The interplay of empirical and deductive reasoning in proving “if” and “only if” statements in a Dynamic Geometry environment

This paper is situated within the ongoing enterprise to understand the interplay of students’ empirical and deductive reasoning while using Dynamic Geometry (DG) software. Our focus is on the relationships between students’ reasoning and their ways of constructing DG drawings in connection to directionality (i.e., “if” and “only if” directions) of geometry statements. We present a case study of a middle-school student engaged in discovering and justifying “if” and “only if” statements in the context of quadrilaterals. The activity took place in an online asynchronous forum supported by GeoGebra. We found that student's reasoning was associated with the logical structure of the statement. Particularly, the student deductively proved the “if” claims, but stayed on empirical grounds when exploring the “only if” claims. We explain, in terms of a hierarchy of dependencies and DG invariants, how the construction of DG drawings supported the exploration and deductive proof of the “if” claims but not of the “only if” claims.     

Odds and evens: mathematical reasoning and informal proof among high school students

Ten high school algebra students were asked to judge simple statements about combining odd and even numbers, stating whether they were true or false. They were also asked to give justifications or explanations for their decisions. All of the students initially reasoned inductively or empirically, appealing to specific cases and justifying their answers with additional examples. On being prompted for any further explanations, seven of the students attempted to formulate some type of non-empirical rationale. However, only three students were able to create fairly coherent arguments, none of which used standard algebraic notation. Instead, two of these original, idiosyncratic arguments were based on visual representations of odd and even numbers, and the third consisted of an informal and partial argument by cases.

Intuition comes to us much earlier and with much less outside influence than formal arguments … Therefore, I think that in teaching high school age youngsters we should emphasize intuitive insight more than, and long before, deductive reasoning. —Polya, George (1981, pp. 2–128)

     

Enhancing language-responsive meaning-making processes as an epistemic catalyst for developing multiplicative reasoning in young children

Multiplicative reasoning involves the ability to coordinate bundled units on a more abstract level (“unitizing”; Lamon, 1994). As it is considered a “cutoff point” for students’ future mathematical learning, teachers must provide equitable access to mathematical conceptual understanding for all students on all mathematical achievement levels. The study presented in this paper investigates to what extent a preventive and a language-responsive instructional approach can have an effect on the outcome of students on different mathematical achievement levels. Three German second grade teachers introduced multiplication to students (n = 66, aged 7–8 years) in their classes using meaning-related phrases (e.g., “6 times 4 means 6 fours”), while teachers in the control group (n = 58) did not focus on using these phrases. Analyses of both a multiplication posttest and a follow-up test showed significant differences between the intervention and control groups on all achievement levels for both conceptual and procedural items.     

Proof in School Mathematics: Insights from Psychological Research into Students' Ability for Deductive Reasoning

There are currently increased efforts to make proof central to school mathematics throughout the grades. Yet, realizing this goal is challenging because it requires that students master several abilities. In this article we focus on one such ability, namely, the ability for deductive reasoning, and we review psychological research to enhance what is currently known in mathematics education research about this ability in the context of proof and to identify important directions for future research. We first offer a conceptualization of proof, which we use to delineate our focus on deductive reasoning. We then review psychological research on the development of students' ability for deductive reasoning to see what can be said about the ages at which students become able to engage in certain forms of deductive reasoning. Finally, we review two psychological theories of deductive reasoning to offer insights into cognitively guided ways to enhance students' ability for deductive reasoning in the context of proof.     

Examining students’ variational reasoning in differential equations

In this study, we explored how a sample of eight students used variational reasoning while discussing ordinary differential equations (DEs). Our analysis of variational reasoning draws on the literature with regard to student thinking about derivatives and rate, students’ covariational reasoning, and different multivariational structures that can exist between multiple variables. First, we found that while students can think of “derivative” as a variable in and of itself and also unpack derivative as a rate of change between two variables, the students were often able to think of “derivative” in these two ways simultaneously in the same explanation. Second, we found that students made significant usage of covariational reasoning to imagine relationships between pairs of variables in a DE, and that mental actions pertaining to recognizing dependence/independence were especially important. Third, the students also conceptualized relationships between multiple variables in a DE that matched different multivariational structures. Fourth, importantly, we identified a type of variational reasoning, which we call “feedback variation”, that may be unique to DEs because of the recursive relationship between a function’s value and its own rate of change.     

Integer comparisons across the grades: Students’ justifications and ways of reasoning

This study is an investigation of students’ reasoning about integer comparisons—a topic that is often counterintuitive for students because negative numbers of smaller absolute value are considered greater (e.g., −5 > − 6). We posed integer-comparison tasks to 40 students each in Grades 2, 4, and 7, as well as to 11th graders on a successful mathematics track. We coded for correctness and for students’ justifications, which we categorized in terms of 3 ways of reasoning: magnitude-based, order-based, and developmental/other. The 7th graders used order-based reasoning more often than did the younger students, and it more often led to correct answers; however, the college-track 11th graders, who responded correctly to almost every problem, used a more balanced distribution of order- and magnitude-based reasoning. We present a framework for students’ ways of reasoning about integer comparisons, report performance trends, rank integer-comparison tasks by relative difficulty, and discuss implications for integer instruction.     

Teaching practices aimed at promoting meta-level learning: The case of complex numbers

In this paper we seek to promote a conceptualization of “teaching toward meta-level learning” based on theoretical and empirical aspects. We adopt the commognitive distinction between object- and meta-level learning, and relate to meta-level learning as involving changes in the metarules that govern the discourse. Specifically, we refer to changes in the discourse on numbers emerging in the shift in discourse from real to complex numbers. We applied implications from the commognitive theory about meta-level learning to the planning and teaching of a lesson about complex numbers. Then, we analyzed the lesson to identify teaching practices that could promote meta-level learning. We found that these teaching practices can be clustered into three theory driven sub-sets: those referring to students’ current discourse on numbers, those referring to their new discourse and those referring to the transition between the two.     

Proof in School Mathematics: Insights from Psychological Research into Students' Ability for Deductive Reasoning

There are currently increased efforts to make proof central to school mathematics throughout the grades. Yet, realizing this goal is challenging because it requires that students master several abilities. In this article we focus on one such ability, namely, the ability for deductive reasoning, and we review psychological research to enhance what is currently known in mathematics education research about this ability in the context of proof and to identify important directions for future research. We first offer a conceptualization of proof, which we use to delineate our focus on deductive reasoning. We then review psychological research on the development of students' ability for deductive reasoning to see what can be said about the ages at which students become able to engage in certain forms of deductive reasoning. Finally, we review two psychological theories of deductive reasoning to offer insights into cognitively guided ways to enhance students' ability for deductive reasoning in the context of proof.     

Evaluating efficacy of environmental education programming

The Clark Fork Watershed Education Program (CFWEP) goals are: (a) increasing students’ understanding of the nature of ecological impacts within their watershed as related to historic mining damage; and (b) increasing students’ sense of stewardship of newly restored landscapes. Data from 2012 to 2016 were evaluated for student knowledge gains (46 trials representing 2,395 student pre‐surveys; 2,409 student post‐surveys). Data from 2013 to 2016 were evaluated for students’ attitudes toward science and disposition toward caring for the environment (38 trials representing 1,479 pre‐surveys; 1,460 post‐surveys). The results of this study support that the program’s goals are being achieved. Students achieved statistically significant gains on knowledge surveys with a 33.4% overall gain pre‐ to posttest (p < 0.0001). Students also moved toward greater positive responses in both attitudes toward science and disposition toward caring for the environment with Cohen’s d effect sizes of “medium effect” for caring toward the environment (d = 0.52) and “small effect” of positive disposition toward science (d = 0.24).     

Revisiting decimal misconceptions from a new perspective: The significance of whole number bias in the Chinese culture

The aim of this study was to investigate Hong Kong Grade 4 students’ understanding of the decimal notation system including their knowledge of decimal quantities. This is a unique study because most previous studies were conducted in Western cultural settings; therefore we were interested to see whether Chinese students have the same kinds of misconceptions as Western students given the Chinese number naming system is relatively transparent and explicit. Three hundred and forty-one students participated in a written test on decimal numbers. Thirty-two students were interviewed to further explore their mathematical reasoning. In summary, the results indicated that many students had mastered reasonable knowledge of decimal notation and quantities, which may be attributed to the Chinese linguistic clarity of decimal numbers. More importantly, the results showed that some students’ construction of decimal concepts have been adversely affected by persistent misconceptions arising from whole number bias. Two kinds of whole number misconceptions, namely “-ths suffix error” and “reversed place value progression error”, were revealed in this study. This paper suggests that a framework theory approach to conceptual change may be an alternative approach to addressing students’ learning difficulties in decimals.     

Bridging incommensurable discourses – A commognitive look at instructional design in the zone of proximal development

There are points in the mathematics curriculum where the “rules of the game” change, for example, the meaning and method of multiplication when negative numbers are introduced. At these junctions the new mathematical discourse may be in conflict with learners’ current discourse. Learners may have little intrinsic motivation to accept new rules whose productiveness they cannot yet appreciate, hence, their first steps in the emerging discourse may need to be ritualized - socially motivated by the teacher’s approval. In this article we ask how careful crafting of task situations can support teachers in leading learners into a new discourse. We propose interdiscursivity – the blending of discursive elements from different discourses – as a mechanism for designing task situations to support learners in taking their first steps in an emerging discourse. On the basis of three examples, we suggest that this mechanism may support participation that is intrinsically motivated (explorative).     

Identifying Kinds of Reasoning in Collective Argumentation

We combine Peirce’s rule, case, and result with Toulmin’s data, claim, and warrant to differentiate between deductive, inductive, abductive, and analogical reasoning within collective argumentation. In this theoretical article, we illustrate these kinds of reasoning in episodes of collective argumentation using examples from one teacher’s practice. Examining different kinds of reasoning in collective argumentation can inform how students engage in generating and examining hypotheses using inductive and abductive reasoning and move toward the deductive reasoning required for proof. Mathematics educators can build on their understanding of these kinds of reasoning to support students in reasoning in productive ways.     

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.     

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.     

Standing on each other’s shoulders: A case of coalescence between geometric discourses in peer interaction

In this paper we draw on the commognitive theory to examine novice students’ transition from familiar mathematics meta-rules to less familiar ones during peer interaction. To pursue this goal, we focused on a relatively symmetric interaction between two middle-school students given a geometric task. During their dyadic problem-solving, the students transitioned from configural procedures to deductive ones. We found that this transition included an interactive coalescence pattern in which one student “borrowed” her partner’s configural sub-procedures and built on them to develop a new deductive procedure. Furthermore, we found that during their peer interaction, the students oscillated between configural, coalesced and deductive procedures. Several patterns in the students’ interpretation of the task-situation contributed to these oscillations. We discuss the contribution of our findings to commognitive research, to geometry learning research and to peer learning research.     

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.     

Do students confuse dimensionality and “directionality”?

The aim of this research is to understand the way in which students struggle with the distinction between dimensionality and “directionality” and if this type of potential confusion could be a factor affecting students’ tendency toward improper linear reasoning in the context of the relations between length and area of geometrical figures. 131 9th grade students were confronted with a multiple-choice test consisting of six problems related to the perimeter or the area of an enlarged geometrical figure, then some interviews were carried out to obtain qualitative data in relation to students’ reasoning. Results indicate that more than one fifth of the students’ answers could be characterized as based on directional thinking, suggesting that students struggled with the distinction between dimensionality and “directionality”. A single arrow showing one direction (image provided to the students) seemed to strengthen the tendency toward improper linear reasoning for the area problems. Two arrows showing two directions helped students to see a quadratic relation for the area problems.     

Covers,matching and odd cycles of a graph

We link some well-known theorems and prove some new ones, each on one or more of the items of the title, and together illustrating their close relationship. The main tools are well-known similar conditions on maximum stable sets and maximum matchings, by which we prove theorems on the existence of odd cycles, including a generalization of Konig's equality between the matching and covering numbers of a bipartite graph. We deal with the question “How nearly bipartite is a graph?” and conjecture an inequality involving the matching and covering numbers and the number of disjoint odd cycles.     

Clusters die hard: Time-correlated excitation in the Hamiltonian mean field model

The Hamiltonian mean field (HMF) model has a low-energy phase where N particles are trapped inside a cluster. Here, we investigate some properties of the trapping/untrapping mechanism of a single particle into/outside the cluster. Since the single particle dynamics of the HMF model resembles the one of a simple pendulum, each particle can be identified as a high-energy particle (HEP) or a low-energy particle (LEP), depending on whether its energy is above or below the separatrix energy. We then define the trapping ratio as the ratio of the number of LEP to the total number of particles and the “fully-clustered” and “excited” dynamical states as having either no HEP or at least one HEP. We analytically compute the phase-space average of the trapping ratio by using the Boltzmann–Gibbs stable stationary solution of the Vlasov equation associated with the N → ∞ limit of the HMF model. The same quantity, obtained numerically as a time average, is shown to be in very good agreement with the analytical calculation. Another important feature of the dynamical behavior of the system is that the dynamical state changes transitionally: the “fully-clustered” and “excited” states appear in turn. We find that the distribution of the lifetime of the “fully-clustered” state obeys a power law. This means that clusters die hard, and that the excitation of a particle from the cluster is not a Poisson process and might be controlled by some type of collective motion with long memory. Such behavior should not be specific of the HMF model and appear also in systems where itinerancy among different “quasi-stationary” states has been observed. It is also possible that it could mimick the behavior of transient motion in molecular clusters or some observed deterministic features of chemical reactions.