The Journal of Online Mathematics and Its Applications

Volume 7. June 2007. Article ID 1405

An Evolutionary Text Book--Evolving by Student Activity

Håkan Lennerstad, David Erman, Maria Salomonsson

Abstract

This article describes a project that involves an undergraduate course in calculus. The course textbook is systematically revised by means of student "dialogues" conducted through a specially designed web site. The project thus extends the students' role in their education, and the textbook evolves to better adapt to the environment for which it is intended-- namely student learning.

Keywords

Main Contents

  1. Introduction
  2. Goals, plans, and practical realization
  3. Results and generalizations

Ancillary Materials

1. Introduction

1.1 Main goals, methods and applicability

This article describes a Swedish project that involves an undergraduate course in calculus. The course textbook is systematically revised by means of student "dialogues" conducted through a specially designed web site.

The goal of our project is to increase student learning by systematizing the way that teachers and students communicate, and to allow this dialogue to reach textbook authors. The project has the following three main objectives, ordered by importance:

  1. To improve the readability of a textbook from the students' perspective. This is student-to-author feedback, rather than student-to-teacher.
  2. To allow systematic improvement of the teachers' understanding of the students' subject knowledge, with all students, not just the most active ones.
  3. To encourage students to take a more active and responsible role in their education.

Textbook improvement is essential since textbooks often play a larger role in student learning than the teacher, particularly in mathematics. A large part of this paper consists of evaluations, which can be summarized as follows:

  1. Students were not used to posing questions in mathematics, rather they were strongly inclined to answering questions. However, they had little problem asking questions after the initial difficulty. This led to a view of the subject that was not so strictly task-oriented.
  2. Several mathematical difficulties in the book, pointed out by the students, were corrected after revision by the author.
  3. Students appreciated the teacher feedback and the fact that their work and remarks could affect the book.

The main principle of this paper is that the most important source of textbook improvement should be student use. The book should change and evolve as a result of this process, so that it may be improved as much as possible for future students.

Of course, it is impossible to directly and immediately implement a student suggestion. The modifications need to be consistent with the ideas of the book and must be mediated by the textbook author. In the model suggested in this paper, learning dialogues are carried out on a web site, where they are recorded, and are later used carefully by the author for textbook revision.

This process is a type of learning for the teacher. One example of an underestimated problem in our project, which led to revisions of the textbook, was how to handle integration constants that appear when solving separable differential equations.

The project also shifts responsibility to the students. It allows a student's work to be considered potentially valuable, not only for that student, but also for future students.

Or course, it would be possible to have a completely electronic text, one that could be revised almost continuously. However, our project concerns an ordinary printed textbook. Naturally, if there are too few revisions, there is no need to print a new book. It is up to the author and the publisher, considering pedagogical needs and economic constraints, to decide when the improvements justify a reprint. For some authors there may be a point where no further revisions are relevant, and the process ends. However, it is also possible that an author continues to finds fundamental improvements as a result of student feedback. In this case, the process can continue indefinitely.

Although we cannot claim statistical validity, here are the main findings of this project:

  1. Teachers and authors can enhance the quality of teaching and of textbooks by listening carefully to the students' ways of working and to their views of the subject.
  2. Students tend to adopt a more responsible view of the subject if they are asked to pose questions and if their work is valued in a long term perspective.
  3. A project of this type can be rather smoothly implemented by a web site that is designed to support dialogue and which mirrors the pages in the textbook for easy reference.

The paper reports the results of the project Student influence on text books, which was funded by the Swedish Council of Higher Education (Rådet för Högre Utbildning, RHU) and carried out at Blekinge Institute of Technology (BIT) during the period from July 2003 to June 2004. The project involved two faculty members, including the project leader, five advanced students who answered questions, and 39 students who asked questions. In this report, the advanced students and the teachers responsible for answering questions on the website are referred to as answerers. The project leader was also the textbook author.

1.2 Previous work

In the past, mathematics textbooks have been developed by students and teachers together, but a continuing, systematic revision process seems to be very unusual. The following paragraphs describe briefly several projects concerned with undergraduate mathematics. The literature on projects of this type is vast, of course, so these are just representative samples. Improving an existing textbook was not the main goal in any of these projects.

In Larson (1999), the author summarizes the design and development of a participatory calculus textbook offered as a subscription site on the Internet. As an interactive multimedia textbook, it integrates text, graphics, animation, simulation. The text also features collaborative environments similar to the familiar online chat and news. The convergence of interactive multimedia course materials with context-sensitive collaborative environments is called a participatory document. The structure of this project represents over four years of design research while implementation represents several person-years. Hundreds of students have used the text and based partly on their feedback, additional projects are being prepared.

The project An Interactive Text for Linear Algebra, (Porter (1995)), involved a computer-based interactive text for linear algebra using guided discovery in a laboratory based course, emphasizing active learning, collaborative learning, and the use of writing.

The article by Frith et al. (2004) describes a study of learning with students using interactive spreadsheet-based computer tutorials in a mathematical literacy course. The project concerns the role of computer technology (and specifically spreadsheets) as a mediator for the learning of mathematics. The data seem to reveal real differences between the learning experiences of students in the lecture sessions and those in the computer laboratories. In some respects the computer tutorials have been more effective in conveying the concepts than the lecture session.

Beyond this introduction, this report is divided in two sections. Section 2 formulates the goals, plans, and the practical implementation, and Section 3 gives evaluations and results.

2. Goals, plans and practical realization

2.1 Text books--requirements and roles

Despite the rise of new learning tools, textbooks continue to play a central role in higher education, particularly in mathematics. Any textbook has to meet two very basic requirements:

  1. The exposition of the subject must be correct, well-organized, and well-written.
  2. The exposition must appropriate for the existing knowledge and learning habits of the students who will use the book.

The second aim is often underestimated in higher education. Thus there is a large potential for improvement in university education, both in terms of textbooks and teacher understanding of the subject knowledge of the students.

We propose a model for ongoing textbook improvement, which may be used for textbooks in any subject. Of course, the intention of writing a book is that the student should learn the subject by reading and working with it. Thus, the learning processes of actual students are obviously valuable material for the author.

One may compare the situation with the concept of evolution in biology. The natural environment of a textbook is student use. Thus, the term evolutionary textbook may be appropriate for a textbook that is continuously improved based on student practice. A natural time scale here is that the book is revised each year. Currently there appear to be very few text books that may be called evolutionary. This reflects the fact that student knowledge is generally neglected, and that student-teacher dialogue rarely is two-way communication.

Mathematics has a tradition of relatively weak teacher-student communication. Silent work with the textbook is predominant, more so than in other subjects. Accordingly, feedback from students to teachers is usually limited. There are several possible reasons for this: The subject is abstract and far from the everyday experiences of students. Students often feel uncertain about the meaning of statements, concepts and the use of symbolic language. In Lennerstad & Mouwitz (2004) the linguistic problems with the symbolic language are discussed--particularly how this language may limit the use of the student's native language. Many researchers (see Ernest (1999), for example) have noted that mathematics involves many tacit or implicit types of knowledge. Thus, student-teacher communication has the potential of transforming implicit knowledge into more articulated knowledge for both students and teachers.

A teacher's knowledge of the students' knowledge in the subject is a very different kind of knowledge than the subject knowledge itself (see Huber & Hutchings (2005)). This knowledge is learned mostly by respectful communication with students. In short, a good teacher needs three qualities:

  1. Subject knowledge.
  2. Pedagogical skill
  3. Understanding of the students' knowledge in the subject.

Traditionally, the first two qualities are regarded in a serious way. However, without the third quality, lectures may be both brilliant and popular, but ineffective in terms of student learning. The fundamental goal of this project is to recognize and develop the third kind of knowledge, at least in terms of textbook improvements.

2.2 Questions

This project tries to give answers to the following questions, which are discussed and evaluated in Section 3:

  1. Can a textbook be significantly improved by student feedback?
  2. If so, what kinds of improvements can the student contribute?
  3. Is it possible to extend this feedback model to other subjects?
  4. Does this project provide other benefits for student' learning?
  5. How much extra resources in time, personnel and equipment are necessary?

2.3 Prerequisites

This project assumes certain obvious fundamental practical requirements. First, the course must have a textbook and a website. Furthermore, funding is required for the teachers and for the advanced students who answer questions. Less funding is required if the students in the course themselves are encouraged to answer questions. In this case answering questions can count towards the course grade, since this process involves learning new aspects of the subject.

The attitude and pedagogical knowledge of the author is also important. The author must have an active interest in the students' ways of working as well as a pedagogical understanding of how their questions can be transformed into textbook improvements. The author must be open to different ways in which the subject can be understood and described, and a well developed view about the main ideas of the course. For example, a proof can be written in different ways, such as starting with the assumption or starting with the result. It may be preceded by a typical example where the calculation is very similar to that of the proof. Also, many proofs can be described geometrically. The compromise between what students need and how to write mathematics without "losing the soul of the subject" is often a delicate task, requiring a specific competence and interest in student knowledge, and a realization of the limits of what a textbook can do.

2.4 The textbook

This section briefly presents the textbook that was used in the project, and which covers the course content. The textbook Envariabelanalys med dialoger (English: Calculus with dialogues) consists of 774 pages. The content is representative of Swedish standards. However, the large number of pages is due to the associated student dialogues on the website. Also, the book contains many examples with full and detailed solutions.

A translation of the table of contents is given below. The 17 chapters are collected in four parts. The first three chapters are somewhat unusual, so we include short descriptions of these.

Part I. Numbers and functions

  1. What is university mathematics? This chapter discusses mathematical branches, aims, goals, methods, graphics and the summation symbol. For example, functions are described as being at the heart of calculus since the main problems of calculus concern either properties of functions or equations with functions.
  2. Numbers and calculations. This chapter describes the five types of numbers, from integers to complex numbers, and the calculations that are possible with each type.
  3. The function concept. This chapter introduces functions formally, geometrically, and intuitively, both with abstract concepts and very simple examples. The main issues for the sequel are composition of functions and inverse functions. The main function classes are presented in the following three chapters.
  4. Polynomials and rational functions
  5. Power functions, exponentials and logarithms
  6. Trigonometric functions

Part II. Limits and derivatives

  1. Limits and continuity
  2. Derivatives
  3. Derivatives and mathematical modeling
  4. A little about McLaruin series

Part III. Integrals

  1. The integral concept
  2. Indefinite integrals and primitive functions
  3. Definite integrals and generalized integrals
  4. Integrals and mathematical modeling

Part IV. Differential equations

  1. General and separable differential equations
  2. Linear differential equations
  3. Differential equations and mathematical modeling

2.5. Software functionality

The URL for the web site is www.bth.se/analysmeddialoger. The web site is constructed using PHP with a MySQL database backend. There is one entry on the web site for each page of the book. A student can read previous dialogues about questions on a specific page in the book, and learn from these dialogues. Questions that do not belong to a specific page have been collected into a single entry.

It is possible to post questions anonymously. Each questioner chooses an alias and password. A password is only required for the purpose of posting messages on the website, not for browsing the site. The subjects of the messages are displayed in a tree structure, where related messages are in the same branch. By clicking a subject, the full message is displayed. When browsing an answer, the previous question is also visible.

The following table gives a short list of translations into English of Swedish words that occur on the home page.

Swedish English
Analys Calculus
Matematikkarta Mathematics map
Artiklar Articles
Butik Shop
Logga ut Log out
Debattinlägg Debate post
Alla inlägg All posts
Om boken About the book
Tyck till Express a view
Registration Registration
Gå till Go to
Nästa Next
Inlägg Post, contribution
Uppg Task
Replikera Replicate
Status Status
Alias Alias
Skapad Created
Öppna Open
Besvarad Answered
Svar Answer

When a question has been posted, an email is sent to each answerer. After the question has been answered, an email is sent to the questioner. The emails contain a clickable link to the website with the question and the new answer.

Mathematical formulas are expressed in plain text with a calculator-like input syntax. Sometimes a little imagination is required, for example to understand an expression such as "integral(from 0 to 2) x^2/(x^2 + 1) dx". One may argue that this is confusing because students need to learn a new notation system. It is also possible to argue that this simplified notation actually helps in the understanding of standard notation. In any event, there have been no complaints about the notation system. Nevertheless, as new tools develop, future projects may perhaps easily use standard notation.

2.6 The mathematics map.

The inner sleeve of the book contains a mathematics map, that gives a metaphorical overview over the entire mathematical content in the book. A mathematics map may contain mountains, forests, rivers, lakes, cities, countries, etc. with all names on the map being mathematical words or formulas. The most important mathematical concepts in the course are dominating features on the map, such as large cities. Of course, there are many possible ways to use geographical constructs to represent mathematical concepts, techniques, and connections. In a sense, a mathematics map is a geographical taxonomy. The sequence of pages in the book can be said to correspond to a certain walk in this landscape.

We next describe briefly how this particular map describes the calculus content in this course, as given in the table of contents above. The map is divided in eight countries: Numbers, Function, Sets, Limit, Continuity, Derivative, Integral, and Differential Equation. The smallest country is Sets, and the largest is Function. The country Function consists in turn of seven provinces representing the main classes of elementary functions. In each region, main items are represented, such as the Fifth Degree Mountains for the impossibility of solving the general fifth degree polynomial equation explicitly. Function also has areas outside these provinces, where general properties are represented. Differential Equation is divided in Linear and Non-Linear. At the border between these two parts is the city Separable, since there are both linear and non-linear separable equations.

The map is dominated by two large rivers: Equation Solving in the west and Composition in the east. Equation Solving has its source in Numbers in the north, and passes by the villages N (natural numbers), Z (integers), Q (rational numbers), R (real numbers) and C (complex numbers), since the five types of numbers arise when attempting to solve polynomial equations. For example, x2 = 2 prompted real numbers, and equations such as x2 = −1 gave rise to the complex numbers. The river Equation Solving is important also in Differential Equation in the south. This river does not pass through Integral, since in this context, equations are not studied in this calculus course.

The river Composition bifurcates in Equation Solving at the city Inverse, since inverse functions concerns both equation solving and composition. The functions cos x and sin x have their inverses arccos x and arcsin x at opposite sides of this river. The river passes the cities Chain Rule (differentiation of a composite function), Variable Substitution (integration of a composite function) and Separable Differential Equation (an equation where the solution is composite). The proofs of the last two are only applications of the chain rule--they are merely reformulations in different contexts. Actually, the idea of the map came from this connection with composition: composition as a red thread between major items was replaced by a blue river. The river metaphor allowed the growth of a landscape also away from the river.

Figure 1. The Mathematics Map
Mathematics Map

Figure 1, the only part of the map currently in English, depicts the central small country Limit with its volcano ∞, and the two villages e and d at its foot. The River of Calculations leads through many curves and bends over the Plains of Standard Limits, out of Limit, into the Plains of Standard Derivatives in Differentiation. After passing through the large city Fundamental theorem of Calculus, the same river traverses the Plains of Standard Integrals. There is also the northbound river Approx, passing Asymptote, Normal and Tangent. The red dots mark walks on the map defined by the narrative of the book--its word by word linear order. Numbered squares and circle are starting and ending points, respectively, for the chapter with that number. The full map can soon be found at mathematics.lennerstad.se.

The map in the book published in 2002, is the first mathematics map, to the best of our knowledge. Since then we have used the basic idea as a pedagogical tool by having student groups design their own maps about mathematics. During mathematical map making, students discuss connections between the mathematical concepts and calculations they know, and produce an overall picture. Mathematical maps have been made at all levels, from kindergarten to university. The geographical metaphor is both familar and flexible; students find many innovative similarities between the geographical and mathematical domains. In this type of mathematics learning, numbers and quantity are not the objects of thought--it is mathematical ideas themselves. Instead of correct answers, focus is on the overall picture of the mathematics. Mathematics teachers have reported that they learn about students' mathematical reasoning by listening to ongoing dialogues during map making. See Lennerstad (2007) for a thorough discussion of mathematics maps.

2.7 Integration in courses

In some of the early courses, website activity was voluntary and the course was not given by the project leader. In general, students did not use the web site sufficiently in these courses. The teachers generally claimed that the students already had enough material and opportunities; the website was not necessary for them; and that students were not used to writing questions. Thus, the website was not sufficiently integrated in these courses. The need for integration has been pointed out by the visiting Council representatives. Following these comments, changes to the use of the website were made. In 2003, students were rewarded by extra exam credit for asking and answering questions, and in the distance course conducted in early 2004, questions were made compulsory.

2.8 Integration at the department

A practical issue regarding the project was the fact that the project leader was not able to meet the mathematics teachers on a daily basis. The mathematics teachers worked in the town of Karlskrona, whereas the project leader had his office in a computer science group (for research reasons) in the town of Ronneby. The project would also have benefited from a more thorough introduction by mathematics teachers at BIT, thus encouraging the use of the web site.

3. Results and generalizations

In this section we describe the form of dialogues (3.1), the evaluations (3.2), comparisons to goals (3.3), conclusions and improvements (3.4) and generalizations to other contexts (3.5)

3.1 Types of Questions

We will describe the different types of questions that were posted on the website. In total 29 students contributed questions and there were seven answerers (two teachers and five advanced students). The notes and comments on the site are categorized into four categories: Questions on problem solving, questions on ideas, words and concepts, questions on applications and history and general feedback to the author. The following table summaries the results. The last two columns indicate the number of comments, and the number of comments that resulted in a direct change in the book (yes or no). In total, 569 comments or questions were posted on the web site.

Question type Yes No
Questions on problem solving
Understanding the problem situation. What is being asked   13
Full calculation but wrong answer. Typically the question is "what's wrong?"   50
Full calculation - check of correctness. Similar to the previous question, but with a correct answer.   9
Little calculation and unclear question. Diffuse questions on how a certain problem could be solved.   32
Clear method question. A question where different methods are mentioned and compared. 43 123
Questions ideas, words, and concepts.
Questions on ideas and concepts: Questions on prime numbers, open and closed intervals, dense set, inverse function, meaning of mathematics, relevance of figures 49 101
Theoretical remark. Discussion about a theorem. Sometimes a correction of the book, sometimes a further explanation is added. 4 6
Numerical aspects.   3
Geometrical interpretation   8
Notations and words. Example: Why is limit denoted by lim? Can the Greek letters be used for anything in mathematics, or should they be used for something special only? 5 44
Questions on applications and history
Questions on applications. Examples: In the limit 20(1 − et) when t → ∞, perhaps representing the temperature in a house, do we ever reach the temperature 20°?   6
Mathematical historical questions   13
Comparisons with other books   5
Feedback to author
Suggestions for improvement 5 4
Suggestions for corrections 27  
Questions about graphs   10
Appreciation of an explanation in book   7
Superlatives on beauty or insight. A few parametric curves are in the book just for their aesthetic value, (cos t + esin t, ecos t) and (cos t + ecos t, esin t), which resembled rocks at a shore. These have triggered the comment "Mathematics is beautiful! And not only in an abstract way understandable by the professional mathematician!"   3

There are also many improvements that occurred as a result of comments from the answerers to the author. These comments sometimes point out errors or unclear explanations in the book or exercise book.

3.2 Evaluation by students

We first describe the grading scheme used in the distance course. The course had four written assignments where group work was encouraged, (at least) 42 compulsory mathematical questions on the website, and an individual examination at the end of the course.

Students used two web sites in the course: one containing the course schedule and all other information, the other was devoted to the book and the mathematical dialogues. Thirty-two registered, and 11 passed the course. Recorded video lectures on the main topics were available on the course web site. In these recorded lectures a student sees pages from the book and a pointer, while the audio is of the author commenting and explaining what is written. The audio comments complement the written text in the sense that the main points are emphasized in less formal language.

The student evaluation is divided in three parts: the course, the book, posting questions and the web site. Twelve students participated, but of course not all answered all questions. Each question allows five degrees of evaluation. The columns contain the number of students giving a particular answer.

A. The Course
Query Reply range 5 4 3 2 1
The course content appears to be Very relevant - not relevant 5 5 1 0 0
Course difficulty level Easy - very difficult 0 3 3 4 0
The amount of material is Too little - too much 0 1 9 2 0
The course plan is comprehensible Very - not at all 1 6 3 1 0
The course staff replied rapidly Always - never 4 6 2 0 0
The technical parts worked Very well - not well at all 4 6 1 0 0
The course gave me valuable knowledge of mathematics Very much - not at all 1 8 2 0 0
The course made it easier for me to solve mathematical problems and work with mathematics Yes - no 1 6 4 0 0
The course made it easier for me to see and understand mathematical connections and concepts Yes - no 2 6 2 0 0
The course made it easier for me to see connections between mathematics and applications Yes - no 0 4 6 2 0
The course was compatible with previous courses within the program Yes - no 0 7 2 1 0
I used the recorded lectures on the web Very often - never 3 2 6 5 0
From the point of view of problem solving, the recorded lectures were: Very useful - worthless 0 3 3 4 0
In order to understand mathematical connections and concepts, the recorded lectures were Very useful - worthless 1 4 5 2 1
B. The Book
Query Reply range 5 4 3 2 1
Readability and ease of understanding Very clear and easy - very unclear and difficult 2 8 1 0 0
Knowledge (text, figures, etc.). In terms of understanding mathematics, the book makes it Very easy - not at all easy 3 4 4 0 0
Learning to solve problems Very easy - not at all easy 0 6 4 0 0
The book has improved my ability to use mathematical formulas. Very much - not at all 0 10 0 0 0
Overview. The book gives an overview of the mathematics that is: Very good - not good at all 0 3 7 1 0
Priority. The book emphasizes what is most important Too much - too little 0 3 7 1 0
Long term knowledge. The book explains so that mathematics I meet in the future will be Much more easy - not at all easy 1 9 1 0 0
Level. The book appears to be Very mathematically advanced - not at all mathematically advanced 0 7 4 0 0
Creates interest. The book makes mathematics and mathematical problems Much more interesting - much more boring 3 6 2 0 0
Attitude. Using the book is Actually fun - not fun at all 0 8 2 1 0
Working time. I have been working with the book
  1. More than 20 hours each week
  2. 10-20 hours each week
  3. 5-10 hours each week
  4. 1-5 hours each week
  5. Less than 1 hour each week
1 2 7 0 1
C. Posting questions and the web site
Query Reply range 5 4 3 2 1
In order to learn mathematics, I think that regularly writing mathematical questions on a web site is: Very good - confusing 0 4 4 1 1
Having compulsory questions is: OK - not OK at all 0 0 5 3 2
The answers to my questions have arrived: Rapidly and always - slowly or never 1 5 3 1 0
The answers to my questions have been: Very good - confusing 2 6 1 0 0
From the answers I have learned: Very much - nothing 0 8 2 0 0
I have been reading other questions and answers at the web site. Often - never 0 2 4 4 0
From questions and answers other than mine I have learned: Very much - nothing 0 2 3 4 0
Posting questions on the web site has changed my attitude about mathematics. Very much positively - very much negatively 0 2 8 0 0
On-line-help. A web site where you can post questions on the book and receive answers is: A very good idea - not useful 5 5 0 0 0
Student influence. That student problems affect future versions of the book is: A very good idea - not useful 5 4 1 0 0
The fact that my questions may affect the next version of the book is Very positive - very negative 2 3 4 0 0

3.3 Evaluation by Student Answerers

The following comments are from David Erman and Maria Salomonsson, the advanced students who served as principle answerers. The focus of the comments is on their own mathematics learning.

As senior students, the opportunity of being involved in the answering of student queries has been very beneficial in several ways. Firstly and foremost, it forces us to write as lucidly and explanatory as our ability allows, and naturally hones our skills as conveyors of information. Secondly, it is a humbling experience, as we are reminded of our own fallibility and lack of knowledge in the areas covered by the book. Several questions posed by students have forced us to return to our own notes and books to find answers for ourselves, just to be able to give satisfactory explanations to the query. Thirdly, we gain a fair amount of new views of the mathematical concepts by reading the questions. This of course gives a wider and more complete grasp of the topics covered. Fourthly, many of the student questions have been regarding the history and stories behind the mathematical notations and people. It is highly unlikely that this information would have been sought out independently by us without the incentive of the CwD site. Not that the interest isn't there, but rather that once you consider yourself "done" learning a topic, you stop wondering. Mathematics has been made a living entity again, in a sense.

3.4. Evaluation by the project leader

Firs we describe the types of textbook revisions that resulted from the student dialogues.

Generality. One may claim that the most fundamental character of mathematics is its generality. This is reflected in student questions, since a very common question in all chapters has been: "Is this general?" Several revisions have been made that essentially state "This is only an example, illustrating a concept." Formulas are often extremely compact; their applicability and scope are seldom obvious to students.

Informal descriptions: Many revisions have included additional informal descriptions motivated by the opinions of the students. This has been limited, however, since the book was already written in a relatively informal way.

Further explanations of calculations: Calculations are often abundantly commented in the textbook. However, more comments on particular steps have been added, based on student feedback. Such changes are not necessarily improvements, since the overview of the calculation (the "larger picture") may decrease.

Constants appearing in integration: Many questions concern the constants that appear in integration, including what the expression "they differ at most in an additive constant" means. This may be an easy question to pose, in the light of the fact that 42 questions were mandatory for each student. Revisions were added that connect the undetermined constant in integration to the fact that a derivative only concerns the change in a function, independently of the level (height) at which the change occurs.

Constant appearing in solving differential equations: Many questions also concern the constants that appear in differential equations. In separable equations, usually two constants appear from x-integration and y-integration, which may be merged into one constant. Furthermore, in solving linear, constant coefficient, second order differential equations, two kinds of unknown coefficients often appear--in the homogenous solution and in attempting to find a particular solution. Further explanations were added to clarify this.

The following gives a summary of types of revisions of the book that occurred from student feedback, sorted in order of importance, as rated by the author.

  1. Errors in the book have been found and corrected.
  2. Explanations have been added on fundamental ways that mathematics works, what mathematical results attempt to say, and the meaning of concepts. These explanations concern the issues described in the subsections on Generality and Informal Descriptions above, and are very important for students. They provide an overview of the ideas, putting detail problems in a more relevant perspective.
  3. Logic in the arguments. Questions often indicate that the students do not understand the goals of various calculations. That is, the context and purposes of the calculations are unclear. Additional explanations have been added, often in more informal language.
  4. Further explanations have been added concerning details in calculations.

Some good questions did not result in revisions since they would have led too deeply into the subject for a basic calculus course. In some cases a reference to another text was included.

There is another possibly significant long term effect. We now have a better understanding of the way students experience the mathematical material, and this may lead to later revisions. or even to entirely new books with a very different structure.

3.5. Limitations

The project suffered from the fact that the project leader was not able to meet the other mathematics teachers on a daily basis. With closer contact, the teachers would probably have been more familiar with the book and the project, and would have better supported the use of the web site.

Another limitation was the lack of an external evaluator. The project leader was also the textbook author.

Only 29 students participated in the project. A larger number of students would have resulted in higher reliability in the questionnaire statistics.

3.6. Comparison to goals

In Section 2.2, several questions were formulated. We next discuss some possible answers, suggested by this project.

1. What kind of improvements of a textbook can students contribute towards?

The statistics in the previous evaluation sections suggest the following types of improvements:

Teachers who engage in dialogues with students can expect disclosures of this kind, thus extending the teacher's student body knowledge.

2. Can a textbook be significantly improved by student feedback?

This is a question with a very large scope, as it depends on an even more basic question upon which teachers do not always agree: what is a good textbook?

As every student is different--in both learning ability and method--a good textbook needs to provide different ways of explaining a given topic. It should be written so that it is easy to study mathematics in different ways, e.g., by problem solving, basic ideas, theorems and proofs, applications, or a combination of these. A long term project of this kind could aim at categorizing the different ways that students work, and produce one version of the book for each learning style, with the same mathematical content. This might require that students be aware of their learning style, which is seldom the case. However, as part of a more holistic education, this is a matter that could be resolved early on in the student's education and provide benefits for the student beyond that of learning mathematics.

Different textbook versions, catering to different learning strategies and processes are probably necessary to be able to achieve significant improvements, and to use long term student feedback efficiently. Without such separate versions, a textbook may improve mostly for the majority of the "average" students, while students with unconventional learning processes may be hampered. An additional problem with such a textbook may be that the book becomes so unconventional in style that teachers not familiar with the project or pedagogical issues may choose to reject the book in favor of a more traditional one.

The process of feedback-based improvement may have several other significant effects on the teaching of mathematics. The learning of the author or teacher may be the most important, as this will benefit future students. Feedback-based improvement may thus be worthwhile not only from textbook improvement perspective, but also from an extension of the teacher's or author's own subject knowledge. Such feedback can be expected to be more natural for students and give more valuable results if the students have learned to take responsibility for their own education.

While not formally provable, the results of the project described in this report indicate that long term student-teacher cooperation can effectively improve text books and educations.

3. Is it possible to extend this feedback model to other subjects?

The fact that dialogues are less common in mathematics than in other subjects make them all the more important, but also more difficult. In most other subjects dialogues can be expected to flow easier. Also, in most other subjects it is not necessary to find ways to write formulas with the usual text symbols. In the humanities and social sciences, verbal argumentation is considered to be part of the subject skill, which is reflected in the activities in these courses. A similar project, tailored more to the specifics of these areas, may give such activities more weight because of the potential changes in the textbook. The idea of systematic student feedback is a fundamental idea which can be expected to be useful for textbook improvement in any subject.

4. Does this project provide other benefits for the students' learning?

An impression while reading the student questions is that the process of formulating and posing questions fosters a more mature relation to mathematics. A common student attitude is to very strictly focus what is expected to be important for the examination, which may lead to a narrow knowledge of the subject. This may be a problem since it is very difficult or expensive to construct examinations that encourages a broad knowledge of the subject. In posing and answering questions, the student partly adopts a teacher role, which may develop a broader kind of knowledge.

Initially, many students have disliked posing questions. They have preferred to produce answers to specific questions before posing their own. Superficially, the former may seem easier. It is a common student habit of answering questions, rather than asking them--particularly in mathematics.

One point of the mandatory posing of questions is that it is a fairly reliable sign of activity. Questions can be hard to find, but once the mathematical work actually starts, questions of different kinds seem to appear. Having access to the web site, students can quickly and easily enter questions and answers--addressing a common problem for teachers of distance courses: to measure the activity of the students.

The fact that the questions of the students may alter a book read by future students may provide a sense of responsibility. This may increase the general responsibility for the education, as well as making mathematics a subject more open to questions.

Many questions have concerned the use of mathematical methods. Answers to these are important for mathematics knowledge, as they strongly increase the ability to succeed in mathematical calculations and problem solving. Such method-related questions rarely appear in a dialogue-free learning environment.

Occasionally, students have exercised cooperation on the web site. Furthermore, in the evaluation students say that they did learn from reading other communications on the web site.

5. How much extra resources in time, personnel and equipment is necessary?

As is described in the comments by David Erman and Maria Salomonsson, answering questions involves a great deal of learning for the students involved. Students can use this as a teaching merit, and often their knowledge, interest and self-confidence in the subject increase. It should not be difficult to find students that to some extent may participate at a relatively low cost. It is also a means of involving talented students in department work.

Most answers typically take from 1 to 10 minutes to complete. Some answers, though, require verifying long calculations or more involved information gathering (historical questions, for example). If web communication is compared to classroom communication, one might expect class room dialogues to be more efficient. However, there are also advantages to web dialogues over class dialogues. First of all, web dialogues are naturally recorded, allowing them to affect the textbook, and be available for other students. The timing is different, and better in some respects. It is possible to give more thought to an answer. In a class room, a student often accepts an argument that is not understood in order not to appear ignorant. In direct dialogue, a teacher can respond to a question before it is properly formulated. On the web, the student is forced to fully formulate a question, and has time to do this. Some students feel more comfortable posing questions anonymously on a web site than they do in a classroom. Both kinds of dialogues have specific advantages, and should be exercised.

When calculating the time resources needed, web communication can be compared to classroom help. If students also are asked to answer questions, not only to pose them, the resources needed become smaller. Monitoring and completing dialogues becomes more important for the teacher, rather than answering direct questions.

3.6. Conclusions and improvements

The project described in this report is an attempt to enhance scientific maturity among students, and simultaneously improve textbooks in a long term perspective. Students are able to participate easily in both asking and replying to questions. A certain initial doubt regarding not only answering questions, but also posing them, is evident. Motivated students who take their education seriously adapt faster to this new process. The project provides teachers (and authors) with better knowledge about students and their needs in acquiring (and retaining) the contents of a given course, resulting in improvement in both teaching and textbooks. Additionally, the project allows students to exercise several competencies: formulation of arguments and scientific questions, student responsibility and cooperation. Through feedback, education and textbooks evolve through teaching processes. A feedback project breaks the potential isolation between teachers and students by encouraging or enforcing student participation, thus encouraging the development of the students' scientific maturity.

Acknowledgements

The project relied on several persons and institutions. Many thanks to Johan Erlandsson and Niklas Säfström who, together with the authors of the present report, have patiently answered questions. Inger Lindström at BIT has processed financial aspects. Last but not at all least we would like to sincerely thank the Council of Higher Education, Sweden for financing and supporting the project in many ways.

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