EXPERIENCES IN DEVELOPING A QUANTITATIVE REASONING PROGRAM FOR STUDENTS AT ZAYED UNIVERSITY

byNakhshin Karim, John Wakefield

(Department of Natural and Quantitative Sciences)to

Learning Technologies and Mathematics, Middle East Conference,Sultan Qaboos University

Muscat, Oman31/03/07 – 2/04/07

ABSTRACT. This paper will take the form of a work in progress. It will trace the development of a quantitative reasoning program, comprising three courses, which all students in Zayed University must take, either wholly or in part. We started from the initial belief, held by several faculty members, that there needed to be a change in the content and style of the mathematics courses that had been taught at the University from its beginning. These courses were the same courses that can be seen in almost every university or community college catalogue in the US as required general education courses. It was felt that, since, for the majority of our students, the last of these courses would be a terminal course, a new approach was needed. We wanted students to understand that mathematics was more important than many of them had been led to believe by their experiences in high school and that mathematics had relevance to their lives. This would require, not only changing the content of the courses, but also adopting a new philosophy of teaching, different from the methodology of Theory–Examples–Exercises, to a more constructivist approach, in which theory was arrived at after concrete, ‘real life’ examples were explored. The paper will outline the rationale behind the original program, the reaction to the philosophy and content of the program within the department and describe the status of the program at present. In conclusion we will report the findings of three US mathematicians/educators who visited the University, acting as consultants, charged with evaluating the program and advising the university on ways of improving it. The result of two of the consultants second visit has been the development of a new program, which is expected to be in place by Autumn 2008.

IntroductionZayed University was established in the United Arab Emirates in 1998, with campuses in Abu Dhabi and in Dubai. The stated aims of the University were to seek to be a leading university in the region and to prepare Emirati women for meaningful and successful 21st century personal and professional lives.

As part of this mission, students in their first two years of their baccalaureate study join a General Education Program (Colloquy On Integrated Learning ) prior to enrolment on any majors program. The Colloquy’s mission is to help to form the students’ understanding of their relation to the world and to prepare students for work in their major field of study. The Colloquy program has several learning goals, one of which is Critical Thinking and Quantitative Reasoning.

To achieve this goal a sequence of Sciences, Mathematics and Technology courses has been designed.

HistoryBy 2003, a few Mathematics teachers at the University were coming around to the belief that the mathematics they were teaching was not doing very much to improve students’ knowledge or understanding of mathematics and its importance. The majority of students were engaged in what has been called “futile learning”: the memorization of context-free symbol manipulation, performed by them in the slim chance that it might be needed at some point in their future studies.(Appendix 1).At this time the Colloquy program was in its early stages of development and after talks with the new Provost and the Dean of Colloquy it was agreed that the style, content and philosophy of teaching mathematics at this level should be reviewed. A group of Mathematics faculty and a Mathematics Education specialist were charged with the task of developing courses which would better suit our students’ needs. The result was the development of three courses of mathematics using a constructivist methodology, which would seek to allow students to learn and understand mathematics in such a way that they might feel was valuable.

Zayed University (ZU) students and MathematicsThe students at ZU are the same as students all over the world, subject to the same responses to learning and doing Mathematics. As with other countries, the UAE suffers from the what is now widely known as the “Math Problem”. Put simply, many students are not good at mathematics; they are turned off (as in no other subject!); they are frightened of mathematics; they see no relevance to their world in the mathematics they are taught; and, as one consequence, their quantitative reasoning skills are poor. However, in the UAE, and probably in much of this part of the world, it is not surprising when the way in which students have been taught mathematics in schools in the past is examined. Rote learning, carefully structured exams and a classroom ethos of regimentation and non-questioning regurgitation of abstract concepts seems prevalent. Only a few of the more mathematically orientated students arrive at the University with a positive attitude towards mathematics.This is changing in the UAE and a new look at Mathematics and Science teaching should have a significant impact on how these subjects are taught in schools. The initiative, known as SAMI, is heavily geared toward a student-centered, constructivist methodology where teachers will be expected to engage students on a journey of discovery.

First VersionStudents enter Zayed University with a range of backgrounds in mathematics; some have already ‘done’ Calculus, while others have no concept of number at all. Worst of all, of course, some hate and fear mathematics.

For this reason it was decided that some ‘remedial’ mathematics was needed - but not in the way they had been taught in schools. In these courses, emphasis would be on number sense, discovery, pattern-seeking and generalizing from specialized situations. Two Developmental Mathematics courses were envisaged to take students from the lowest entry level to preparation for Colloquy. The Developmental Math courses would be run while students were in the University’s Readiness program, which comprises a sequence of courses for students whose English language scores were not high enough for them to be allowed to enter the Colloquy program.Once students entered the Colloquy program, they would embark on a course which would seek to illustrate the importance of mathematics in the world in two main ways: by modelling situations using elementary functions and by understanding data using descriptive statistics.The courses were named:

Developmental Mathematics I MTH001Developmental Mathematics II MTH002Explorations in Mathematics COL111

MethodologyIt was felt that if the students attitudes towards mathematics were to be improved then the way in which they were to be taught must change also. It was decided to adopt a constructivist approach, in which students and teachers would engage in a dialogue in class in order to involve students more in the learning process. Using this method students would be introduced to a range of topics through real world problems which would exemplify certain mathematical concepts. Only after concrete examples had been investigated would generalizations be explored and theoretical aspects of the concepts introduced.This approach requires a much greater subject matter knowledge and a greater interaction with students than does the teacher-centered, rote and memorization methodology. We were lucky that so many of our faculty actually welcomed this new method, despite negative inputs from certain faculty, not all of whom were mathematicians.

Course outlines are appended (Appendix 2)

Reaction to the Courses from within the DepartmentA series of meetings were held in which the rationale, philosophy and methodology for the proposed courses were discussed. Faculty opinions were divided on the philosophy and methodology from the start and have remained so. A common complaint from those faculty opposed to the changes was that it was not “real mathematics”. That is, it was not algebra delivered in the Theory – Example – Exercises format. It was said that by not teaching in this way, students would not see “the beauty of mathematics.” Such views have proved impossible to shake.However, it has been gratifying to see how many mathematics teachers at the University have embraced the philosophy and senior management are no longer swayed by the arguments of the dissenters. At present prospective candidates for positions in the department are asked for their opinions on quantitative reasoning courses and on teaching methods with a view to attracting and engaging the ‘right’ faculty.

Course ReviewAfter the program had been running for a number of years it was decided to invite a number of consultants to critique the courses and advise the University on the way ahead. Three eminent Mathematicians/Mathematics Educators from the U.S. were engaged and visited the University for a period of about one week in which they read the course material, sat in on classes, interviewed faculty from the department itself and faculty in other colleges. At the end of their visit a frank and open report was sent to the University and distributed to faculty. The result of the visit strengthened the University’s belief in the program and senior management in the University resolved to implement the recommendations of the consultants in full.

A second visit from two of the original consultants, Joan Leitzel and Deborah Hughes-Hallet, involved intense, in-depth scrutiny of the programs and resulted in firm agreement on the next steps.

The Next StepsAfter the first visit of the consultants a revised sequence of courses was drawn up, the plan conveyed to the consultants and a group of faculty members (including the original group) were charged with the task of rewriting the new courses.The new sequence essentially combined DMI and DMII and split COL111 into two course, one concentrating on modelling using data and the other modelling using functions.The suggested sequence has three courses:

1) Developmental Math: is designed to introduce students to the nature of mathematics and the applications of mathematics through simple real world problems. The goal is to improve the quantitative reasoning ability of the students as well as improve computational skills.This course will lead students from work with numbers to the next stage which is the development of the function concept and an introduction to algebraic reasoning. The function concept is a cornerstone of mathematics and without this idea students cannot fully appreciate mathematics and its applications.

2) Quantitative Reasoning I: (Modelling with Data) is designed to be based on real data related to Zayed University majors and/or to UAE and the region. Ideally a case study would be presented, then management, analysis and presentation would follow.Emphasis would be placed on having students do their own interpretation of the data and come to their own conclusions. The course will involve extensive use of Excel.

3) Quantitative Reasoning II: (Modelling with Functions) This course is designed to provide students with a broad general education in quantitative reasoning and critical thinking. It will also provide a foundation for the development of their ability to function competently and confidently in majors’ programs. The course will focus on analytical reasoning and thinking to solve real world problems in business, finance, economics, computer science, education and the natural sciences.

The content of the course will be delivered through classroom activities and projects to introduce the students to the various topics. For some topics or case studies, data can be acquired from primary sources connected with other courses, such as Environmental Science, Health Science and QR1 and it is also planed to use data or case studies from the region. In each area, knowledge, analytical skills, critical thinking and understanding will be developed using relevant examples for discussion, analysis and interpretation in class with follow up exercises or assignments of a similar nature to be done individually or in groups outside the classroom.

Draft course outlines are appended (Appendix 3)

The Role of TechnologyZayed University is a “Laptop University”. All students are issued with a laptop on entry, take courses in using technology appropriately and are expected to bring laptops to every class (in Mathematics at least).It was decided from the outset that the new Mathematics program would use this technology extensively to provide a powerful tool for solving problems and analyzing data. Mathematics students are expected to be proficient in Word, have sufficient knowledge to use Excel in data analysis and are taught to use CAS software to investigate and interpret the models they encounter.The future courses are being designed to take more advantage of Excel’s features and new graphing software is being reviewed.

ConclusionFrom its beginning, the Program has relied on dedicated, committed faculty to defend it against those opposed to it, who see this way of teaching, and the content, as a downgrading of their subject. Fortunately for the program, those teachers who have been in favour of this approach have not buckled under the weight of criticism from those opposed to it.The findings of the consultants provided much needed affirmation of our belief that the path we were forging was heading in the right direction.One last comment needs to be added to address a repeated criticism of the program from within the department. This program is not, and has never claimed to be, a preparation for a major in Mathematics. It is a general education program. The University does not have a Mathematics major, but, if and when the time comes, it is generally agreed that a course of a mathematically more rigorous nature would be needed in addition to the existing program.The new course is planned to be fully operational by Autumn 2008. Much now has to be done now to develop materials and gather data which will be seen as relevant, important and interesting to our students in order to fulfill a stated aim of Zayed University: to prepare Emirati women for meaningful and successful 21st century personal and professional lives.

And one last word: it has been the experience of faculty who have taken on board this new approach that the students have gained more understanding of the role of mathematics in the world and that it has been fun to teach.

Appendix 1 I Had A Dream Last Night (or The Math Of Sisyphus)I had a dream last night. I was going to be teaching some students Basic College Mathematics in a classroom suffused with a golden glow. It was our first day together and we were getting to know each other. I asked them about their past experiences in Mathematics classes. A couple of girls at the front said they enjoyed mathematics and found it interesting and beautiful. What a joy to have them in my class, I thought. Not so for the rest of the class. They had so many horror stories about their encounters with Math teachers and the way they were made to feel when they couldn’t do the mathematics that I soon became depressed and decided not to ask them any more questions on that subject.

We started off with some simple arithmetic. Whole number addition seemed ok for them; only one or two had difficulty. Whole number subtraction caused more problems; some of their algorithms for performing the operation seemed almost haphazard, subtracting from top or bottom depending on which was the larger digit, with tiny marks scribbled in the most unlikely places. Multiplication was not too bad except for the occasional zero appearing on the end of one row of digits or another, like an uninvited guest at a wedding trying to get in on a photograph. I gave up trying to understand the division algorithms they had in their heads within seconds! The students looked longingly at the calculators on their desks. No way! I thought. What these students need is ME teaching them THE BASICS!

We started with The Place Value System for the natural numbers and were soon cruising through the rules of positive numbers, moving on to integers and the rules of integer arithmetic. Order of operations came next. Week One gone, five classes, 8 algorithms already – lovely! I thought to myself. If I carry on like this everything is going to be great.

And so it went on: fractions, lowest common multiple, prime factorization, mixed numbers, ratio and proportion; the algorithms were piling up: 12 algorithms; 18 algorithms; 25 algorithms. Then I went onto percentages. I was really enjoying myself. I tested the students on the algorithms after each week or two and everything was fine. They learned their algorithms for the tests and performed them quite well on the whole. I was pleased. This is ‘real’ mathematics teaching I thought.

At the end of the course the students had 45 algorithms to learn and the final exam would test about 10 of them. No problem! “Just learn all the algorithms and do these 120 problems and you’ll be ok.”, I told them.

I was so happy with these students that I asked my supervisor if I could take them again for the next level course - Algebra. “No problem”, she said.

Next semester we started off with some simple collection of ‘like terms’, followed by examples with parentheses. But negative numbers and minus signs were a problem here. So we had to review the rules of integer arithmetic. Multiplication and division were a little bit difficult for some. They haven’t learn their algorithms properly, I thought.

So … we started again with some simple arithmetic. Whole number addition seemed ok for them; only one or two had difficulty. Whole number subtraction caused more problems; some of their algorithms for performing the operation seemed almost haphazard, subtracting from top or bottom depending on which was the larger digit, with tiny marks scribbled in the most unlikely places. Multiplication was not too bad except for the occasional zero appearing on the end of one row of digits or another, like an uninvited guest at a wedding trying to get in on a photograph. I gave up trying to understand the division algorithms they had in their heads within seconds! The students looked longingly at the calculators on their desks. . . .

And that’s when the nightmare began!

Appendix 2 Original Programs

DEVELOPMENTAL MATHEMATICS ICourse Outline

UNIT 1 INTRODUCTION TO NUMBER SENSEPatterns and Relationships with Whole NumbersOperations with whole numbers

UNIT 2 DATA ANALYSIS AND PROBABILITY

UNIT 3 MEASUREMENT AND FRACTIONAL NUMBERS

UNIT 4 SHAPE AND SPACE GEOMETRY

DEVELOPMENTAL MATHEMATICS IICourse Outline

UNIT 1 INTRODUCTION TO ALGEBRAGrowing PatternsRecursive RelationshipsFunctional Relationships

UNIT 2 INTRODUCTION TO FUNCTIONSGraphing Number Patterns

UNIT 3 FROM FUNCTIONS TO EQUATIONSVariables and EquationsThe Meaning of Solving Equations

UNIT 4 STUDYING FUNCTIONSLinear Functions

Slope, intercepts Writing Linear Equations

Quadratic Functions (time permitting )

COL111 EXPLORATIONS IN MATHEMATICSCourse Outline

UNIT 1 INTRODUCTION TO MATHEMATICAL MODELING

Section1 1.1 What is Thinking Mathematically1.2 Using Technology For Understanding

1.3 What is Mathematical Modeling

UNIT 2 CONNECTING DATA, GRAPHS AND FUNCTIONS

Section 1 Data RelationshipsSection 2 Linear FunctionsSection 3 Quadratic Functions Section 4 Exponential Functions

UNIT 3 SETS

UNIT 4 CHOICE AND CHANCE (Probability) UNIT 5 DESCRIPTIVE STATISTICS

Appendix 3 New Proposals (DRAFT ONLY)

DEVELOPMENTAL MATHEMATICS Course Outline

UNIT 1 Number Sense

Introduction to Number Sense Patterns and Relationships with Whole NumbersOperations with whole numbersEstimation, mental arithmetic and determination of reasonableness of answersMeasurement, Fractional Numbers, negative numbers and Percent Exponents

UNIT 2 Introduction to Algebra

Growing PatternsRecursive RelationshipsFunctional Relationships

UNIT 3 Introduction to Functions

Graphing Number PatternsIntroducing the Concept of FunctionRepresentations of Functions

QUANTITATIVE REASONING I (MODELING WITH DATA)Course Outline

1.Excel Basics (introduced as needed)

Downloading data from the internet Sorting and filtering Arithmetic operations –(+,-,*,/,^,%) -order of operations Common Excel functions (Sum, Count, Max, Min) Excel functions for mean, median, quartile, standard deviation Making charts Scatter plots, trendlines; regression equations, projections and R2

Formulas – relative and absolute cell referencing Using IF, SUMIF and COUNTIF functions and the logic involved

2.Single Variable Data

Information from data

Graphical Presentation

Numerical descriptorso Central tendency – mean, mediano Spread – range, quartiles, standard deviation

Notation and Algebrao Subscripted variableso Sigma notationo Some algebra review – solving equations given with sigma notation and subscripted

variables

3. Multi (two) variable data

Graphical Presentationo Scatter plot

Information from the datao Association (positive/negative)o Trends (linear, non linear)o Outliers

Numerical descriptorso Correlationo Regression line and formula – from Excelo Correlation and causation (hidden variables)

Predictionso Interpolation and extrapolation – extending trendlineso Building spreadsheet using formulas

QUANTITATIVE REASONING II (MODELING WITH FUNCTIONS)Course Outline

UNIT 1 INTRODUCTION TO MATHEMATICAL MODELINGThinking mathematically to solve some simple problems involving, for example percentages, series and number patterns. This will involve the use of specializing, conjecturing, generalizing and using simple methods of proofs, simple deductive reasoning. Introduction to modelling through everyday experience, hence finding functional relationships for simple(linear) problems.

UNIT 2 DATA RELATIONSHIPS AND ANALYSIS OF DATAUsing data in Excel to check rates of change, to look for trends (increasing/decreasing/both/neither), patterns and relationships.

UNIT 3 LINEAR FUNCTIONSReview of linear functions and their graphs from tables of data; Checking rates of change of a function for linearity; Alternative forms for the equation of a straight line using simple problems related to majors ( linear constraints, linear inequalities). Systems of linear equations as a means of solving simple real problems involving up to three variables (profit/loss/breakeven) .

UNIT 4 EXPONENTIAL FUNCTIONSUsing data to introduce exponential functions – Modeling exponential data. Examples from environmental science (population growth – increasing and decreasing/simple and logistic), health science (bacterial growth, safe cooking ranges) , finance (interest rates: comparing simple with compound and using continuous compound interest to introduce e); Growth rates and growth factors (amortization, inflation rates, magazine circulation); General formula for the family of exponential functions; Further explorations. Solving inverse problems of the exponential function using technology (doubling time of populations, half life). Logarithmic function as the inverse of an exponential function.

UNIT 5 QUADRATIC FUNCTIONS

Using film clips of sport/projectiles/decorative features (fountains) to introduce the parabola. Using data in Excel to check for quadratic functions. Modelling quadratic functions from case studies (maximum area of enclosures, projectiles, minimum cost models). Modeling general power functions (Examples from environmental science – species-area relation (Bio-geographical theory))

Appendix 4

Examples from Current Courses

Current DMII

This unit introduces students to functions through patterns – a very strong theme throughout all the courses.

This exercise is taken from

Unit 1: Introduction to Algebra: Growing Patterns – Recursive Functions

Exercise

Draw the next two shapes in the pattern. How many matchsticks are there in each pattern?

Complete the table

Term Number (n) 1 2 3 4 5 6 … 10 … 20 …Number of Matchsticks (Un)

u1 = …………..

u2 = …………..

u3 = …………..

u4 = …………..

Write down a recursive formula for the pattern. ………………………………..

This exercise is taken from Unit 1: Introduction to Algebra

Growing Patterns – Functional Relationships

Exercise

Draw the next two shapes in the pattern. How many matchsticks are there in each arrangement?

Complete the table

Term Number (n) 1 2 3 4 5 6 … 10 … 20 …

Number of Lines (un)

u1 = …………..

u2 = …………..

u3 = …………..

u4 = …………..

Write down a functional formula for the pattern. ………………………………..

Current COL111

Exponential FunctionsThe Zayed University cafeteria has a brick oven in which the cheese manaqish is cooked.

The oven is allowed to cool down overnight and is then preheated in the morning before the students arrive. When the cook arrives in the morning he heats the oven by turning on the gas. The temperature of the bricks in the oven B(t) is given by the relationship:

B(t) = 150 –

where B is the brick temperature (in degrees Celsius) and t is the time in seconds after turning on the gas to warm the oven.

1. Using Maple, plot the graph of the function B(t) over the range of t = 0.. 40 seconds.

2. What is the temperature of the kitchen in the cafeteria in the morning?

Explain your answer.

3. Use functional notation to express the temperature of the bricks in the oven for t = 20 seconds after the gas has been turned on.

4. Use Maple to find how much the temperature increased in the first 10 seconds.

5. Use the graph to estimate how much the temperature increased in the period from 20 seconds to 40 seconds after being turned on.

6. Between manaqish baking’s, the oven temperature is allowed to fall. Use Maple to find how long will it take for the temperature to rise back to 140oC, if it is allowed to fall to 120 oC.