Classifying Combinations: Investigating Undergraduate Students’ Responses to Different Categories of Combination Problems

Lockwood, Elise; Wasserman, Nicholas H.; McGuffey, William Holmes

doi:10.1007/s40753-018-0073-x

Cited by 13 publications

(7 citation statements)

References 7 publications

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“…This also seems to hold true in reverse, i.e., students who have a positive attitude to mathematics are more successful at solving mathematical problems [53][54][55][56][57]. The results of several authors show that success in solving mathematical problems is influenced by students' procedural skills, including their ability to use (dominantly mathematical) tools productively and to choose an appropriate representation in the mathematization of problem situations [58][59][60][61]; their level of control of processes related to mathematical activity, such as reasoning, communication, generalization, or mathematical modeling [62][63][64][65][66][67][68]; and the level of their conceptual understanding of mathematical concepts [69][70][71][72][73].…”

Section: Mathematical Problem-solvingmentioning

confidence: 99%

Relation between Pupils’ Mathematical Self-Efficacy and Mathematical Problem Solving in the Context of the Teachers’ Preferred Pedagogies

et al. 2020

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In research focused on self-efficacy it is usually teacher-related phenomena that are studied, while the main aspects related to pupils are rather neglected, although self-efficacy itself is perceived as a belief in one’s own abilities. Evidently, this strongly influences the behavior of individuals in terms of the goal and success in mathematical problem-solving. Considering that alternative teaching methods are based on the principle of belief in one’s own ability (mainly in the case of group work), higher self-efficacy can be expected in the pupils of teachers who use predominantly the well-working pupil-centered pedagogies. A total of 1133 pupils in grade 5 from 36 schools in the Czech Republic were involved in the testing of their ability to solve mathematical problems and their mathematical self-efficacy as well. Participants were divided according to the above criteria as follows: (i) 73 from Montessori primary schools, (ii) 332 pupils educated in mathematics according to the Hejný method, (iii) 510 pupils from an ordinary primary school, and (iv) 218 pupils completing the Dalton teaching plan. In the field of mathematical problem-solving the pupils from the Montessori primary schools clearly outperformed pupils from the Dalton Plan schools (p = 0.027) as well as pupils attending ordinary primary schools (p = 0.009), whereas the difference between the Montessori schools and Hejný classes was not significant (p = 0.764). There is no statistically significant difference in the level of self-efficacy of pupils with respect to the preferred strategies for managing learning activities (p = 0.781). On the other hand, correlation between mathematical problem-solving and self-efficacy was confirmed in all the examined types of schools. However, the correlation coefficient was lower in the case of the pupils from the classes applying the Hejný method in comparison with the pupils attending the Montessori schools (p = 0.073), Dalton Plan schools (p = 0.043), and ordinary primary schools (p = 0.002). Even though the results in mathematical problem-solving are not consistent across the studies, the presented results confirm better performance of pupils in some constructivist settings, particularly in the case of individual constructivism in the Montessori primary schools. The factors influencing lower correlation of self-efficacy and performance in mathematical problem-solving ought to be subject to further investigation.

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Section: Mathematical Problem-solvingmentioning

confidence: 99%

Relation between Pupils’ Mathematical Self-Efficacy and Mathematical Problem Solving in the Context of the Teachers’ Preferred Pedagogies

et al. 2020

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“…The ability to solve complex open-ended problems is anchored in some various different factors including the following: (i) metacognitive knowledge [6,7], (ii) positive attitude towards mathematics [8][9][10][11][12], (iii) mastering mathematics processes as are reasoning, generalization, communicating the results or mathematical modeling [13][14][15][16][17][18] and (iv) high level of mathematical proficiency in both, procedures [19,20] and conceptual understanding [21][22][23], including arithmetic, algebraic and combinatorial thinking. These aforementioned factors are not isolated, but they rather support each other mutually in the influence of the ability to solve complex open mathematical problems [24,25].…”

Section: Introductionmentioning

confidence: 99%

“…Combinatorics has a special place in the field of mathematics and mathematics education. It is considered as a source of frequent and hard difficulties of students at various levels [15]. The fact is that just the combinatorial thinking presents the process of creating various possible combinations of ideas and cognitive operations [36].…”

Section: Introductionmentioning

confidence: 99%

Relations between Generalization, Reasoning and Combinatorial Thinking in Solving Mathematical Open-Ended Problems within Mathematical Contest

2020

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Algebraic thinking, combinatorial thinking and reasoning skills are considered as playing central roles within teaching and learning in the field of mathematics, particularly in solving complex open-ended mathematical problems Specific relations between these three abilities, manifested in the solving of an open-ended ill-structured problem aimed at mathematical modeling, were investigated. We analyzed solutions received from 33 groups totaling 131 students, who solved a complex assignment within the mathematical contest Mathematics B-day 2018. Such relations were more obvious when solving a complex problem, compared to more structured closed subtasks. Algebraic generalization is an important prerequisite to prove mathematically and to solve combinatorial problem at higher levels, i.e., using expressions and formulas, therefore a special focus should be put on this ability in upper-secondary mathematics education.

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“…Researchers have implicitly highlighted the accessibility of combinatorics by demonstrating student reasoning about counting problems in their studies. In studies about combinatorial learning (e.g., Batanero, et al, 1997;English, 1991English, , 1993Eizenberg & Zaslavsky, 2004;Fischbein & Gazit, 1988;Fischbein & Grossman, 1997;Fischbein, Pampu, & Manzat, 1970;Lockwood & Gibson, 2015;Lockwood & Purdy, 2019a, 2019bLockwood, Wasserman, & McGuffey, 2018;Maher & Martino, 1996a;Maher & Martino, 1996b;Tillema, 2013Tillema, , 2014Tillema, , 2018Tillema & Gatza, 2016), counting problems often arise from every day contexts. This tends to be true of the field of combinatorics in general, but it is certainly true of the set of problems explored in such research.…”

Section: Combinatorial Tasks Are Accessible and Require Little Mathematical Background Knowledgementioning

confidence: 99%

“…Third, in response to some of these claims about (and calls for) ways in which combinatorics can foster rich mathematical thinking, the combinatorics education community has demonstrated very thoroughly that students can reason richly and deeply within combinatorics. One way in which they have done this is to demonstrate sophisticated student understanding of particular combinatorial topics, such as the multiplication principle (e.g., Lockwood & Caughman, 2016;Lockwood, Reed, & Caughman, 2017;Lockwood & Purdy, 2019a, 2019b, bijections and isomorphism (e.g., Mamona-Downs & Downs, 2004;Muter & Maher, 1998;Powell & Maher, 2003;Tarlow, 2011), combinations and the binomial theorem (Maher & Speiser, 1997;Lockwood, Wasserman, & McGuffey, 2018;Speiser, 2011;Tillema & Burch, 2020;Wasserman & Galarza, 2019), combinatorial proof (e.g., Engelke & CadwalladerOlsker, 2010;Maher & Martino, 1996a, 1996bLockwood, Reed, & Erickson, in press;Tarlow & Uptegrove, 2011), and equivalence (Lockwood & Reed, in press). Take, for instance, the multiplication principle (accessible even to elementary students); it is perhaps the most basic counting principle, and yet, in research studies, undergraduate students wrestled with its use in combinatorial problems, taking several hours over the course of a teaching experiment to articulate subtle nuances in the principle (Lockwood & Purdy, 2019a, 2019b.…”

Section: Combinatorics Problems Provide Opportunities For Challenging Rich Mathematical Thinking For All Studentsmentioning

confidence: 99%

A case for combinatorics: A research commentary

Lockwood

Wasserman²,

Tillema

2020

The Journal of Mathematical Behavior

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In this commentary, we aim to make a case for the explicit inclusion of combinatorial topics in the curriculum -both in K-12 classrooms and in introductory postsecondary mathematics courses -where it is currently essentially absent. To do so, we suggest ways in which researchers might inform the field's understanding of combinatorics and its potential role in curricula. We reflect on five decades of research that has been conducted since a call by Kapur (1970) for a greater focus on combinatorics in mathematics curricula. We offer five assertions about combinatorics, including three existing assertions and two new assertions that relate to increasingly relevant trends in mathematics education. Specifically, we discuss the following in making our case for combinatorics: 1) Combinatorics is accessible, 2) Combinatorics problems provide opportunities for rich mathematical thinking, 3) Combinatorics fosters desirable mathematical practices, 4) Combinatorics can contribute positively to issues of equity in mathematics education, and 5) Combinatorics is a natural domain in which to examine (and develop) computational thinking and activity. As we discuss each of these ideas, we summarize and synthesize existing research and offer new ideas for research. Ultimately, we hope to make a case for the valuable and unique ways in which combinatorics might effectively be leveraged within K-16 curricula, and we hope to elevate its status in the mathematics education research community.

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Classifying Combinations: Investigating Undergraduate Students’ Responses to Different Categories of Combination Problems

Cited by 13 publications

References 7 publications

Relation between Pupils’ Mathematical Self-Efficacy and Mathematical Problem Solving in the Context of the Teachers’ Preferred Pedagogies

Relation between Pupils’ Mathematical Self-Efficacy and Mathematical Problem Solving in the Context of the Teachers’ Preferred Pedagogies

Relations between Generalization, Reasoning and Combinatorial Thinking in Solving Mathematical Open-Ended Problems within Mathematical Contest

A case for combinatorics: A research commentary

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