Creative and algorithmic mathematical reasoning: effects of transfer-appropriate processing and effortful struggle

Two separate studies, Jonsson et al. (J. Math Behav. 2014;36: 20–32) and Karlsson Wirebring et al. (Trends Neurosci Educ. 2015;4(1–2):6–14), showed that learning mathematics using creative mathematical reasoning and constructing their own solution methods can be more efficient than if students use algorithmic reasoning and are given the solution procedures. It was argued that effortful struggle was the key that explained this difference. It was also argued that the results could not be explained by the effects of transfer-appropriate processing, although this was not empirically investigated. This study evaluated the hypotheses of transfer-appropriate processing and effortful struggle in relation to the specific characteristics associated with algorithmic reasoning task and creative mathematical reasoning task. In a between-subjects design, upper-secondary students were matched according to their working memory capacity.

The main finding was that the superior performance associated with practicing creative mathematical reasoning was mainly supported by effortful struggle, however, there was also an effect of transfer-appropriate processing. It is argued that students need to struggle with important mathematics that in turn facilitates the construction of knowledge. It is further argued that the way we construct mathematical tasks have consequences for how much effort students allocate to their task-solving attempt.     

Leveraging interdiscursivity to support elementary students in bridging the empirical-deductive gap: the case of parity

Research has recognized deductive reasoning as challenging but not impossible for young mathematics learners. In this paper, we present a learning environment developed to assist elementary-school students to bridge the empirical-deductive gap in the context of parity of numbers. Using the commognitive framework, we construe the empirical-deductive gap as part of a broader divide between two discourses that abide by different rules of a “mathematical game”: a discourse on specific numbers and a discourse on numeric patterns. Interdiscursivity is leveraged as a mechanism for instructional design, where students’ familiar routines with specific numbers are teased out and advanced to make sense in the new discourse. We mobilize this mechanism to create opportunities for students to play an active role in recognizing issues with empirical reasoning and generating deductive arguments to establish the validity of universal statements. The environment is illustrated with a small group of 8-year-olds who learned to justify deductively that “odd + odd = even”.     

The co-construction of arguments by middle-school students

This paper highlights the value of student collaboration in doing mathematics, demonstrates how urban, middle-school students, working together, co-constructed justifications for their solutions, and shows that certain conditions are associated with the promotion of a culture of reasoning. It is documented that students collaboratively built arguments that took the form of proof, challenged each others’ arguments, and justified these arguments in small groups and whole class discussions. In producing their mathematical justifications, students included the input of others. Finally, the way in which students, by expanding on the arguments of others, also used alternative forms of reasoning which in many cases led to even more refined arguments is discussed.     

Validating proofs and counterexamples across content domains: Practices of importance for mathematics majors

Validating proofs and counterexamples across content domains is considered vital practices for undergraduate students to advance their mathematical reasoning and knowledge. To date, not enough is known about the ways mathematics majors determine the validity of arguments in the domains of algebra, analysis, geometry, and number theory—the domains that are central to many mathematics courses. This study reported how 16 mathematics majors, including eight specializing in secondary mathematics education, who had completed more proof-based courses than transition-to-proof classes evaluated various arguments. The results suggest that the students use one of the following strategies in proof and counterexample validation: (1) examination of the argument's structure and (2) line-by-line checking with informal deductive reasoning, example-based reasoning, experience-based reasoning, and informal deductive and example-based reasoning. Most students tended to examine all steps of the argument with informal deductive reasoning across various tasks, suggesting that this approach might be problem dependent. Even though all participating students had taken more proof-related mathematics courses, it is surprising that many of them did not recognize global-structure or line-by-line content-based flaws presented in the argument.     

Learning mathematics through algorithmic and creative reasoning

There are extensive concerns pertaining to the idea that students do not develop sufficient mathematical competence. This problem is at least partially related to the teaching of procedure-based learning. Although better teaching methods are proposed, there are very limited research insights as to why some methods work better than others, and the conditions under which these methods are applied. The present paper evaluates a model based on students’ own creation of knowledge, denoted creative mathematically founded reasoning (CMR), and compare this to a procedure-based model of teaching that is similar to what is commonly found in schools, denoted algorithmic reasoning (AR). In the present study, CMR was found to outperform AR. It was also found cognitive proficiency was significantly associated to test task performance. However the analysis also showed that the effect was more pronounced for the AR group.     

Types of reasoning required in university exams in mathematics

Empirical research shows that students often use reasoning founded on copying algorithms or recalling facts (imitative reasoning) when solving mathematical tasks. Research also indicate that a focus on this type of reasoning might weaken the students’ understanding of the underlying mathematical concepts. It is therefore important to study the types of reasoning students have to perform in order to solve exam tasks and pass exams. The purpose of this study is to examine what types of reasoning students taking introductory calculus courses are required to perform. Tasks from 16 exams produced at four different Swedish universities were analyzed and sorted into task classes. The analysis resulted in several examples of tasks demanding different types of mathematical reasoning. The results also show that about 70% of the tasks were solvable by imitative reasoning and that 15 of the exams could be passed using only imitative reasoning.     

Assessments accompanying published textbooks: the extent to which mathematical processes are evident

Assessments accompanying published textbooks are often used by teachers in the USA as a primary means to evaluate students’ mathematical knowledge. In addition to assessing content knowledge, assessments should provide insight into students’ ability to engage with mathematical processes such as reasoning, communication, connections, and representations. We report here an analysis of the extent to which the assessments accompanying published textbooks in the USA at the elementary, middle grades, and high school levels provide opportunities for students to engage with these mathematical processes. Results indicate that in elementary grades, communication, connections, and graphics are not consistently emphasized across grade levels and publishers. In middle grades, students are rarely asked to record their reasoning or translate among representational forms of a concept. In high school geometry, students are given many opportunities to interpret and create graphics, but the same is not true for algebra. With the exception of connections, the results suggest that inconsistent emphasis is placed on the mathematical processes within assessments accompanying commercial textbooks in the USA.     

CHANGES IN SCHOOL STUDENTS’ OBSERVATIONS AND REASONING DURING A 4-PHASE TEACHING SEQUENCE

We examine the written responses of fifteen students (aged about 14½ years) to a homework task and their responses to the same task in a subsequent lesson. Students were asked to make observations about the sum of three consecutive numbers and to explain why they think these are true, thereby giving students the opportunity to engage in structural reasoning. The teaching sequence had four phases designed to allow students to make, share and develop their observations and reasoning, and we found a clear improvement in the quality of students’ responses. As far as students’ reasoning is concerned, this suggests limitations may stem at least in part from a lack of familiarity with the nature of mathematical reasoning.     

Levels of arithmetic reasoning in solving an open-ended problem

This paper presents the results of an experimental teaching carried out on 12-year-old students. An open-ended task was given to them and they had not been taught the algorithmic process leading to the solution. The formal solution to the problem refers to a system of two linear equations with two unknown quantities. In this mathematical activity, students worked cooperatively. They discussed their discoveries in groups of four and then presented their answers to the whole class developing a rich communication. This study describes the characteristic arguments that represent certain different forms of reasoning that emerged during the process of justifying the solutions of the problem. The findings of this research show that within an environment conducive to creativity, which encourages collaboration, exploration and sharing ideas, students can be engaged in developing multiple mathematical strategies, posing new questions, creating informal proofs, showing beauty and elegance and bringing out that problem solving is a powerful way of learning mathematics.     

Elements in accepting an explanation

This paper addresses the question of what criteria influenced the acceptance of two “explanations” by grade 5 students. The students accepted the use of deductive reasoning as explanatory, as well as using reasoning by analogy in their own explanations. The “explanations” can be interpreted as proofs by mathematical induction. The main weakness of mathematical induction as a form of explanation was the arbitrariness of the initial step. The induction step did not seem to trouble these students. Other elements in their acceptance of explanations were concreteness, familiarity, and opportunities for multiple interpretations.     

The role of operating premises and reasoning paths in upper elementary students’ problem solving

The validity of students’ reasoning is central to problem solving. However, equally important are the operating premises from which students’ reason about problems. These premises are based on students’ interpretations of the problem information. This paper describes various premises that 11- and 12-year-old students derived from the information in a particular problem, and the way in which these premises formed part of their reasoning during a lesson. The teacher’s identification of differences in students’ premises for reasoning in this problem shifted the emphasis in a class discussion from the reconciliation of the various problem solutions and a focus on a sole correct reasoning path, to the identification of the students’ premises and the appropriateness of their various reasoning paths. Problem information that can be interpreted ambiguously creates rich mathematical opportunities because students are required to articulate their assumptions, and, thereby identify the origin of their reasoning, and to evaluate the assumptions and reasoning of their peers.     

Assessing Students' Mathematical Communication

Assessment of students' mathematical communication through the use of open-ended tasks and scoring procedures is addressed, as is the use of open-ended tasks to assess students' mathematical communication by providing students opportunities to display their mathematical thinking and reasoning. Also, two scoring procedures (quantitative holistic scoring procedure and qualitative analytic scoring procedure) are described for examining students' communication skills.     

The development of reasoning skills during compulsory 16 to 18 mathematics education

The belief that studying mathematics improves reasoning skills, known as the Theory of Formal Discipline (TFD), has been held since the time of Plato. Research evidence supports this idea, at least in the context of students who had chosen to study post-compulsory mathematics. Here we examined the development of reasoning skills in 16- to 18-year-old Cypriot students, who are required to study mathematics until age 18. One hundred and eighty-eight students, studying high- or low-intensity mathematics, completed the abstract Conditional Inference Task and the contextual Belief Bias Syllogisms task at ages 16, 17 and 18. While the high-intensity group improved on the conditional inference task and showed a reduction in belief bias, the low-intensity group did not change on either measure. This is promising for the TFD, but suggests that a certain level of mathematical study may be necessary for students' general reasoning skills to develop.     

Mathematical poetic structures: The sound shape of collaboration

This paper outlines a new method of mathematical discourse analysis focused on identifying poetic structures in students’ mathematical conversations. Following the linguistic anthropology tradition inspired by Roman Jakobson, poetic structures refer to any conversational repetition of sounds, words or syntax; this repetition draws attention to the form of the message. In mathematical conversations, poetic structures can express patterns, rhythms, similarities or dissimilarities associated with a task. Methodological dilemmas associated with identifying and representing poetic structures and pragmatic responses are highlighted. An analysis of a nine minute algebraic problem-solving conversation revealed eight types of mathematical poetic structures that collectively assisted all of the students’ vital mathematical insights. The paper aims to demonstrate that poetic analysis of mathematical conversations can bridge the illusory distinction between mathematical discourse and mathematical reasoning.     

When Two Circles Determine a Triangle. Discovering and Proving a Geometrical Condition in a Computer Environment

Visualization of mathematical relationships enables students to formulate conjectures as well as to search for mathematical arguments to support these conjectures. In this project students are asked to discover the sufficient and necessary condition so that two circles form the circumscribed and inscribed circle of a triangle and investigate how this condition effects the type of triangle in general and its perimeter in particular. Its open-ended form of the task is a departure from the usual phrasing of textbook’s exercises “show that…”.     

Using Student Reasoning to Inform the Development of Conceptual Learning Goals: The Case of Quadratic Functions

Despite the proliferation of mathematics standards internationally and despite general agreement on the importance of teaching for conceptual understanding, conceptual learning goals for many K-12 mathematics topics have not been well-articulated. This article presents a coherent set of five conceptual learning goals for a complex mathematical domain, generated via a method of systematic empirical analysis of students' reasoning. Specifically, we compared the reasoning of pairs of students who performed differentially on the same task and inferred the pivotal intermediate conceptions that afforded one student deeper engagement with the task than another student. In turn, each pivotal intermediate conception framed an associated conceptual learning goal. While the empirical analysis of student reasoning is typically used to understand how students learn, we argue that such analysis should also play an important role in determining what concepts students should learn.     

The effect of explanations on mathematical reasoning tasks

Studies in mathematics education often point to the necessity for students to engage in more cognitively demanding activities than just solving tasks by applying given solution methods. Previous studies have shown that students that engage in creative mathematically founded reasoning to construct a solution method, perform significantly better in follow up tests than students that are given a solution method and engage in algorithmic reasoning. However, teachers and textbooks, at least occasionally, provide explanations together with an algorithmic method, and this could possibly be more efficient than creative reasoning. In this study, three matched groups practiced with either creative, algorithmic, or explained algorithmic tasks. The main finding was that students that practiced with creative tasks did, outperform the students that practiced with explained algorithmic tasks in a post-test, despite a much lower practice score. The two groups that got a solution method presented, performed similarly in both practice and post-test, even though one group got an explanation to the given solution method. Additionally, there were some differences between the groups in which variables predicted the post-test score.     

Problem solving heuristics,affect, and discrete mathematics

It has been suggested that activities in discrete mathematics allow a kind of new beginning for students and teachers. Students who have been “turned off” by traditional school mathematics, and teachers who have long ago routinized their instruction, can find in the domain of discrete mathematics opportunities for mathematical discovery and interesting, nonroutine problem solving. Sometimes formerly low-achieving students demonstrate mathematical abilities their teachers did not know they had. To take maximum advantage of these possibilities, it is important to know what kinds of thinking during problem solving can be naturally evoked by discrete mathematical situations—so that in developing a curriculum, the objectives can include pathways to desired mathematical reasoning processes. This article discusses some of these ways of thinking, with special attention to the idea of “modeling the general on the particular.” Some comments are also offered about students' possible affective pathways and structures.     

An epistemic model of task design in dynamic geometry environment

Dynamic geometry environment (DGE) has been a catalytic agent driving a paradigm shift in the teaching and learning of school geometry in the past two decades. It opens up a pedagogical space for teachers and students to engage in mathematical explorations that niche across the experimental and the theoretical. In particular, the drag-mode in DGE has been a unique pedagogical tool that can facilitate and empower students to experiment with dynamic geometrical objects which can lead to generation of mathematical conjectures. Furthermore, the drag-mode seems to open up a new methodology and even a new discourse to acquire geometrical knowledge alternative to the traditional Euclidean deductive reasoning paradigm. This discussion paper proposes an epistemic model of techno-pedagogic mathematic task design which serves as a theoretical combined-lens to view mathematics knowledge acquisition. Three epistemic modes for techno-pedagogic mathematical task design are proposed. They are used to conceptualize design of dynamic geometry tasks capitalizing the unique drag-mode nature in DGE that opens up an explorative space for learners to acquire mathematical knowledge.