Education Research International

Volume 2016, Article ID 5240683, 9 pages

http://dx.doi.org/10.1155/2016/5240683

## What Is Known about Elementary Grades Mathematical Modelling

^{1}University of Nevada, Las Vegas, NV 89154, USA^{2}Universitat Autònoma de Barcelona, Cerdanyola del Valles 08193, Spain

Received 11 April 2016; Revised 24 May 2016; Accepted 5 June 2016

Academic Editor: Shu-Sheng Liaw

Copyright © 2016 Micah S. Stohlmann and Lluís Albarracín. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Mathematical modelling has often been emphasized at the secondary level, but more research is needed at the elementary level. This paper serves to summarize what is known about elementary mathematical modelling to guide future research. A targeted and general literature search was conducted and studies were summarized based on five categories: content of mathematical modelling intervention, assessment data collected, unit of analysis studied, population, and effectiveness. It was found that there were three main units of analysis into which the studies could be categorized: representational and conceptual competence, models created, and student beliefs. The main findings from each of these units of analysis are discussed along with future research that is needed.

#### 1. Introduction

Mathematical modelling has mainly been emphasized at the secondary level, but for students to become more adept modellers the elementary grades need to be given more attention. We know that mathematical modelling abilities improve over time [1]. It then makes sense then to start mathematical modelling at earlier ages. There are many benefits to mathematical modelling that elementary students are missing if they are unable to participate in mathematical modelling: developing mathematical understandings [2, 3], coming to appreciate mathematics more and see it as more real life and applicable [4, 5], and developing communication and life skills [6].

At the elementary level, real world mathematics problems are often traditional word problems where teachers may instruct students on finding key words. Students come to believe that they should identify the numbers in the problem and do some operations with these numbers [7, 8]. For example, a study that was done in the elementary grades involved telling students this story: “*Mr. Lorenz and three colleagues started at Bielefeld at 9 AM and drove the 360 kilometers to Frankfurt with a rest stop of 30 minutes*.” There is no question, it is just a story. The story was told to kindergartners and they just say, “*Thank you for the story*.” The story was told to first graders and a few of them combine the numbers to get an answer. The story was then told to 2nd graders all the way up to 6th graders. Every year, more students than the previous grade level combine numbers and give an answer [9].

We want students to have the opposite effect, where each year they become more adept at reasoning with real world situations. If mathematical modelling is integrated more in the elementary grades students will be more used to situations that can be solved, making assumptions and approximations and identifying which is the most important information in a problem and what more information needs to be known. There is good work being done in the elementary grades in mathematical modelling, but more can be done in conducting research with this age group. This paper describes what is known about mathematical modelling at the elementary grade level (age 10 and under) in order that future research can be identified and situated in this literature.

#### 2. Essential Elements of Mathematical Modelling

Our definition of mathematical modelling is an iterative process that involves open-ended, real world, practical problems that students make sense of with mathematics using assumptions, approximations, and multiple representations. Other sources of knowledge besides mathematics can be used as well. Mathematical modelling curricula should have multiple acceptable models that can be developed.

Mathematical modelling begins with a key question that stems from the real world problem. A key question can guide the solution and work of a mathematical modelling activity. An example of a key question is how big is someone based on his or her footprint and stride length? [10] A key question can serve in focusing on work and is often the way that people approach problems in their jobs.

Both clear verbal and written communication are paramount while students work on a mathematical modelling task and detail their solution. Students must also reflect on the modelling process in order to make explicit the mathematics that they used and how well they understood it. In addition, modelling activities should also be open-ended [11].

A modelling cycle (Figure 1) that appears often in the literature is from Blum and LeiB [12] and connects to several of our essential elements. There is a distinction between the real world and mathematics. It can be seen that students must make sense of the problem with mathematics involving assumptions and approximations, often called mathematizing, and then ensure that the model developed makes sense in the realistic context. Though not shown in the cycle it is well known that the modelling process is iterative in nature [13].