Mathematical Connections Pdf

[Ezekiel Tulagan]

Whatis a mathematical connection? Why are mathematical connections important? Why are they considered part of the Exemplars rubric criteria? And how can I encourage my students to become more independent in making mathematical connections?

This blog represents part one of a four-part series (parts two, three, and four here) that explores mathematical connections and offers guidelines, strategies and suggestions for helping teachers elicit this type of thinking from their students. We find many students enjoy making connections once they learn how to reflect and question effectively.

The Common Core State Standards for Mathematics (CCSSM) support the need for students to make mathematical connections in problem solving. Reference to this can be found in the following Standards for Mathematical Practice:

When students apply the criteria of the Exemplars rubric, they understand that their solution is more than just stating an answer. Part of that solution is taking time to reflect on their work and make a mathematical connection to share.

When students begin to explore mathematical connections, teachers should take the lead by providing formative assessment tasks that introduce new learning opportunities and provide practice, so they may become independent problem solvers. As part of this process, teachers will want to focus on five key areas to help students develop an understanding of mathematical connections.

Using formative problem-solving tasks to introduce and practice new strategies and representations is part of the problem-solving process. Teachers should provide formal instruction so that students may grow to independently determine and construct strategies or representations that match the task they are given. An example of this can be seen in the primary grades when many teachers introduce representations in the following order: manipulative/model, to diagram (including a key when students are ready), to five/ten frames, to tally charts, to tables, to number lines. This order allows students to move from the most concrete to the more abstract representations.

(2) Encourage students to continue their representations. Mathematical connections may be made when students continue a representation beyond the correct answer. Examples of this can be seen when a table or linear graph is continued from seven days to 14 days or when two more cats are added to a diagram of 10 cats to discover how many total ears a dozen cats would have. Another example includes adding supplemental information to a chart such as a column for decimals in a table that already has a column indicating the fractional data. In this case, the student extends his or her thinking to incorporate other mathematics to solve the task. It is important to note that connections must be relevant to the task at hand. In order to meet the standard, a connection must link the math in the task to the situation in the task.

(4) Incorporate inquiry into the problem-solving process. Asking students to clarify, explain, support a part of their solution to a math partner, the whole class, or a teacher, not only helps develop independent problem solvers but also leads to more math connections. In your discussions, use verbs from Depth of Knowledge 2 (identify, interpret, state important information/cues, compare, relate, make an observation, show) and from Depth of Knowledge 3 (construct, formulate, verify, explain math phenomena, hypothesize, differentiate, revise). By asking students questions that provide them the opportunity to show and share what they know, connections become a natural part of their solutions.

(5) Encourage self- and peer-assessment opportunities in your classroom. Encourage students to self-assess their problem-solving solutions either independently, with a math partner or with the support of their teacher. The more opportunity students have to use the criteria of the Exemplars assessment rubric to evaluate their work, the more independent they become in forming their solutions, which will include making mathematically relevant connections.

Although it is important to think about the connections among concepts within the grade level or courses that we teach, it is also important to reflect on the connections across grade levels. This work involves thoughtful discussions with other colleagues about the way that concepts are taught and the potential linkages among those ideas. Many of us learned mathematics as isolated pieces of information. Taking a mathematical concept and considering how it originates, extends, and connects with other concepts across the grades will help teachers to develop a deeper understanding. It is then that we can plan instruction that ensures that our students regularly make connections to help them make sense of the mathematics they are learning.

This consensus on the importance of connections in the profound understanding of mathematics is also recognised by many curricular documents that emphasize the importance of the making of mathematical connections in the classroom (e.g. Department of Education U. K., 2013; National Council of Teachers of Mathematics, 2000). In particular, the connection standard (NCTM 2000) refers to connections between mathematics and other contexts and/or connections between mathematical ideas that include broad connections, such as the connection between rational numbers, proportionality and linear relationships, and more specific connections such as the connection between two specific representations of the same concept.

Although there are many empirically based classifications of connections, some types of connections reported in those classifications are particularly relevant in the context of mathematics teaching and learning. First, connections between mathematics and other contexts or disciplines are highlighted by curricular guidelines (e.g. NCTM, 2000). They are also at the heart of mathematical modelling at school and play an important role in STEM education when mathematics is brought into play. Moreover, they have been subject to study in some research studies about mathematical connections (e.g. Dolores-Flores & Garcia-Garcia, 2017; Evitts, 2004). Second, connections between representations are the kind of mathematical connection that appears in most theoretical models for connections from any perspective. One of the most relevant characterisations of connections between representations is the one proposed by Duval (2006), who introduces the idea of conversion when the connection involves a change of register of semiotic representation (a register of semiotic representation is a semiotic system that permits transformations of representations), and treatment when the register is maintained. The identification of this kind of connection is consistent in other relevant characterisations of connections (e.g. Adu-Gyamfi et al., 2017; Businskas, 2008; Dolores-Flores & Garcia-Garcia, 2017; Dreher et al., 2016; Rodrguez-Nieto, 2021) and in some of our previous results (De Gamboa & Figueiras, 2014). Third and finally, procedural connections are also a type of connection identified consistently in several studies. They were introduced by Businskas (2008), and refer to the use of rules, algorithms or formulas. This last kind of connections is also identified by Dolores-Flores & Garcia-Garcia (2017) and Eli et al. (2011).

Considering the broadness of the notion of connection reported above and using the relational and pragmatic perspective proposed by the OSA, we focus on discursive interactions between the teacher and the students. Therefore, we understand mathematical connections in the context of classroom practice as an explicit relation between two objects (Fig. 1).

The data used to perform the analysis are video and audio recordings of eight regular class sessions of 60 min and their corresponding transcriptions. The eight sessions covered topics such as basic properties of natural numbers (g.c.d and l.c.m), the introduction of negative numbers, the basic properties of the operations with integers, combined operations with integers, powers, roots, and some word problems. To collect the information, non-participant observation was used (Caldwell & Atwal, 2005; Cohen et al., 2007). Therefore, the class group was observed, listened to and video-recorded and notes were taken, without intervening in the design of the sessions, the activities proposed or the development of the sessions. The data were processed in two phases: the first was aimed at identifying connections and establishing episodes, and the second was aimed at characterising and classifying the connections identified in the first phase.

To characterise and classify the connections identified in the first phase, we follow 4 stages that allow us to progressively refine the analysis of connections and their structure. The categories were developed and refined inductively as grounded in the data.

In stage 1, when the three elements that determine each connection defined in Fig. 1 are identified (\(O_1\), \(O_2\) and relation), the analysis revealed that, in some cases, the utterance that defined the connection may trigger (in the context of a classroom dialogue) other utterances that were related to the relation that defined the connection. These utterances created new relations between objects that enriched the original connection. These relations are called links. Each link can be a reformulation of the relation between O1 and O2, or a relation between either O1 or O2 and another object. Therefore, connections revealed as networks that can be made of one or more links. When a connection has only one link (i.e. it is defined by a single relation defined by a single utterance), the link coincides with the connection. When a connection is made of more than one link, the relation is defined by the coordination between the links involved in the network. The identification of this internal structure of connections motivated two parallel analyses of each connection in stages 2, 3 and 4 (Table 1): one global analysis focused on the role of each connection in the classroom context where it was made, and a specific analysis focused on the nature of each of the links that made up each connection.

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