MTH14

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IT IS A STATEMENT ARRIVED AT INDUCTIVELY OR BASED ON EXISTING DATA. THEY MAY BE IN THE FORM OF A STATEMENT OR JUST A NUMBER.

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ONCE THE STUDENT HAS FAMILIARIZED HIMSELF WITH THE PROBLEM AND DECIDED A LINE OF ACTION TO FOLLOW, HE STARTS TO GENERATE EXAMPLES AND TO EXPLORE. HE CAN MAKE USE OF ANY STRATEGY LIKE DRAW A DIAGRAM, MAKE A MODEL OR MAKE AN ORGANIZED LIST.

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A situation can be considered a problem

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According to National __________, “_____________

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Doing mathematics involves

.

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It is the hallmark of mathematical activity and a major means of developing mathematical knowledge (Standards 2000, p. 116).

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It suggests that problem solving should be the

.

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When a statement is thought to be true but remains unproved, it is called

.

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It is the most important part of solving any problem.

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is a major pathway to discovery

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What are the stages in solving problems?

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They make the mistake of “doing something” before understanding (or at least thinking they understand) the problem.

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selected strategy or set of strategies.

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It is defined as the algorithmic and deductive approach to solving a problem

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In implementing the chosen strategies, one might not be able to solve the problem, but

.

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As he refines his understanding of the problem,

.

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This process begins with a clearly defined goal or objective after which there is a great reliance on relevant past experience, referred to as repertoire

.

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He defined design as the algorithmic and deductive approach to solving a problem

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He defined design to produce possible options that will lead towards a solution of the problem

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These options are then examined through a process of conscious evaluations (Dewey 1933) to determine their suitability for advancing the problem towards the

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It is the process of deducing the solution from that which is already known.

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is all that a solver has to draw on when first attacking a problem.

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They state that prior knowledge is a key element in the problem solving process.

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It influences the problem solver’s understanding of the problem as well as the choice of strategies that will be called upon in trying to solve the problem.

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What is George Polya's book?

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It incorporate the resource of past experiences and prior knowledge into their initial attack on a problem.

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One of the heuristics that refine, none is more influential than the one created by

.

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What are the four phases of Problem Solving?

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In his book How to Solve It (1949) Pólya discussed a

that relies heavily on a repertoire of past experience.

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It is a general problem-solving framework (Tversky & Kahneman, 1974).

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It is a problem-solving formula that provides you with step-by-step instructions used to achieve a desired outcome (Kahneman, 2011).

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He said that there are many logical ways to solve problems.

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It is a plan of action used to find a solution.

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It is another type of problem solving strategy.

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It is slightly more sophisticated than guess and check.

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It is a well-known strategy in solving problem.

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It must be followed exactly to produce a correct result

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It can help mathematicians to picture the problem and find the solution.

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Generally speaking, any object that can be used in some way to represent the situation the students are trying to solve, is

.

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These are particular types of diagrams that can be used in solving a problem.

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Students enjoy this type of strategy. They themselves take the role of things in the problem.

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When does Act It Out (and Use Equipment/Manipulatives) effective?

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The information that has been set out in the

will hopefully lead students to the correct solution.

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It is a strategy related to Act it Out.

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your materials to find solution to the problem.

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Using it is a good way to sort out and organize the information that has been given in the question.

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It is a strategy that will help students sort out the information that has been given in the problem.

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Once, the students can see

for the solution, they can then attempt to solve the problem more easily.

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These are the two aspects of working systematically. It helps students to bring a logical and systematic development to their mathematics if they begin to organize things systematically as they go.

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Once the

has been identified, the students can predict what will happen next and then continue the pattern to find the correct solution.

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Polya, (1957) suggested this strategy: “If you can’t solve a problem, then there is an easier problem you can solve: find it.” He also said: “If you cannot solve the proposed problem, try to solve first some related problem.

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In this strategy, students look for a

from the information that has been given in the problem.

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This strategy requires students to use the information they have been given in the question to eliminate possible solutions to finally discover the correct solution.

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It is a useful heuristic in which you begin solving the problem by focusing on the end result.

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It make things easier because they tell us how a group of objects acts in the same way. Once you see this, you have much more control over what you are doing.

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It is a pattern involving an ordered arrangement of numbers.

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It is an excellent strategy to use when the final outcome of the problem has already been given.

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It is very important to

of your work. Students have to know where they have been and where they are going or they will get hopelessly muddled.

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Being

may mean making a table or an organized list but it can also mean keeping your working in some order so that it is easy to follow when you have to go back over it.

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It means following an idea for a while to see where it leads, rather than jumping about all over the place chasing lots of possible ideas.

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Why Mathematical Investigation?

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Steps in Logical Reasoning

1. Organize your information

2. Examine relationships, and think about your information to form a logical answer.

3. Generalize each relationship to reach a logical conclusion.

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Steps in Make a Table, List or Chart

1. decide what information needs to go on your chart.

2. List all the possibilities in a table or chart to make sure pieces of the problem don't get lost.

3. Organize and record the information so it makes sense.

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Who says that real-life or life-like learnings that are open ended and provide opportunities for students to use multiple pathways to investigate the situation/problem.

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I is an approach to learning that is focused on the exploration of a topic in depth and make connections between different representations.

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real-life or life-like learnings that are

and provide opportunities for students to use multiple pathways to investigate the situation/problem.

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They stated that mathematical investigation is an approach to learning that is focused on the exploration of a topic in depth and make connections between different representations.

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involve a search for pattern and structure

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What is THE ESSENCE OF MATHEMATICAL INVESTIGATION?

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It may be framed variously as: a problem to be solved; a question to be answered; a significant task to be completed, or an issue to be explored

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IT IS A CAREFUL EXAMINATION OF A MATHEMATICAL TASK IN ORDER TO DISCOVER SOME UNDERLYING MATHEMATICAL FACTS OR STRUCTURES.

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EXPLAINED THAT “THE ESSENCE OF MATHEMATICAL INVESTIGATION IS TO INQUIRE INTO SITUATIONS WHICH ARE OF A MATHEMATICAL NATURE”.

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INCORPORATE A RANGE OF SKILLS AND CONCEPTS, POSSIBLY CONNECTING SEVERAL CONCEPTS PREVIOUSLY TAUGHT AS A CULMINATING ACTIVITY OR AS AN ONGOING TEACHING TOOL THROUGHOUT THE TERM THAT GRADUALLY SCAFFOLDS THE STUDENTS’ UNDERSTANDING.

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What are the two perspectives on the process in a mathematical investigation?

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IT CONSIST OF SIMPLE OPEN-ENDED QUESTIONS THAT REQUIRE THINKING OF MULTIPLE SOLUTIONS, OR IT COULD MEAN FINDING THE SAME SOLUTION USING DIFFERENT PATHWAYS.

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It is a style of teaching or learning that involves a pupil in the use of some of the features of mathematical investigations to learn certain facts in mathematics, to acquire certain skills and increase the pupils knowledge of certain concepts

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INVOLVE A SIMILAR OPEN-ENDED QUESTION BUT IN A MORE COMPLEX CONTEXT THAT MIGHT INVOLVE WORKING IN PAIRS OR GROUPS FOR A WHOLE LESSON OR THE BEST PART OF ONE.

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Poses problem, or chooses situation with goal in mind.

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Chooses starting situation (or approves students choice).

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It takes a real-world situation described in words and describes that situation using algebra.

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Poses Problem. Leaves method of solution open.

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Finds own way to solve problem.

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Defines own problems within situation. Attempts to solve them. (Ernest 1984)

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What are the 4 skills needed in mathematical investigation?

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FROM THE EXAMPLES HE GENERATED, HE COLLECTS NECESSARY DATA OR INFORMATION AND HE

THEM IN SUCH A WAY THAT HE CAN LOOK FOR PATTERNS. HE CAN MAKE USE OF A TABULAR PRESENTATION.

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It is a procedure, a description of a set of steps that can be used to solve a mathematical computation

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It is a diagram showing the relationship of quantities, especially such a diagram in which lines, bars, or proportional are as represent how one quantity depends on or changes with another.

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An examination of something to identify causes, key factors, relationships and possible results

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THIS CONSISTS OF CHECKING THE CONSISTENCY OF THE CONJECTURE AGAINST EXISTING CASES FOR WHICH DATA ARE AVAILABLE, OR PREDICTING RESULTS OF UNTRIED CASES AND THEN OBTAINING THE RELEVANT DATA. THE DATA MAY SUPPORT THE CONJECTURE OR PROVIDE A COUNTER-EXAMPLE INDICATING THE NEED TO REVISE OR REJECT THE CONJECTURE.

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sequences of methods and mathematical operations& procedures carried out in order to arrive at specific objectives.

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What are the 4 skills developed in mathematical investigation?

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THE STUDENT MAY TAKE A BREAK WHEN HIS MIND GETS TIRED, HE FEELS THERE IS NO PROGRESS IN HIS INVESTIGATION OR HE CANNOT SEE A WAY TO CONTINUE. THIS IS THE BEST TIME FOR HIM TO RELAX, TO GATHER OTHER RELEVANT INFORMATION OR TO THINK OF OTHER LINE OF ACTIONS NECESSARY TO PURSUE THE INVESTIGATION. THERE MAY BE SEVERAL SUCH BREAKS DURING THE INVESTIGATION DEPENDING ON THE MODE AND ABILITY OF THE STUDENT.

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It used to assess the credibility of the claims people make or post and to assess the quality of the reasoning people display when they make arguments or give explanations

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DEPENDING ON THE ABILITY OF THE STUDENT, THE INVESTIGATION CAN BE EXTENDED BY CONSIDERING OTHER RELEVANT SITUATIONS THAT START WITH, “WHAT IF .” OR “WHAT WOULD HAPPEN IF”

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It build relation in every element by relating information which has been known with accumulation, multiplication, division, and subtraction operation.

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WHILE DATA ARE ALREADY GENERATED AND ORGANIZED, THE STUDENT MAY NOTICE SOME PATTERNS OR RELATIONSHIPS. THESE PATTERNS SUGGEST GENERALIZATION WHICH APPEAR TO APPLY TO OTHER CASES UNDER CONSIDERATION.

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Any process of arranging items systematically.

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ONCE A CONJECTURE HAS BEEN TESTED AGAINST AND SUPPORTED BY THE DATA, STUDENTS SHOULD BE ENCOURAGE TO _______ WHY THE CONJECTURE HOLDS FOR NEW CASES. SUCH AN _________ SHOULD CONSIST OF MORE THAN THE OBSERVATION THAT THE CONJECTURE HAS HELD FOR THE CASES EXAMINED SO FAR. AS STUDENTS BECOME MORE SOPHISTICATED, IT MAY ALSO BE POSSIBLE FOR SOME STUDENTS TO PROVIDE DEDUCTIVE JUSTIFICATION, OR A FORMAL PROOF, FOR A GENERALIZATION.

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AS THE INVESTIGATION PROGRESSES, IT MAY BECOME NECESSARY TO

THE APPROACH IN ORDER TO MAKE IT MORE SIMPLIFIED AND MORE SYSTEMATIC OR MORE GENERAL OR OTHERWISE IMPROVED SO THAT IT IS EASIER FOR THE STUDENT TO MAKE CONJECTURES. THIS CAN ALSO BE DONE AFTER TESTING THE CONJECTURE ESPECIALLY IF IT DOES NOT HOLD TRUE FOR SOME OTHER CASES.

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AT THIS STAGE THE STUDENT IS ENCOURAGED TO WRITE A IT OR A BRIEF ACCOUNT OF THE INVESTIGATION. THIS HIGHLIGHTS THE MAJOR IDEAS AND THE PHASES OF THE INVESTIGATION INCLUDING ALL CONJECTURES, TESTS AND PROOFS CARRIED OUT

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THESE INVESTIGATIONS ARE CHARACTERIZED BY BEING ABLE TO ACHIEVE SOME SUCCESS THROUGHOUT THE INVESTIGATION. IN OTHER WORDS, SUCCESS IS NOT DEPENDENT ON A VITAL BREAKTHROUGH.

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IMPORTANCE OF MATHEMATICAL INVESTIGATION

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THESE ARE PROBLEMS WHICH ARE RELATED TO REAL PROBLEMS (I.E. PROBLEMS ORIGINATING OUTSIDE MATHEMATICS), BUT WHICH EITHER HAVE ALREADY BEEN TURNED INTO A MATHEMATICAL PROBLEM OR IT IS AT LEAST CLEAR WHAT THE MATHEMATICAL PROBLEM INVOLVES. 9 THERE IS NO ELEMENT OF CRITICAL JUDGEMENT IN DECIDING WHAT FACTORS SHOULD BE INCLUDED IN THE PROBLEMS OR WHAT WEIGHTING SHOULD BE GIVEN TO THE FACTORS.

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IN THIS CATEGORY, WE ARE DEALING WITH PROBLEMS OF DIRECT RELEVANCE. IN THE FORM IN WHICH THEY ARE POSED THEY HAVE LITTLE TO DO WITH MATHEMATICS AND MATHEMATICAL ANALYSIS MAY OR MAY NOT HELP TO SOLVE THEM. 8 INDEED, OFTEN THE MOST DIFFICULT PART OF THE INVESTIGATION IS TO TRANSLATE THE REAL PROBLEM INTO A MATHEMATICAL ONE, I.E. THE FORMULATION OF A MATHEMATICAL MODEL.

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THIS TYPE OF INVESTIGATION IS CHARACTERIZED BY HAVING A 'BREAKTHROUGH' TO SOLVE THE PROBLEM. USUALLY LITTLE USEFUL PROGRESS CAN BE MADE UNTIL THE BREAKTHROUGH HAS BEEN MADE _ THE SOLUTION THEN BECOMES OBVIOUS. (D. BURGHES 1984)

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MI CAUSES

OF CONCEPTS LEARNED

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AT THIS STAGE THE STUDENT NEEDS TO ATTAIN FAMILIARITY WITH THE PROBLEM OR SITUATION. HE STARTS FROM THE SIMPLEST OR WHATEVER IS INTERESTING, OFTEN IN AN UNORGANIZED WAY. DIFFERENT WAYS OF INTERPRETING THE PROBLEM MAY BE EXPLORED, ONE OR MORE PROBLEMS OR ASPECTS INHERENT IN THE SITUATION MAY BE IDENTIFIED AND ONE OR MORE PRODUCTIVE LINES OF ACTION MAY EMERGE.