مناقشة كتاب When the problem is not the question and the solution is not the answer: Mathematical Knowing and teaching

When the problem is not the question and the solution is not the answer: Mathematical Knowing and teaching

From the article on mathematical presentation when The Problem Is Not the Question and the Solution Is Not the Answer, many questions pop up in the mind of a reader. Implication of standardized and revolutionary models of mathematical content deliveries sparks interest in discussion and thus debate.  This leads to contentions in presentation models of mathematical contents.  Propositions of various models considered suitable in content delivery are thus left as the only new channels of thinking mathematically.

This article has presented challenging thoughts on mathematical presentations. By believing that mathematical challenges are not resident in the questions and solutions not in answers, Lampert (1990) provides controversial models of learning mathematics as evident in the analysis and critique provided above. It is true that there is proposition that mathematical justification need deductive proofing through engagements in teacher-learner argumentations. He also proposes that there is need to understand teaching practice for evaluation of effective teaching through research design and interpretive responses.

           

Let’s startJ

1.      When you read the topic of the article... What was your thinking?!

2.       What is the main thing of this article? (A research project in which students dealing with exponents)..

3.       In the article there is some guessing in attempts to solve mathematical problems.. Can you remember through what? (modesty and use of prescribed models)

4.       In the Article Lampert argues –as discussed by Lakatos- that mathematical pathways are unclear. .. Can you think why it is unclear?

5.      What is an axiom in mathematics?! (Something is knowing and being accepted as true without needing of any proof).. Can you think of an example?!

6.       What is this zig-zag process to which Lampert refers?

It is a process in which one revises conclusions and revises assumptions in an attempt to come to know (mathematics), thus prompting some mathematicians to re-question previously held conclusions

7.    In Page 31, what are the qualities of doing mathematics based on Polya's moral?

a). Intellectual courage: We should be ready to revise any one of our beliefs

b). Intellectual honesty: We should change a belief when there is good reason to change it

c). Wise restraint: We should not change a belief wantonly, without some good reason, without serious examination

8.    How do these moral qualities stand in contrast to how mathematics is done in the classroom?... (We seem to be quite obsessed with certainty in the classroom)

9.    What are the authorities in the classroom? (Teachers and textbooks)

10.     Should the teacher take full authority in teaching mathematics?! Why?!

(When teachers take full authority in teaching mathematics hence overlooking the chance of engagement from their students. This model of teaching has been considered wrong since it does not cater for openness to revision of proposed axiomatic conclusions. )

11.    Telling the students if their answers are right or wrong does it scales up or down the engagement of the learning process? (scales down)

12.   What is Mathematical conjectures??? Should student do that in class room??
      (Mathematical conjectures demand that students explain their answers, reasoning and decision arrival thus proposing for a more engaging model of teaching mathematics.)

13.   Explaining answers is good because we know that they understand?!.. but what is the critique of the model?!

(The critique of this model of teaching is that it requires a lot of time to implement and thus students might just have the basic foundation of mathematical argumentation but lack a wider mathematical understanding since they will only be subjected to few topics in mathematics during a given period of time)

14. What is the best solution for this model??

15. What is mean “ interactive platforms in mathematical teaching and learning in a classroom environment.” And what is weakness?

 This is projected to ensure understanding of classroom environment for every party to be engaged in the argumentative discussion. This approach is considered to be one of the best approaches to mathematical discourse because of the interactivity it provides in understanding of mathematical axioms.  The weakness of this approach is the assumption that teacher and student should concentrate on augmenting  اجمتنق (زيادة)mathematical axioms while some of the learners might have low abilities of engagement or short-lived attention for lack of or distraction of interest.

16.   What is "evaluation of effective teaching"?? Do we need it???

Yes we do.  This implies that educational discharge will be accurately done since understanding and analyzing what had been previously taught will automatically increase the quality of teaching since there will be improvement each time a lessons ends implying that before the commencement of another lesson, weak points have been identified and points of future avoidance made and thus delivery of quality and accurate work done. 

17. Through action research, there is understanding of effective lesson deliveries and with the augmentation of interpretive research, desired skills are isolated and delivered to respective and desired students hence maximizing their understanding of mathematics… do you know why is that??

18.     She said that this approach is unusual… Why?p.35

-          First, theory testing and development of practice were conducted simultaneously.

-          Second, the practice and its subsequent analysis draw on social science approaches to educational research and epistemological arguments that characterize the subject being taught.

-          Third, the practice under study is deliberately transformative.

19. In her second point she wrote the word “epistemological” so, what does the word epistemology mean? (It refers to the acquisition of, or how one acquires, knowledge)

20. Two methodology have contributed in this approach.. what are they?P.36

(Action research and interpretive social science)

21.     What is the purpose of interpretive research??

Is not to determine whether general propositions about learning or teaching are true or false but to further our understanding of the character of these particular kinds of human activity

22. What is a caveat of action research?

Some feel that it may not be significant enough evidence to substantiate rather bold claims because the teacher and the researcher are one in the same. This precipitates several questions about conflict of interest as well as how thorough of a job the individual can do given two widely disparate responsibilities simultaneously.

23.   What control does a teacher have over meaning in the mathematics classroom and how is meaning impacted?

Quite a bit of control and the type of the learning environment created impacts the level/type of meaning

24.  Did Dr. Lampert do anything unique in the lesson?

Personally, I think putting responses on the board and putting names next to them and also calling all of them hypotheses was quite innovative

25.   Through involved processes of induction and deduction.. How this can help student thinking?

Through involved processes of induction and deduction, students are able to make conclusive claims that are preceded with sufficient reason or premises.  This way, mathematical teaching becomes quite accurate and effective.  Besides, consideration of content and context is proposed to familiarize the learners with content in relation to their context.

26.                 When a new thinking in mathematics can be present?

When preceded with a mathematical challenge and this might imply overworking students with mathematical tasks. 

27.                 How effective will the model of presenting mathematics in an argumentative environment reach desired results?

This question poses many answers, both in the affirmative and the opposite dimensions.  Answering these questions will only depend on the length of time taken into research into the new methods of mathematical presentation. Some biasness though is thought to spark another interest in the methods of research to be used in carrying out investigative actions in trying to understand efficiency of these models.

28.       What is the role of a teacher in teaching mathematics in classroom setting?

The second question that arises after reading this article concerns the role of teachers in teaching mathematics in a classroom (Knott, 2009). Due to proposition of a mathematical argumentative environment, the role of the teacher is limited to raising or substantiating arguments with learners in the classroom. The question on the role of a teacher in classroom arises- Thus, the function of a teacher as the leader and one responsible for showing students mathematical solution methods disappears. This creates a lot of confusion in the essence of a teacher in classroom environment and the purpose of student in learning mathematics if they should engage in mathematical argumentative models of learning. 

29. If students should learn mathematics through argumentative learning processes, then the role of teacher as the captain disappears?? Is it correct??

 (students will sail their own ships in trying to reach mathematical axiomatic conclusions. Secondly, the argumentation provided in the paper requires that the teacher be simply a guide of mathematics without correspondence of tactical reasoning. Student engagement simply implies getting their understanding perspectives and improving method of content delivery.)

30. How would one make an exponential content familiar to the teaching context? Is it the teacher Job or the student job to do that??

This question concerns contextualizing mathematical contents in classrooms. Some mathematical contents do not have directly similar contexts in real life and thus contextualizing them in classroom setting might prove strenuous. While considering this approach in teaching mathematics, there is observable hardship on the side of the teacher in presenting some content due to the fact that the suitability of the context is often evasive. It is considered the role of a teacher to show direction to learners and thus his role, although this might be considered conservative, must clearly sail the learning boat while students listen, understand and only engage in clarification question researches).

31.   . How much time would really be required in implementing this type of approach such that questions have been solved and are no longer the problems?

This gives implication of larger amounts of time required in implementation of this type of learning. Availability of such large amounts of time becomes a concern especially when handling large volumes of content to be delivered within short durations.  Impeding of provision or delivery of the required content within the required duration is challenging since more time is allocated to demanding involvement of the teacher and student in the learning process

32.   Has anyone ever seen a version of figure 1?Page 45

Yes/no

33. Was  Dr. Lampert's question on finding the last digit in 54, 64, 74 ostensibly very complex?

No, not really

34. If not , why was it a question worthy of investigating?

She made them do the problem without being able to do the computation (and without a calculator). Hence, they had to basically create a mathematical model to answer the question

35.  How did such an ordinary problem become such a rich discussion? 

It became a rich discussion because of Dr. Lampert's prompts or charge in answer the question. She didn't just ask them to answer the question(s), she asked them to generalize and prove.

36. (When) should teachers tell students if their answers are correct?

Various answers

37.   Is a climate that fosters creativity apparent in this classroom? Justify your answer.

Yes, mainly because she lets students explore various solution paths and encourages individual responses

38.     Lastly, there is association of rules to arguments in the article. It is obvious that rules are statement prone to rigidity giving conditions of action. Arguments on the other hand are open-ended statements accommodating diverse views of the involved parties.  Rules do not take into acceptance argumentative statements but and they do precede argumentative statements. Rather rules are antecedents of argumentative statements where they are considered as the end of arguments. The article thus proposes that students take argumentative approaches in justification of their correct answers. The essence of learning mathematics is to understand the underlying rules of giving solutions to problems. Argumentum on the rules of solution arrival implies disputation of the rules and thus weakening the rules. Thus, the question on why rules should be argued amongst students or between learners and teachers cannot escape the attention of the reader. Presentations of mathematical models are based on the premises of universalities. When the teacher presents a mathematical content to students, there is assumed correctness in the teacher which strengthens the confidence of learners in acquisition of mathematical understanding. Proposition of rule-challenging dimension in the learning process will jeopardize this confidence and hence greatly affecting the learning process of mathematical concepts. Thus, the question why rules should be subjected to challenge arises for further discussion.

           

           

                                                   References
Knott, L. (2009). The Role of Mathematics Discourse in Producing Leaders of        Discourse. Charlotte: Information Age Pub.
Lampert, M. (1990). “When the Problem Is Not the Question and the Solution Is Not the             Answer: Mathematical Knowing and Teaching.” American Educational Research    Journal, 27(1) 29-63.

Book Review: Enhancing Adult Motivation to Learn

Book Review

Wlodkowski, Raymond. J.

Enhancing Adult Motivation to Learn, 2nd Ed

San Francisco: Jossey-Bass, 1999,376 pp. ISBN 0-7879-0360-4

Overview

Enhancing Adult Motivation to Learn explores a new and different approach on the issues surrounding adult learning with a key focus on the impact of cultural diversity on adult motivation. The main theme or purpose of the book is to identify and show how instructors can awaken the desire to learn in adult learners by understanding and appreciating their different cultural backgrounds. Based on various theories of motivation, the author proposes different motivational strategies that can be used by instructors to; create, enhance and maintain the learners’ interests.

According to Wlodkowski the author of the book, motivation refers to the energy that drives adults to become naturally inclined to compete for things they consider very important to them. The author further adds that, there is a very close relationship between motivation and the learner’s cultural background which in turn affects the ability and willingness to learn. Adult learners come from diverse backgrounds in terms of their culture, ethnic origin, race and gender, and instructors must clearly understand and appreciate this fact if they are to motivate and encourage adult learning.

The author supports his view on the impact of motivation on learning using his own personal experiences, assumptions and facts from reviewed literature and research studies that explore the use of motivation as an enhancement tool for learning.

The book is a revised edition of the same and it mainly focuses on the effect of ethnic differences, racism and gender on the motivation of adult learners. According to the author, every learner posses some intrinsic motivation characteristics which emanates from within and can thus become positively motivated for as long as they are able to view themselves as the center (locus) of the causes and effects of their learning. In simple terms, adults are able to learn as much as they are willing to learn.

Based on this presumption that learning motivation must come from within, the book seeks to explores way in which instructors can elicit and stimulate the already present intrinsic motivation characteristics in order to encourage learning among adult learners. Reviewed from edition one which was published in 1985, this revised edition is more comprehensive and responsive to the issue of ethnic, racial and cultural diversity and its impact on adult learning.

The issue of cultural diversity is carefully interwoven with the issue of motivation in each and every chapter of the book giving rise to a comprehensive and practical guide for instructors involved in adult learning programs.

The book is carefully organized into eight comprehensive chapters. The first three chapters are somehow delineated from the other five chapters as they address the issue of motivation from a multi-disciplinary perspective. The remaining five chapters mainly focus on practical examples on motivation of adult learners and they also give workable strategies on how the issue can be addressed more conclusively.

Evaluation

Enhancing Adult Motivation to Learn is one of the few books in the field of Adult Education that directly address the issue of motivation so clearly and comprehensively. The book is scholarly written and it serves as a rich source of reference for instructors and trainers in the field of adult learning. With eight comprehensive chapters, the book is organized in a logic manner which is easy to read and understand.

The first chapter of the book reviews the background information surrounding the term motivation. This chapter also provides a summary of the issues surrounding adult learning in relation to motivation and cultural diversity. In this chapter, the author uses the term motivation to refer to a learner’s willingness to be attentive in class and to learn in some cases while in other instances, he uses it to refer to a situation where a learner willingly applies what is learnt in class to situations outside the classroom.

In chapter two, the author explores the five key characteristics of an effective instructor that enhance learner motivation. These qualities include; enthusiasm, expertise, clarity, empathy and responsiveness to the learners’ cultural diversity. He further develops these five key qualities that every effective instructor must possess in order to be able to motivate the students based on the motivational theory and literature from cultural and ethnic studies. According to Wlodkowski, an instructor should first be an expert in the field that he or she is teaching so as to be able to share the knowledge to the students in a way that all the learners will understand clearly. Moreover, the instructor should be able to understand each of the learners and be able to apply different teaching approaches to meet each learner’s individual learning needs. To be able to capture and maintain the learners’ attention, the instructor should be enthusiastic and should also communicate with clarity in a way that every learner is able to understand.

In chapter three, the author explores the factors to consider when designing a motivating learning model. These factors include; attitude, stimulation, competence, emotional state, needs and reinforcement. Based on these factors, Wlodkowski proposes a “Motivational framework for culturally responsive teaching” aimed at assisting instructors to enhance the motivation of their students. The framework outlines four conditions for culturally responsive adult teaching which include; cultivating positive attitudes, enhancing meaning, establishing inclusion and engendering learners’ competence. According to the author, these four conditions form a common culture base on which all the needs of the learners are valued and addressed equally.

The next four chapters; 4, 5, 6 and 7 form the main body of the book as they continue exploring the four conditions of the motivational framework in detail. These four chapters are consistent in format with each chapter covering one of the conditions mentioned above. In each chapter, the author first provides the background information on the condition to be discussed before giving practical examples and workable motivational strategies on how the condition can be incorporated into the actual teaching process. Throughout the four chapters, the author proposes various motivational strategies which are aimed at enhancing adult learner motivation in each area discussed.

A total of 60 motivational strategies are discussed in chapters 4, 5, 6 and 7. One example of such a learning strategy is Strategy 1 which proposes that, for an instructor to cultivate a positive learning attitude in the students, he or she should constantly share valuable information with the learners. This will make the learning process more entertaining and the learners will be more interested hence motivated to learn.

The last chapter; chapter 8 serves as a summary for the entire book where the author explains how all the various motivational strategies discussed in the four previous chapters can be applied in enhancing adult learner motivation. This chapter also provides additional examples for instructional planning and guidelines for assessment by on adult learners.

Strengths and weaknesses

The book is well organized and the ideas easily flow from one chapter to the other making it a comprehensive and interesting guide for instructors. Having practical examples to illustrate the key points, the author’s ideas are explicitly presented with excellent clarity which makes them easy to understand and apply across all settings. The sixty proposed motivational strategies are particularly helpful for instructors as they contain wisdom and practical skills which if properly translated and applied can guarantee excellent results in terms of enhancing adult learners’ motivation and interest in learning.

In addition, the author does not disappoint in delivering the proposed ideas and concepts and despite being a great scholar in the field of psychology, he avoids the use of psychobabble and jargon words which would complicate the text. In addition, the author employs humor and many practical examples which help drive the point home by attracting and maintaining the interest of the reader.

However, the book is too a bit too choppy. The author spends too much time explaining and justifying the motivational strategies and some of the learning strategies even tend to overlap. For instance, Strategy 42 which states that; “when possible, the instructor should state and demonstrate to the learners the possible advantages of learning” has a more or less similar meaning as Strategy 67 which states that; “the instructor should help the learners to be aware of the natural consequences of learning and the impact of such consequences.”

Recommendation

Enhancing Adult Motivation to Learn is a great read for instructors in the field of adult learning. It not only provides excellent resources on motivation but it also provides a practical framework and strategies for motivation of adult learners. I would therefore strongly recommend this book to any instructor who is already involved or interested in developing a new course in the field of adult learning as the book serves as an excellent instructional tool regardless of the stage of their teaching career. However, most of the techniques and strategies proposed in the book are more applicable to small groups and would not work very well on large groups of learners. I would highly recommend this book to teachers, instructors, trainers and anyone who is interested in motivating adult learners.

Author

The author Dr. Raymond Wlodkowski has a rich background in the field of psychology and is a licensed psychologist. He is currently a professor at Regis University in Denver where he is also the Founding Director of the commission for accredited programs.

Reference:

Wlodkowski, Raymond. (1999) Enhancing Adult Motivation to Learn: A Comprehensive Guide

For Teaching All Adults (Revised Edition). San Francisco: Jossey-Bass Publishers