Ali bin Abi Thalib’s (ra) Mathematical Brilliance

Imam Ali bin Abi Thalib (karramallahu wajhah) was endowed with a quick, sharp, incisive, mathematical mind. Here are a few interesting stories in which Imam Ali’s mathematical brilliance revealed itself.

Whole Number and not a Fraction

One Day a Jewish person came to Imam Ali (krw), thinking that since Imam Ali thinks he is too smart, I’ll ask him such a tough question that he won’t be able to answer it and I’ll have the chance to embarrass him in front of all the Arabs.

He asked “Imam Ali, tell me a number, that if we divide it by any number from 1-10 the answer will always come in the form of a whole number and not as a fraction.”

Imam Ali (krw.) looked back at him and said, “Take the number of days in a year and multiply it with the number of days in a week and you will have your answer.”

The Jewish person got astonished but as he was a polytheist (Mushrik), he still didn’t believe Imam Ali (krw.). He calculated the answer Imam Ali (krw.) gave him.

To his amazement he came across the following results:

- The number of Days in a Year = 360 (in Arab)

- The Number of Days in a Week = 7

- The product of the two numbers = 360×7=2520

Now…

2520 ÷ 1 = 2520

2520 ÷ 2 = 1260

2520 ÷ 3 = 840

2520 ÷ 4 = 630

2520 ÷ 5 = 504

2520 ÷ 6 = 420

2520 ÷ 7 = 360

2520 ÷ 8 = 315

2520 ÷ 9 = 280

2520 ÷ 10= 252

The Five Loaves of Bread

Zarr Bin Hobeish relates this story: Two travelers sat together on the way to their destination to have a meal. One had five loaves of bread. The other had three. A third traveler was passing by and at the request of the two joined in the meal. The travelers cut each of the loaf of bread in three equal parts. Each of the travelers ate eight broken pieces of the loaf.

At the time of leaving the third traveler took out eight dirhams and gave to the first two men who had offered him the meal, and went away. On receiving the money the two travelers started quarrelling as to who should have how much of the money. The five-loaf-man demanded five dirhams. The three-loaf-man insisted on dividing the money in two equal parts (4 dirhams each).

The dispute was brought to Imam Ali (krw.) (the Caliph of the time in Arabia) to be decided.

Imam Ali (krw.) requested the three-loaf-man to accept three dirhams, because five-loaf-man has been more than fair to you. The three-loaf-man refused and said that he would take only four dirhams.

At this Imam Ali (krw.) replied, “You can have only one dirham. You had eight loaves between yourselves. Each loaf was broken in three parts. Therefore, you had 24 equal parts. 8×3=24

Your 3 loaves made 9 parts out of which you have eaten 8 portions, leaving just 1 piece to the third traveler. (3×3)-8=1

Your friend had 5 loaves which divided into 3 made 15 pieces. He ate 8 pieces and gave 7 pieces to the guest.(5×3)-8=7

As such the guest shared 1 part from your loaves and 7 from those of your friend. So you should get one dirham and your friend should receive seven dirhams.”

Dividing Inheritance (Warits)

What is a wife’s share?

Imam Ali (krw) was once interrupted while he was delivering a sermon from the pulpit by someone who asked him how to distribute the inheritance of someone who had died leaving a wife, his parents and two daughters. The Imam instantly answered:

“The wife’s share becomes one ninth.”

How?

This answer is in fact the result of a long analysis with a number of steps. Ordinarily, we have to decide on the original share of each of these heirs, in the following way:

The wife takes one eighth, in view of the presence of an inheriting child. [Holy Quran 4:12]

The deceased’s father and mother take one sixth each. [Holy Quran 4:11]

The two daughters take two thirds of the inheritance. [Holy Quran 4:11]

So the total will be: 1/8 + 1/6 + 1/6 + 2/3 = 3/24 + 4/24 + 4/24 + 16/24 = 27/24

This means the share becomes less than 1/8 in view of the increase of the total of the shares which are so fixed and prescribed. So the one eighth, the original share due to the wife out of twenty-four total shares, has become three shares out of a total of twenty-seven, which is one ninth.

Imam Ali’s mind went through this complex mathematical process in a second!

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Design Research from a Learning Design Perspective

Summary

EDUCATIONAL DESIGN RESEARCH

Chapter 3. Design Research from a Learning Design Perspective.

Koeno Gravemeijer and Paul Cobb


>  This chapter elaborates on an approach to design research that is categorized as falling within the broader category of design research that aims at: creating innovative learning ecologies in order to develop local instruction theories on the one hand, and to study the forms of learning that those learning ecologies are intended to support on the other hand.

> This approach has its roots in the history of the two authors: one has a background in socio-constructivist analysis of instruction. The socio-constructivist approach was inspired by a desire for understanding; the other has done work on realistic mathematics education (RME) that is carried out in the Netherlands. The RME approach is built because of a need for educational change.

> This chapter defines what design research is by discussing the three phases of conducting a design experiment: preparing for the experiment; experimenting in the classroom, and conducting retrospective analyses.

> This book uses an experiment on statistics to illustrate the various phases in a concrete design experiment.

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Introduction Educational Design Research

EDUCATIONAL DESIGN RESEARCH

Chapter 1. Introducing educational design research

Jan van den Akker, Koeno Gravemeijer,

Susan McKenney and Nienke Nieveen

Origins of this book

Design research has been gaining momentum in recent years, particularly in the field of educational studies. This is shown in the topics about this on prominent journal articles, book chapters, books, special issue of journals dedicated specifically to the topic, and the more general need to revisit research approaches, including design research. All of these give influence toward the definition of the approach that now start to solidify, but also to differentiate. The need of methodological guidelines and promising examples also begin to surface. Besides it, there is the educational research community to seriously reflect on setting standards that improve the quality of this approach. This book offers such a reflection. Most of its chapters are revised, updated, and elaborated versions of presentations given at a seminar held in Amsterdam, organized by NWO/ PROO.

Motives for design research

1. The desire to increase the relevance of research for educational policy and practice.

Educational research has long been criticized for its weak link with practice. This is especially criticized by those who view educational research as a vehicle to inform improvement. In order to increase more practical relevance of design research, researchers and practitioners construct increasingly work- able and effective interventions.  As a result the success of this work gives influence on improving policy.

2. Scientific ambitions.

Along- side directly practical applications and policy implications, design research aims at developing empirically grounded theories through combined study of both the process of learning and the means that support that process.  As the thrust to better understand learning and instruction in context grows, research must move from simulated or highly favorable settings toward more naturally occurring test beds.

3. The aspiration of increasing the robustness of design practice.

Many educational designers energetically approach the construction of innovative solutions to emerging educational problems, yet their understanding often times remains implicit in the decisions made and the resulting design. From this perspective, there is a need to extract more explicit learning that can advance subsequent design efforts.

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Variables and Hypotheses

Summary

HOW TO DESIGN AND EVALUATE RESEARCH IN EDUCATION

Fraenkel J.R. Wallen, N.E

Chapter 3: Variables and Hypotheses

THE IMPORTANCE OF STUDYING RELATIONSHIPS

-    Identifying relationships among variables enhances understanding where we may learn what happened, or where or when (and even how) something happened, not only why it happened.

-    Understanding of relationships helps us to explain the nature of the world in which we live and how parts of it are relate by detect connection between them.

VARIABLES

What is a Variable?

-    A variable is any characteristic or quality that varies within a class of objects. The individual members in the class of objects must differ or vary to qualify the class as variable.

-    A constant is any characteristic or quality that is identical within a class of objects. Individual members in the class are held constant and not allowed to vary.

Quantitative versus Categorical Variables

-    A quantitative variable is a variable that varies in amount or degree (rather than all or none) along continuum from less to more, but not in kind. Two obvious examples are height and weight. Quantitative variable can often (but not always) be subdivided into smaller unit, for example: length. Besides it, we can assign numbers to different individuals or objects to indicate how much of the variable they posses, for example: variable “interest” of students toward a subject.

-    A categorical variable is a variable that varies only in kind (qualitatively different), not in degree, amount or quantity. Examples: Eye color, gender, religious preference, occupation, position on a baseball team, political party, teaching method, and most kinds of research “treatments” or “methods.”

-    Researchers in education often study the relationship between (or among) either (1) two (or more) qualitative variables; (2) one categorical and one quantitative; or (3) two or more categorical variable.

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The Research Problem

Summary

HOW TO DESIGN AND EVALUATE RESEARCH IN EDUCATION

Fraenkel J.R. Wallen, N.E

CHAPTER 2: THE RESEARCH PROBLEM

Research problem

Research problem is a problem that someone would like to research.

Research Question

-    Usually many research problems are stated as questions which serve as the focus of research’s investigation.

-    What makes research questions be researchable is data of some sort of information can be collected to answer the questions.

Characteristics of Good Research Questions

Four essential of good research questions: feasible, clear, significant, and ethical.

1. Feasible

That is can be investigated with available resources. Lack of feasibility often seriously limits research effort.

2. Clear

That is unambiguous, specifically define terms used, operational needed, so people can understand what the terms used in a research question mean.

Defining Terms in Research

There are essentially three ways to clarify important terms in a research question:

-      Constitutive definition (dictionary approach, using other words to make what is meant clearer).

-      Definition by example

-      Operational definitions (researchers specify the actions or operations necessary to measure or identify the term)

3. Significant

That is worth investigating. The values of intended research are the possibility of the research question to advance knowledge, to prove education practice and to prove the human condition.

4. Ethical

Research questions frequently (but not always) suggest a relationship to be investigated. A suggested relationship means that two qualities or characteristics are tied together or connected in some way.

 

Nature of Educational Research

Summary

HOW TO DESIGN AND EVALUATE RESEARCH IN EDUCATION

Fraenkel J.R. Wallen, N.E

CHAPTER 1: NATURE OF EDUCATIONAL RESEARCH

WAYS OF KNOWING: Sensory experience (incomplete/undependable), Agreement with others (common knowledge wrong), Experts’ opinion (they can be mistaken) and Logic/reasoning things out (can be based on false premises)

WHY RESEARCH IS OF VALUE: Scientific research (using scientific method) is more trustworthy than expert/colleague opinion, intuition, etc.

SCIENTIFIC METHOD

Scientific Method (testing ideas in the public arena)

-       Put guesses (hypotheses) to tests and see how they hold up

-       All aspects of investigations are public and described in detail so anyone who questions results can repeat study for themselves

-       Replication is a key component of scientific method

Scientific Method (requires freedom of thought and public procedures that can be replicated): Identify the problem or question, clarify the problem, determine information needed and how to obtain it, organize the information obtained, and interpret the results.

TYPES OF RESEARCH

  • Some of the most commonly used scientific research methodologies in education are experimental research, correlational research, causal-comparative research, survey research, content analysis research, qualitative research, and historical research.
  • Experimental: Researcher tries different treatments (independent variable) to see their effects (dependent variable). In simple experiments compare 2 methods and try to control all extraneous variables that might affect outcome. Need control over assignment to treatment and control groups (to make sure they are equivalent). Sometimes use single subject research (intensive study of single individual or group over time)
  • Correlational Research: Looks at existing relationships between 2 or more variables to make better predictions.
  • Causal Comparative Research: Intended to establish cause and effect but cannot assign subjects to treatment/control; Limited interpretations (could be common cause for both cause and effect); Used for identifying possible causes; similar to correlation.
  • Survey Research: Determine/describe characteristics of a group; Descriptive survey in writing or by interview; Provides lots of information from large samples. Three main problems:  clarity of questions, honesty of respondents, return rates.
  • Ethnographic research (qualitative): In depth research to answer WHY questions. Some are historical (biography, phenomenology, case study, grounded theory). Ethnographic research is one form of qualitative research. Another common form of qualitative research involves case studies.
  • Case Studies: a detailed analysis of one or a few individuals.
  • Historical Research: Study past, often using existing documents, to reconstruct what happened, Establishing truth of documents is essential.
  • Action Research (differs from above types). Not concerned with generalizations to other settings. This is a type of research by practitioners designed to help improve their practice.
  • Content analysis research involves the systematic analysis of communication.
  • Each of the research methodologies described constitutes a different way of inquiring into reality and is thus a different tool to use in understanding what goes on in education.

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Didactical Background of a Mathematics Program for Primary Education

This is a summary from the other article in Mathematics  written by A.Teffers “Didactical Background of a Mathematics Program for Primary Education”

1. Introduction

This contribution outlines an instruction-theoretical framework of realistic mathematics education for primary school that was developed in The Nedherlands in the period between 1970 and 1990. At the outset a learning-instruction structure is described. Next it is illustrated how this structure is inter-twined in the various learning strands. Then the conclusion is how these strands of learning give concrete shape at lesson level.

2. Instruction – Theoretical Framework of Realistic Mathematics Education

The instruction-theoretical framework is outlined on the basis of column on the basis of column arithmetic for division. A realistic course for ‘long division’ is reached through some steps, from the simple problem until the complex problem. Students will solve the problems using their own ways that might be various from the simple method until they can learn to carry out the long division procedure. The five major learning and teaching principles that lie at the basis of such “realistic” courses will be described in the context of the previously mentioned learning strands for long division.

2.1  Construction and Concretising

The first learning principle in learning mathematics is constructive activity. This is clearly visible in the outline coursed where students can discover the division procedure for themselves because a concrete orientation basis is laid for the skill to be learned. Finally, they come to realize what the arithmetic operations are leading up to.

2.2  Levels and Models

The learning mathematics is a process which moves at various levels of abstraction. This level-characteristic can be seen from the solutions methods that students used. To be able to achieve the raising in level from informal to formal arithmetic the students must have at his disposal the tools to help bridge the gap between the concrete and the abstract.  Materials, visual models, model situations, schemes, diagrams and symbols serve this purpose.

2.3  Reflection and Special Assignments

The learning of mathematics and in particular the raising of the level of the learning process is promoted through reflection. In this third instruction principle the pupils must constantly have the opportunity and be stimulated at important junctions in the course, to reflect on learning strands that have already been encountered and to anticipate on what lies ahead. Important assignments through which one and other can be achieved are the previously mentioned free productions and the conflict problems.

2.4  Social Context and Interaction

Learning is not merely a solo activity but something that occurs in a society and is directed and stimulated by that socio-cultural context. Students can find solution for a problem through discussion among them. Consequence of this principle is that mathematics education should by nature be interactive, which besides room for individual work, it must also offer opportunity for the exchange of ideas, the rebuttal of arguments, and so forth.

2.5  Structuring and Interviewing

Learning Mathematics does not consist of absorbing a collection of unrelated knowledge and skill elements, but is the construction of knowledge and skills to a structured entity. This fifth principle means that learning strands must be intertwined with each other. Besides it, pure arithmetic and the making of applications must from the very start be connected with each other.

2.6  The Structure of the Learning-Instruction Principle

In the foregoing the learning principles and instruction principles that were connected are: the concept of learning  as construction with the lying of a concrete orientation basis; the level-character of learning seen on the long term with the previous of models, schemes, and symbols; the reflection aspect learning with the assigning of special tasks, with as main categories free production items and conflict problems; learning as a social activity through interactive instruction; and the structural of learning with the intertwining of learning strands.

3. Four Directions in Mathematics Education

Beside the realistic direction in mathematics education, there are three others that can be distinguished, namely: the empiristic, the mechanic and the structuralistic.

3.1  The Empiristic Approach to Long Division

The empiric approach to long division because, in general, it is one that is not thought. Instead, the short cut of informal arithmetic methods is pursued, in part via mental arithmetic. At this approach what stands out especially is the difference in regard to the use of the model in order to break away from informal to formal arithmetic.

3.2  The Mechanistic Approach to Long Division

In tens of lessons the diffusion algorithm is practiced from simple case to complex case. The degree of complexity is especially determined by the magnitude of the numbers (dividend and divisors); regrouping actions for multiplications and subtraction; and ‘bothersome’ zeroes. Divisions with remainders do not appear until near the end of the course.

3.3  The structuralistic approach to long division

In the structuralistic approach the emphasis in the teaching long division lies very much on the place value system. The problem with the structuralistic set up of the learning strand for long division is that the algorithm is taught primarily at the formal arithmetic level. The sucject-systematic final algorithm is not pursued in a direct manner, but developed gradually via context problems and context restricted arithmetic.

3.4  Direction in the 5 by 5 Learning-Instruction Structure

The structuralistic method does not fit into the learning-instruction structure well either. For instance, the construction principle can only be applied to a limited degree because a solid concrete orientation basis is lacking. Observe the contracting background of the realistic 5 x 5 education-learning structure have more relief as in the example of long division. Therefore, the build up of elementary skill can take a place via a process of reinvent, yet, must imperatively take place if the mathematical rules and structures are to be widely applicable.

3.5  Summary and Generalisation

The four mentioned directions in mathematics education can be disguised according to the presence or absence of the components of horizontal and vertical mathematisation like below:

Horizontal Vertical
Mechanistic - -
Empiristic + -
Structuralistic - +
Realistic + +

Horizontal mathematising is the modeling of problem situations thus that these can be approached with mathematical means (it leads from the perceived world to the world of symbols). Vertical mathematising is directed at the perceived building and expansion of knowledge and skills within the subject system, the world of symbols.

4. Examples of realistic Learning Strands with an Emphasis on Vertical Mathematising

4.1  Counting

Counting is a realistic approach as a reaction to the structural concept. The expense of counting activities is to develop number concept by way of practising logical forms of reasoning. Elementary arithmetic is founded on counting movement, in short synchronous counting. When young children are asked to count how many number, they are not yet able to count resultatively. There is a third educational path to help them understand about counting. That is playing the game of counting and doing arithmetic by rolling the dice, dominoes, and a dot card on which tokens are placed in circle.

4.2  Automatiing and Memorizing of Addition and Subtraction up to Twenty

Working with dice or a dot charts has helped many children automatise and even memorize ever so many elementary additions, subtractions and ‘split-ups’ to twelve. Expansion and completion of that process of automatising and memorizing to twenty demands a new model situation. The models that can be chosen are: a string of beads with a five-structure for arithmetic to ten; and the arithmetic rack with five-structure for arithmetic to ten and especially twenty, a cardinal model.

4.3  Addition and Subtraction to a Hundred

In the realistic viewpoint addition and subtraction to a hundred is thereof not immediately algorithmised, but there is room for all sorts of varied strategies of efficient (mental) arithmetic. So, students can use their own ways in various strategies from informal level until formal level. Place value or empty number line can be used in this case. When students use corresponding subtraction or addition strategies, it can facilitate memorization.

4.4  Learning the Tables

In general there are two methods by which to learn tables: reproduction and reconstruction didactics. These methods are useful for reproduction of memorized knowledge, reconstruction knowledge via skilled arithmetic and connecting to informal working method. When student use table, they also can learn about sums, multiplication and division.

4.5  Mental Arithmetic

Mental arithmetic is considered as doing arithmetic mentally where standard procedure is carried out mentally and the calculation is made in the head. Various forms of mental arithmetic are distinguished, namely: estimating, varied and standard mental arithmetic.

4.6  Column Arithmetic

Characteristic for column is that the arithmetic is done with the individual digits, while with mental arithmetic the numbers that are operated with retain their own ‘value’.

4.7  Ratio

The most realistic feature of the realistic approach of ratio is that it does not steer directly to the so called rule of three in working out the fourth term (a : b = c : ? , maka ? = bc/a). If numbers are used in ratios then the double number line can help in solving of ratio problems, namely: determining the ratio relationship, comparing equivalent ratios, making equivalent ratios, and determining the fourth proportional.

4.8  Fractions

The double number line can also be used for operation with fractions. The comparison with fraction can take place by switching to a different unit of measure where one another can be depicted on the double number line.

5. An Example of Realistic Mathematics Education at Classroom Level with an Emphasis on Horizontal Mathematising

Students are given a problem and they work in groups. There is discussion among them and the teacher guide them by asking some questions that will lead them get better understanding about the problem.

6. Conclusion

One could relate the described series of lessons which horizontal mathematising or the vertical learning strands to the learning-instruction structure. One could also investigate how the number line fulfils the bridging function in these courses between the formal context-bound level and the formal level, or make a detailed analysis of the issues and indicate their function in the learning-instruction structure.