A case study on the impact of automated assessment in engineering mathematics

Abstract

A rapidly increasing number of modules in degree programmes now utilise virtual learning environments. A main feature of virtual learning environments, to which their adoption by many educators is attributed, is automated assessment. Automated assessment is particularly attractive where the number of students taking a module is large. As mathematics is an essential pre-requisite for those studying engineering courses and is compulsory in almost every engineering programme, classes tend to be large. However, there is ongoing scrutiny of the various aspects of virtual learning environments, including automated assessment. This paper is based on a case study that highlights aspects of the impact of automated assessment by studying the performance of students taking an engineering mathematics module where automated assessment is utilised in the form of quizzes.

Introduction

Networks have now become commonplace, computer technology more readily available and affordable to the public and, as reported in many studies (Brohn, 1986; Gbomita, 1997; McDowell, 1995; Stephens et al., 1998; Thelwall, 2000 and Zakrzewski and Bull, 1998), virtual learning environments (VLEs) are now generally reliable. As a result of such advances in information technology, VLEs are fast becoming an integral part of many taught courses.

VLEs provide a number of advantages compared to conventional face-to-face learning. They can utilise repositories that can be accessed at all times, enabling the effective use of time and making catching up easier. They also enable automatic assessment and instant feedback that makes self-assessment possible and results in an improved learning experience (Brohn, 1986; Gbomita, 1997; McDowell, 1995; Stephens et al., 1998; Thelwall, 2000 and Zakrzewski and Bull, 1998).

Automated assessment is one of the main benefits provided by utilising VLEs, and has made them appealing to many educators. Automated assessment drastically reduces time spent on assessment by instructors to as low as 30%. (Where no electronic assessment is used, about 75% of the time given to a module is normally spent on assessment (Smaill, 2005)). Revision without using valuable lecture time is made possible, lowering running costs and increasing efficiency (Griffin and Gudlaugsdottir, 2006). The use of automated assessment is therefore expected to increase the quality of education, as instructors should have more time to concentrate on improving and updating the delivery and content of courses (Juedes, 2003).

Given that that a decline in mathematics skills has been reported (Davis et al., 2005), automated assessment can play a vital role in improving student learning in engineering mathematics. However, varied opinions have been presented on the suitability and effectiveness of automated assessment, with some studies presenting no improvement and negative feedback while others have reported positive results (Smailes, 1998). It is in this light that this work evaluates automated assessment by studying the performance of students taking an engineering mathematics course with VLE-based automatically assessed quizzes. Quiz questions requiring single answers were utilised that fulfil the evaluation and procedural knowledge of the revised Bloom’s taxonomy (Coleman et al., 1998). Single answer questions also eliminate the effects of other pedagogical factors as much as possible in the evaluation of automated assessment.

The details of this study are presented in the next section, followed by the results of the study and analysis and finally the conclusions reached by the study.

The study

This study is based on a group of first year students on a foundation degree programme in electrical and mechanical engineering in 2006. This is a two year programme for students who do not qualify for the conventional degree programme. At the end of the programme students who perform exceptionally well may transfer to one of the conventional electrical and mechanical engineering degree programmes. Students should have a minimum UCAS tariff of 140 points at GCE/A-level or equivalent and passes equivalent to GCSE grade C or above in at least four subjects, including English and mathematics.

The study was carried out on an introductory mathematics module in the first semester of the first year which covered the fundamentals of engineering mathematics. The topics covered included algebra, co-ordinate geometry and statistics. The pedagogical goals were for students to acquire knowledge of and skills in the listed topics and, as a result, to be able to identify and apply appropriate methods to solve different problems or obtain information.

The group constituted of 52 students, of which 42 (81 %) were male and ten (19 %) were female. Face-to-face lectures and tutorials were provided every week, each lasting two hours, throughout the duration of the course. In addition, the students had a two hour supervised laboratory session every week in which they utilised software tools to solve mathematical problems.

The E-learning component was employed using a commercial VLE package, WebCT (now Blackboard – see http://www.blackboard.com). The web-based resource constituted learning resources that included tutorials, lecture slides, notes, laboratory exercises, quizzes and self-test exercises. It also had auxiliary facilities such as a module handbook, calendar, tutorials, bulletin board and email. Such a combination of face-to-face lectures and the use of VLEs has been reported to be students’ preferred learning environment (Smailes, 1998 and Davis et al., 2005). The module was assessed by a series of online quizzes and computer-based laboratory exercises (solving problems utilising software tools and then submitting reports electronically) and a hand-written test at the end of the module. The mark distribution for the different assessments making up the module, namely laboratory exercises, quizzes and the written test, were 10, 25 and 65% respectively.

Quizzes and self-test exercises

There were four assessed quizzes in total. These were held fortnightly from the third week of the course and were intended to provide a stimulus for learning and revising material learnt in lectures. The quizzes were supervised (to authenticate the identity of the students and to minimise cheating by ensuring that students worked independently at their consoles) and were to be completed within an hour. The number of questions per quiz varied. A password was provided at the beginning of each quiz as a further security measure. There were two dyslexic students in the class who did not require any additional time to complete the quizzes.

Self-test exercises covering material similar to that in the next quiz were posted at least a week prior to each quiz. These provided the opportunity for students to test themselves and practise at their own pace without drawing on the instructor’s time. The quizzes are also ideal for students who find it difficult to participate in face-to-face sessions and they also cater for different learning styles, for instance those with a kinesthetic and reading inclination. Studies have shown that through the use of automated quizzes, performance improves rapidly at each attempt of an exercise (Brown, 2004). There were five self-test exercises. The material in the last exercise was not tested in a quiz. The questions in the quizzes and self-test exercises had randomly generated variables. A scheme similar to that of Chen (2004) was utilised, where each time a student attempted a quiz or exercise the variables were set to different values. In this instance the variables were one or more numbers representing some quantity in a question such as angles, length, data values or constants in equations. As a result each student had different variable values for the same question. This made copying and assistance from fellow students more difficult. An example question is shown in Figure 1.

Figure 1. Example question

As the questions in the exercises simply required a single answer this enabled the effectiveness of the automated system to be assessed, rather than other pedagogical factors. The feedback provided by the system was the correct answer only -additional feedback and a discussion of the problems were provided in the next class. In this way students were not tempted to enter wrong answers so as to obtain the worked solutions of the problem, which in some cases can be detrimental (Chen, 2004). Indeed, students did exhibit this behaviour, especially in the first self-test exercises. This behaviour greatly diminished in subsequent exercises presumably because variables had different values each time the exercises were accessed. Feedback was provided in weekly tutorial sessions.

Statistics on the students’ use and grades on both of the quizzes and self-test exercises were automatically logged by the VLE (WebCT). The VLE also logged other activities such as the frequency of accessing resources.

Written test

The written test was held at the end of the semester when all VLE based quizzes had been completed. It consisted of 23 questions. Of these, nine were from topics covered in the quizzes and the first four self-test exercises (algebra, trigonometry, vectors and geometry). Six questions were from statistics, the topic covered by the last self-test exercise, which did not have an accompanying quiz. Eight questions were in areas not covered in the quizzes. These questions were on formulae, functions and algebraic expressions. However, it should be noted that all topics covered in the written test had been covered in lectures and tackled in the weekly tutorial sessions. The duration of the written test was two hours and it was closed book.

Results

The confidence intervals for means in the results presented here were determined using the analysis of variance (ANOVA) statistical test. The results show the average mark and its respective confidence interval. The ANOVA F figure and probability (p) of the null hypothesis are also given. The implied null hypothesis in this instance is that the effect being tested for is not true. This corresponds to an ANOVA F figure of one. The confidence interval level shown in all cases in this work is 95%. Figure 2 shows the distribution of students as a function of the number of quizzes and self-test exercises they took.

Figure 2. Distribution of students as a function of number of quizzes and self-tests taken

Figure 3 shows the average mark and their respective 95% confidence intervals for grades obtained in the written test by students as a function of the number of assessed quizzes they attempted. Only results for students who took between two and four quizzes are shown, as the students who took fewer than two quizzes are small in number and hence may distort the results. The probability of these results assuming null hypothesis is 0.028. This is less than 0.05, implying that students performed better the larger the number of quizzes they attempted.

Figure 3. Average test score and 95% confidence interval for mean as a function of quizzes taken

The performance as a function of the number of self-test exercises is shown in Figure 4. The probability of this result assuming null hypothesis is 0.058 (F = 2.473), implying that the number of self-tests taken by a student may be a factor as it is close to 0.05. This result is attributed to the observed behaviour of students during self-test exercises, for instance entering wrong answers so as to obtain the worked solutions of problems. This behaviour was mainly exhibited for the first self-test exercise. Students were discouraged from this practice in subsequent exercises when they realised that the variables changed each time they accessed a question. As a result some students only viewed questions without attempting to enter answers. Similar behaviour is reported by Smail (2005) where students attempted to win-out by memorising answers.

Figure 4. Average test score and 95% confidence interval for mean as a function of self-test exercises taken

Figure 5 shows a comparison of the performance in the written test on the nine questions relating to topics covered in the quizzes of students who completed the quizzes and those who did not. Although the probability for the null hypothesis is 0.093 (F=1.49) it should be noted that students who completed the quiz performed better on average in six of the nine questions.The results in the three questions that suggest otherwise are attributed to a combination of the fact that the number of students who did not complete the quizzes was much smaller than those that did (about 10 compared to about 40) and that two students in the group who did not complete any quizzes scored 93 and 74% in the written test, much higher than the class average of 50%. The lower limit of the 95% confidence interval is lower for students who did not complete corresponding quizzes for two (questions 1 and 7) of the three questions. Furthermore, question 3 was on fractions, a topic most students would already have been conversant in, hence rendering the influence of completing the quizzes of little consequence.

Figure 5. Average test score per question on topic covered in quizzes and 95% confidence interval for mean. C: completed respective quiz; N: did not take quiz

Figure 6. Average test score per question on topic covered in statistics self-test and 95% confidence interval for mean. C: completed selftest; N: did not take self -test

Group B is not shown in the results due to the small amount of data in this category. Here the probability for the null hypothesis for the mean is 0.063, however, the mean is observed to be highest for group D who had the highest number of quizzes and self-test exercises. Figure 7b shows results for the same groups for questions in the written test in the following categories: (A) topics covered in the quiz; (S) topics covered in self-test exercises and (N) other topics. It is evident that the performance in topics covered in quizzes and self-tests is significantly greater than that in topics not covered.

Figure 7a. Average test score per group for questions on topics covered in quizzes and 95% confidence interval for mean

Figure 7b. Average test score per group for questions on topics covered in quizzes and 95% confidence interval for mean. A: questions on topics covered in quiz; S: questions on topics covered in statistics self test; N: questions on topics not covered in quiz or statistics self-test

The overall performance by all students in questions covered in quizzes, self-tests and other topics is shown in Figure 8. The results clearly show better performance in questions that were previously covered in quizzes and self-test exercises. There is also no overlap of the confidence intervals of questions covered in either quizzes or the statistics self-test with questions not covered.

Figure 8. Overall average test score per group for questions on topics covered in quizzes and 95% confidence interval for mean. A: questions on topics covered in quiz; S: questions on topics covered in statistics self test; N: questions on topics not covered in quiz or statistics self-test

A survey of students on this course indicated that their feelings were generally amicable towards the VLE, with 78.5% of the 46 respondents finding nothing negative about automated assessment. Similar attitudes to automated assessment have been recorded with students preferring computer quizzes to conventional tests (Reinhardt, 1995; Chirwa, 2006 and Mastascusa, 1997).

For the results in Figure 7a the class population is grouped into the categories listed in Table 1.

Table 1. Definition of Groups Group Definition A number of self-test <= 3; number of quiz <= 2 B number of self-test >3 ; number of quiz <= 2 C number of self-test <= 3; number of quiz > 2 D number of self-test > 3; number of quiz > 2 Conclusion

The results presented here suggest that students performed better in topics that were covered in quizzes and self-test exercises. Taking into account that areas not covered in automated assessments were covered in tutorials it can be concluded that automated assessment contributes to more effective learning. Here, too, there are no indications that automated assessment significantly deters performance, which is in agreement with some studies (Coleman et al., 1998).

Although savings in time when using automated assessment are negative for low numbers of students in some studies (Smaill, 2005), the benefit of allowing students to test themselves and learn from their mistakes privately (Chen, 2004) is, however, not diminished. They are therefore an effective means of implementing assessment driven learning. However, they are inhibited by the limitations of the complexity of solutions that can be implemented compared to those possible in the conventional handwritten form. Another positive feature of VLE based quizzes is the ability to generate different variable values in the same questions for different students, minimising cheating and resulting in more effective learning of the principles required to solve problems.

In view of the above, a programme incorporating automated assessment is not likely to be detrimental. The programme can further be improved by the implementation of automated assessment in appropriate pedagogical contexts, for instance as a component of a larger problem. Once automated assessment has been set up it is no longer labour intensive and becomes easier to implement than conventional ‘manual’ assessment exercises, especially for larger groups of students.

It should be noted that, although the sample size in this study is too small to be conclusive in its own right, this work contributes to existing studies in this area and provides a clearer understanding of the effects of automated testing. The results from the self-test exercises and the quizzes indicate that students apply themselves more to assessed work (Race, 2001) and, as a result, the use of automated assessment is likely to have a positive pedagogical effect.

References

Blackboard Inc. http://www.blackboard.com [accessed 18 July 2008].

Brohn, D.M. (1986) The use of computers in assessment in higher education, Assessment and Evaluation in Higher Education, 11 (3), 231-239.

Brown, R.W. (2004) Undergraduate Summative Assessment Experiences. 34th ASEE/IEEE Frontiers in Education Conference, 20-23 October 2004, Savannah, GA, USA.

Chen, P.M. (2004) An automated feedback system for computer organization projects. IEEE Transactions on Education, 47 (2), 232-240.

Chirwa, L.C. (2006) Use of E-learning in Engineering Mathematics. International Conference on Innovation, Good Practice and Research in Engineering Education, 24-26 July 2006, Liverpool, UK.

Coleman, J.N., Kinniment, D.J., Burns, F.P., Butler, T.J. and Koelmans, A.M. (1998) Effectiveness of Computer-Aided Learning as a Direct Replacement for Lecturing in Degree-Level Electronics. IEEE Transactions on Education, 41 (3), 177-184.

Davis, L.E., Harrison, M.C., Palipana, A.S, Ward, J. P. (2005) Assessment Driven Learning of Mathematics for Engineering Students. International Journal of Electrical Engineering Education, 42 (1), 63-72.

Gbomita, V. (1997) The adoption of microcomputers for instruction: implication for emerging instructional media implementation. British Journal of Educational Technology, 28 (2), 87-101.

Griffin, F. and Gudlaugsdottir, S. (2006) Using Online Randomised Quizzes to Boost Student Performance in Mathematics. 7th international Conference on Information Technology Based Higher Education Engineering, 10-13 July 2006, NSW.

Juedes, D.W. (2003) Experiences in Web-Based Grading. 33rd ASEE/IEEE Frontiers in Education Conference, 5-8 November 2003, Boulder, CO, USA.

McDowell, L. (1995) Effective teaching and learning on foundation and access courses in engineering, science and technology. European Journal of Engineering Education, 20 (4), 417-425.

Mastascusa, E.J. (1997) Incorporating “Computer-Graded” Components Into Electronic Lessons. Frontiers in Education Conference, 5-8 November 1997, Boulder, CO, USA.

Moura Santos, A., Santos, P.A., Dion´ýsio, F.M., Duarte, P. (2002) On-line assessment in undergraduate mathematics: An experiment using the system CAL for generating multiple choice questions. 2nd International Conference on the Teaching of Mathematics, 1-6 July 2002, Crete, Greece. Available online from http://www.math.uoc.gr/~ictm2/Proceedings/pap139.pdf [accessed 19 July 2008].

Race, P. (2001) The Lecturer’s Toolkit: A practical guide to learning, teaching and assessment, 2nd edition. London: Kogan Page.

Reinhardt, A. (1995) New Ways to Learn. BYTE, 20, 50-72.

Smailes, J. (1998) CALculating Success? Available from http://www.icbl.hw.ac.uk/ltdi/evalstudies/essuccess.htm [accessed 18 July 2008].

Smaill, R.C. (2005) The Implementation and Evaluation of OASIS: A Web-Based Learning and Assessment Tool for Large Classes. IEEE Trans. Education, 48 (4), 658-663.

Stephens, D, Bull, J. and Wade, W. (1998) Computer-assisted assessment: suggested guidelines for an institutional strategy. Assessment and Evaluation in Higher Education, 23 (3) 283-294.

Thelwall, M. (2000) Computer based assessment: a versatile educational tool. Computers and Education, 34 (1), 37-49.

Zakrzewski, S. and Bull, J. (1998) The mass implementation and evaluation of computer-based assessments. Assessment and Evaluation in Higher Education, 23 (2), 141-152.

About the author

Lawrence Cloepass Chirwa BEng., MSc., PhD., PgDip., PgCHEP, Lecturer, School of Electrical and Mechanical Engineering, University of Ulster, Jordanstown, Newtownabbey BT37 0QB. Tel: 028 90368213 Email: l.chirwa@ulster.ac.uk

This is the html version of the file http://www.maa.org/cupm/crafty/Chapt8.pdf.

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Page 1 65

Engineering:

Electrical Engineering

CRAFTY Curriculum Foundations Project

Clemson University, May 4-7, 2000

Ben Oni, Report Editor

Kenneth Roby and Susan Ganter, Workshop Organizers

Summary

This report focuses on establishing the foundation mathematics needed to support the study and practice

of electrical engineering with emphasis on the undergraduate level.

To strengthen communication between communities of mathematicians and electrical engineers, we

have prepared this document to highlight the areas of mathematics that are most applicable to the study

and practice of electrical engineering.

For outcome objectives, we propose that the mathematics taught to undergraduate electrical engineering

students should help them in developing skills to:

1. Formulate problems in electrical engineering from real life situations,

2. Conceptualize the outcomes of electrical problems,

3. Simplify complex problems and estimate the reasonableness of solutions,

4. Visualize solutions graphically from inspection of their mathematical descriptions,

5. Visualize the form of a time function by inspection of the poles and zeros of its frequency transform,

6. Be able to mathematically model physical reality,

7. Perform rudimentary analysis in electrical engineering,

8. Validate solutions to electrical engineering problems.

Narrative

Introduction and Background

Electrical engineering deals with the manipulation of electrons and photons to produce products that benefit

humanity. The design of these products is based on scientific principles and theories that are best described

mathematically. Mathematics is thus the universal language of electrical engineering science.

Undergraduate electrical engineering education must provide students with the conceptual skills to for-

mulate, develop, solve, evaluate and validate physical systems. Our students must understand various

problem-solving techniques and know the appropriate techniques to apply to a wide assortment of problems.

We believe that the mathematics required to enable students to achieve these skills should emphasize con-

cepts and problem-solving skills more than emphasizing repetitive mechanics of solving routine problems.

Students must learn the basic mechanics of mathematics, but care must be taken that these mechanics do

not become the primary focus of any mathematics course.

Page 2 66

The Curriculum Foundations Project

More generally, it is vitally important that electrical engineering students recognize the importance and

beauty of mathematics in their chosen profession. We feel strongly that students will appreciate the power

of mathematics if each mathematics course clearly states its objectives at the outset. Students should be

told what they are going to study, why they are going to study it, and how it fits into the engineering pro-

fession. This motivation will need to be repeated throughout each course.

Many undergraduate mathematics curricula currently supporting electrical engineering programs could

be modified to better meet the needs of these programs. What follows are common weaknesses (from the

viewpoint of electrical engineering) seen in many mathematics curricula.

1. Too much time and emphasis are placed on topics that are not widely used while topics that have wide-

spread use often receive cursory treatment. One example is the excessive time and attention spent on

various solution techniques for ordinary differential equations. Although understanding the structure of

solutions for first- and second-order, constant coefficient differential equations is important for electri-

cal engineering problems, more useful and widely used are Laplace transforms and related techniques.

Yet these latter topics are often given cursory treatment in favor of more general structure theory.

2. There is often a disconnect between the knowledge that students gain in mathematics courses and their

ability to apply such knowledge in engineering situations. Perhaps, the use of more engineering or real

life examples will reduce this disconnect. Based on current learning theory, efforts to focus on under-

lying principles (not necessarily abstract statements of mathematical concepts) that are applicable in

many different contexts are effective in helping students to transfer knowledge.

3. Current mathematics curricula for engineering are front-end loaded. Consequently, as a matter of tim-

ing, many topics are presented too early and cannot be reinforced soon enough through engineering

applications before students forget the topics

4. Too often, mathematics is taught as a list of procedures or as theorem-proof exercises without ground-

ing the mathematics in reality. While we do not expect mathematics instructors to be well versed in all

engineering applications, we would like examples of mathematical techniques explained in terms of the

reality they represent. We strongly urge that team taught mathematics courses be considered. Teams

would consist of mathematics and electrical engineering professors. We feel that team-teaching could

better motivate and enthuse our students.

5. Failure to utilize appropriate technological tools while continuing to focus on mastery of symbolic manip-

ulation often encourages memorization and rote algorithm practice at the expense of conceptual and

graphical comprehension. Introducing symbolic manipulation programs, e.g., MathCAD, Mathematica,

Maple, would be valuable to subsequent electrical engineering courses whose instructors choose to

allow/encourage students to perform routine symbolic and numerical manipulations using such programs.

6. The first two years of mathematics that support instruction in electrical engineering should present stu-

dents with conceptual understanding of mathematical disciplines other than just single variable calculus,

multivariable calculus and ordinary differential equations. Other mathematical subjects that are important

for electrical engineering students include linear algebra, probability and stochastic processes, statistics,

and discrete mathematics.

Electrical engineering is an exciting and creative profession. Those engineers possessing an understanding

and facility of mathematics have an opportunity to be among the most creative of designers. Students need

to know and to feel how important, how useful, and how meaningful mathematics is. Many courses stress

the drudgery, not the beauty. This needs to be changed.

Electrical Engineering Subdisciplines

To describe our mathematics recommendations in sufficient detail, the undergraduate electrical engineering

curriculum is broken down into the following broad areas:

Page 3 Engineering: Electrical Engineering

67

1. Electrical Circuits

2. Electromagnetics

3. Systems, including Controls, Linear and nonlinear Circuits, and Power

4. Signals

5. Design

6. Microprocessor/Computer Engineering

What follows are summaries of the proposed mathematics requirements for each subdiscipline.

1. Electrical Circuits

The electrical circuits course is the passageway to electrical engineering. Of critical interest are the logical

thinking skills to analyze electric circuits. In this course, students are introduced to the application of phys-

ical laws, e.g., Ohm’s, Faraday’s and Kirchoff’s, in electrical engineering. Students are also introduced to

the electrical engineering foundation elements: resistor, inductor and capacitor, and their response (voltage,

current and power profiles) to DC, steady state AC, and transient stimuli respectively. In most institutions,

the circuits course is a two-part series with DC circuit analyses and transient response offered in the first

semester and AC circuits and steady state response offered the second semester.

A. DC Circuits. Typical problems in this section involve the simplification of series, parallel and mesh

circuits. Analyses of these circuits require setting up, manipulating, and obtaining solutions to alge-

braic equations. Subtle mathematical skills in the understanding of the circuit problems also include

direct and inverse proportionality to enable students to understand voltage and current divider rules

respectively.

In the circuit areas dealing with power, and power transfer, knowledge of integral calculus and

basic differentiation is required especially for maximum power transfer analysis.

B. AC Circuits. This part of circuit analysis deals with the response of different circuit configurations

and elements to steady state sinusoidal inputs. Different mathematical techniques are necessary to

simplify the circuits before gaining understanding of the response. The foundation mathematics

necessary for the analyses include:

Concept of functions, especially sinusoidal functions. Students need to understand and visualize

profiles of basic functions. Use of common real life examples is strongly suggested in teaching

this topic.

Application of trigonometric identities to sinusoidal analyses.

Manipulation and representation of sinusoidal functions in Euler, polar, and rectangular coordi-

nates.

Complex algebra.

C. Transients. This topic deals with response of discrete circuit elements, or combinations thereof, to

electrical stimuli. At the DC stage, the typical stimulus is the step. The mathematical background

required in the analysis includes exponential functions and introductory differential equations. In the

latter subject the primary focus should be on standard solutions to first and second order differen-

tial equations with constant coefficients rather than on more general techniques for solving differen-

tial equations. The preferred and more useful approach to solving differential equations in electrical

engineering is via the Laplace transform. Laplace transform methods reduce differential equation

problems to algebraic formats with which students feel more comfortable. This topic should be pre-

sented during the first year of undergraduate mathematics.

At the AC stage, the typical stimuli are sinusoidal, triangular and square functions. Usually, the

interest in this setting is steady state rather than transient. The Laplace transform still provides the

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The Curriculum Foundations Project

preferred method of analysis because, in addition to reducing differential equation problems to

algebraic equation problems, it also incorporates initial conditions in the solution.

2. Electromagnetics

Study of electromagnetic fields and waves is a crucial area in electrical engineering for which understanding

of vector algebra and vector calculus is required. The basic laws of electromagnetics are summarized in

Maxwell’s equations:

Faraday’s Law:

Ampere’s Law:

Gauss’ Law:

No Isolated Magnetic Charge:

These are partial differential equations that require deep conceptual understanding of vector fields and

operations related to vector fields: gradient, divergence, and curl. With the increasing power and avail-

ability of software, e.g., Mathematica, Maple, and Matlab, to perform the actual manipulations, it is crucial

that students develop a conceptual understanding of vector fields and related operations.

It is less important to emphasize the actual manipulations. For example, it is less important that a student

be given a scalar-valued function of several variables and be asked to compute the gradient. It is more

important that students be able to start with a contour plot (topographic map) of a scalar-valued function

of several variables and draw the gradient function.

It is less important that students start with a vector field and be able to compute the divergence or curl.

It is more important that students be able to interpret verbally and graphically pictures of vector fields.

Students should be able to identify regions in which the magnitudes of the divergence or curl will be

large or small. To support conceptual understanding, graphical interpretation, and verbal description it is

helpful to connect students of vector calculus with applications such as electromagnetic fields, fluid

mechanics and heat transfer.

The study of electromagnetics requires a conceptual understanding of partial differential equations and

their solutions, and the power and limitations of numerical solutions techniques. The study of specific par-

tial differential equations that permit closed-form solutions is less important than the development of this

conceptual picture.

Since most students in electrical engineering do not begin studying electromagnetic fields and waves until

their junior year, it is important that the relevant topics of vector calculus and partial differential equations

not be taught before the second semester of the sophomore year. Timing of the topics is important to help stu-

dents connect their studies in mathematics with their study of electromagnetics. Individual schools should

encourage conversations between faculty in electrical engineering and mathematics to prepare a mathemat-

ics curriculum that is responsive to the specific requirements of the electrical engineering department.

3. Systems

Control Systems-Linear and NonLinear Circuits. One purpose of systems analysis is to represent

reality mathematically. At the undergraduate level, linear time-invariant systems are discussed, studied and

designed. The systems may be continuous or discrete and may have one or more inputs and one or more

outputs. The system is modeled as a “box,” a device that modifies the signals entering it resulting in an

output according to the transfer function of the system:

0

B

?· =

K K

D ?

?· =

K K

D

H J

t

?

?× = +

?

K

K

K

K

B

E

t

?

?× =

?

K

K K

Page 5 Engineering: Electrical Engineering

69

Within electrical engineering, the systems problem has the following forms:

1. Find the transfer function of a SISO (single input-single output) system,

2. Find the transfer function of a MIMO (multi input-multi output) system,

3. Given the transfer function of a SISO system, what is the output of the system if the input to the sys-

tem is a specified function?

4. Given the transfer function of a SISO system, how must the system be modified to satisfy given spec-

ifications?

5. Given the state equation matrices for a MIMO system, what are the outputs of the system if the input

to the system is a specified vector function?

6. Given the state equation matrices for a MIMO system, how must the system be modified to satisfy

given specifications?

In undergraduate courses the systems studied are linear and time-invariant. Continuous systems can be

modeled by ordinary differential equations although the order of these equations might be quite high.

Software packages such as MATLAB are used extensively in most systems courses.

For continuous systems the mathematical tools needed consist of:

1. Laplace transforms and techniques such as partial fraction expansions and residues

2. State variable techniques including eigenvalues/eigenvectors, interpretation of the matrices, etc.

3. Basic differential equations, focused on standard solutions to common problems.

For discrete systems difference equations are used instead of differential equations and the discrete-

time state model is used. The mathematical tools needed are:

1. Difference equations

2. The state transition matrices and solutions to discrete-time state models

3. Z-transforms

4. Discrete Fourier transforms

5. Fourier analysis.

For both continuous and discrete systems it is important to be able to use the poles and zeros of trans-

formed time functions to visualize the system’s time response to various inputs. For continuous systems

the s-plane is important. Students should be able to plot the poles and zeros of the transfer function and

from this plot know the form of the impulse response by inspection. They should understand how the loca-

tions of the poles affect the output of the system. They should see how the locations of zeros affect the var-

ious modes of the system. In other words, they need to see the time response in the s-plane and understand

the physical realities encoded in the poles and zeros. For discrete systems the same holds true for the z-

plane.

The mathematical courses supporting systems are primarily linear algebra and ordinary differential

equations. As stated above, ordinary differential equations courses tend to overemphasize the development

of numerous solution methods for first- and second-order differential equations. In truth, systems deal with

higher order differential equations, and for SISO continuous systems Laplace transforms are almost always

the preferred method of solution. Discrete systems use the method of z-transforms. Both continuous and

discrete MIMO systems use state variable techniques.

Input

Output

SYSTEM

Page 6 70

The Curriculum Foundations Project

Consequently, we would prefer courses that place more emphasis on Laplace transforms, z-transforms,

and state variable techniques for solving ordinary differential and difference equations.

Linear algebra courses usually attempt to teach state variable techniques. We recommend that these

courses further develop the concepts taught in the “new” differential/difference equations course. State

transition matrices and their properties could be studied. Eigenvalues and eigenvectors could be explained.

The relation between the roots of the characteristic equation and eigenvalues could be stated. Other topics

might include:

1. Techniques for computing the matrix exponential and its integral,

2. Eigenvalue-eigenvector methods for computing matrix exponentials,

3. Decomposition of time-invariant state matrices,

4. Jordan forms,

5. Singular value decompositions and state space applications.

The elements of linear systems could be taught during the sophomore year.

Power Systems comprise the study of the transmission and distribution of electric power. The study of

power systems depends upon a firm mathematical grounding in the use and manipulation of trigonomet-

ric functions as well as algebraic manipulation of complex numbers. The use of phasor notation (an appli-

cation of polar co-ordinates) plays a central role in power systems analysis. Students must also know

Euler’s formula and be facile in going from polar to rectangular co-ordinates and vice-versa.

Power systems analysis requires not only algebraic manipulation but also recognition of the changes a

signal undergoes and the form of the signal that results. While a solution having the form

volts,

may be correct, the form

volts

is more useful. The student can visualize a sinusoidal wave with an amplitude of 1.414 volts that lags the

input signal by 45. Next, the student can visualize a sinusoid that decays exponentially. Thus, the wave-

form with its phase angle can be easily visualized, whereas the sum of sinusoids gives little information

about the phase angle. Power systems may be taught as early as the student’s 5th semester of undergrad-

uate studies.

4. Signals/Communications

One of the most fundamental applications in electrical engineering is the transmission, modification and

reception of signals. Communication systems is concerned with:

1. The transmission of signals through electric networks

2. The modulation and demodulation of signals

3. Sampling

4. Noise

5. Statistical methods of information transmission systems

Digital signal processing is an important area within electrical engineering. The digitization, modula-

tion, transmission, demodulation, and reception of signals is vital to modern communications. Image pro-

cessing and pattern recognition techniques fall within the purview of digital signal processing.

Communications and digital signal processing are taught in depth usually during the last two or three

semesters of the student’s undergraduate studies. The understanding of mathematical concepts is essential

2

3

( ) 5

5

[1.414cos( 45 )]

t

t

f t

e

e

t

–

–

=

–

– °

2

3

( ) 5

5

(cos sin )

t

t

f t

e

e

t

t

–

–

=

–

+

Page 7 Engineering: Electrical Engineering

71

within the communications area. Of particular importance are:

1. Basic algebraic techniques

2. Basic trigonometric identities

3. Integration techniques, including partial fractions and integration by parts

4. Taylor Series Expansion (i.e., linear approximation, expansion out to two or three terms)

5. The Fourier Transform and its use

6. Fourier Series

7. The use of the Laplace Transform

8. The use of the Z Transform

9. Probability and Stochastic Processes

5. Design

Design and modeling are two generic tasks in which engineers participate after completing their under-

graduate degrees. In addition to preparation in mathematics for the other disciplines within electrical engi-

neering, there are additional areas in mathematics that are necessary to support learning and growth in

design and modeling. Such areas include statistics, empirical modeling, parameter estimation, system

identification, model validation and design of experiments. Demand from industry for expertise in these

areas appears to be much stronger than demand within electrical engineering curricula. That may explain

why these areas are not prerequisites for courses in electrical engineering. However, expertise in these

areas is increasingly important for electrical engineering graduates.

Topics from these areas that will be valuable for engineering graduates include the concept of a random

variable, analysis of sets of data, concepts of sample means, sample variances and other sample statistics

as random variables, and hypothesis testing. To illustrate why these topics are important here are some

examples of applications.

First Example, Simple Parameter Estimation: Construct a circuit containing a resistor (resistance = R)

and a capacitor (capacitance = C). If the capacitor is initially charged and then discharged through the cir-

cuit, voltages and currents decay exponentially. Data on a particular voltage can be taken at various points

in time. Students then must estimate the time constant ( = RC) using the accumulated data. There are a

variety of techniques through which estimates of the time constant can be obtained. Students need to be

familiar with the techniques as well as the supporting concepts and broader applications.

Second Example, Design of Experiments: Design a feedback controller that meets several specifications

and minimizes percent overshoot. There are a number of parameters that may be adjusted. Students should

be able to design a set of experiments that will help determine narrow intervals for the parameter values

in order to optimize the design.

C

R

V

+

_

v(t)

i(t)

v(0)=Vo

Page 8 72

The Curriculum Foundations Project

Third Example, Model Validation: Develop an empirical model for a complex physical process. Once

the model is produced, students should be able to develop a set of experiments to help them understand

the validity of the model.

6. Computer Engineering/Microprocessors

Digital logic design and microprocessors require a mathematical background that is fundamentally different

than the background necessary for the areas discussed above. Circuits, electromagnetics, signals, and systems

require mathematics in which the variables can be any real number, i.e., continuous mathematics. Digital

logic design and microprocessors require mathematics in which the variables can only assume values in a

finite set, so-called discrete mathematics. Students need instruction that emphasizes the fundamental dif-

ference between continuous and discrete mathematics.

More specifically, students need Boolean algebra and finite state systems. For Boolean algebra, they

need to understand truth tables for the basic operators: NOT, AND, OR, NAND, and NOR. They need to

analyze combinational networks constructed from these basic operators and methods by which the net-

works may be simplified. Examples that help students relate combinational networks to actual applications

will help build motivation and understanding.

For finite state systems, students need to understand the concepts of a finite state machine and a state

transition diagram. Understanding these concepts can be strengthened by examining Mealy and Moore

realizations and the equivalence between the two realizations. In addition, connections between finite state

machines and regular expressions should be explored. Finally, students need to start with a description of

a physical situation, synthesize a state transition diagram, and then design the combinational logic that

together with memory can realize the state transition diagram.

Understanding of these concepts from Boolean algebra and finite state machines will provide students

with the necessary mathematical background to study computer engineering and microprocessors.

Understanding and Content

What follows are brief summaries of our responses to the specific questions posed by the Curriculum

Foundations Organizing Committee.

What conceptual mathematical principles must students master in the first two years?

The mathematics required for electrical engineering students should emphasize concepts and problem

solving skills more than emphasizing repetitive mechanics of solving routine problems. Students must

learn the basic mechanics of mathematics, but care must be taken that these mechanics do not become the

primary focus of any mathematics course.

What mathematical problem solving skills must students master in the first two years?

There is often a disconnect between the knowledge that students gain in mathematics courses and their

ability to apply such knowledge in engineering situations. Perhaps, the use of more engineering or real life

examples will reduce this disconnect. Too often mathematics is taught as a list of procedures or as theo-

rem-proof exercises without grounding the mathematics in reality. While we do not expect mathematics

H

K

X

Y

+

–

Page 9 Engineering: Electrical Engineering

73

instructors to be well versed in all engineering applications, we would like examples of mathematical

techniques explained in terms of the reality they represent.

Students of electrical engineering need to be skillful at mathematically modeling physical reality. They

need to be able to simplify complex problems, estimate the reasonableness of solutions, and visualize solu-

tions graphically from inspection of the mathematical descriptions.

What broad mathematical topics must students master in the first two years?

The first two years of mathematics that support instruction in electrical engineering should present stu-

dents with conceptual understanding of mathematical disciplines other than just single variable calculus,

multivariable calculus and ordinary differential equations. Other mathematical subjects that are important

for electrical engineering students include linear algebra, probability and stochastic processes, statistics,

and discrete mathematics.

Listed below are the most important mathematical topics that we believe students in electrical engi-

neering should learn during the first two years of undergraduate studies. All of these topics were discussed

earlier in this report. The Appendix provides another summary of topics, this one organized by the six sub

disciplines of electrical engineering that were identified in the previous section.

Manipulation, solution, and analysis of real and complex algebraic equations

Basic differential and integral calculus

Standard solutions for basic differential equations, in particular first- and second-order differential

equations with constant coefficients

Laplace, Fourier and Z transforms

Vector calculus

Taylor series

State variables and finite state systems

Difference equations

Probability and stochastic processes

Statistics

Model validation

Parameter estimation-techniques and application

Boolean algebra-analysis and application

Technology

How does technology affect what mathematics should be learned in the first two years?

New engineering and mathematical software only reduce the dependency on routine, excessive and repeti-

tive mathematical computations. Software should not be used to replace the necessity to teach students how

to pose and formulate mathematical questions and how to evaluate answers obtained for these questions.

What mathematical technology skills should students master in the first two years?

The use of mathematical software is necessary. In this regard, mathematics departments are strongly

encouraged to routinely use common math software tools and to promote students’ use of them. Failure to

utilize appropriate technological tools while continuing to focus on mastery of symbolic manipulation

often encourages memorization and rote algorithm practice at the expense of conceptual and graphical

comprehension.

Page 10 74

The Curriculum Foundations Project

What different mathematical technology skills are required of different student populations?

We did not identify any difference in requirements.

Instructional Techniques

What are the effects of different instructional methods in mathematics on students in your discipline?

We are not aware of differing effects.

What instructional methods best develop the mathematical comprehension needed for your discipline?

Hands-on, interactive engagements and project-based learning methods have been observed to promote

students’ learning.

Team teaching of mathematics courses should be considered. Teams could consist of faculty members

from both mathematics and electrical engineering. We believe that team-teaching could better motivate

and enthuse electrical engineering students.

Each mathematics course should clearly state its objectives. Students should be told what they are going

to study, why they are going to study it, and how it fits into the engineering profession.

What guidance does educational research provide concerning mathematical training in your discipline?

We are not aware of the relevant educational research.

Instructional Interconnections

What impact does mathematics education reform have on instruction in your disciplines?

We are not aware of or familiar with current directions in mathematics education reform

How does education reform in your discipline affect mathematics instruction?

We did not have a response to this question.

How can dialogue on education issues between discipline and mathematics best be maintained?

One effective method for continuing this dialogue between mathematics and electrical engineering is to

have more extensive contact between individuals in each discipline and the major professional associations

in each discipline. Professional engineering associations which should be contacted include:

The American Society for Engineering Educators (ASEE)

The National Electrical Engineering Department Heads Association (NEEDHA)

Similar organizations for mechanical, civil and chemical engineering

Page 11 Engineering: Electrical Engineering

75

WORKSHOP PARTICIPANTS

Ben Oni, Associate Professor of Electrical Engineering and Acting Dean of Engineering, Tuskegee

University

Brian Butz, Professor of Electrical Engineering, Temple University

Jeffrey Froyd, Research Professor in Electrical Engineering, Dwight Look College of Engineering, Texas

A&M University

Ronald Talley, Department Head and Professor of Electronic and Engineering Technolgy, Tri-County

Technical College, South Carolina

Mathematics Participants

William Barker, Professor of Mathematics, Bowdoin College

Robert Fennell, Professor of Mathematical Sciences, Clemson University

Page 12 76

The Curriculum Foundations Project

APPENDIX: Specific Issues on Understanding and Content

1. Electrical Circuits

Algebraic equations, manipulations and solutions

Differential and integral calculus

Concept of functions

Exponential functions

Sinusoidal functions, representation and manipulation in Euler, polar, rectangular coordinates

Application of trigonometric identities

Algebra of complex numbers

Introductory differential equations – focus on standard solutions to problems with basic inputs including

step, sinusoids, triangular, square functions (teach by end of first year)

Laplace transform (emphasize this topic over differential equation techniques in the first year)

2. Electromagnetics

Vector calculus (Do not teach before the 2nd semester of the sophomore year)

Conceptualization

Operation

Gradients

Divergence

3. Systems

Continuous Systems

Laplace transforms and techniques

Integration by parts

Partial fraction methods

State variables

Eigenvalues and eigenvectors

Basic differential equations-focus on standard solutions to common problems

Discrete Systems

Difference equations

Systems of first order differential equations

Z-transforms

Discrete Fourier transform

Fourier analysis and techniques

Power Systems

Sinusoidal functions

Algebra of complex numbers

Page 13 Engineering: Electrical Engineering

77

4. Signals/Communications

Fourier analysis, transform and techniques

Integration by parts

Partial fraction methods

Probability and stochastic processes

Z-transform

Taylor series (linear approximation-interested only in the first two terms)

5. Design

Statistics

Data collection

Sampling

Analysis (distribution, graphical techniques)

Concepts of random variables

Model validation

Parameter estimation

System identification

6. Computer Engineering/Microprocessors

Boolean algebra

Finite state systems

This is the html version of the file http://math.berkeley.edu/~wu/ICMtalk.pdf.

Google automatically generates html versions of documents as we crawl the web.

Page 1 How mathematicians can contribute to K-12

mathematics education

H. Wu

February 26, 2006

“To overcome the isolation of education research,

more effective links must be created between

educational faculties and the faculties of universities.

This could allow scholars of education better

acquaintance with new developments in and across

the disciplines and other professional fields of

the university, while also encouraging discipline-based

scholars with interests in education to collaborate

in the study of education.”

Lagemann, 2000, p. 241.

I would like to make a general disclaimer at the outset. I think I should only talk

about things I know firsthand, so I will limit my comments to the K-12 mathematics

education in the U.S. rather than take a more global view. Such a restriction is not

necessarily fatal since a friend of mine observed that what takes place in the U.S.

tends also to take place elsewhere a few years later. For example, in France there is

now a Math War that resembles the American Math Wars of the nineties (Education

Week, 2005). We live in a global village after all.

Let me begin with a fairy tale. Two villages are separated by a hill, and it was

decided that for ease of contact, they would drill a tunnel. Each village was entrusted

with the drilling of its own half of the tunnel, but after both had done their work, it

was discovered that the two halves didn’t meet in the middle of the hill. Even though

1

Page 2 a connecting tunnel between the two lengths already built could be done at relatively

small expense, the two villages, each in defense of its honor, prefer to continue the

quarrel to this day.

This fairy tale is too close to reality for comfort when the two villages are re-

placed by the education and mathematics communities, with the former emphasizing

the overriding importance of pedagogy and the latter, mathematical content.1 Math-

ematics education rests on the twin pillars of mathematics and pedagogy, but the

ongoing saga in mathematics education is mostly a series of episodes pitting one

against the other. There is probably no better proof of the disunity between these

communities than the very title of this article. Indeed, if someone were to write

about “How chemists can contribute to chemical engineering”, that person would be

considered a crank for wasting ink on a non-issue. Chemical engineering is a well-

defined discipline, and chemical engineers are perfectly capable of doing what they

are entrusted to do. They know the chemistry they need for their work, and if there

is any doubt, they would freely consult with their colleagues in chemistry in the spirit

of cooperation and collegiality. Therefore, the fact that we are going to discuss “How

mathematicians can contribute to K-12 mathematics education” in the setting of

the International Congress speaks volumes about both mathematics education and

mathematicians.

The title of this article implicitly gives away the power structure of mathematics

education in the academic world: Educators hold the rein. Since education research

is thriving and research funding is ample, it is not surprising that educators want

to protect their intellectual independence in the university environment. Rumblings

about how mathematically unqualified teachers or deficient curricula are undercutting

mathematics learning do surface from time to time, but we have not witnessed the

expected aggressive action agitating for collaboration with mathematicians. Other

troubling issues related to mathematics content, such as the presence of incorrect

assessment items in standardized tests, likewise fail to arouse genuine concern in the

mathematics education community. To an outsider, the protection of the “education”

1In writing about sociological phenomena, especially education, it is understood that all state-

ments are statistical in nature unless stated to the contrary, and that exceptions are part and parcel

to each statement. In fact, there are striking (though isolated) exceptions in the present context.

The reader is asked to be aware of this caveat for the rest of this article.

2

Page 3 enclave seems to matter more to university educators than collaboration with the

research mathematics community that could strengthen K-12 mathematics education.

By contrast, if the department of chemical engineering consistently produces engineers

with a defective knowledge of chemistry, or if accidents occur in its laboratories with

regular frequency, would the chemical engineering faculty not immediately spring to

action? This question prompts the thought that maybe we no longer know what

mathematics education is about, and it is time for us to take a second look.

One meaning of the word “engineering” is the art or science of customizing sci-

entific theory to meet human needs. Thus chemical engineering is the science of

customizing chemistry to solve human problems, or electrical engineering is the sci-

ence of customizing electromagnetic theory to design all the nice gadgets that we have

come to consider indispensable. I will put forth the contention that mathematics ed-

ucation is mathematical engineering, in the sense that it is the customization of

basic mathematical principles to meet the needs of teachers and students.2

I will

try to convince you that this is a good model for the understanding of mathematics

education before proceeding to a discussion of how mathematicians can contribute

to K-12 mathematics education. The far-from-surprising conclusion is that, unless

mathematicians and educators can work as equal partners, K-12 mathematics edu-

cation cannot improve.

Regarding the nature of mathematics education, Bass (2005) made a similar sug-

gestion that it should be considered a branch of applied mathematics.3 What I would

2After the completion of this article, Skip Fennell brought to my attention the article “Access

and Opportunities to Learn Are Not Accidents: Engineering Mathematical Progress in Your School”

by William F. Tate, which is available at:

http://www.serve.org/ downloads/publications/AccessAndOpportunities.pdf

Tate is concerned with equity and uses “engineering” as a metaphor to emphasize the potential for

designing different educational policies and pedagogical activities to promote learning, but without

addressing the mathematics. On the other hand, the present article explains why mathematics

education is the engineering of mathematics.

3Hy Bass lectured on this idea in December of 1996 at MSRI, but [5] seems to be a convenient

reference. After the completion of this article, Zalman Usiskin informed me that in the Proceedings

of the U.S.-Japan workshop on the mathematics education of teachers in 2000 that followed ICME-

9 in Japan, he had written that “‘Teachers’ mathematics’ is a field of applied mathematics that

deserves its own place in the curriculum.” Along this line, let it be mentioned that the paper of

Ferrini-Mundy and Findell [8] made the same assertion and, like Bass, it does not touch on the

engineering aspect of mathematics education. The need for mathematicians and educators to work

3

Page 4 like to emphasize is the aspect of engineering that customizes scientific principles to

the needs of humanity in contrast with the scientific-application aspect of applied

mathematics. Thus, when H. Hertz demonstrated the possibility of broadcasting

and receiving electromagnetic waves, he made a breakthrough in science by mak-

ing a scientific application of Maxwell’s theory. But when G. Marconi makes use

of Hertz’s discovery to create a radio, Marconi was making a fundamental contribu-

tion in electrical engineering, because he had taken the extra step of harnessing an

abstract phenomenon to fill a human need.4 In this sense what separates mathemat-

ics education as mathematical engineering from mathematics education as applied

mathematics is the crucial step of customizing the mathematics, rather than simply

applying it in a straightforward manner to the specific needs of the classroom. There

is no better illustration of this idea of customization than the teaching of fractions in

upper elementary and middle schools, as I now explain.

Students’ failure to learn fractions is well-known. School texts usually present

a fraction as parts of a whole, i.e., pieces of a pizza, and this is the most basic

conception of a fraction for most elementary students. However, when fractions are

applied to everyday situations, then it is clear that there is more to fractions than

parts-of-a-whole, e.g., if there are 15 boys and 18 girls in a classroom, then the ratio

of boys to girls is the fraction 15

18

, which has nothing to do with cutting up a pizza

into 18 equal parts and taking 15. In the primary grades, it is not a serious problem

if students’ knowledge of fractions is imprecise and informal, so that a fraction can be

simultaneously parts-of-a-whole, a ratio, a division, and an operator5, and a number.

Children at that age are probably not given to doubts about the improbability of an

object having so many wondrous attributes. At some stage of their mathematical

development, however, they will have to make sense of these different “personalities”

of a fraction. It is this transition from intuitive knowledge to a more formal and

abstract kind of mathematical knowledge that causes the most learning problems.

This transition usually takes place in grades 5-7.

on equal footing in mathematics education is likewise not mentioned by these educators.

4The invention was actually due to N. Tesla, but like many things in life, popular preception

displaces the truth. I am indebted to S. Simic for pointing this out to me.

5For example, the fraction 3

4

can be regarded as a function (operator) which associates to each

quantity three-quarters of the same quantity.

4

Page 5 There is by now copious mathematics education research6 on how to facilitate

children’s learning of the fraction concept at this critical juncture in order to op-

timize their ability to use fractions efficiently. At present, what most children get

from their classroom instruction on fractions is a fragmented picture of a fraction

with all these different “personalities” lurking around and coming forward seemingly

randomly. What a large part of this research does is to address this fragmentation

by emphasizing the cognitive connections between these “personalities”. It does so

by helping children construct their intuitive knowledge of the different “personali-

ties” of a fraction through the use of problems, hands-on activities, and contextual

presentations.

This is a good first step, and yet, if we think through students’ mathematical

needs beyond grade 7, then we may come to the conclusion that establishing cognitive

connections does not go far enough. What students need is an unambiguous definition

of a fraction which tells them what a fraction really is. They also need to be exposed to

direct, mathematical, connections between this definition and the other “personalities”

of a fraction. They have to learn that mathematics is simple and understandable, in

the sense that if they can hold onto one clear meaning of a fraction and can reason

for themselves, then they can learn all about fractions without ever being surprised

by any of these other “personalities”.

From a mathematician’s perspective, this scenario of having to develop a concept

with multiple interpretations is all too familiar. In college courses, one approaches

rational numbers (both positive and negative fractions) either abstractly as the prime

field of characteristic zero, or as the field of quotients of the integers. The problem

is that neither is suitable for use with fifth graders. This fact is recognized by math-

ematics education researchers, as is the fact that from such a precise and abstract

definition of rational numbers, one can prove all the assorted “personalities” of ra-

tional numbers. If I have read the research literature correctly, these researchers

despair of ever being able to offer proofs once they are forced to operate without an

abstract definition, and that is why they opt for establishing cognitive, rather than

mathematical connections among the “personalities” of rational numbers. The needs

6Here as elsewhere, I will not supply explicit references because I do not wish to appear to be

targeting specific persons or works in my criticism. I will be making generic comments about several

general areas.

5

Page 6 of the classroom would seem to be in conflict with the mathematics. At this point,

engineering enters.

It turns out that, by changing the mathematical landscape entirely and leaving

quotient fields and ordered pairs behind, it is possible to teach fractions as mathe-

matics in elementary school, by finding an alternate mathematical route around these

abstractions that would be suitable for consumption by children in grades 5-7. With-

out going into details, suffice it to say that at least the mathematical difficulties can

be overcome, for example, by identifying fractions with certain points on the number

line (for this systematic development, see, e.g., Jensen 2003, or Wu, 2001c). What is

of interest in this context is that this approach to fractions is specific to the needs

of elementary school and is not likely to be taught, ever, in any other situation. In

addition, the working out of the basic properties of fractions from this viewpoint is

not quite straightforward, and it definitely requires the expertise of a research math-

ematician. As to the further pedagogical implementation to render such an approach

usable in grades 5-7, the input of teachers and educators would be absolutely indis-

pensable.7 We therefore get to witness how mathematicians and educators are both

needed to turn a piece of abstract mathematics into usable lessons in the school class-

room. This is customization of abstract theory for a specific human need, and this is

engineering at work.

Through this one example of fractions, we get a glimpse of how the principles

of mathematical engineering govern the design of a curriculum. Less obvious but of

equal importance is the fact that even mathematics education research cannot be

disconnected from the same principles. If, for example, a strong mathematical pres-

ence had been integral to the research on fractions and rational numbers, it would

be very surprising that the research direction would have developed in the direction

it did. Compare the quote by Lagemann at the beginning of this article as well as

Lagemann, 2000.

An entirely analogous discussion of customization can be given to any aspect of

mathematics education, but we single out the following for further illustrations:

(a) The design of an “Intervention Program” for at-risk students. Up to this

7Some teachers who have worked with me are trying out this approach with their students in San

Francisco.

6

Page 7 point, the methods devised to help these students are largely a matter of

teaching a watered-down version of each topic at reduced pace; this is poor

engineering from both the theoretical and the practical point of view. In

Milgram-Wu, 2005, a radically different mathematical engineering design

is proposed to deal with this problem.

(b) The teaching of beginning algebra in middle school. The way symbols

are usually handled in such courses, which necessitates prolix discussions

in the research literature of the subtlety of the equal sign, and the way

variable is introduced as the central concept in school algebra are clear

indications that the algebra we teach students at present has not yet been

properly customized for the needs of school students. See the Preface and

Sections 1 and 2 of Wu, 2005d, and also Wu, 2006, for a more detailed

account of both the problems and their proposed solutions.

(c) The writing of mathematics standards at the national or state level. This

is an example of what might be called “practical optimization problems”,

which customize the mathematics to meet diverse, and at times conflict-

ing, needs of different clientele. Cf. Klein et al., 2005.

The concept of mathematics education as mathematical engineering also sheds

some light on Lee Shulman’s (1986) concept of pedagogical content knowledge. There

has been a good deal of interest in precisely describing the kind of knowledge a teacher

should possess in order to be effective in teaching. In the field of mathematics, at

least, this goal has proven to be elusive thus far (but cf. Hill-Rowan-Ball, 2004),

but Shulman’s intuitive and appealing formulation of this concept crystallizes the

diverse ideas concerning an essential component of good teaching. From the point

of view of mathematical engineering, one of the primary responsibilities of a teacher

is to customize her mathematical knowledge in accordance with the needs of each

situation for students’ consumption. This particular engineering knowledge is the

essence of pedagogical content knowledge. Although this approach to pedagogical

content knowledge does not add anything new to its conception, it does provide a

framework to understand this knowledge within mathematics, one that is different

from what one normally encounters in educational discussions. It makes explicit at

7

Page 8 least three components to effective teaching: a solid mathematical knowledge, a clear

perception of the setting defined by the students’ knowledge, and the flexibility of

mind to customize this mathematical knowledge for use in this particular setting

without sacrificing mathematical integrity.

The idea of customizing mathematics “without sacrificing mathematical integrity”

is central to mathematical engineering. In engineering, it is obvious that, in trying to

customize scientific principles to meet the needs of humanity, we cannot contradict

nature regardless of how great the human needs may be. In other words, one respects

the integrity of science and does not attempt anything so foolish as the construction

of anti-gravity or perpetual-motion machines. Likewise, as mathematical engineering,

mathematics education accepts the centrality of mathematics as a given. Again using

the example of teaching fractions, a mathematics educator would know that no matter

how one tries to teach fractions, it must be done in a way that respects the abstract

meaning of a fraction even if the latter is never used explicitly. If, for instance,

an educator catches himself saying that children must adopt new rules for fractions

that often conflict with well-established ideas about whole numbers, then he knows he

is teaching fractions the wrong way because, no matter what efforts one puts into

making fractions intuitive to children, one cannot do violence to the immutable fact

that the rational numbers contain the integers as a sub-ring. The need to teach the

arithmetic of fractions as a natural extension of the arithmetic of whole numbers has

gone unnoticed for far too long, with the result that too many of our students begin to

harbor the notion that, after the whole numbers, the arithmetic of fractions is a new

beginning. Such bad mathematical engineering in curricular designs is unfortunately

a common occurrence.

The only way to minimize such engineering errors is to have both mathemati-

cians and educators closely oversee each curricular design. In fact, if we believe in

the concept of mathematics education as mathematical engineering, then the two

communities must work together in all phases of mathematics education: Any educa-

tion project in mathematics must begin with a sound conception of the mathematics

involved, and there has to be a clear understanding of what the educational goal

is before one can talk about customization. In this process, there is little that is

purely mathematical or purely educational; almost every step is a mixture of both.

Mathematics and education are completely intertwined in mathematical engineering.

8

Page 9 Mathematicians cannot contribute to K-12 mathematics education if they are treated

as outsiders.8 They have to work alongside the educators on equal footing in the plan-

ning, implementation, and evaluation of each project. But this is far from the reality

at present.

For at least three decades now, the mathematics and K-12 education communities

in the U.S. have not been on speaking terms in the figurative sense. (Cf. Washington

Post, 1999.) The harm this communication gap has brought to K-12 mathematics

education can be partially itemized, but before doing that, let me point out three

general consequences of a philosophical nature. The first one is that the isolation

of the education community from mathematicians causes educational discussions to

over-focus on the purely education aspect of mathematics education while seemingly

always leaving the mathematics untouched. The result is the emergence of a subtle

mathematics avoidance syndrome in the education community, and this syndrome will

be seen to weave in and out of the following discussion of the specific harmful effects of

this communication gap. Given the central position of mathematics in mathematical

engineering, it would be noncontroversial to say that this syndrome should vanish

from all discussions in mathematics education as soon as possible.

The fact that many mathematicians teach mathematics and design mathematics

courses throughout their careers seems to escape the attention of many educators.

Here is a huge reservoir of knowledge and experience in mathematical engineering on

tap. The chasm between the two communities in effect denies educators access to

this human resource at a time when educators need all the engineering help they can

get.

The final consequence can best be understood in terms of the Darwinian dic-

tum that when a system is isolated and allowed to evolve of its own accord, it will

inevitably mutate and deviate from the norm. Thus when school mathematics ed-

ucation is isolated from mathematicians, so is school mathematics itself, and, sure

enough, the latter evolves into something that in large part no longer bears any re-

semblance to mathematics. Correct definitions are not given, or if given, they are

not put to use (Milgram-Wu, 2005, Wu, 2001a, 2005a and 2005c). The organic co-

8This only tells half the story about mathematicians. See the comments near the end of this

article.

9

Page 10 herence of mathematics is no longer to be found (Wu, 2002), or when “mathematical

connections” are intentionally emphasized, such “connections” tend to be the trivial

and obvious kind. Logical deduction becomes an afterthought; proofs, once relegated

to the secondary school geometry course, were increasingly diluted until by now al-

most no proofs at all are found there, or anywhere else in the schools (Wu, 2004).

And so on. This development naturally brings down the quality of many aspects of

mathematics education.

The absence of dialog between the two communities has led to many engineering

errors in mathematics education, one of them being the unwelcome presence of math-

ematically incorrect test items in state and other standardized tests (Milgram 2002,

2003). The same kind of defective items also mar many teachers’ credentialing tests

(Askey 2006a, 2006b). A more subtle effect of the absence of mathematical input on

assessment is the way test scores are routinely misinterpreted. The low test scores

have been used to highlight students’ dismal mathematical performance, but little

or no thought is given to the possibility that they highlight not necessarily students’

achievement (or lack thereof) but the pervasive damage done by defective curricular

materials, or even the chronic lack of effective teaching. Such a possibility may not

be obvious to anyone outside of mathematics, but to a mathematician, it does not

take any research to confirm the fact that when students are taught incorrect math-

ematics, they learn incorrect mathematics. Garbage in, garbage out. If the incorrect

mathematics subsequently shows up in students’ test scores, how can we separate

the errors due to the incorrect information students were given, from the errors due

to students’ own misconceptions? A more detailed examination of this idea in the

narrow area of school algebra is given in Wu, 2006. The need for mathematicians’

participation in all phases of assessment is all too apparent.

The lack of collaboration between mathematicians and mathematics educators

affects professional development as well. The issue of teacher quality is now openly

acknowledged and serious discussions of the problem are beginning to be accepted

in mathematics education (cf. Ma, 1999, and Conference Board of the Mathematical

Sciences, 20019). As a result of the inadequate mathematics instruction teachers

receive in K-12, their knowledge of mathematics is, by and large, the product of the

9Whatever reservations one may have concerning the details of its content, it is the fact that such

a volume could be published under the auspices of a major scientific organization that is important.

10

Page 11 mathematics courses they take in college.10 In very crude terms, the number of such

required mathematics courses is too low, and in addition, these courses are taught

either by mathematicians who are not in close consultation with teachers, and are

unaware as to what is needed in the school classroom, or by mathematics educators

who are not professional mathematicians. The former kind of course tends to be

irrelevant to the classroom, and the latter kind tends to be mathematically shallow

or incorrect. It is only natural that teachers coming out of such an environment turn

out to be mathematically ill-prepared.

Similar woes persist in in-service professional development, thereby ensuring that

teachers have little access to the mathematical knowledge they need for their pro-

fession. For example, the last decade has witnessed the appearance of case books

consisting of actual records of lessons given by teachers.11 The idea is to invite teach-

ers to analyze these lessons, thereby sharpening their pedagogical sensibilities. In too

many instances, however, blatant mathematical flaws in the cited cases are overlooked

in the editors’ commentaries. This raises the specter of bringing up a generation of

teachers who are proficient in teaching school students incorrect mathematics. In

this instance, it would appear that the need to respect mathematical integrity in

mathematical engineering has been all but forgotten.

The most divisive outcome of the noncommunication between the two communi-

ties in the U.S. is undoubtedly the conflict engendered by the new (reform) curricula

written in the past fifteen years. I take up this discussion last, because it brings

us face to face with some subtle issues about mathematicians’ participation in K-12

mathematics education. The prelude to the writing of these curricula is the unchecked

degeneration in the mathematical integrity of the existing textbooks from major pub-

lishers over the period 1970-1990, a fact already alluded to above. This degeneration

triggered the reform spearheaded by NCTM (National Council of Teachers of Math-

ematics, 1989). Rightly or wrongly, the new curricula were written under the banner

of the NCTM reform, and the manner in which some of the reform texts were imposed

10It may be useful to also take note of what may be called “the second order effect” of university

instruction: teachers’ knowledge of mathematics is also conditioned by their own K-12 experiences,

but these teachers’ teachers were themselves products of the mathematics courses they took in the

university.

11Let it be noted explcitly that I am discussing the case books in K-12 mathematics education

only.

11

Page 12 on public schools led eventually to the well-known Math Wars (Jackson, 1997). The

root of the discontent over these texts is the abundance of outright mathematical

errors,12 as well as what research mathematicians perceived to be evidence of a lack

of understanding of the mathematics. An example of the latter was the promotion of

children’s invented algorithms at the expense of the standard computation algorithms

in the elementary mathematics curriculum. Although the promotion was partly an

overreaction to the way the standard algorithms were often inflicted on school children

with nary a word of explanation, it also reflected a lack of awareness of the central

importance of the mathematical lessons conveyed by the reasoned teaching of these

algorithms.

The “subtle issues” mentioned above stem from the fact that the writing of some

of the new reform curricula actually had the participation of a few mathematicians.

The first thing to note is that the latter are the rare exceptions to the general non-

communication between the mathematics and education communities. The noncom-

munication is real. At the same time, these exceptions seem to point to an apparent

contradiction: How would I reconcile my critical stance toward these reform curricula

with the principal recommendation of this article, namely, that mathematicians be

equal partners with educators in the mathematics education enterprise? The answer

is that there is no contradiction at all. The participation by mathematicians is, in

general terms, a prerequisite to any hope of success in K-12 mathematics education,

but in no way does it guarantee success. It is helpful in this context to recall similar

discussions that routinely took place some eight years ago when some mathemati-

cians first went public with the idea that mathematics teachers must have a solid

content knowledge. The usual rejoinder at the time was that “knowing mathematics

is not enough (to be a good teacher)”. This is a common confusion that mistakes a

necessary condition for a sufficient condition.13 There is no quick fix for something

as complex as mathematics education. Getting mathematicians to fully participate

is only the beginning; the choice of the mathematicians and the hard work to follow

will have a lot to say about the subsequent success or failure.

It is appropriate at this point to recall what was said at the beginning of the

12These errors tend to be different from the earlier ones to be sure, but errors they are.

13And need I point out, there are some who intentionally use this confusion to reject that mathe-

matical content knowledge is important for teachers, or that getting mathematicians to participate

in mathematics education is critical for its success.

12

Page 13 article about the power structure of mathematics education: thus far, educators get

to make the decisions. Granting this fact, I should amplify a bit on the difficulties of

choosing the right mathematicians for education work. Mathematicians have a range

of background and experiences and, consequently, often have a range of opinions on

matters of education as well. It is important that the range of these opinions be

considered in all aspects of education. Many of the less happy incidents of the recent

past in K-12 mathematics education were the result of choosing mathematicians of

a particular persuasion. In addition, educators must make their own judgement on

which among the mathematicians interested in K-12 are knowledgeable about K-12.

Among the latter, some possess good judgment and leadership qualities while others

don’t. Educators must choose at each step. If there are algorithms for making the

right choices, I don’t happen to know them.

Every mathematician potentially has something to offer in K-12 mathematics

education: even an occasional glance at textbooks to check for mathematical cor-

rectness can be very valuable. However, if mathematicians want to participate in

serious educational work in K-12, what must they bring to the table? I believe the

most important thing is the awareness that K-12 mathematics education is not a

subset of mathematics, and that there is quite a bit to learn about the process of

customization that distinguishes K-12 mathematics education from mathematics. In

particular, much (if not most) of the mathematics they teach in the university cannot

be brought straight to the school classroom (Wu, 1997; Kilpatrick et al., 2001, Chap-

ter 10 and especially pp. 375-6), but that it must first go through the engineering

process to make it suitable for use in schools. If I may use the example of fractions

once again, mathematicians interested in making a contribution to K-12 may find it

instructive to get to know the reason that something like “equivalence classes of or-

dered pairs of integers” is totally opaque to students around the age of twelve. They

would also want to know the reason that students of that age nonetheless need a

definition of a fraction which is as close to parts-of-a-whole as possible. They should

also get to know the appropriate kind of mathematical reasoning for students in this

age group, because they will ultimately be called upon to safeguard such reasoning

in the curriculum and assessment for these students.

Mathematicians may regard school mathematics as technically primitive (in the

sense of skills), but they must take note of its conceptual sophistication (Jensen, 2003;

13

Page 14 Wu, 2001b, 2001c, and 2005d; cf. also Aharoni, 2005). Above all, they must know

that school mathematics is anything but pedagogically trivial: There is absolutely

nothing trivial about putting any material, no matter how simple, into a correct

mathematical framework so that it may be profitably consumed by school students.

Mathematicians who want to contribute to K-12 mathematics education have to be

constantly on the alert to ensure that the minimum requirements of their profession

– the orderly and logical progression of ideas, the internal cohesion of the subject,

and the clarity and precision in the presentation of concepts, – are still met in math-

ematics education writings. This is no easy task. If mathematicians want to enter

K-12 mathematics education as equal partners with educators, then it is incumbent

upon them to uphold their end of the bargain by acquiring this kind of knowledge

about mathematical engineering.

The concept of mathematics-education-as-mathematical-engineering does not sug-

gest the creation of any new tools for the solution of the ongoing educational problems.

What it does is to provide a usable intellectual framework for mathematics education

as a discipline, one that clarifies the relationship between the mathematics and the

education components, as well as the role of mathematicians in mathematics educa-

tion. For example, it would likely lead to a better understanding of why the New

Math became the disaster that it did. Most importantly, this concept lays bare the

urgent need of the mathematical presence in every aspect of K-12 mathematics ed-

ucation, thereby providing a strong argument against the self-destructive policy of

keeping mathematicians as outsiders in mathematics education. The chasm between

mathematicians and educators must be bridged if our children are to be better served.

I am cautiously optimistic14 that there are enough people who want to rebuild this

bridge (cf. Ball et al., 2005), all the more so because the indications are that the

NCTM leadership is also moving in the same direction. I look forward to a future

where mathematics education is the joint effort of mathematicians and educators.

Acknowledgement I am first of all indebted to my colleague Norman E. Phillips for

providing a critical piece of information about chemistry that got this article off the ground.

The suggestion by Tony Gardiner to re-organize an earlier draft, and the penetrating com-

14In January of 2006.

14

Page 15 ments on that draft by Helen Siedel, have left an indelible imprint on this article. Tom

Parker, Ralph Raimi, and Patsy Wang-Iverson gave me very detailed corrections. David

Klein also made corrections and alerted me to one of the references. In addition, the fol-

lowing members of the e-list mathed offered suggestions for improvement: R. A. Askey,

R. Bisk, E. Dubinsky, U. Dudley, T. Foregger, T. Fortmann, K. Hoechsmann, R. Howe,

W. McCallum, J. Roitman, M. Saul, D. Singer, A. Toom. Cathy Seeley and Skip Fennell

also made similar suggestions.

It gives me pleasure to thank them all.

References

Aharoni, R. (2005). What I Learned in Elementary School. American Educator, Fall issue.

http://www.aft.org/pubs-reports/american educator/issues/fall2005/aharoni.htm

Askey, R. A. (2006a). MSRI presentation in 2004, to appear.

Askey, R. A. (2006b). Mathematical content in the context of this panel. In: Mogens Niss

et al. (Eds). Proceedings of the Tenth International Congress on Mathematical Education.

Ball, D. L., Ferrini-Mundy, J., Kilpatrick, J., Milgram, J. R., Schmid, W., and Schaar, R.

(2005). Reaching for common ground in K-12 mathematics education. Notices Amer. Math. Soc.

52, 1055-1058.

Bass, H. (2005). Mathematics, mathematicians, and mathematics education. Bulletin

Amer. Math. Society, 42, 417-430.

Conference Board of the Mathematical Sciences. (2001). The Mathematical Education of

Teachers, CBMS Issues in Mathematics Education, Volume 11. Providence, RI: American

Mathematical Society.

Education Week. (2005) A Purge at the French High Committee for Education (HCE).

Education Week, November 27, 2005. http://www.educationnews.org/A-Purge-at-the-French-

High-Committee-for-Education-HCE.htm

Ferrini-Mundy, J. and Findell, B. (2001). The mathematics education of prospective teach-

ers of secondary school mathematics: old assumptions, new challenges. In: CUPM Dis-

cussion Papers about Mathematics and the Mathematical Sciences in 2010: What Should

Students Know? Washington DC: Mathematical Association of America.

Hill, H., Rowan, B., and Ball, D. L. (2004). Effects of teachers’ mathematical knowledge

for teaching on student achievement,

http://www-personal.umich.edu/ dball/BallSelectPapersTechnicalR.html

Jackson, A. (1997). The Math Wars: California battles it out over mathematics education

reform. Notices of the American Mathematical Society, Part I, June/July, 695-702; Part II,

August, 817-823.

15

Page 16 Jensen, G. (2003). Arithmetic for Teachers. Providence, RI: American Mathematical Soci-

ety.

Kilpatrick, J. Swafford, J. and Findell, B., eds. (2001). Adding It Up. Washington DC:

National Academy Press.

Klein, D. et al. (2005). The state of State MATH Standards. Washington D.C.: Thomas

B. Fordham Foundation.

http://www.edexcellence.net/foundation/publication/publication.cfm?id=338

Lagemann, E. C. (2000). An Elusive Science: The Troubling History of Education Research.

Chicago and London: The University of Chicago Press.

Ma, L. (1999). Knowing and Teaching Elementary Mathematics, Mahwah, NJ: Lawrence

Erlbaum Associates.

Milgram, R. J. (2002) Problem solving and problem solving Models for K-12: Preliminary

Considerations. http://math.stanford.edu/ftp/milgram/discussion-of-well-posed-problems.pdf

Milgram, R. J. (2003) Pattern recognition problems in K – 12.

http://math.stanford.edu/ftp/milgram/pattern-problems.pdf

Milgram, R. J. and Wu, H. (2005) Intervention program, http://math.berkeley.edu/~wu/

National Council of Teachers of Mathematics. Curriculum and Evaluation Standards for

School Mathematics. Reston, VA: National Council of Teachers of Mathematics.

Shulman, Lee. (1986). Those who understand: Knowledge growth in teaching, Educational

Researcher, 15, 4-14.

Washington Post. (1999). An Open Letter to United States Secretary of Education, Richard

Riley. November 18. http://mathematicallycorrect.com/nation.htm

Wu, H. (1997). On the education of mathematics teachers (formerly entitled: On the

training of mathematics teachers). http://math.berkeley.edu/~wu/

Wu, H. (2001a). What is so difficult about the preparation of mathematics teachers?

http://math.berkeley.edu/~wu/

Wu, H. (2001b). Chapter 1: Whole Numbers (Draft). http://math.berkeley.edu/~wu/

Wu, H. (2001c). Chapter 2: Fractions (Draft), http://math.berkeley.edu/~wu/

Wu, H. (2004). Geometry: Our Cultural Heritage – A book review. Notices of the American

Mathematical Society, 51, 529-537. http://math.berkeley.edu/~wu/

Wu, H. (2005a). Key mathematical ideas in grades 5-8. http://math.berkeley.edu/~wu/

Wu, H. (2005b). Must content dictate pedagogy in mathematics education?

http://math.berkeley.edu/~wu/

Wu, H. (2005c). Professional development: The hard work of learning Mathematics.

http://math.berkeley.edu/~wu/

16

Page 17 Wu, H. (2005d). Introduction to School Algebra (Draft), http://math.berkeley.edu/~wu/

Wu, H. (2006). Assessment in school algebra. To appear.

Hung-Hsi Wu

Department of Mathematics, #3840

University of California at Berkeley, CA 94720-3840

wu@math.berkeley.edu

17

This is the html version of the file http://elib.mi.sanu.ac.rs/files/journals/tm/23/tm1221.pdf.

Google automatically generates html versions of documents as we crawl the web.

Page 1 THE TEACHING OF MATHEMATICS

2009, Vol. XII, 2, pp. 51-56

LONG TERM EFFECTS IN LEARNING MATHEMATICS IN

FINLAND-CURRICULUM CHANGES AND CALCULATORS

Olli Martio

Abstract. Two similar tests to measure the skills of the Finnish school children in

mathematics took place in 1981 and 2003. The tests are compared to a test measuring

the knowledge of basic concepts in mathematics after the student examination. The

results of the tests reflect the changes in the mathematics curriculum and teaching

practices in Finland.

ZDM Subject Classification: D63, D64; AMS Subject Classification: 00A35.

Key words and phrases: Student’s assessment; calculators; PISA; TIMMS.

1. Introduction

Curricula changes in the Finnish school system have taken place in 8-10 year

intervals. The official curriculum texts are rather short in details. Schools are

free to choose their textbooks and there is neither an official inspection nor an

official approval of the textbooks in Finland. The free market principle prevails.

Hence textbooks and teaching practices should be studied in order to understand

the mathematics curriculum. A similar system is used in many countries. A rather

detailed description of mathematics and science teaching in Finland can be found

in [1]. This collection of articles also contains an account of the teacher training

system used in Finland.

Almost everybody finishing the high school (gymnasium) participates in the

matriculation (student) examination at the age of 18. Hence this test provides

an opportunity to study the final effectiveness of the Finnish school system. The

mathematics test is not obligatory although most students take it. The matric-

ulation test is 150 years old and its mathematics part has essentially remained

the same, except for the problems, for the last hundred years. In mathematics a

student may choose a basic or an advanced test independently of which courses

the student has followed at school. The basic test is more common. Both tests

consist of 15 problems written on an A4 sheet. A student can choose at most 10

problems out of 15. In practice, solving two problems, or slightly less, he or she

is able to pass the test. Eight or nine correctly solved problems is the standard

requirement for the highest grade but this varies annually. The students are graded

using seven grades whose distribution is the same each time. Because of this the

grades in matriculation tests cannot be used to compare changes in mathematical

skills of the students. The test problems have changed considerably during the

Page 2 52

O. Martio

last decades. The problems are based on the aforementioned, rather loosely stated,

official curriculum.

A survey of the Finnish matriculation test in mathematics is in [2].

The purpose of this article is to study the changes in curriculum and teaching

practices that have had the most serious long term effects in learning mathematics

in Finland.

2. Mathematics curriculum-changes and effects

The changes in the mathematics curriculum in Finland have followed the in-

ternational trends. Since 1970 three major revisions have taken place. The first

was influenced by the so-called New Math. This created a lot of discussion but

had a relatively small effect. The second revision can be labelled “Back to basics”.

The last change “Problem solving” had a much greater impact. It was very much

influenced by the demand that the applications of mathematics are all important-

mathematics as such has little value. The influence of calculators was also profound.

It was thought unnecessary to teach those skills which can be performed by a cal-

culator. Similar changes were experienced in other OECD countries.

In Finland these trends had the following effects on the mathematics curricu-

lum.

• Mathematics at school became descriptive – exact definitions and proofs were

largely omitted.

• Geometry was neglected.

• Computations were performed by calculators and numbers and not on a more

advanced level.

Students also experienced difficulties when moving from elementary school

mathematics to secondary school mathematics and especially to high school math-

ematics. Little has been done to ease this friction.

A rather recent test problem in a basic mathematics matriculation examina-

tion demonstrates these effects. “Why is the sum of the angles in a triangle 180

degrees?” Nobody knew although the problem was explained in some textbooks

(a line cuts two parallel lines in equal angles). This shows that teaching of mathe-

matical principles has declined, at least on the basic course, and replaced by a list

of facts given without reasoning. Many teachers are content to demonstrate this

property of all triangles with scissors and paper.

L. Näveri [3] has studied the effects of the curriculum changes in Finland. Two

similar tests were performed in mathematics in 1981 and in 2003. Participants

belonged to the age group 15-16 year old (9. grade); this corresponds to the age

group in the PISA survey since the school starts at the age of seven in Finland. The

tests were participated by more than 350 students. The problems were identical

and supposed to be solved without a calculator. In the following only samples of

the test questions are presented.

The first samples of questions concerns multiplication and the percentages

show the correct answers.

Page 3 Long term effects in learning mathematics in Finland

53

Multiplication

1981

2003

5 · 5 · 5 · 5=54

95.2%

90.1%

(-3)2 = 9

67.8%

47.5%

18 · 4 · 32 · 15 = 15 · 32 · 4 · 18

93.2%

85.9%

0,015 · 248 = 0,15 · 24,8

66.8%

62.3%

0 · 8436 = 0 · 0,536

79.0%

65.6%

In the questions concerning rational numbers the performance drop from 1981

to 2003 was the highest, 20%.

Rational numbers

1981

2003

26 + 17 =

98.5%

89.8%

(1/2) · (2/3) =

56.4%

36.9%

(4/3) · 5 =

66.3%

44.4%

(1/6) · (1/2) =

56.5%

28.3%

(1/5) : 3 =

49.2%

27.5%

1278/2 =

55.1%

36.8%

Also in the algebra section the results did not give a healthy picture of the

effects of the curriculum changes.

Algebra

1981

2003

103 · 102 =

72.5%

43.3%

x4 · x5 =

71.7%

47.3%

(592)3 = (593)2

61.1%

31.7%

If calculators were allowed in the test, the results would have most likely shown

different figures.

In the 2003 survey it was also asked: Explain with your own words the meaning

of (4/5) · 5. The results were as follows:

Correct

6.5%

Almost correct

5.4%

Correct computation but explanation incorrect

8.8%

No explanation but computation corect

31.5%

Incorrect computation and explanation

31.0%

No answer

16.8%

Page 4 54

O. Martio

Rather few reliable international surveys have been made to compare the

changes of the students performance in elementary and secondary school math-

ematics in the time scale of 2-3 decades. It is difficult to separate the effects that

are due to the changes in the curriculum from those which are due to changes in

teaching practices. The survey [3] certainly shows that these effects exist. It would

be interesting to survey the situation in other countries and to look for a general

pattern behind the results. There is at least one reason behind the above results.

It is the use of calculators.

Finland was the best country among the OECD countries in the PISA 2003 sur-

vey. This survey concentrated to the 14 year old age group. The type of questions

asked in [3] were rare in the PISA test. In the TIMSS 1999 report the performance

of the Finnish pupils was also above average. In the latter test the problems were

closer to the questions asked in [3]. The reasons for the Finnish PISA success are

analyzed in [1]. A more critical discussion can be found in the Finnish electron-

ic journal Solmu [4] of school mathematics (http://solmu.math.helsinki.fi/)

where two special issues have been devoted to the Pisa survey.

3. After the matriculation examination

Students, who have passed the matriculation test, do not only go to universities

to study. Many of them go to professional schools (training schools for nurses,

various engineering colleges etc.)-usually, but not necessarily, they are students

who have got low grades in the matriculation test.

During the last ten years teachers in professional schools, and not only math-

ematics teachers, have complained on the level of the mathematical skills of the

new students. The following sample from [5] shows that these complaints are not

without basis. The test was performed for freshmen in an engineering college and

the figures indicate the percentages of those who correctly answered the problems

on the left hand side. “Basic test” and “Advanced test” mean students who have

passed the corresponding matriculation examination in mathematics. Calculators

were not allowed.

Basic test

Advanced test

v

32 + 42 =

55%

78%

(1/3 – 1/7)/4 =

25%

54%

a2 – (a + 1)2 + 2a =

17%

50%

Find R from the formula U = E – IR

26%

68%

ln(x2) – 2 ln x =

7%

34%

The test shows that formula handling, rational numbers, logarithms and alge-

braic operations by hand are difficult for those who have passed the basic matric-

ulation examination. Among the students who have passed the advanced test and

Page 5 Long term effects in learning mathematics in Finland

55

who have had much more mathematics lessons at school there are many who have

not learnt basic algebraic operations.

4. Conclusions

The most serious drawbacks in the Finnish mathematics curriculum are the

order and time allocated to different concepts and skills. It is outside the scope

of this report to analyze the situation in detail. Some typical examples can be

mentioned. In the advanced course probability is taught before the concept of an

integral and sequences and series are left to the very end. As the above reports

indicate there are serious defects in the secondary school mathematics curriculum.

In Finland a customer cannot any more ask for 3/4 kilogram meat in a butcher’s

shop since the meaning is not known to a shop-assistant. The right expression is

750 g since this can be fed to a computer. Although the changes in the mathematics

curriculum were made to help people to use mathematics in everyday life, this aim

has badly failed. The problems now considered at school are not those people meet

later on. Problem solving has been overestimated in all levels of the mathematics

curriculum. Teachers at professional schools have learnt this in a hard way.

From the studies [3] and [5] a serious defect in mathematics teaching emerges.

This is the incorrect use of calculators in teaching. Although number handling is

learnt by pen and paper at the elementary school, many students later completely

forget this skill because they have got used to calculators. This does not concern

so much the best 15-20% of the students as results show in the matriculation

examination. The use of calculators is overemphasized since nowadays their use is

extremely limited in everyday life. Professional users of mathematics almost never

use them. Hence the time spent with calculators does not follow the idea that the

skills obtained at school should have some practical value later.

Calculators came to schools in 1975-1995. After a slow beginning they are now

used more and more. No doubt this has been the most essential change in teaching

mathematics and the effects can be seen in the reports [3] and [5].

Mathematics does not concern professional mathematicians only. Mathemat-

ics is used more and more in ordinary professions and the problems involved are

different from those in the PISA survey. In Finland, as in many countries, the

mathematics curriculum includes concepts and skills which once have been put

there because somebody has thought them useful. In most cases time has shown

that these special skills do not meet the demands of the society any more. The

Finnish curriculum architecture and teaching practices require considerable changes

to meet the challenge. Here Finland is not alone.

REFERENCES

[1] How Finns Learn Mathematics and Science, Editors: E. Pehkonen, M. Ahtee, J. Lavonen,

Sense Publishers 2007, 278 pp.

[2] Lahtinen, A., The Finnish Matriculation Examination in Mathematics, In: Nordic Presen-

tations (eds. E. Pehkonen, G. Brandell & C. Winslw), 2005, 64-68. University of Helsinki.

Department of Applied Sciences of Education. Research Report 262.

Page 6 56

O. Martio

[3] Näveri, L., Understanding computations, Dimensio 3/2005, 49-52 (in Finnish).

[4] SOLMU, 2 special volumes on the PISA survey 1/2005-2006, 2/2005-2006: http://solmu.

math.helsinki.fi/2005/erik1/ (in Finnish) and http://solmu.math.helsinki.fi/2006/

erik2/ (in Finnish).

[5] Tuohi, R. et al., Fact or fiction-mathematical skills of new engineering students, Turun

ammattikorkeakoulun raportti 29, Turku 2004 (in Finnish).

Department of Mathematics and Statistics, P.O.Box 68 (Gustaf Hällströmink. 2b), FI 00014

University of Helsinki, Finland

E-mail: olli.martio@helsinki.fi