Discussion, analysis, and critique
In this article, there is conscious guessing in attempts to solve mathematical problems through modesty and the use of prescribed models. Initially, Lampert argues –as discussed by Lakatos- that mathematical pathways are unclear. The accepted process of mathematical justification needs deductive proofing.
With increased growth in mathematics over time in openness to revision of axiomatic conclusions, there has been deductive thinking as people engage in proving tactics and later discover shortcomings in initially proposed axioms (Lampert, 1990).
Conscious guessing has thus been referred to be risk-taking since openness in the proposed axiomatic conclusions or acceptance of possibilities of inappropriateness in the process of conscious guessing is inaccurate. The argumentation on what mathematical knowledge should comprise of has become a question of debate in the mathematical world. As evident in the article, major argumentative responses have largely focused on the role of teachers in teaching mathematics. Lampert (1990) argues that 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. Informing students whether their answers are right or wrong simply scales down the engagement of the learning process required in mathematical teaching.
Mathematical conjectures demand that students explain their answers, reasoning, and decision arrival thus proposing a more engaging model of teaching mathematics. 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 a few topics in mathematics during a given period.
Activity variations have been suggested as interactive platforms in mathematical teaching and learning in a classroom environment. This is projected to ensure understanding of the 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 the understanding of mathematical axioms. The weakness of this approach is the assumption that teacher and student should concentrate on augmenting mathematical axioms while some of the learners might have low abilities of an engagement or short-lived attention for lack of or distraction of interest.
As Lampert (1990) argues, there is a need to understand teaching practice for the evaluation of effective teaching. 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 lesson 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.
Through action research, there is an 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.
Although this method of action research has been proposed in the teaching of mathematics, the examiner and the actor or an individual to be examined happens to be the same person. This might present some biasness since a researcher might give information as self-pleasing as possible hence contributing to a lack of openness in the teaching process. Therefore, this means that for this approach to take effect, the researcher must be different and such indicating the recruitment of another professional into the education sector to give attention to mathematical classroom teaching practices. This is bound to incur education expenses in the teaching process alone and thus overshadow other resource funding hence decreasing the level of mathematic abilities in students.
When the problem is not the answer presents an inventive way of mathematical content delivery. Through involved processes of induction and deduction, students can 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 about their context.
Teaching that focuses on continuous thinking is also proposed whereby, the final answer should not determine the end of mathematical thinking. Although these methods are greatly effective in mathematical discourse, new thinking in mathematics can only be present when preceded with a mathematical challenge and this might imply overworking students with mathematical tasks.
The question of argumentative engagement of students in mathematics is overshadowed in providing platforms for mathematical thinking. While Lampert (1990, p.45) proposes that thinking should be kept private, it implies that mathematical thinking should not take an argumentative dimension. This contradicts with the major premise of this paper whereby the proposition of interactive, action research and interpretive responses are significant in mathematic content delivery. Although he notes later that influence of group thinking in a mathematical environment, which implies affected and non-authentic mathematical presentation of reasons to mathematical arguments, he does not however put the comparison to the previous proposition. This means that the whole of the proposed mathematical teaching and thinking is likely going to be affected to reach ineffective levels. Noting of stubbornness in individuals is also likely going to influence final mathematical closes as evident in assertive conclusions containing rigidity.
Questions to be discussed
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. The 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. The question of effectiveness in the new proposed model sparks debate. 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 business though is thought to spark another interest in the methods of research to be used in carrying out investigative actions in trying to understand the efficiency of these models.
The second question that arises after reading this article concerns the role of teachers in teaching mathematics in a classroom (Knott, 2009). Due to the 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 the classroom arises- what is the role of a teacher in teaching mathematics in a classroom setting? 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 a classroom environment and the purpose of students in learning mathematics if they should engage in mathematical argumentative models of learning. If students should learn mathematics through argumentative learning processes, then the role of the teacher as the captain disappears making it obvious that students will sail their 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 the method of content delivery.
Thirdly, how would one make exponential content familiar to the teaching context? This question concerns contextualizing mathematical content in classrooms. Some mathematical contents do not have directly similar contexts in real life and thus contextualizing them in a 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 because 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 sail the learning boat while students listen, understand, and only engage in clarification question researches (Knott, 2009).
The fourth question that arises from this reading concerns the amount of time required in implementing this model of teaching mathematics. How much time would be required in implementing this type of approach such that questions have been solved and are no longer the problems? This gives the implication of larger amounts of time required in the 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 the demanding involvement of the teacher and student in the learning process.
Lastly, there is an association of rules to arguments in the article. Rules are statements 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 the 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 of 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 mathematical content to students, there is assumed correctness in the teacher which strengthens the confidence of learners in the acquisition of mathematical understanding. The proposition of the rule-challenging dimension in the learning process will jeopardize this confidence and hence greatly affect the learning process of mathematical concepts. Thus, the question of why rules should be subjected to challenge arises for further discussion.
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. There is indeed a proposition that mathematical justification needs deductive proofing through engagements in teacher-learner argumentations. He also proposes that there is a need to understand teaching practice for the evaluation of effective teaching through research design and interpretive responses.
- 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.