When people think of elementary school mathematics, they usually bring to mind number facts, calculations, and algorithms. This isn’t surprising, as these topics tend to dominate classroom work in many elementary schools internationally.
There is little doubt that elementary students should know the multiplication tables, be able to do simple calculations mentally, develop fluency in using algorithms to carry out more complex calculations, and so on. Indeed, these topics are fundamental to students’ future learning of mathematics and important for everyday life. Yet, is elementary students’ engagement with these topics in itself engagement with “real” mathematics?
I suggest that classroom discourse in an elementary school classroom where students engage with “real” mathematics should satisfy two major considerations. First, it should be meaningful and important to the students. Elementary students’ engagement with the topics I mentioned earlier can offer a productive context in which to satisfy this first consideration, especially if students’ work is characterized by an emphasis not only on procedural fluency but also on conceptual understanding.
Second, the classroom discourse in an elementary school classroom where students engage with “real” mathematics should be a rudimentary but genuine reflection of the broader mathematical practice. One might interpret the second consideration as asking us to treat elementary students as little mathematicians. That would be a misinterpretation. The point is that some aspects of mathematicians’ work that are fundamental to what it means to do mathematics in the discipline should also be represented, in pedagogically and developmentally appropriate forms, in elementary students’ engagement with the subject matter.
In its typical form, classroom discourse in elementary school classrooms fails to satisfy the second consideration. A main reason for this is the limited attention it pays to issues concerning the epistemic basis of mathematics, including what counts as evidence in mathematics and how new mathematical knowledge is being validated and accepted. The notion of proof lies at the heart of these epistemic issues and is a defining feature of authentic mathematical work. Yet the notion of proof has a marginal place (if any at all) in many elementary school classrooms internationally, thus jeopardizing students’ opportunities to engage with “real” mathematics.
Consider, for example, a class of eight–nine-year-olds who have been writing number sentences for the number ten and have begun to develop the intuitive understanding that there are infinitely many number sentences for ten when subtracting two whole numbers (e.g., 15-5=10). In most elementary school classrooms the activity would finish here, possibly with the teacher ratifying students’ intuitive understanding thus giving it the status of public knowledge in the classroom. However, in a classroom that aspires to engage students with “real” mathematics, new mathematical knowledge isn’t established by appeal to the authority of the teacher, but rather on the basis of the logical structure of mathematics. Thus the teacher of this classroom may help the students think how they can prove their intuitive understanding.
Given appropriate instructional support, students of this age can prove that there are infinitely many number sentences for ten when subtracting two whole numbers. For example, a student called Andy in a class of eight–nine-year-olds I studied for my research generated an argument along the following lines:
To generate infinitely many subtraction number sentences for ten, you can start with 11-1=10. For each new number sentence you can add one to both terms of the previous subtraction sentence. This looks like this: 12-2=10, 13-3=10, 14-4=10, 15-5=10, and so on. This can go on forever and will maintain a constant difference of ten.
Andy’s argument used mathematically accepted ways of reasoning, which were also accessible to his peers, to establish convincingly the truth of an intuitive understanding. This argument illustrates what a proof can look like in the context of elementary school mathematics. The process of developing this argument contributed also a powerful element of mathematical sense making to Andy’s work with number sentences for ten: As he carried out calculations to write the various number sentences, he thought deeply about key arithmetical properties (e.g., how to maintain a constant difference) and he put everything together in a coherent line of reasoning. Thus an elevated status of proof in elementary students’ work can play a pivotal role in students’ meaningful engagement with mathematics. This presents a connection with the first consideration I discussed earlier.
To conclude, elementary school mathematics as reflected in typical classroom work internationally falls short of being “real.” Yet it has the potential to become “real” if the learning experiences currently offered to elementary students are transformed. A major part of this transformation needs to concern the epistemic basis of mathematics, with more opportunities offered for students to engage with proof in the context of mathematics as a sense-making activity. The teacher has an important role to play as the representative of the discipline of mathematics in the classroom and as the person with the responsibility to induct students into mathematically acceptable ways of reasoning and standards of evidence. This is a complex role that cannot be fully understood without a strong research basis about the kind of teaching practices and curricular materials that can facilitate elementary students’ access to “real” mathematics.
Featured image credit: Math by Pixapopz. Public domain via Pixabay.