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.