- 1. 1 MATHEMATICAL SKILLS I) ARITHMETIC SKILLS Arithmetic “Mathematics is the queen of sciences and Arithmetic is the queen of Mathematics“ . Greek word The science of numbers and art of computing Oldest branch of the subject Early Math Skills
- 2. 2 Before starting school, most children develop an understanding of addition and subtraction through everyday interactions. Key Math Skills for School • Number Sense (forward and backward) • Representation (mathematical ideas using words, pictures, symbols, and objects ) • Spatial sense (Geometry, ideas of shape, size, space, position, direction and movement.) • Measurement (finding the length, height, and weight of an object ) • Estimation (more, less, bigger, smaller, more than, less than) • Patterns (learn to make predictions, to understand what comes next, to make logical connections ) • Problem-solving (using past knowledge and logical thinking skills to find an answer )
- 3. 3 Top 4 Basic Math Skills Students Should Learn • Problem solving (to develop analytical thinking) • Applied Math (applying math in everyday situations) • Estimation and approximation (to use almost every day) • Necessary computational skills Fluency • Math fluency is defined as the “ability to recall basic math facts.” • By enhancing fluency, one can develop more confidence and less dependence. Tips for speed and accuracy • Do speed addition and subtraction tests (50 problems in 3 minutes) • Memorize multiplicationtables up through 15 (extend to 15)
- 4. 4 • Recall your multiplicationtables for division • Simplify calculations into smaller,easier calculations Fundamental Arithmetic Ideas • Addition • Subtraction • Multiplication • Division • Fractions Latin word ‘break’ • Decimal Fractions Simon Stevin,Belgium,1584 Decimal point by Napier,1617 • Ratio • Proportion
- 5. 5 • Percentage • Simple interest • Compound interest • Area • Square /Rectangle • Parallelogram • Triangle • Cross roads • Circular roads • Circle • Volume • Cylinder
- 6. 6 II) GeometricSkills Geometry Greek word ‘earth measurement’ The science of lines and figures (or) The science of space and extent Deals with the position ,shape and size of bodies. It is the picturized arithmetic / algebra 3 stages of teaching Geometry • Practical Stage common geometrical concept • Stage of Reasoning to prove theorems and exercises • Systematic Stage acquisition of mastery in reasoning
- 7. 7 Traits of mathematically able children Ability to make and use generalizations often quite quickly. Rapid and sound memorization of mathematical material. Ability to concentrate on mathematics for long periods. An instinctive tendency to approach a problem in different ways Ability to detect unstated assumptions in a problem.
- 8. 8 III) Interpretationof graphs and charts Graphs Graphs are especially useful for presenting quantitative data. A graph is a visual form of data from a table. Graphs are ideal for communicating scientific information. A graph can make it easier to analyze and interpret the information you have collected. Features of a graph • title • grid • horizontal axis or X-axis • vertical axis or Y-axis
- 9. 9 Differentkinds of graphs 1)Line graphs To show how one variable affects another. Line graphs are useful in that they show the relationship between two variables. 2)Bar and column graph Both used to show categories of data that has been counted. In a column graph, the height of the column shows the number of individuals. In a bar graph, the length of the horizontal bar represents the number of individuals. 3)Histograms 1) Histograms look similar to column graphs but are different because each column represents a group of related data. 2) The height of the column shows the number of individuals counted, like a column graph.
- 10. 10 4) Pie graphs 1) Pie graphs are useful for showing the percentage composition of various categories that are unaffected by each other. 2) Computer programs can present pie graphs in many different, interesting shapes 5)Scattergrams 1) Scattergrams are graphs that are used to find patterns in some kinds of data. 2) The information about each individual or test is plotted as a separate point. Choosing a graph type Each time you need to draw a graph, think about the different kinds of graphs. Decide which kind of graph will make your data the easiest to analyses and interpret. Sometimes there are several kinds of graphs that would be suitable.
- 11. 11 For example, Usually you can use a bar graph, a column graph or a pie graph to present the same kinds of information. A line graph and a histogram can show the same data.
- 12. 12 IV) H O T Skills ‘H O T Skills ‘ - Higher Order Thinking Skills History Human thinking skills can be classified into two major groups: Lower order thinking skills (LOTS) Higher order thinking skills (HOTS). LOTS have three aspects Remembering Understanding Applying. (The first 3 of Bloom’s taxonomy) HOTS have three aspects Analyzing Evaluating Creating. (The last 3 of Bloom’s taxonomy )
- 13. 13 Why teach Higher Order Thinking Skills(HOTS)? • “Higher-order thinking skills are valued because they are believed to better prepare students for the challenges of adult work and daily life and advanced academic work. “ - Pogrow • Higher-order thinking may also help raise standardized test scores. • The revised secondary mathematics curriculum has shifted its emphasis to the fostering of HOTS. Higher Order Thinking Skills • Higher-order thinking is thinking on a higher level than memorizing facts or telling something back to someone exactly the way that it was told to you. • HOTS is the highest part in Bloom’s taxonomy of cognitive domain.
- 14. 14 Five fundamental HOTS 1.Problem solving skills 2. Inquiring skills 3. Reasoning skills 4. Communicating skills 5. Conceptualizing skills. 1.Problem solving skills • According to National Council of Teachers of Mathematics (NCTM), “ problem solving is a process of applying previously acquired knowledge to new and unfamiliar (or unforeseen) situations.” • Four phases are , - understanding the problem - devising a plan of solving the problem - carrying out the plan - examining the reasonableness of the result and making evaluation.
- 15. 15 2. Inquiring skills • Inquiring involves discovering or constructing knowledge through questioning or testing a hypothesis. • The essential elements are, Observation analysis summarizing verification 3. Communicating Skills • Communication involves receiving and sharing ideas and can be expressed in the forms of numbers, symbols, diagrams, graphs, charts, models and simulations. • 2 types : oral and written • 3 stages : construct refine
- 16. 16 consolidate 4. Reasoning Skills • Reasoning is drawing conclusions from evidence, grounds or assumptions. • 2 types : inductive reasoning deductive reasoning 5. Conceptualizing Skills • Conceptualizing involves organizing and reorganizing of knowledge through perceiving and thinking about particular experiences in order to abstract patterns and ideas and generalize from the particular experiences. The formation of concepts involves classifying and abstracting of previous experiences Three components in HOTS (1) critical thinking skills (2) creative thinking skills (3) systems thinking skills.
- 17. 17 Nine factors that comprise HOTS • the use of mathematical concepts • the use of mathematical principles • impact predicting • problem solving • decision-making • working in the limits of competence • trying new things • divergent thinking • imaginative thinking. Uses of Higher-order thinking • to manipulate information and ideas • to synthesis, generalize, explain, hypothesis (or) arrive at some conclusion or interpretation. • to solve problems and discover new meanings and understandings. • i.e., the teacher is not certain what will be produced by students.
- 18. 18 Note the following 1)There is no simple, clear and universally accepted definition of HOTS. 2) The five HOTS cannot be easily isolated from each other in mathematicalwork. 3) HOTS can be taught in isolation from specific contents, but incorporating them into content areas seems to be a popular way of teaching these skills. 4) Computer provides an excellent tool for teaching HOTS Questions and investigations • Closed question - 6 + 4 = ? • Open question - What numbers could you add together to make 10? • Investigation - There are chickens and calves in the farmers paddock. If there are 24 legs altogether, how many of each animal could there be?
- 19. 19 HOT and investigation Example : Noah counted 24 legs as the animals walked into the ark. What types of animals might they have been?
- 20. 20 Levels of thinking Learning of rules and facts Applying rules and formulae in standard situations Solving specific problems in novel situations Investigating issues and problem situations Relationshipbetween HOTS and student performance in Mathematics • There is a linear, positive and strong relationship between HOTS and the GPA of students. • Students with high level of HOTS are expected to succeed in their next study in study program of mathematics education. • Students who have high HOTS tend to get high GPA in mathematics instruction. • Therefore, the value of HOTS can be used as an indicator in the selection of new students.
- 21. 21 • In order to thrive in learning mathematics, students should have high level of HOTS. To improve student’s HOTS • To improve student’s HOTS , revise textbooks used in mathematics learning in primary and secondary schools. • The mathematics textbooks used in Indonesia should promote student’s critical and creative thinking. • Examples and practice tests provided should be able to train students to think critically and creatively by using open-ended test. • The open-ended test is a test used as an instrument in this study to measure students' HOTS
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