Literary articles - Lewis Carroll 2024

Logic in Wonderland: Alice's Adventures in Wonderland as the Context of a Course in Logic for Future Elementary Teachers

Nitsa Movshovitz-Hadar and Atara Shriki

The teaching experiment described in this chapter assumes at the outset that children's literature can be a useful context for teaching elementary ideas of logic while bridging the gap between the abstractness of formal logic and its expression in a real world context. Alice's Adventures in Wonderland, by Lewis Carroll (a unique combination of a logician and a story teller) was chosen for this purpose, based upon a careful examination of its potential. Inspired by The Annotated Alice (Carroll 2000), over 75 additional annotations to Carroll's book were developed, having in mind their employment in an introductory course in logic for prospective elementary school teachers specializing in mathematics. These annotations are in four categories: Logic, Mathematics, General education, and Science. Sample annotations are included. This chapter describes the tasks and activities developed for the course. Data collection instruments were interwoven in the teaching materials development. A sample is included as well. Several results are reported and discussed.

1 The Problem
The importance of connectivity and of making mathematics relevant to the learner as vehicles for improving learner's understanding and motivation to delve into more depth in mathematics, are emphasized in many major documents (e.g. NCTM 2000). If this is true of any area in mathematics, it must be true for formal logic, which, from our experience with prospective teachers, is among the least favorite topics in teachers colleges. This topic is perceived as not useful, dry, uninteresting, and definitely not directly applicable to the school curricula. Nevertheless, an introductory one-semester course in logic follows an elementary course in set theory, and both are mandatory for accreditation, in Israel. It should be noted that in various other countries an elementary course in formal logic is not compulsory at college level, yet many of them engage students with topics that are related to logic, such as quantifiers, producing proofs and inferring from statements (Dubinsky & Yiparaki, 2000; Selden & Selden, 1995; Moore, 1994).

How can the teaching of logic for prospective teachers be made more “juicy”? Is there an intriguing way to expose students to “the game” of drawing conclusions? – These were the questions we struggled with as we started preparing for the new academic year. It occurred to us that it might be useful to employ Lewis Carroll's masterpiece for this purpose. Would students benefit from it? Enjoy it? That remained to be seen. To find out, we set up an empirical intervention study that is described in this chapter.

2 A Brief Theoretical Background
2.1 Difficulties of Students in Reasoning and Inferring Logically
Promoting children's reasoning abilities has been widely recognized as one of the pillars of mathematics education, for many years (e.g. Gregory & Osborne, 1975). Clearly, teachers cannot be expected to promote their students' reasoning properly, unless they themselves understand logical implications and are able to employ them reliably. Teachers colleges requiring an introductory course in logic and set-theory assume these courses to serve the purpose of constituting the foundation for developing future-teachers' logical thinking, or at least their ability to distinguish between valid and fallacious inferences.

Nonetheless, research points to various difficulties students encounter while requested to infer logically. Dubinsky and Yiparaki (2000) studied difficulties students have with mathematical statements that involve quantifiers. They believe that it is not reasonable to expect undergraduate students to learn much mathematics if they do not know how to read and interpret the language of mathematics. They argue that in order to understand a complex statement there is a need to analyze the statement based on the syntax of the language in which the statement is given. These researchers showed that university students meet various difficulties when they are requested to relate to logical statements involving quantifiers and that they are much more capable of handling the natural language statements than mathematics statements. Moore (1994) found that undergraduate mathematics majors have difficulties in producing even apparently trivial proofs. The students in Selden and Selden's (1995) study could not reliably determine the logical structure of common mathematical statements, and had difficulties in determining the correctness of their proofs. In addition, the students had trouble with transforming informally written mathematical statements into equivalent formal versions using symbols. Many students have difficulties in producing formal arguments and manipulating symbols in a formal way without having a deeper understanding of what the symbols really mean (e.g. Schoenfeld, 1991).

Part of these difficulties can be attributed to the traditional approaches to teaching logic, in particular to the language that is used within these courses (Epp, 2003), and to the manner in which the various aspects of logic are presented in textbooks (Alibert & Thomas, 1991). In the vast majority of students' textbooks, statements are not written in formal language but in “natural” language (Dubinsky & Yiparaki, 2000). Moreover, Dreyfus (1999) believes that students are unable to give satisfactory explanations and answers because most of them had never learned what counts as a mathematical argument.

Logic, by its nature, is abstract, and the manner in which words and statements are used in the spoken everyday language is not always consistent with their meaning in the context of logic. Furthermore, students have difficulties in managing logical statements in both contexts (Dubinsky & Yiparaki, 2000).

Cheng, Holyoak, Nisbett, and Oliver (1986) found that integrating concrete examples into logic courses and explaining logical principles by referring to analogies in everyday life improve students' ability to reason.

This suggests that a possible way to overcome the difficulties is teaching logic through children's literature. Indeed, Cotti and Schiro (2004) view the use of children's literature as “one of the powerful tools available to help in teaching mathematics”, as it provides “a rich, meaningful, real world or fantasy context that can stimulate and motivate children's learning” (p. 339).

2.2 Using Children's Literature as a Means for Teaching Mathematics
The use of children's literature presents “a natural way to connect language and mathematics” (Midkiff & Cramer, 1993, p. 303). Integrating children's literature in mathematics lessons enables students to be actively engaged with the learning materials (Conaway & Midkiff, 1994), and provides a base for establishing understanding of concepts (Midkiff & Cramer, 1993). It can also serve as an important vehicle for exploring mathematical ideas, as the natural context and the fact that mathematics is naturally embedded in familiar situations offer opportunities for discussing and highlighting mathematical ideas (Whitin, 1994). Moreover, “Stories can help students understand the meaningful contexts that support mathematical thinking. They will see mathematics not as a prescribed set of algorithms to master, but as a way of thinking about their world. Children's literature presents a nonthreatening avenue to test out current notions about important mathematical concepts” (Whitin & Gary, 1994, p. 394). “And, most important” Whitin (1994) claims “children's literature powerfully demonstrates that mathematics is a way of thinking. For these reasons children's literature deserves a prominent role in the mathematics curriculum for all learners” (p. 441). Furthermore, Whitin and Gary (1994) perceive the use of children's literature as an open invitation for students to connect their own interests and experiences to various mathematical concepts. In this way mathematics is no longer regarded as limited to workbooks, but becomes a purposeful tool for solving problems and making decisions.

Considering the above, we believe that for teaching elementary ideas of logic, the use of children's literature in general, and Alice in particular, can bridge between the abstractness of logic and its expression in everyday life. In addition, it might also help to reject the belief of many mathematics educators as to the fact that logic is too dry to capture students' interest (Epp, 2003). Teaching and learning logic through Alice convey an educational message too. Assuming that future teachers have no previous experience with learning or teaching in that style, it provides them with an opportunity to become familiar with this method and recognize its benefits for their future students. This is in accord with Cotti and Schiro's (2004) findings that most mathematics teachers are in favor of using children's books during mathematics instruction, believing that it might help in creating situations in which children can construct their own mathematical meanings.

Encouraged by these findings, and inspired by The Annotated Alice (Carroll, 2000) we developed additional annotations to this book, having in mind its implementation in training prospective elementary school teachers in general, and in particular those specializing in mathematics. These annotations are in four categories: Logic, Mathematics, General education, and Science.

The rest of this chapter details the strategy for integrating Alice's Adventures in Wonderland, with our annotations as mentioned above, in the learning environment for an introductory logic course for prospective elementary school teachers.

It also discusses the sequencing of a series of appropriate tasks, and the underlying rationale for their design, as the backbone of the empirical course, illustrated by a sample of task-worksheets.

In addition this chapter presents the research method and instruments employed to collect data pertaining to the impact of this empirical course, as interwoven in the teaching-learning process to minimize interference between the two.

Several results related to students' achievements and motivation are also discussed within space limitation.

3 The Process
We developed a series of 90 minutes lesson-plans based upon reading quotes from Alice's Adventures in Wonderland, and discussing them with our students. The lesson plans include handouts for independent study and group work. Designing the learning environment took five steps:

a. The first step was a careful examination of Lewis Carroll's text, highlighting quotes that have a potential for becoming starters for our students' activities or discussions. We found 75 quotes and annotated them. These annotated quotes were classified into four categories: (i) Logic (ii) Mathematics, (iii) General issues of education, and (iv) Science. Here are a few examples in each category.

i. Logic. E.g. It was all very well to say ‘Drink me,' but the wise little Alice was not going to do that in a hurry. ‘No, I'll look first,' she said, ‘and see whether it's marked “poison” or not'; for she had read several nice little histories. . . such as,

that a red-hot poker will burn you if you hold it too long; and that if you cut your finger very deeply with a knife, it usually bleeds; and she had never forgotten that, if you drink much from a bottle marked ‘poison,' it is almost certain to disagree with you, sooner or later.

However, this bottle was not marked ‘poison,' so Alice ventured to taste it, and finding it very nice, (it had, in fact, a sort of mixed flavour of cherry-tart, custard, pine-apple, roast turkey, toffee, and hot buttered toast,) she very soon finished it off (Carroll 2000, Page 17, top half).

Discuss with students the sentence – it's marked “poison” or not – in particular the meaning of the word “or”. Is there a third alternative?

Discuss with students the “if...then...” statements in this paragraph, as well as their corresponding inverse, converse and contra-positive. Could Alice be sure that the bottle was safe to drink? (See also Appendix 1, example ii.)

ii. Mathematics. E.g. ‘What a curious feeling!' said Alice; ‘I must be shutting up like a telescope.' And so it was indeed: she was now only ten inches high, and her face brightened up at the thought that she was now the right size for going through the little door into that lovely garden' (Carroll 2000, Page 17, lines 9–10 from bottom).

Discuss with students – measurement of length: ideas such as various units, measurement tools, comparison, estimation – show by hands how long is 10 inches;

iii. General issues of education.E.g.What an ignorant little girl she'll think me for asking. . . (Carroll 2000, Page 14, lines 7–9).

Discuss with students – the issue of refraining from asking “stupid” questions. Should we hurry to ask every question? Should we try and figure out things for ourselves first? Or look for a written reference (as in a scientific enquiry), or some posted information (as in finding your way in a new place).

iv. Science. E.g. First, however, she waited for a few minutes to see if she was going to shrink any further: she felt a little nervous about this; ‘for it might end, you know,' said Alice to herself, ‘in my going out altogether, like a candle. I wonder what I should be like then?' And she tried to fancy what the flame of a candle is like after the candle is blown out, for she could not remember ever having seen such a thing (Carroll 2000, Page 17, line 12 from bottom).

Discuss with students – the three states of aggregation and the law of conservation of material

The annotations revealed that an opportunity to discuss a specific subject in logic, such as exclusive and inclusive “or”, occurred more than once in the text, obviously. It also revealed that the first chapter is the richest in resources (22 annotations of total of 75) for raising logic issues. This had an impact on the next steps in our work.

b. The second step was to compare the essence of these annotations with the syllabus of a traditional introductory course in logic for future-teachers, and to examine the prospect of covering the syllabus. The results showed that the first chapter is not only the richest, but includes pointers to everything we intended to employ in the syllabus. Namely, Logic and everyday language: A simple sentence and the truth value.

Negation. A compound sentence. Binary sentential connectives and their truth tables: Conjunction and “and”; Disjunction and inclusive/exclusive “or”; De Morgan's Laws; Implication and conditional sentences; Necessary and sufficient conditions; Equivalence and biconditional sentences. Inference rules: Affirming the Antecedent (Modus Ponendo Ponens) and Denying the Consequent (Modus Tollendo Tollens); Tautologies and equivalence; Quantifiers: Existential and Universal ones. Proof: Direct and indirect (by negation), by exhaustion, by counterexample.

c. At this point we faced a dilemma: which of the two lesson-plan sequencing options is more effective? The two options with conflicting rationale being (i) Sequencing along the logic syllabus and integrating suitable quotes from Alice, vs. (ii) Re-sequencing the logic syllabus along page by page reading of Alice, namely stopping at suitable quotes for raising issues from the syllabus in the order they occur. The former seemed right from the mathematics teaching aspect we were accustomed to, giving it a new spark, on the account of breaking the flow of the story into somewhat disconnected pieces. The latter seemed more revolutionary, as it implied breaking the commonly accepted order of teaching the subject matter, due to constraints imposed by the literal context. Having considered the pro and cons, we decided to face the challenge and struggle with re-sequencing our logic syllabus as dictated by the flow of the story. Because the first chapter was found to be the richest (as mentioned in the first step above) we naturally focused the exposition on the first chapter from Alice, and hence decided to use the rest of the story as resource for practice activities. At the end of this chapter we return to this dilemma and look into our decision in retrospect to examine its justification in view of the result.

d. We then proceeded by outlining a series of fourteen 90-min lessonplans based upon reading through the text of Alice's Adventures in Wonderland (Carroll 2000), designating the stops at certain quotes to be used as a leverage for discussing certain themes in logic, mathematics (mostly set-theory), or sometimes science or general issues of education. These lesson-plans served as the backbone for the course.

e. Finally, we prepared a detailed design of tasks for each of the 14 lessons. Our approach took into consideration our desire to collect data for this empirical study. However, we strictly adhered to our top priority of the course, namely providing students with a series of learning experiences leading to an understanding of the main themes of the course. This was the guideline we adopted as we designed the learning activities and tasks as instruments for data collection, integrated into the teachinglearning process. The next section gives the details of the task development.

4 Activities and Tasks Development
The general rule we followed in the lesson-plan detailed design was a variation on the old maxim: First tell them what you are going to tell them, then tell them, then tell them what you have told them. Rather than “tell them”, we took the constructivists' approach and attempted to provide the students with learning activities.

To this end, the framework for material development for each lesson included the following eight stages:

(1) Individual activity (handout): A short opening question–answer activity, through which students initially wonder about their elementary concepts and pre-conceptions related to the major themes of that lesson. This activity sets up the stage for teaching and clarifying the concepts and the ideas (see stages 3 and 6 below). Follow up on this activity is deferred (see stage 8 below).

(2) Whole class activity: Reading aloud a section from Alice (Chapter 1), followed by, when relevant, a short whole class discussion of educational/mathematical/ scientific issues triggered by the section just read.

(3) Individual activity (handout): A guided study of selected quotes from the section of Alice just read, aimed at introducing a theme in logic. This activity makes students work on the connection between the use of the concepts in the text and their intuitive pre-concepts as revealed in the initial activity (see stage 1 above). This handout is revisited later (see stage 6 below.) Two sample handouts appear in Appendix 1.

(4) Small-group/whole class activity: Discussion of the individual ideas surfaced through the previous individual activity, giving time for students, exchange of their ideas, attempting to convince one another in case of disagreements.

(5) Expository teacher action: Course instructor presents the concepts and the ideas coherently, resolving any misunderstandings that may have occurred previously.

(6) Whole class discussion: Students take a second look at their writings in the handout of stage 3 resolving any points that remained unclear.

(7) Practice and take-home assignment (handout): This handout is based upon quotes from chapters other than the first one in Alice, quotes which bring up the topics in logic, discussed in class. Two sample handouts appear in Appendix 2.

(8) Individual activity (handout): A second attempt to address the questions in the opening handout (stage 1). This activity intends to make students aware of their knowledge development and bring up yet unclear issues. A sample handout appears in Appendix 3.

Materials developed were revised twice. Once after the first experimental implementation of the course, and again at the end of the second semester. The full range of materials is included in Shriki and Movshovitz (2008).

5 The Study
Our study took the nature of action-research using the following as data collecting instruments:

a. Students' response to the lesson-opening handouts. (see section 4 “Activities and Task Development”, stage 1 above, and a sample handout in Appendix 3);

b. Students' response to the lesson-end handouts. (see section 4 “Activities and Task Development”, stage 8 above, and a sample handout in Appendix 3);

c. Students' work on selected homework assignment handouts. (see section 4 “Activities and Task Development”, stage 7 above, and sample handouts in Appendix 2)

d . Informal 5–7 minute individual conversational interviews with 2–3 students conducted on the way out of each lesson; the affective domain was at the focus of these conversations, hence students' feelings about the knowledge gained through the lesson that had just ended were probed.

e. Written lesson reports documenting the implementation of each lesson-plan and detailing critical events and the informal interviews; Reports were written in an accumulative diary immediately after the end of each lesson; (Due to the lack of sufficient funding for this project it was impossible to videotape the sessions, so instead we used these records as evidence for analysis of the outcome.)

f . Students' response to end-of-term evaluation in two parts: ( i ) Final 120 minute ordinary end-of course written exam based upon the syllabus (A sample test-item appears in Appendix 4.) (ii) Essay writing: Students were asked to reflect upon the development of their personal knowledge within the particular set up of the course and to discuss pro and cons for integrating children's literature in school mathematics and in teachers college mathematics.

6 Results and Discussion
The results obtained by administering the instruments mentioned in section 5 above, served as a data base for analyzing student's progress and attitudes as well as for revising the materials.

At the beginning of the first lesson we distributed copies of Alice, telling the prospective teachers that it was going to be our textbook for the logic course. Many students turned the pages of the book and expressed a slight embarrassment, wondering how it would be possible for “a book with no numbers and no formulas” to serve as a textbook.

Eighteen students participated in the second experimental implementation of the course. Comparing their final grades using instrument f (i) with previous years groups of students who took this course without Alice, yielded no significant difference.

Due to space limitations we focus in the rest of this section on the results yielded by the Essay Writing instrument f(ii).

As will be detailed below, students' written reflections referred mainly to their benefits from the entire learning environment that encompassed reading from Alice and working on the activities and the tasks. They also related to possible implications of their course experience to their future school teaching. A few students pointed to some disadvantages of learning logic through Alice or any other subject through children's books. Here are a few excerpts from their essays.1

1 The excerpts are translated from Hebrew, as it is the tongue in which teaching took place.

6.1 Benefits from Learning Through Reading Alice
Referring to the benefits of learning logic through Alice, most of the students described their experience using affective as well as cognitive expressions:

. . . reading Alice made me realize that logic is not merely a ‘mathematical matter', but also relates to one of the most amazing children's book on which I was raised, and probably to manyotherstoo...;

...Alice is a book full of imagination and humor. It deals with ‘children's matter' in children's language. It is most enjoyable to learn logic, which appears to be boring, in that manner.Itmademefeelveryspecial...;

Like Wonderland, logic also seems to me as fantasy. Thus it was fascinating to learn logic through Alice.

The topics in logic are very complex. I believe that reading Alice changed the entire atmosphere that might have been created in the lessons otherwise. I suppose it made me and the other students more relaxed;

A few students mentioned that reading from Alice helped them understand some specific topic:

...for example, the empty set. The story discusses the subject in a very concrete manner; It was interesting to learn topics like ‘if...,then...', ‘for all' through reading the story. I suddenly found myself looking for conditional sentences or quantifiers in other texts like the news paper as well.

6.2 The Learning Environment
Integrating Alice in the logic course established a new and untried learning environment for the students. They expressed it as follows:

Using the book of Alice for teaching logic enabled us to experience different learning environment, an enjoyable one.

No more books full of formulas, no more conventional exercises and other standard teaching aids that are used in other math courses. It was really a new and satisfying experience for me; Learning through Alice urged me to attend the lessons. I don't remember feeling like that before, actually I tend to miss many lessons and learn from books. I enjoyed the lessons very much. There was one lesson I could not attend, but I was very curious to know what parts of the book you read that day and what issue came out of it;

Reflecting on my experience with this original learning environment, I realize that reading Alice motivated me to learn logic not just because I was forced to, but mainly because I enjoyed it.

6.3 Implication for School Teaching
Part of the students related not only to their own new experience with the new environment, but also to its potential for implementation in school teaching:

We all know that children don't read much these days. They prefer watching television and chatting through the computer. I believe that as educators we should strive to find ways to show them how fun reading is. Teaching mathematics through children's books can serve both purposes – enjoying reading books and enjoying the learning of mathematics.

I love the idea of teaching mathematics using children books. It is very interesting. It made me more attentive. I even implemented the method (using Five Balloons) in one of the classes in which I am practicing my math teaching, and I felt that it was easier for the children to learn the subject as I connected it to their own world;

It is most advisable for teachers to integrate children's books in mathematics lessons or any other lessons. In this way we, the adults, can enter the children's life, speak their language, and actually experience being a child again;

I think it is a great way to teach children. Stories help attract their attention, and the children would wait for the next lesson to hear the rest of the story, like I did.

Teaching logic based on Alice showed me that there are other ways to teach mathematics, moving away from the routine. I believe teachers should be able to implement various teaching methods. They should learn how to surprise their students; otherwise the students might get bored;

Integrating Alice or other children's books in mathematics lessons provides an unusual stimulus to learn the subject matter. It motivates the students and inspires their learning. From my experience with Alice, I believe that enjoyment encourages learning;

Children, especially the young ones, cannot concentrate during the entire lesson period. They need frequent breaks. Reading stories that are relevant to the topic of the lesson, can be used for ‘letting the mind rest', and enable the children to be more attentive.

6.4 Disadvantages of Learning Through Alice or Other Children's Book
Four of the eighteen students referred to the difficulties they experienced with learning logic through Alice and commented also about the possible limitations of integrating children's books in mathematics lessons:

There were times that reading the book disturbed be. Although the sentences and the described situations seemed strange to me, I insisted on considering them as ‘normal', namely-things that might happened in my everyday life. It distracted my mind from the lesson;

Many times I found myself confused. I could not decide whether I should refer to the sentence in its logical sense or to its meaning in the context of the story;

Alice's story is a fairy tale, and is not relevant to our topic. You could provide us other examples, not from the book, using much more simple words, and consequently we could have used more time for learning logic;

Using children literature might also be ‘dangerous'. The story may possibly distract the students' attention, and they would be busy with the plot instead of the subject matter.

In summary, it appears that most of the prospective teachers' utterances refer to affective aspects and to their satisfaction from the learning environment. Additional research is needed in order to be able to determine whether prospective teachers who experience learning logic through Alice perceive and understand the various topics differently or better than those who learn it in a more traditional manner.

7 Concluding Remarks
Reflecting upon the teaching experiment we conducted, we feel that the bold move of breaking the common order of an introductory course in logic had a justifiable payoff. Students as well as us came out of the course with a sense of fulfillment, having coped with challenging tasks that had meaning, and appreciating the power of logic as putting language matters in order. Nevertheless, further in-depth analysis of the data collected is needed in order to reveal the subtleties of students understanding of logic gained through such a different course and possibly their ability to cope with a higher level course in logic following such an introduction. It would be interesting also to attempt logic in wonderland method in school and examine its impact on school students’ language and reasoning ability.