# Promoting Interdisciplinary Research Collaboration among Mathematics and Special Education Researchers

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## Abstract

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## 1. Introduction

#### 1.1. Shared Mission: The Importance of Mathematical Concepts for Student Learning

#### 1.2. Theoretical Framing of Mathematical Concepts

## 2. Overview of Concept Definitions from the Research Literature

#### 2.1. Concepts Stated/Defined (Relationship among Mathematical Objects)

“refers to understanding that the relation between the fraction’s numerator and denominator determines its magnitude rather than either number alone; that fraction magnitudes increase with the numerator size and decrease with the denominator size; that the closer the numerator is to the denominator, the closer the fraction is to 1; that a fraction with a numerator larger than the denominator is always > 1 and vice versa; and that all fractions can be represented on the number line”.(p. 627)

“Partitioning a unit into n equal parts creates parts one of which will iterate n times to make the whole. Iterating a small quantity n times produces a specific large quantity that is n times as large. So partitioning a unit into n equal parts creates parts of a particular size”.(p. 129)

#### 2.2. Concepts as a Mathematical Topic (e.g., Reference to the Common Core)

“In terms of content, the CCSS define equivalence as the understanding of two fractions that are the same size or at the same point on a number line. Evidence of a student’s growing understandings of rational number equivalence also includes the ability to recognize and generate simple equivalent fractions (e.g., 1/2 = 2/4, 4/6 = 2/3) and explain why the fractions are equivalent (e.g., by using a visual fraction model)”.(pp. 135–136)

#### 2.3. Concepts Characterized by What Students Can/Cannot Do

For example, students who have constructed an iterative fraction scheme view 3/7 as 3 times 1/7 and 7/5 as 7 times 1/5, which means these students have constructed fractional numbers (Steffe & Olive, 2010). One situation of an iterative fraction scheme is a request to make a length that is seven-fifths of a given length. For this request to be sensible, students have to be able to posit a length that stands in relation to the given length yet is freed from relying on being part of a whole for meaning. Students who have constructed an iterative fraction scheme can make the posited length by partitioning the given length into five equal parts, disembedding one part, and iterating it seven times. For these students, the result is a multiple of a unit fraction (7/5 is 7 times 1/5) as well as one whole (5/5) and ⅖.(p. 204)

“Specifically, these students demonstrate difficulties with rank-ordering fractions and identifying equivalent fractions (Grobecker, 2000; Hecht & Vagi, 2010; Mazzocco & Devlin, 2008), signifying a lack of conceptual understanding of fractions that can lead to misconceptions with fractions that impinge on the successful performance of fractions. For example, students incorrectly name fractions when equal parts are not shown in the figure (e.g., rectangular), failing to visualize equally sized parts in the whole (Barnett-Clarke, Ramirez, & Coggins, 2010). Also, lack of conceptual understanding of fractions can, in turn, limit students’ ability to apply routine computational procedures involving fractions (Siegler et al., 2010)”.(p. 375)

“Generalized learning about fractions. Our third measure indexed generalized learning about fractions, with a strong focus on fraction concepts. This measure was comparably different from the focus of instruction in both conditions and addressed fractions magnitude understanding and the part-whole interpretation of fractions with equal emphasis. We administered 19 released items from the 1990–2009 National Assessment of Educational Progress (NAEP): easy, medium, or hard fraction items from the fourth-grade assessment and easy from the eighth-grade assessment”.(p. 634)

“The test included four items related to fractions concepts and vocabulary and six items each on adding, subtracting, and comparing fractions. All problems consisted of proper fractions with one- or two-digit numerators and denominators. This test was administered prior to the beginning of baseline fractions instruction”.(p. 404)

#### 2.4. Reflecting on Mathematics Education and Special Education Researchers’ Concept Definitions

## 3. Operationalizing Fraction Concepts for Designing Our Future Study

Partitioning a unit into n equal parts (each 1/n of the unit) creates parts, one of which will iterate n times to make the whole (called n/n or 1). Iterating a subunit (1/n of a unit) m times produces a specific quantity that is m times as large (m/n of the unit), and partitioning m/n of the unit into m equal parts will create the subunit (1/n of a unit).

## 4. Implications for Effective Interdisciplinary Research in Special Education and Mathematics Education

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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## Share and Cite

**MDPI and ACS Style**

Lannin, J.K.; Rodrigues, J.; van Garderen, D.; Lei, Q.; Singell, E.L.; Karim, S.
Promoting Interdisciplinary Research Collaboration among Mathematics and Special Education Researchers. *Educ. Sci.* **2023**, *13*, 1150.
https://doi.org/10.3390/educsci13111150

**AMA Style**

Lannin JK, Rodrigues J, van Garderen D, Lei Q, Singell EL, Karim S.
Promoting Interdisciplinary Research Collaboration among Mathematics and Special Education Researchers. *Education Sciences*. 2023; 13(11):1150.
https://doi.org/10.3390/educsci13111150

**Chicago/Turabian Style**

Lannin, John K., Jessica Rodrigues, Delinda van Garderen, Qingli Lei, Emily L. Singell, and Salima Karim.
2023. "Promoting Interdisciplinary Research Collaboration among Mathematics and Special Education Researchers" *Education Sciences* 13, no. 11: 1150.
https://doi.org/10.3390/educsci13111150