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## A Method for Using Adjacency Matrices to Analyze the Connections Students Make Within and Between Concepts: The Case of Linear Algebra

The central goals of most introductory linear algebra courses are to develop students' proficiency with matrix techniques, to promote their understanding of key concepts, and to increase their ability to make connections between concepts. In this article, we present an innovative method using adjacency matrices to analyze students' interpretation of and connections between concepts. Three cases provide examples that illustrate the usefulness of this approach for comparing differences in the structure of the connections, as exhibited in what we refer to as dense, sparse, and hub adjacency matrices. We also make use of mathematical constructs from digraph theory, such as walks and being strongly connected, to indicate possible chains of connections and flexibility in making connections within and between concepts. We posit that this method is useful for characterizing student connections in other content areas and grade levels.

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## Leveraging Read Alouds for Mathematical Connections

season or holiday, examining differing genres, or making crosscurricular connections. These connections often align with science and social studies content. For example, a kindergarten teacher might introduce a unit on prominent female figures in history

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## Multiple Connections

What makes a good scenario for a fruitful experience with connections? The following example illustrates a web of connections that students might make in a real-world scenario that begins with familiar and fundamental concepts: perimeter and area of rectangles.

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## The Area Model: Building Mathematical Connections

that do not always expose the existing relationships between concepts and across grade levels. Keeping the focus on overarching topics and models in mathematics can assist teachers in building connections and developing students as sense makers who use

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## Making Mathematical Connections

Mathematically literate students should view mathematics as a way of looking at their environment that aids understanding and adds insight This attitude toward mathe matics can be fostered in the daily routines of the classroom. Mathematical experiences need not be restricted to the “math period” but can be incorporated throughout the school day. The importance of making mathematical connections, both within mathematics and between mathematics and other curriculum areas, is emphasized by the inclusion of “mathematical connection” as one of the curriculum standards for school mathematics (NCTM 1989). This article how how a simple manipulative device useful for taking attendance can be used to exercise mathematical thinking processes in a variety of contexts at different grade levels throughout elementary school.

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## The Mathematics-Children's-Literature Connection

The Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) promotes mathematical power for all students so that they can function a informed citizens in a rapidly changing and technologically complex society. A way of working toward this goal is by investigating connections within mathematics and between mathematics and other instructional areas. The mathematic— children's-literature connection is examined in this article.

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## Introducing Volume 16: Make Connections!

As you begin this exciting new year, fresh from summer, I am sure that many of you are wondering how you can connect to the lives of the young adolescents you are about to work with. Volume 16 of Mathematics Teaching in the Middle School will offer articles and departments that will help you make this connection in your classroom.

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## Mathematics Concepts Using STEM Connections

Introduce two topics in math, volume and triangle types, by using connections to STEM. Contributors to the iSTEM (Integrating Science, Technology, Engineering, and Mathematics) department share ideas and activities that stimulate student interest in the integrated fields of science, technology, engineering, and mathematics (STEM) in K–grade 6 classrooms.

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## Mathematics + Social Studies = Learning Connections

All too often the school day of a six-year-old is fractured into thirty-minute segments, of compartmentalized learning. When each subject is dealt with in isolation, children are often robbed of the opportunity to make connections from one part of the curriculum to another. Since establishing such relationships lies at the heart of the educational process, early childhood teachers face the daily challenge of building bridges from one subject area to another.

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## Mathematics concept maps: assessing connections

Assessment is an ongoing process integrated within instruction (NCTM 2000). Assessment activities can take on a variety of forms, one being performance tasks. Concept mapping is a technique that offers students opportunities to demonstrate learning through performance. In this example, elementary school students designed math concept maps, allowing their teachers to assess the children's concept connections and other valuable, relevant skills before determining the steps necessary for further learning.