In this post I propose to focus on the meta-selfie, in other words the text and language accompanying the mathematical selfie, which students are asked to provide as an explanation for the mathematics represented in their pictures. The meta-selfie allows us to dive into the the world of the students and understand what they saw in their picture: it is this translation of the visual into words or symbols that helps us further immerse ourselves into another's mathematical world. Let's take a look at how mathematical selfies help create opportunities for students to demonstrate that they can present their mathematical thoughts precisely and accurately, while giving teachers a way to assess students' proficiency in sharing their understanding of mathematical ideas. During workshops, the meta-selfie takes an oral form, therefore giving an interactive spin to the selfie activity.
In our secondary mathematics teacher preparation programs, our students have to complete an EdTPA portfolio during their student teaching clinical experience. Developed by, and in consultation with, a number of education stakeholders, "EdTPA is a performance-based, subject-specific assessment and support system used by teacher preparation programs throughout the United States to emphasize, measure and support the skills and knowledge that all teachers need from Day 1 in the classroom." (EdTPA). In particular, language is an important component of the EdTPA portfolio, one which prospective teachers tend to struggle with as they reflect on their teaching. According to EdTPA, the language demands in Secondary Mathematics include: function, vocabulary, discourse, syntax and mathematical precision. Here we borrow some of the definitions provided by the EdTPA handbook in order to organize our thread.
1) Language function: Every mathematical activity involves at least one language function determined by the expected learning outcomes. As part of their planning, teachers should be clear about which language function(s) they wish to emphasize and have students demonstrate in any given lesson. Examples include comparing measurements, or explaining strategies for solving a problem, or describing properties of a shape, or proving a theorem... etc.
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| Picture 1: Definition versus description |
2) Vocabulary: When discussing mathematics, students use a vocabulary specific to the discipline, words, phrases and symbols that may not have the same meaning in other fields of study or in everyday life such as "table, ruler, square, face, chord, digit, times, set". Some other general academic vocabulary might also be used across academic disciplines such as "compare, analyze, evaluate, describe, sequence, classify". Finally, subject-specific words and/or symbols are defined for use exclusively in the discipline (exponent, numerator, denominator, equilateral, multiple, ÷, ≥, ×).
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| Picture 2: Symbolic representations |
On the other hand, Picture 3 highlights the difference between everyday uses of mathematical terms and their specific meanings in mathematics. Indeed the door-knob reflection across the y-axis is an isometry, the two door knobs being congruent in the picture. However the reflection of the student when compared to the person actually taking the picture and being reflected in the door knob, would not qualify as a mathematical reflection since distances are not preserved. I'll let you reflect on this for a little while...
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| Picture 3: Mathematical versus everyday reflection |
3) Discourse: "Mathematical discourse refers to how members of the discipline talk, write, and participate in knowledge construction, using the structures of written and oral language." In the classroom, discipline-specific discourse has distinctive features or ways of structuring oral or written language, or representing knowledge visually.
Picture 4 provides a good visual for a parallelogram, in addition to introducing us to the beginning of an explanation for developing the area formula for parallelograms. Indeed, shifting the cards back into a neat pile rearranges the stack into a rectangular configuration, the area of which will be equal to the area of the original parallelogram, base*height. Even though the student mentions the rectangle, they may not have reached that level of knowledge creation yet with the picture, therefore it is the teacher's role to capitalize on the opportunity.
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| Picture 4: Parallelogram versus rectangle |
According to EdTPA, other examples of discourse may take the form of constructing an argument such as a two-column proof, interpreting graphical representations in the form of graphs or diagrams, and making and supporting a conjecture.
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| Picture 5: Mathematical conventions versus contextual interpretation |
When mentioning discourse, one also considers conventional ways of representing and speaking about mathematical objects, which can sometimes seem implicit. In Picture 5 the student implicitly assumes that we are going to read the graph from left to right, therefore concluding that the slope is negative. This is not always obvious to students who are first introduced to graphing functions, and it may not be obvious to someone going up this flight of stairs. Furthermore it is not always the case in mathematics that we read from left to right. Indeed when performing operations on whole numbers, the standard algorithms for addition and subtraction favor an organization of the procedure going right to left in order to regroup by place values efficiently.
4) Syntax: Syntax is defined as "the rules for organizing words or symbols together into phrases, clauses, sentences or visual representations." One of the main functions of syntax is to organize language in order to convey meaning. Examples include mathematical sentences such as the linear equation in Picture 2, or verbal sentences found in word problems (There are 5 times as many apples as oranges), conditional sentences (If a dress is $45 after a 50% discount, then what was its original price?), or the use of logical connectors such as "and", "or", "if, then"... etc.
Pictures 6 and 7 illustrate how mathematical syntax can be used within a selfie to include information using symbols.
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| Picture 6: Syntax of tesselations |
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| Picture 7: Symbolic of mathematical representations |
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| Picture 8: A square... or is it? |
5) Mathematical precision: Last but not least, students should be "precise and accurate with definitions and symbols in labeling, measurement, and numerical answers". This involves correctly labeling the axes of a graph, specifying units of measure during calculations, calculating accurately and expressing numeric answers with appropriate precision for the context of a problem. Precision sometimes seems to lack when using selfies to represent mathematical concepts, because often when looking for mathematical models in our world, these are only approximate representations. Picture 8 is an example of such dilemma. However the student was keen at recognizing the limitations of their representation of a square, and pointing them out, which is also evidence of attending to mathematical precision.
To conclude, let us consider the social dimension of mathematical language in the mathematics classroom: Through group work and class discussions students are encouraged to present and argue their reasoning, they learn to refine their way of expressing themselves logically in order to convince others of what they believe is correct. Using a common language, common representations, and the conventions of communicating mathematically, they learn to explain problem solving strategies they have used, reflect on their efficiency, and critique the reasoning of others in constructive ways. These acquired skills do not stay in the math classroom and should be qualities of every active member of a society. During one of my workshops in South Africa for math club facilitators, one participants seemed to be confused when presented with the term "triangle". Not knowing the English term, she had difficulties completing the picture assignments. However with some help from other participants, the assignment provided her with visuals that she could rely upon the next time she encounters the term, thus empowering her in her ability to communicate mathematically. In other words, looking for and discussing selfies provided her with a non-threatening opportunity to improve her communication skills, as well as her knowledge of mathematical representation, all seemingly important pre-requisites for a math club facilitator.
