147_Making Connections through Promoting Mathematical Thinking

Making Connections through Promoting Mathematical Thinking Lim-Teo Suat Khoh, MME, NIE Mathematics Teachers Conference June 2011 Overview Mathematical Thinking and Connections in the Singapore Mathematics Curriculum Connections within mathematics strands Connections across strands Connections with the real world Mathematical Thinking and Making Connections Aims of Mathematics Education in Schools Develop the mathematical thinking and problem solving skills and apply these skills to formulate and solve problems Recognise and use connections among mathematical ideas, and between mathematics and other disciplines Do these sound familiar? Mathematical Thinking and Making Connections Students should use various thinking skills and heuristics to help them solve mathematical problems Thinking skills are skills that can be used in a thinking process, such as classifying, comparing, sequencing, analysing parts and wholes, identifying patterns and relationships, induction, deduction and spatial visualisation Connections refer to the ability to see and make linkages among mathematical ideas, between mathematics and other subjects, and between mathematics and everyday life This helps students make sense of what they learn in mathematics Do these sound familiar? Singapore mathematics curriculum The Singapore mathematics curriculum promotes making connections and thinking skills Think for 30 sec and tell your neighbour which mathematical thinking skill or skills you most often encourage in your classes Then tell him/her what motivates you to encourage those thinking skills or what value you see in encouraging those mathematical thinking skills Mathematical thinking skills are the essence of mathematics A person does mathematics when he/she engages in such thinking processes Mathematical thinking skills provide connections which makes mathematics topics meaningful – otherwise we just have a repertoire of disconnected facts and rules Topics in the Syllabus Data Topic Where are the connections between topics within or across strands? Connections within Strand Making Connections Within Strand As Mathematics is largely hierarchical, it is necessary to build new concept on those previously established and learned Making such connections are essential as they give justification for the new concept learned Euclidean Geometry as a field of study is the natural strand to develop connections since the whole structure of concepts, theorems and properties are connected via logical reasoning bridges In topics of other strands, building of one concept upon another is also necessary Example 1: Constructions Compass constructions of angle bisector and perpendicular bisector of line segment Think through the steps of the construction Do you simply tell your students the steps? Do they know why the steps work? Example When teaching quadratic expressions ax2 + bx + c: Get students to give a few quadratic expressions – try to have more diversity in the coefficients (negative, non-integers) Ask them to change their quadratic expressions slightly so that they are no longer quadratic When teaching polynomials, students can be asked to give examples of polynomials they have already encountered Always include non-examples, tricky examples etc when establishing concepts Example 6: Using processes learned earlier In teaching processes, it is also important to make connections so that students know we are using processes learnt earlier Substitution is a very powerful tool in mathematics because it allows the person transform what he is working with into a form he can work with because of processes learned earlier So here the skill is the altering process and the connection is to a solution method already mastered Example Solving simultaneous equations y2 + (2x + 3)2 = 10 2x + y = Solving equations where unknown is in the exponent e-x(2e-x + 1) = 15 Integration Differentiation (chain rule) Summary of learning theories Piaget, Bruner, Dienes, Gagne, Skemp, Marton Concept Mapping Use of topic maps across years could be useful for teachers Concept maps within a topic cluster can provide learner with overview of what (s)he has learned and what the connections are Preferably, students can draw their own maps after each topic cluster Functions Average rates of change Graphs Intuitive understanding of tangent as instantaneous rates of change Derivative Limits Connections across strands Across domains of Number, Algebra Algebra as generalised arithmetic, to provide a language to articulate rule Letters as pattern generaliser Algebra as processes to solve problems Letters as unknowns Sec teachers to link algebra to model method Number Across domains Algebra and Geometry Algebra as a language to articulate relationships Letters as variables Cartesian geometry expresses relationships in terms of points in space across dimensions Use of Algebraic processes to solve equations in geometry, trigonometry – meaning of solution as values that satisfy the relationship which is expressed as one or a set of equations Across Data, Algebra and “Geometry” Algebraic language used to express relationships from data Visual representation to illustrate data relationships in a different mode e.g scatter plots Connections to Reality The applicability of Mathematics Usefulness of basic mathematics is obvious for maths at primary level but not secondary Applications are at higher levels beyond what the secondary students see and they regard attempts to apply as impractical, unnecessary or irrelevant to their lives May need to go for novelty effect or link to what they are interested in e.g sports, how points are awarded in games and how these are linked to strategies, mathematics in nature Going broader – Points to watch out for Reality check – complexity of real life, nonlinearity of relationships, information gathering, extraneous information, different perspectives Using data to make informed choices, decision making skills from multiple perspectives should be encouraged Letting students choose their own problems, which could depend on the school context, latest fashion Work with teachers of other subjects for cross-discipline projects Going deeper – beyond syllabus Tickle their mathematical fancy – go further, beyond curriculum demands e.g a result has been established in a 2-dimensional plane; what about in 3-dimensions or on a curved surface? Challenge them to use deeper mathematics e.g Calculus for growth models Start with a problem and allow students to find the mathematics necessary (which they may not have learned yet or may never learn) Thinking skills fostered/encouraged Hypothesising Checking hypothesis Making inferences Explaining Convincing/justifying (not just mathematical proofs) Giving examples Making generalisations Conclusion Content-wise Connections are the essence of mathematical structures Pedagogy-wise Connections enhance learning (better and deeper learning) ...Overview Mathematical Thinking and Connections in the Singapore Mathematics Curriculum Connections within mathematics strands Connections across strands Connections with the real world Mathematical Thinking. .. connections among mathematical ideas, and between mathematics and other disciplines Do these sound familiar? Mathematical Thinking and Making Connections Students should use various thinking skills... him/her what motivates you to encourage those thinking skills or what value you see in encouraging those mathematical thinking skills Mathematical thinking skills are the essence of mathematics