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How to Address Difficulties in Math

The third chapter deals with the analysis of the instruments for addressing difficulties in mathematics.

How to address Difficulties and Learning Disabilities in Math

Table of Content

1. Designing the Intervention Tools – a Theoretical Framework

Karagiannakis’s and colleagues (2016), propose a model classifying mathematical skills involved in learning mathematics into four domains: Core number, Memory, Reasoning, and Visual-spatial (the frame is presented in Table 1). Their findings support the hypothesis that difficulties in learning mathematics can have multiple originsand they provide a means for sketching students’ mathematical learning profiles.

The frame helps to characterize students’ difficulties in mathematics.

Table 1: Karagiannakis’s and colleagues’ frame: domains of the four-pronged model and sets of mathematical skills associated with each domain

We recall that the model framed also the design of Questionnaire B2, aimed at better understanding students’ profiles of difficulty. When constructing B2, we chose questions that were related to the cognitive areas as well to three mathematical domains: arithmetic, geometry, algebra (core number is not related to all cognitive areas). As a result, we proposed questions that were located in some cells of the following table (Table 2).

Table 2: Double relation between cognitive areas (memory, reasoning and visuo-spatial) and mathematical domains (arithmetic, geometry, algebra).

The same frame is used for the design of the intervention tools. Here we present additional theoretical references that support the design of the intervention tools.

First of all, we refer to the Universal Design for Learning (UDL) principles (Table 3), a framework specifically conceived to design inclusive educational activities ( ).

Table 3: UDL guidelines

The Center for Applied Special Technology (CAST) has developed a comprehensive framework around the concept of Universal Design for Learning (UDL), with the aim of focusing research, development, and educational practice on understanding diversity and facilitating learning. UDL includes a set of Principles, articulated in Guidelines and Checkpoints (For a complete list of the principles, guidelines and checkpoints and a more extensive description of CAST’s activities, visit ). The research grounding UDL’s framework is that “learners are highly variable in their response to instruction. [...]"

Thus, UDL focuses on these individual differences as an important element to understand and design effective instruction for learning.

To this aim, UDL advances three foundational Principles: 1) provide multiple means of representation, 2) provide multiple means of action and expression 3) provide multiple means of engagement. In particular, guidelines within the first principle refer to means of perception involved in receiving certain information, and of “comprehension” of the information received. The guidelines within the second principle take into account the elaboration of information/ideas and their expression. Finally, the guidelines within the third principle deal with the domain of “affect” and “motivation”, also essential in any educational activity.

For our analyses, we will focus in particular on specific guidelines within the three Principles (The items are taken from the interactive list at ).

Guidelines within Principle 1 (provide multiple means of representation), suggest proposing different options for perception and offering support for decoding mathematical notation and symbols. Moreover, guidelines suggest the importance of providing options for comprehension highlighting patterns, critical features, big ideas, and relationships among mathematical notions. Accordingly, we will propose the use of the software AlNuSet to guide information processing, visualization, and manipulation, in order to maximize transfer and generalization.

Moreover, the guidelines from Principle 2 (provide multiple means of action and expression) suggest to offer different options for expression and communication supporting planning and strategy development. Finally, the guidelines from Principle 3 show how certain activities can recruit students’ interest, optimizing individual choice and autonomy, and minimizing threats and distractions.

In section 4 we will present examples of activities, discussing the type of mathematical learning they address and the cognitive area they support. We will show how these examples have been designed within the frame of the UDL principles in order to make them inclusive and effective to overcame math difficulties identified through B2 questionnaire.

Another theoretical reference we refer to comes from the experience of the European Project FasMed, that focused on formative assessment in mathematics and science ( ).

Formative assessment (FA) is conceived as a method of teaching where “evidence about student achievement is elicited, interpreted, and used by teachers, learners, or their peers, to make decisions about the next steps in instruction that are likely to be better, or better founded, than the decisions they would have taken in the absence of the evidence that was elicited” (Black & Wiliam, 2009, p. 7). FaSMEd project refers to William and Thompson (2007)’s study, that identifies five key strategies for FA practices in school setting: (a) clarifying and sharing learning intentions and criteria for success; (b) engineering effective classroom discussions and other learning tasks that elicit evidence of student understanding; (c) providing feedback that moves learners forward; (d) activating students as instructional resources for one another; (e) activating students as the owners of their own learning. The teacher, student’s peers and the student him- or herself are the agents that activate these FA strategies. The FA strategies are summarized in table 4.

Table 4

According to such conceptualization of formative assessment, the European Project FaSMEd designed and tested several class activities that exploit technology to support formative assessment strategies.

FaSMEd activities are organized in sequences that encompass group work on worksheets and class discussion where selected group works are discussed by the whole class, under the orchestration of the teacher. Taking into account formative assessment strategies and technology functionalities, Cusi, Morselli & Sabena (2017, p. 758) designed three types of worksheets for the classroom activity:

  1. problem worksheets: worksheets introducing a problem and asking one or more questions involving the interpretation or the construction of the representation (verbal, symbolic, graphic, and tabular) of the mathematical relation between two variables (e.g. interpreting a time-distance graph);

  2. helping worksheets, aimed at supporting students who face difficulties with the problem worksheets by making specific suggestions (e.g. guiding questions);

  3. poll worksheets: worksheets prompting a poll among proposed options”.

The authors identified feedback strategies (Table 5) the teacher may adopt to give feedback to students (Cusi, Morselli & Sabena, 2018, p. 3466). These strategies are employed in the class discussion that is organized by the teacher after the group work on worksheets.

Table 5: Feedback Strategies

We draw from the FaSMEd experience the idea of creating classroom activities in the formative assessment perspective that may promote inclusion. An example of FaSMEd activity is the Time-distance graphs activity, that will be presented in section 4

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The SMiLD project (2018-1­IT02­KA201­048274) is funded by the European Commission through the Italian National Agency for the Erasmus+ Programme. This web site reflects the views only of the author, and the Commission cannot be held responsible for any use which may be made of the information contained therein.