Children’s Combinatorial Counting Strategies and their Relationship to Conventional Mathematical Counting Principles

Höveler, Karina

doi:10.1007/978-3-319-70308-4_6

Cited by 7 publications

(5 citation statements)

References 13 publications

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“…We first note we are writing from the perspective of researchers in the United States where the problem we are trying to articulate is especially acute. However, we acknowledge that in some countries, including Germany (Höveler, 2017), Israel (Eizenberg & Zaslavsky, 2003, 2004, Brazil (Borba, Pessoa, Barreto, & Lima, 2011), Spain (Batanero, Navarro-Pelayo, & Godino, 1997;Batanero, Godino, & Navarro-Pelayo, 2005;Godino, Batanero, & Roa, 2005), and Hungary (Vanscó, Beregszászi, Burian, Emese, Stettner, & Szitányi, 2016), to name a few, combinatorics already has a strong presence in school curricula. We do not have space to address how combinatorics is treated in each of these other countries, but we use the fact that many other countries do include combinatorial topics as motivation to enact change in the U.S. We now briefly elaborate on the status and history of combinatorial topics in mathematics education in the U.S., which is part of our motivation for writing this commentary.…”

Section: Context and Motivation -A Brief History Of Combinatorics In The Curriculum In The U Smentioning

confidence: 98%

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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Section: Context and Motivation -A Brief History Of Combinatorics In The Curriculum In The U Smentioning

confidence: 98%

A case for combinatorics: A research commentary

Lockwood

Wasserman²,

Tillema

2020

The Journal of Mathematical Behavior

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“…Combinatorics has received considerable attention in the mathematics education research literature within the past few decades (Batanero et al, 1997;Coenen et al, 2018;English, 1991;Hurdle et al, 2016;Lockwood, 2013Lockwood, , 2014Lockwood, , 2022Schuster, 2004;Tillema, 2013). It is also included, to some degree, in the curricula of many countries including Israel, Brazil, Germany, Spain, and Hungary; because of its role in the curricula, considerable work has been done in investigating the teaching and learning of combinatorial topics among students at a variety of ages in these and other countries (Batanero et al, 2005;Borba et al, 2011;Hart & Sandefur, 2018;Höveler, 2018;Vancsó et al, 2018).…”

Section: Combinatorics and Enumerationmentioning

confidence: 99%

Teaching and learning discrete mathematics

Sandefur

Lockwood

Hart

et al. 2022

ZDM Mathematics Education

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In this paper, we provide an overall perspective on the teaching and learning of discrete mathematics. Our aim is to highlight what research has been conducted in this area and to connect it to existing research ideas for future work. We begin by characterizing discrete mathematics and its role in the school curriculum, highlighting themes, topics, and mathematical practices that distinguish discrete mathematics. We then present potential benefits of focusing on discrete mathematics topics for mathematics education; in particular, we discuss the accessibility of topics in discrete mathematics, the connection to mathematical processes and affect, and the relevance of discrete mathematics in our current society. We also emphasize discrete mathematics from an international perspective, highlighting studies from the US, Italy, France, Chile, and Germany, which are across all school levels–primary, middle, and secondary school, and with some implications for post-secondary education. We particularly discuss discrete topics including number theory, combinatorics, iteration and recursion, graph theory, and discrete games and puzzles; we describe and situate these topics within literature. We also suggest the additional topics of game theory and the mathematics of fairness that we hope to see addressed in future studies.

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“…Students often have difficulties working with combinatorial problems (Eisenberg & Zaslavsky, 2003;Fischbein & Gazit, 1988). Several studies over the years have promoted approaches to enhance students' capabilities in solving combinatorial problems, from primary children (English, 1991;Hoeveler, 2018;Zak, 2020) to middleand high-school students (Ďuriš et al, 2021). In our study, we aimed to address some difficulties with combinatorial problems by creating an educational path that takes graph theory into account as a support in solving the problems.…”

Section: Graph Theory and Combinatorics In Mathematics Educationmentioning

confidence: 99%

Graph theory and combinatorial calculus: an early approach to enhance robust understanding

Ferrarello

Gionfriddo

Grasso

et al. 2022

ZDM Mathematics Education

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The objective of this work is to show an educational path for combinatorics and graph theory that has the aim, on one hand, of helping students understand some discrete mathematics properties, and on the other, of developing modelling skills through a robust understanding. In particular, for the path proposed to middle-school students, we used a connection between k-permutations and colourings of graphs: we indicated a way to solve problems related to counting all the possible arrangements of given objects in a k-tuple under given constraints. We solve this kind of problem by associating a graph with the constraints related to the k-tuple and by using graphs’ colourings, in which every colour is associated with one of the objects. The number of arrangements is given by finding the number of colourings through an algorithm called the Connection-Contraction Algorithm. The educational path is set within the Teaching for Robust Understanding framework and the goal, from the mathematical skills perspective, is to enhance modelling, passing from real situations (the fish problem in our experiment) to mathematical problems (the graph’s colouring in our experiment) and vice versa through the use of technology (the Connection-Contraction Algorithm with yEd editor, in our experiment), by using an extended modelling cycle. The meetings with students were videotaped and some results of the experimentation are given.

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Children’s Combinatorial Counting Strategies and their Relationship to Conventional Mathematical Counting Principles

Cited by 7 publications

References 13 publications

A case for combinatorics: A research commentary

A case for combinatorics: A research commentary

Teaching and learning discrete mathematics

Graph theory and combinatorial calculus: an early approach to enhance robust understanding

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