Developing cognitive skills in the course of mathematics

Alla StolyarevskaThe East-Ukrainian Branch of the International

Solomon University, Kharkov, Ukraine

© Stolyarevska A., 2010

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The annotation

The role of student in educational process has been changed: the weight of educational process transfers from teacher to student. That’s why it is necessary to strengthen the functions of students’ assistant, to help students in organization of the individual learning process.

Therefore, one of the ways of determining the course of mathematics is to define it in terms of intended student behavior.

At this stage we can use the Bloom's Taxonomy, which provides a structured presentation of human cognition from low-level thought processes like simple recall to higher-order thinking skills like synthesis and evaluation.

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The main objectives

Audience: the students of the Computer Science faculty at the Eastern-Ukraine branch of the International Solomon University, Kharkov

Curriculum: the course of «Operations Research and Mathematical Programming»

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Contents

The main aspects of the learning theory The Bloom’s taxonomy The essential elements of a lesson plan The behavioral verbs The examples of activities The question frame Conclusions

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The learning theory

To learn is to acquire knowledge or skill. Learning also may involve a change in attitude or behavior. Learning theory may be described as a body of principles advocated by psychologists and educators to explain how people acquire skills, knowledge, and attitudes.

Various branches of learning theory are used in formal training programs to improve and accelerate the learning process. Key concepts such as desired learning outcomes, objectives of the training, and depth of training may be applied.

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The cognitive theory

Much of the recent psychological thinking and experimentation in education includes some aspects of the cognitive theory. The cognitive theory focuses on what is going on inside the student's mind.

The cognitive theory acknowledges the importance of reinforcing behavior and measuring changes. Positive reinforcement is important, particularly with cognitive concepts such as knowledge and understanding.

The only way to get a clue about what the student understands is to measure behavior that remains.

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Learning is much greater than teaching

“No man can be a good teacher unless he has feelings of warm affection toward his pupils and a genuine desire

to impart to them what he himself believes to be of value”. Bertrand Russell

Learning is the process of struggling to make the outside reality (objects, truths, ideas, concepts, principles, etc.) a part of the way you see the world (perception), reflect it and influence it (skills), which helps you to develop personally.

We can not teach, we can only create the conditions in which students can learn.

We should do our best motivating our students to learn as much as possible and try to make sure that learning does take place.

The more learning our students have, the better teachers we are.

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The goals of learning

There are three main goals of learning: give knowledge, develop a skill, prepare students for the first two.

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The important questions

Do you keep these goals in mind before/during the lesson?

How much of your lesson time do you devote to each of them?

Even if you have the goal, do you choose the adequate way of achieving it?

Do you assess the result of your teaching? And if it’s not the desired one, who do you hold

responsible, and what action do you take?

These questions are very important, because though we are teachers, our goal is not teaching, but learning (on the students’ side).

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Background In the late 1950s into the early 1970s in the USA there

were attempts to dissect and classify the varied domains of human learning - cognitive (knowing, head), affective (feeling, heart) and psychomotor (doing, hand/body).

The resulting efforts yielded a series of taxonomies in each area.

A taxonomy is really just a word for a form of classification. The taxonomies deal with the varied aspects of human learning and are arranged hierarchically proceeding from the simplest functions to those that are more complex.

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Bloom's taxonomy

Professor Benjamin Bloom of Chicago University and co-workers that met from 1948 to 1953, devised a stairway with six steps to learning.

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Levels of learning

The cognitive taxonomy is predicated on the idea that cognitive operations can be ordered into six increasingly complex levels.

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Mastery learning

Benjamin Bloom recognized that what was important in education was not that students should be compared with each other, but that they should be helped to achieve the goals of the curriculum they were studying. The goal attainment was more important.

Mastery learning was an expression of what Bloom believed to be an optimistic approach to the realization of educational goals.

Mastery Learning was initiated in 1963 by John B. Carroll*. Mastery learning is fit for both individual study and group study.

*John Bissell Carroll, an American psychologist, known for his contributions to psychology, educational linguistics and psychometrics.

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Mastery learning

It has been said that 90% of a learning population can master a subject when mastery learning methods are implemented.

Mastery learning rests on "Take your time", that is, take the time you need to learn something well. Time to learn must be adjusted to fit aptitude. No student is to proceed to new material until basic prerequisite material is mastered.

Bloom suggests a pre-test and review at the beginning of a semester of the essential, basic facts, skills, concepts that are necessary to later success.

And at the end of an instructional unit - every two weeks - a test to find out what has been learned.

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Bloom’s revised taxonomy

For the next 40 years, the application of Bloom’s work found its way into many instructional disciplines.

In 2000, Anderson and Krathwohl revised the taxonomy to make the model more appropriate to current audiences (slide No 16)

In 2002, Barbara Clark, a researcher in educational practices of the gifted, adapted the revised taxonomy into the circular graphic (slide No 17).

Anderson, L W, & Krathwohl D R (eds.) (2001). A Taxonomy for Learning, Teaching, and Assessing: A Revision of Bloom's Taxonomy of Educational Objectives. New York: Longman

Clark Barbara (1986). Optimizing Learning: The Integrative Education Model in the Classroom. Merrill Publishing Co.

16The revised taxonomy

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Behavioral verbs

Behavioral verbs are the heart of learning objectives, which are in turn the core component of effective lesson plans.

If defined and used consistently, they are a highly effective way to indicate, and communicate to others, specific, observable student behavior.

Behavioral verbs describe an observable product or action.

Teachers constantly make inferences about student learning on the basis of what a student does or produces.

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Behavioral verbs

It follows then, that one way to define curriculum is in terms of intended student behavior.

Learning objectives based on a set of verbs that have some measure of agreement as to meaning can provide a useful vehicle for the purpose of developing performance-based curriculum.

Consistent use of defined behavioral verbs in composing, rewriting or selecting learning objectives can lead to improvement in efforts to change and reform education in general and curriculum in particular.

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Lesson planThere are only a few essential elements of a lesson plan:

Objectives - what students will be able to do as a result of the lesson

Standards - which state content and developmental standards are addressed in the lesson

Procedures - what the teacher will do to get the students there Assessment opportunities - what the teacher can do to see if the

lesson was taught effectively:  watching students work, assigning application activities, getting feedback, etc.  (Can include both formal and informal assessment and both formative and summative evaluations.)

Modifications/accommodations for any special needs of students in the class

Many lesson plans also include:      Materials needed for the class period and any special equipment      Time estimates     Procedural subpoints

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A Starting Point

(Seven-element format:  just one way to structure a lesson; developed for math classes)

Anticipatory Set (setting the stage) - attention-getter and focuser

Statement of Objectives - tell students what they'll be able to do as a result of the lesson

Instructional Input - lecture, but not necessarily lecture: demo, explanation, instructions

Modeling - demonstrate, show what you tell Check for Understanding - watch faces, ask questions Guided Practice - help students start practicing new skills,

applying new knowledge Independent Practice - turn them loose to work on their

own, homework assignment, etc.

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Course outlineThe linear programming problems

1 History of linear programming 2 Uses 3 Standard form 4 Augmented form (slack form) 5 Duality 6 Theory 6.1 Existence of optimal solutions 6.2 Optimal vertices (and rays) of polyhedra 7 Algorithms 7.1 The simplex algorithm of Dantzig 8 Open problems and recent work 9 Integer unknowns 10 Solvers and scripting (programming) languages

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Brief History The founders of the subject are Leonid Kantorovich, a

Russian mathematician who developed linear programming problems in 1939, George B. Dantzig, who published the simplex method in 1947, and John von Neumann, who developed the theory of the duality in the same year.

The linear programming problem was first shown to be solvable in polynomial time by Leonid Khachiyan in 1979, but a larger theoretical and practical breakthrough in the field came in 1984 when Narendra Karmarkar introduced a new interior point method for solving linear programming problems.

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SoftwareSoftware for this course platform: Maple solver: Excel modeling tools: C++

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The linear programming problem

Minimize Z=15x1+10x2

subject to

0,0

4

3

2

21

12

2

1

xx

xx

x

x

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Examples of Activities

The following examples of student activities are meant to illustrate the uses of the defined behavioral verbs in classroom settings involving the specific subject content areas.

The purpose of these examples is to clarify the meaning of the definitions of behavioral verbs.

With a clear knowledge of the meaning of these verbs, a person should be able to classify the learning behavior of any student he observes, whether or not he knows the learning objective.

It's all about understanding what a student is doing that shows the intended behavior.

For example, a person who observes a student pointing out on a chart of atomic diagrams, the diagrams that represent elements named by the teacher, will be able to classify the behavior as identifying.

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Examples of Activities

Apply a Rule: Construct the contour lines of different values for a given objective function.

Classify: The objective function and the range of permissible values are given. Classify the contour lines.

Define: Give a definition of a linear programming problem.

Demonstrate: Demonstrate that the objective function for the linear programming problem is a set of straight lines.

Describe: Describe the method for finding the maximum and minimum values of objective function under given constraints.

Diagram: Give a geometric interpretation of linear programming problem.

Distinguish: Distinguish between linear and nonlinear objective function.

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Examples of Activities

Estimate: Evaluate the value of the maximum and minimum of the objective function as the points of extremum.

Identify: Specify the half-plane corresponding to the inequality.

Interpret: Interpret a figure with a linear programming problem.

Name: Name the possible meanings area for the linear programming problem.

Predict: Predict the answer if the objective function has the given direction.

Solve: Solve the linear programming problem State a Rule: Why does an inequality Ax<=B correspond

to the left (right) half-plane?

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Question Frames for Developing Higher-Level Questions

Recall:What isDefineIdentify theWho didAnalysis:What is the main idea ofList the main events ofWhat are the parts ofWhat is the topic ofComparison:What is the difference between___and_______Inference:What do you think will happen next in theWhat is the main conclusion fromPredict what ___________-will doWhat would happen ifEvaluation:What is your opinion ofWhat is the best solution of the problem ofEvaluate the writing ofDefend your opinion about

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Question Frames

Recall:What is application of linear programming?Analysis:What do you understand by linear programming problem? What are the requirements of linear programming problem? What are the basic assumptions of linear programming problem? What are the limitations of linear programming?What is the feasible region in linear programming?What is the main idea of simplex method?Can a linear programming problem have multiple optimal

solution?Comparison:What are advantages and disadvantages of linear programming?Evaluation:Can you give me some example for mathematical formulation of

linear programming problems?

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Maple Worksheets Used in Interactive Operations Research with Maple. The author is Mahmut Parlar

http://www.business.mcmaster.ca/OM/parlar/ORMapleBook/download.htm#R51C6http://www.business.mcmaster.ca/OM/parlar/

The linear programming problems:

AircraftIP.mws MixedConstraintsDual.mws AircraftLP.mws RandomLP.mws BasicSols.mws ToyLP.mws CoinTossDual.mws ToyLPCosttheta1.mws CoinTossGame.mws ToyLPManualSimplex.mws degeneracy.mws ToyLPNewVar.mws DualFormula.mws ToyLPRHS70.mws FurnitureDual.mws ToyLPRHSDelta2.mws infeasible.mws ToyLPSimplex.mws MarketingGame.mws transportation.mws MixedConstraints.mws unbounded.mws

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Interactive Operations Research with Maple

Mahmut Parlar is the author of the book Interactive Operations Research with Maple: Methods and Models, Birkhauser, Boston (2000), that describes the effective use of the computer algebra system Maple in operations research modelling.

“The primary focus of the book is on how to use Maple in solving various operations research models. Chapter 4 deals with linear programming. Sensitivity and post-optimality analysis and special linear programs such as the transportation problem and two-person games are discussed. A brief introduction to integer linear programs is also included in this chapter. The book is a very valuable supplement for operations research courses, especially those aimed at students with good mathematical background”.

Abraham Punnen

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Conclusions The main parts of the course were the following: The teacher had to define the goal of the course, the

way to achieve this goal, the method of material representation, the training methods, the types of problems and the questions for discussion, the ways of discussion organizing, the mode of communication.

One of the productive means in the course creation was its defining in terms describing students’ behavior. At this stage the Bloom's Taxonomy was used.

The students with no exposure to Maple could quickly gain sufficient expertise in solving many operations research problems.

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Thank You for attention!