# Mathematical Modelling: The Foundation of Middle School Math

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## Contents

**Overview and Purpose**

Think back of your middle school mathematical learning year, what do you remember? Was it full of memorization and drill? Was it full of tricks that made no sense? This is probably is because the mathematics that we have learned prior to Common Core Math emphasizes on answers and tricks rather than conceptual understanding. With the globalization and automation that is happening, we need to ensure that mathematical learning consists of conceptual understanding that supports the transfer of knowledge to new contexts. This is because the conceptual understanding ensures true mastery of mathematical knowledge and application in real-life contexts.

With that being said, the mini-course will be discussing mathematical models among middle school math teachers. Mathematical models are an effective tool to emphasize conceptual understanding. Research has shown that students who use accurate visual representations are “six times more likely to correctly solve mathematics problems than are students who do not use them.” (Boonen, van Wesel, Jolles, & van der Schoot, 2014). Hence, this course aims to support middle school teachers to:

- Use mathematical models as mathematicians
- Promote the use of models in the classroom
- Provide feedback for students on the use of mathematical models
- Include effective sequence of questions in lesson plans

**Needs Assessment**

**Part 1: Intent**

The NAEP achievement report for New York City reveals that only approximately 33% of 8th graders are at or above proficient at math. This brings up the issue of learning and teaching math for understanding in order for students to successfully solve math problems. Research has shown that students who use accurate visual representations are “six times more likely to correctly solve mathematics problems than are students who do not use them.” (Boonen, van Wesel, Jolles, & van der Schoot, 2014). The math models are also important to students who have learning disabilities or lower-performing students as they are more likely to solve the problems accurately. (Krawec, 2014). My hypothesis is that teachers need to start preparing students to use visual models while also internalizing the models themselves.

**Part 2: Gathering Information**

In order to assess the possible need for a mini-course on mathematical visual models, a survey of middle school math teachers was conducted. Ten surveys were distributed and received. Fifteen surveys were distributed and four responses were received.

**Part 3: Summary**

**SUMMARY OF RESPONSES TO MIDDLE SCHOOL MATH TEACHER SURVEY**

Related to modelling/drawing

Drawing helps me visualize the problems/ideas [2 agree, 2 strongly agree] Drawing is an effective way to answer a question/state an idea [4 agree] Drawing and modelling are similar because they both show the artist’s logical thinking and ideas [3 agree, 1 strongly agree] Related to mathematical modelling

Mathematical modelling is an important practice standard of Common Core Math Standard for middle school students[3 strongly agree, 1 strongly agree] I accept a mathematical model as a strategy in my classroom [4 agree] I have taught at least two of the following: place value chart, ratio table, number line, and/or double number line [4 strongly agree] My students benefit greatly from mathematical modelling [2 agree, 2 strongly agree] My students struggle with mathematical modelling [1 agree, 3 strongly agree] Related to professional development

My students will struggle less with mathematical modelling if I can teach them about the models in greater details [3 agree, 1 depends] All mathematical models progress and can be used in different grades [2 agree, 2 neutral] I understand how the models progress within middle school math contexts and can use them to solve most middle school math problems [2 neutral, 2 disagree] I am well-prepared to teach any of the following models to any of middle school grades: place value chart, ratio table, number line, and/or double number line [1 agree, 2 disagree, 1 neutral] I need targeted professional development about mathematical modelling, including how to use them and how to teach them [3 agree, 1 strongly agree] All respondents indicated positive attitudes towards drawing and modellings as possible ways to express ideas and logical thinking. All respondents acknowledged the importance of mathematical modelling in academic contexts and indicated some knowledge on mathematical modelling, including at least two of the mentioned models. Most respondents, 3 out of 4, feel unprepared to teach mathematical modelling and all respondents indicated the need for professional development in this area.

All of the respondents in this survey have been teaching middle school, from grade 5 to 8, for more than 3, but less than 6 years. Based on the responses to this survey, although all respondents indicated positive attitudes towards and some knowledge on mathematical modelling, it appears that some teachers need support to understand the model themselves and how to teach them to the students. All respondents indicated that they lack knowledge of how mathematical models progress within middle school mathematics. This also explains why half of the respondents showed some confusion about when mathematical models can be used. More specifically, half of the respondents answered “neutral” when asked about whether mathematical models can be used in different contexts and different grades or not.

Therefore, a mini-course on mathematical modelling training that aids the understanding of three core models, number line, ratio table, and place value chart, is needed. This will not only support the teachers in understanding how to use the models to explain their reasoning themselves but allow them to support their learners to successfully use these models. This will aid mathematical visualization and allow teachers to differentiate the content for struggling students and students with IEPs.

**Performance Objectives**

- Learners will be able to reflect on the importance of mathematical visual model and how it connects to the Common Core Math Standards and Practice
- Learners will be able to correctly use a mathematical visual model (number line, ratio table, or place value chart) to correctly solve three out of three problems
- Learners will be able to craft sequential scaffolding questions on the visual models during whole-class discussion to push students' deeper understanding
- Learners will be able to give sample feedback on the use of mathematical model in their classroom

**Course Units**

Unit 1: What is Mathematical Modeling? And Why Should We Teach it in Middle School?

Brief overview

At the end of this unit, learners will be able to:

- Define key benefits of mathematical modeling
- Identify the need for mathematical modeling in middle school mathematics
- Identify the valuable skills students will develop from learning to solve middle school mathematics using models
- Interpret the Common Core Math Practice Standards and identify the standards that require mathematical modeling

Unit 2: Key Mathematical Modellings in Real-Life Word Problems

Brief Overview

At the end of this unit, learners will be able to:

- Identify key mathematical models that students will use in middle school
- Identify at least two models that their students can benefit from in the current grades
- Explain how mathematical models support students’ logical thinking and deeper understanding of new concepts
- Analyze examples of how mathematical models can be used in their respective grade level

Unit 3: Progression of Mathematical Modelling in Middle School Mathematics

Brief Overview

At the end of this unit, learners will be able to:

- Connect specific grade-level content to the mathematical model
- Solve different grade-level mathematical problems using the same model
- Explain how key mathematical models progress within middle school mathematics

Unit 4: Classroom Strategies and Planning for Mathematical Models

Brief Overview

At the end of this unit, learners will be able to:

- Analyze an example model to teach mathematical models using questioning and show-calling
- Identify exemplar and non-exemplar sequence of questions that can support students’ analysis of the new mathematical model
- Collaborate to create a sequence of questions to support students’ analysis of the new mathematical model
- Modify the sequence of questions based on given feedback to meet the criteria of success

Unit 5: Application of Mathematical Modelling in The Classroom

Brief Overview

At the end of this unit, learners will be able to:

- Create their own sequence of questions for a new mathematical model for an upcoming lesson
- Promote the use of mathematical models in their own classrooms
- Provide feedback for sample work on the use of mathematical models

## References

- Akhras, A., (2014). Supporting Students with Mathematical Models. [online] Great Minds. Available at: <https://gm.greatminds.org/math/blog/eureka/supporting-students-with-mathematical-models> [Accessed 6 November 2021].
- AR7271. (2017, Oct 14). Area Model Multiplication vs Old School Method (Common Core Math) [Video]. YouTube. https://www.youtube.com/watch?v=ejqH1ofSx8U&ab_channel=AR7271.
- National Center for Education Statistics. (2017). National Assessment of Educational Progress: An overview of NAEP. Washington, D.C.: National Center for Education Statistics, Institute of Education Sciences, U.S. Dept. of Education.
- Boonen, A. J., van Wesel, F., Jolles, J., & van der Schoot, M. (2014). The role of visual representation type, spatial ability, and reading comprehension in word problem solving: An item-level analysis in elementary school children. International Journal of Educational Research, 68, 15-26.
- Levy, R., (2015). 5 Reasons to Teach Mathematical Modelling. [online] American Scientist. Available at: <https://www.americanscientist.org/blog/macroscope/5-reasons-to-teach-mathematical-modeling> [Accessed 10 November 2021].
- Maclearie, M. (2020). Ratio Table Practice [Video]. YouTube. https://www.youtube.com/watch?v=LFB29oT7Zd0&ab_channel=MorganMaclearie.
- Mathcation. (2020). How to do Ratio Tables [Video]. YouTube. https://www.youtube.com/watch?v=ZgUCbr4fiXM&ab_channel=Mathcation.com
- Math with Mr. J. (2021). Adding Integers Using a Number Line. [Video]. YouTube. https://www.youtube.com/watch?v=3LUTYhmltQY&ab_channel=MathwithMr.J.
- Math with Mrs. Berger. (2020). Double Number Lines [Video]. YouTube. https://www.youtube.com/watch?v=jgPupNt0KCU&ab_channel=MathwithMrs.Berger
- National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards. Washington, DC: Authors.
- Teach and Glow (2020). Place Value Song) [Video]. YouTube. https://www.youtube.com/watch?v=ex2ihP_XdU0&feature=emb_title&ab_channel=TeachandGlow.

## Extended Resources

Author | Types of Resources and Name | Link | Description of Resources |
---|---|---|---|

Sarah Hampton | Blog, "Modeling in the Middle School Math Classroom" | https://my.nctm.org/blogs/sarah-hampton/2018/12/19/modeling-in-the-middle-school-math-classroom | What is and what is not mathematical modelling? |

Nancy Wolf | Book, "Modeling with Mathematics" | https://www.heinemann.com/products/e06259.aspx | Support for and examples of incorporating modeling in the classroom |

Achieve the Core | Interactive Media, "Coherence Map" | https://achievethecore.org/coherence-map/ | All grades' math standards are linked together based on progression. Each standard includes a sample assessment, including the solutions that mostly consist of a supporting visual model} |