qitep 8 oct 2014 problem solving mathematics

Endah Retnowati e.retno@uny.ac.id

 _Problem

 _{Problem solving}

 _{Problem based learning}

 _{Heuristic plan/strategy or heuristics}

 How students solve problem

 _{The problem with PBL}

 _{Where is my name tag?}

 _{What is her name?}

 _{I hope today I will be learning new material.}

What could it be?

 Can I have my lunch break extended 15

minutes?

 _{I could not access the internet.}

A question is categorised a question as a problem if it cannot be answered

immediately using knowledge available at

Suppose five days after the day before

(1) the process: a set of activities, and

an ill-defined problem has multiple acceptable

products and many possible ways for reaching them,

a well-defined problem has only one possible

product and one agreed process for reaching

 _{Write a mathematics problem solving on a}

1. Understand the problem

2. Devise a plan

3. Carry out the plan

1. Draw a diagram if at all possible

2. If there is an integer parameter, look for an

inductive argument

3. Consider arguing by contrapositive or

contradiction

4. Consider a similar problem with fewer

 _{Contrapositive:}

 )nstead of proving the statement )f X is true, then Y is

true, you can prove the equivalent statement )f Y is false, then X must be false

 _{Contradictions:}

 Assume, for the sake of argument, that the statement you would like to prove is false. Using this

assumption, go on to prove either that one of the given conditions in the problem is false, that

(1) identifying the problem,

(2) representing the problem,

(3) selecting an appropriate strategy,

(4) implementing the strategy, and

 _{What is given}

 _{What is unknown}

 _{What operations are allowed}

 _{Determine a general course of attack}

 _{Restate the problem (the unknown) so that it}

is more like a familiar problem

 _{Carry out the computations}

 _{Look over the process you went through}

 _{Try to see how this experience can be helpful}

1. Read and think

2. Explore and plan

3. Select a strategy

4. Find an answer

 _{Identify the fact}

 _{Identify the question}

 _{Visualise the situation}

 _{Describe the setting}

 _{Organise the information}

 _{Is there sufficient information}

 _{Is there too much information}

 _{Draw a diagram or construct a model}

 _{Pattern recognition}

 _{Working backwards}

 _{Guess and test}

 _{Simulation or experimentation}

 reduction./expansion

 _{Organised listing/exhaustive listing}

 _{Logical deduction}

 _Estimate

 _{Use computational skills}

 _{Use algebraic skills}

 _{Use geometric skills}

 _{Check your answer}

 Is the computation correct?

 Is the question answered?

 Is the answer reasonable?

 _{Find alternate solutions}  _{What is....?}

 _{Extend to either:}

 A generalisation or

 A mathematical concept

 _{Discuss the solutions}

Teaching component skills

general problem solving strategy

Teaching within specific domain

For Gilda s party, the (oagie (ouse prepared a

 _{Pattern recognition}

 _{Working backwards}

 _{Guess and test}

 _{Simulation or experimentation}

 reduction./expansion

 _{Organised listing/exhaustive listing}

 _{Logical deduction}

 _{Problem 4: Two points on the surface of the}

unit sphere (in 3-space) are connected by an arc A which passes through the interior of the sphere. Prove that if the length of A is less

than 2, then there is a hemisphere H which does not intersect A.

 _{Problem 5: Let a, b, and c be positive real}

numbers. Show that not all three of the terms

 _{Schema construction and automation}

 _{Transfer knowledge to new/advanced}

The sequence of external events including the organisation and content of instructional

materials and behaviors of the teacher.

The learner s existing knowledge, including

facts, procedures and strategies that may be required in the learning situation

The nature of the learner s memory system,

 _{Learning outcomes}

 The cognitive changes in the learner s knowledge or

memory system, including newly acquired facts, procedures and strategies

 _{Outcome performance}

 The learner s performance & behaviour on tests that

 _{Learner centered approach}

 Learner characteristics

 Learning process and outcomes

 Instructional manipulations affects changes in the

learner s knowledge

 Learner is an active information processor

 Knowledge is constructed by learner

SELECTING

PATTERN RECOGNITION

LEARNING

 _{ATTENTION AND PERCEPTION}

 _{PATTERN RECOGNITION}

SELECTING

PATTERN RECOGNITION

LEARNING PROCESS:

 _{The assignment of meaning to incoming}

stimuli

 Is the detection of incoming stimuli by your senses

 Is the process by which stimuli are perceived, recognised

Detection of a stimuli through senses

Storage of some representation of the stimuli in memory system

Pattern recognition

Detection of a stimuli through senses: stimulus may be seen/heard

Storage of some representation of the stimuli in memory system: stored these into icon/echo

Pattern recognition: circle/writing/sound

information from LTM used to recognise pattern

 _{Nature of stimulus (context of stimulus)}

 _{Background of knowledge}

 _{Pattern recognition occurs when elements}

SELECTING

PATTERN RECOGNITION

LEARNING PROCESS:

Allocation of cognitive resources to a task

Critical for learning – to process information

learners have to pay attention

BUT

(uman s have extremely limited processing

 _{Generally people cannot attend to more a}

few things at once

 _{Under many conditions multi-tasking is not}

very effective because attention is divided

too much, leading to poor executions of tasks (divided attention)

 _{Automation of skills can compesate for}

 _{The type of TASK influences attention}

allocation

 Nature of task

 Nature of need

 Motivations

 _{Attention is allocated differently according to}

RESOURCE LIMITED

A task where

performance will improve if more

attention is shifted to that task

*CONCENTRATION

DATA-LIMITED

Performance is limited by the quality of the

presented task

Some tasks are so

 _{Students are easily distracted}

 Teacher is giving important explanations,

students mind starts to wander –tuning in to other conversations (sounds), looking out the window (visuals), thinking of other matters

internal cognition ….etc

 _{Prior knowledge used to make decisions about} the meaning of the stimuli

 _{When stimuli in the environment are recognised}

as something stored in memory

 _{Two systems for recognising paterns:}

 Parts to whole (Example?)

 Whole to parts (Example?)

 _{Directly affects perception process}

 _{Allows perception occurs}

SELECTING

PATTERN RECOGNITION

LEARNING PROCESS:

 _{Repeat a telephone number}

 _{Repeat an unfamiliar foreign word}

 12 + 13 = ?

 _{What have you been doing just before this?}

 _{Close your eyes and pick up an object in front}

of you.

 Limited in capacity

 Miller s research: the magic number of seven 7±2 chunks

of new meaningful information)

 Cowan s research: 4±1 chunks of new information to be processed

 _{Limited in duration}

 Recalls decay over time unless actively rehearsal occurs

 Information lost very rapidly when people are distracted from rehearsing

 _{Keep it simple, do not overload working}

memory

 _{Draw attention to most important points}

 _{Present information using different}

 Processing unlimited amount of

information that is already familiar.

SELECTING

PATTERN RECOGNITION

LEARNING PROCESS:

 _{Unconcious component of our memory}

 _{Unlimited capacity and duration}

 _{Where cognitive structures are organised}

 _{Explicit memory}

 Conscious recall, recognition of previous knowledge/information/experience

 Implicit memory

 No record of previously remembering events

 _{LTM is actively constructed using schemata}

 _{Activated schemata determine what}

incoming information is relevant

 _{Schemata are continually reconstructed}

 ASSIMILATION

 New information that fits into an existing schema is added

 ACCOMODATION

 _{Schema construction and automation}

 _{Transfer knowledge to new/advanced}

 _{Occurs without intension and conciousness}

 _{Less cognitive effort}

 _{Less error}

 _{Performance is quicker}

 Automated performance

 _{Develop learning} – _{to more difficult task}

 _{The amount of schema constructed in LTM}

Goal free problems discourage students from creating sub-goals and separate the problem state and the problem goal because the

problem goal is not given.

Instead, students are required to work forward

 _{Can you please give an example of goal-free}

 _{Anybody knows what it is?}

Learning from Examples

Examples:

(1) x2 + 5x + 6 = (x + 2)(x + 3) (2) x2 + 7x + 6 = (x + 1)(x + 6) (3) x2 _{+ 8x + 12 = (x + 2)(x + 6)}

(4) x2 + 7x + 12 = (x + 3)(x + 4) (5) x2 + 13x + 12 = (x + 1)(x + 12)

Exercises:

(1) x2 _{+ 11x + 18 = ( )( )}

Learning by Doing

Problems: Identical with examples for learning-from-examples section except that

right-hand sides of equations are shown as:

= (x + __ ) (x + __ )

 _{What research says about this comparison?}

 _{WE only}

 _{Pairing WE and similar PS}

 _{Self-explanation effect}

 _{Split attention effect}

 Redundancy effect

 _{Modality effect}

 _{Process oriented or product oriented}

 _{Mutiple or uniform solutions}

A car moving from rest reaches a speed of 20 m/s after 10 seconds. What is the acceleration of the car?

u = 0 m/s

v = 20 m/s

t = 10 s

v = u + at a = (v – u)/t

 _{Practive improving problem solving based}