachers know because misconceptions
are indicators which students produce their own ideas based on what they learned. It is the opportunity todevelop critical discussions.
 In the classroom communication, understanding others means re-present others’ idea by him/herself andproducing another example by him/herself. If we understand others and produce another example, it will
be possible to say if your saying is true.
 Teaching approach should be preferred depending on the objectives.
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Summary: Basic Principles

Summary: Basic Principles

2. Principle to design Curriculum
Terminology to explain conceptual difference is aimed to explain the task
sequence and objectives for each lesson.
 In mathematics, we internationally share the technical terms. Each term’s concept itself explains
mathematical invariant but it does not well explain mathematics curriculum itself because the meaning of
the same technical term usually change and extend in the curriculum sequence. On mathematics,concepts usually well defined and never change. Thus, in case, we use the word ‘concept’ we should
explain it exactly like mathematicians. Such an explanation is necessary to make clear what is essential
however it is NOT the terminology to explain curriculum and task sequence.
 Instead of explaining mathematical technical terms, on this lecture we more focused on the terminology
which explain conceptual differences to explain how the meaning of mathematical terms change on ourcurriculum. It explains conceptual differences on the task sequence.
 We also discussed levels and phases as terminology to explain conceptual differences and task sequencesuch as Part III, Topic 2 Figure, Topic 3 Measurement and Topic 4 Pattern and Data.

We learned various meaning of mathematics curriculum.
In the case of Philippines, curriculum includes up to the lesson plan.
In many countries, curriculum standards does not include terminology beyond technical terms in mathematics.
93. Principle to develop Mathematical Thinking in Curriculum
Terminology to explain mathematical thinking which aimed to explain
invariant processes of thinking which should be taught through thecurriculum.
 Repetition of the same process is the opportunity to learn mathematical thinking however for‘learning how to’ students need the opportunity to reflect on the process and appreciate it.
 Even though terminology of invariant mathematical thinking never explains each student thinking
itself, it explain how it will be possible to repeat on the task sequence on the curriculum.
 Wide meaning of mathematical thinking include values, attitudes and habits even through narrowmeaning of mathematical thinking is mathematical ideas and ways of thinking.
 Variant Terminology to explain conceptual differences are also include unique idea and ways ofthinking
 Various Mathematical activity such as problem solving, modeling and inquiry always characterized by
specific process such as PPDAC cycle.
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These are final reviews and summary.

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