MATHEMATICAL THINKING RESEARCH GROUP
mathematics, thinking, teaching, learning critical thinking, problem solving, non routine
Development of Mathematical Thinking
What is Mathematics?
Mathematics is so useful that there could be no civilization without it, and it is so beautiful that some theorems and their proofs—those which cause us to gasp, or to laugh out loud with delight — Should be hanging in museums. (Underwood Dudley,1987)
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If math is so great and timeless and beautiful, why do millions of people HATE it so much?
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“We cannot justify teaching mathematics to 19-22 year olds by asserting that they will find it useful,”
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NO people will says I hate Music or I hate Art or I hate Literature.
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Intelligent people will never say that.
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If they do, it will be an insult to their intelligence!
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Yet, it’s ok to hate Math!
If Mathematics was an elective and school and colleges didn’t demand it, fewer students would learn it. But fewer students would fail it, too, and fewer students might drop out of college, and the level of cheating would go down, and the sum total of student misery would be substantially reduced.
Mathematics Miseducation.
“Any implication that mathematical thinking is being taught through schools and College is plainly false,”…
--Parmjit, 2016
Factors Contributing To Students’ Poor Mathematical Thinking
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Majority of students faced difficulties in the fundamentals of arithmetic and pre- algebra.
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Majority of the school and even college students still faced difficulties in conceptualizing word problems.
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There was an over reliance on the usage of calculators in computing problems.
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A low mastery of logical Thinking.
A low level of heuristic strategies repertoire among the students.
What is the Function of Education?
The function of Education is to teach one to think intensively and to think critically…… Intelligence plus character- that is the goal of true education. (Martin Luther King, 1947)
What is the Purpose of Learning Mathematics?
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The purpose of learning mathematics in schools and higher education is to develop the thinking capabilities of an individual, especially in the context of analytical and logical thinking.
What is Mathematics Teaching?
Teaching is an interaction process between teacher and students over content in an environment.
In teaching math, we need:
Curiosity,
Fun and
Entertainment
What is Mathematical Thinking?
Mathematical thinking is an abstraction process of fundamental mathematics concepts using non-routine problems in solving problems. It is a whole way of looking at things, of stripping them down to their numerical, structural, or logical essentials, and of analyzing the underlying patterns.
“Thinking mathematically is about developing habits of mind that are always there when you need them - not in a book you can look up later” Ridgway (2001, p. 1)
Domains of Mathematical Thinking
Reasoning and Proof
Recognize reasoning and proof as fundamental aspects of mathematics.
Make an investigate mathematical conjectures.
Develop and evaluate mathematical argument and proofs.
Select and use various types of reasoning and methods of proof.
Communication
Organise and consolidate mathematical thinking through communication.
Communicate their mathematical thinking coherently and clearly to peers, instructors and others.
Analyse and evaluate the mathematical thinking strategies of others.
Use the language of mathematics to express mathematical ideas precisely.
Connections
Recognize and use connections among mathematical ideas.
Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
Analyse and evaluate the mathematical thinking strategies of others.
Use the language of mathematics to express mathematical ideas precisely.
Representation
Create and use representations to organize, record and communicate mathematical ideas.
Select, apply and translate among mathematical representations to solve problems.
Use representation to model and interpret physical, social and mathematical phenomena
Why teach Mathematical Thinking?
One of the major aims of mathematical learning, especially at tertiary level is towards the development of mathematical thinking. According to Ridgway (2001), "thinking mathematically is about developing habits of mind that are always there when you need them - not in a book you can look up later (p. 1). It is a pre-built thinking in the mind of an individual when solving problems.
Logic in Developing Mathematical Thinking
Logical reasoning is the process of using a rational, systematic series of steps based on sound mathematical procedures and given statements to arrive at a conclusion.
Logical thinking is the process in which one uses reasoning consistently to come to a conclusion.
Problems or situations that involve logical thinking call for structure, for relationships between facts, and for chains of reasoning that “make sense.”
Mathematical Thinking in Schools and Colleges
Research outcome on college mathematics depicts a dichotomous situation where on one hand, it has consistently indicates college students are getting good math grades in their transcript, however it is not being translated into the development of mathematical thinking. On the other hand, it expressed near universal agreement that many students arrive unprepared for the intellectual demands and expectations after high school and struggle during their early years. The consequence from these is high failure rate, not graduating on time, hatred towards mathematics and not prepared for the intellectual demands and expectations of posts-college education.
Mathematical Thinking as a New Course in Schools and Colleges
Thus, we are proposing a Mathematical Thinking course where its application is a way of thinking which allows concepts, processes and their uses to be built up, problems to be explored and solved, conjectures to be made and examined, and complex ideas about the world to be communicated in precise, succinct ways. Indeed, an ability to think with and about mathematics will replaced the memorisation of set procedures and the solution of routine problems as the focus of school and college mathematics learning at current time.
Conclusion
We as Math teachers/instructors are ACTIVELY Teaching BUT Students’ are NOT LEARNING!
The common misconception is that “doing mathematics” is the same as getting involved in “mathematical thinking”.
Mathematics learning for college mathematics should be re-engineered where the focus of doing mathematics should be inclined towards “teaching students’ to think”.
Not easy as it’s a marathon and not a sprint
Short term pain BUT long term gain.
The danger: We always expect instant results!