This article explores a conceptual relationship between learner choice and mathematical sense-making. It argues that when learners can exercise choice in their mathematical activities, mathematical sense-making can be enhanced. The literature around mathematical modelling suggests a link between sense-making and learner choice. A three-tiered conceptual analysis allowed ‘purposiveness to thinking’ from the author through engagement with selected literature. Research questions related to a three-tiered analysis: generic, context-specific, and conditional accounts of sense-making in mathematics classrooms were formulated. The analysis resulted in a framework showing how sense-making may be constrained or enhanced in mathematics classrooms through learner choice. This article may add to our holistic understanding of sense-making in mathematics classrooms. It may contribute to mathematics teacher education by proposing that teachers are resourced to facilitate learners’ conceptual and procedural choice in primary or secondary mathematics classrooms.

Learners often view learning mathematics as non-sense-making (Dienes,

Reusser (

Mathematics teachers would like to see learners engage in mathematical sense-making. The term ‘sense-making’ is often interwoven with ideas of deepening understanding and application of mathematical concepts (Van Velzen,

For Weick, Sutcliffe and Obstfeld (

The quote ‘Any mathematical experience in which students

Since mathematical modelling led to the ideas in this article, a brief definition is necessary at this point. The definition of Lesh and Doerr (

conceptual systems that consist of elements, relations, operations and rules governing interactions that are expressed using external notations system and that are used to construct, describe, or explain the behaviours of other systems – perhaps so that the other system can be manipulated or predicted intelligently. A mathematical model focuses on structural characteristics (rather than for example physical or musical characteristics) of relevant systems. (p. 10)

Model-eliciting problems are reality-based problems where the product that learners are required to create or design will

Modelling is known to increase sense-making (Lesh & Doerr,

The idea of decision-making, freedom or choice is also considered by other scholars in mathematics education. Polya’s (in Kilpatrick,

Terms such as ‘own methods’ are considered to be consistent with learners being allowed to make decisions or have choice when solving problems. Hiebert et al. (

What does learner choice look like in mathematics classrooms? Learner choice entails learners being in the driving seat of the methods, procedures, representations and explanations in the mathematics class. Learners will have the option of entering a problem from their knowledge base, tackling the problem using their own ‘mathematical toolbox’ (Jensen,

This article is structured in the following way. The section on method and research questions provides some discussion of the method followed for the analysis of the concepts under scrutiny. It also sets out the questions that were formulated in undertaking the study. The section that follows the method focuses on the occurrence of sense-making in mathematical classrooms generally. It is followed by a section looking at a specific mathematical activity, that is, modelling, focusing specifically and descriptively on what features of modelling activities support learner choice to enhance learner sense-making. Following this, some basic tenets for sense-making in mathematics classrooms are proposed. The final section concludes with a possible framework provided by the author for understanding sense-making through learner choice in mathematics classrooms.

Bousso, Poles and Da Cruz (

In this article, a generic analysis seeks to answer the question:

According to Soltis (

A context analysis will then answer the question:

In this case a context-specific analysis only is done. This article will focus on sense-making in one particular type of task – mathematical modelling. In this section, mathematical modelling as a specific type of mathematics activity is described and analysed for features of enhanced sense-making.

A conditional analysis will answer the question:

Soltis (

These questions were answered through engagement with selected literature on mathematical modelling and sense-making. This engagement allowed ‘purposiveness to thinking’ (Wilson,

Teaching and learning as explicated by Brousseau (

This section looks at different conceptions of sense-making in mathematics classrooms. The analysis looks at various definitions and conceptions of what sense-making could look like in mathematics classrooms. Schoenfeld (

In mathematics classrooms sense-making can be compromised by learners’ inflexible concept cores (Trzcienieka-Schneider,

Tabachneck, Koedinger and Nathan (

Schoenfeld (

The

In summary, learners should be involved in ‘doing the mathematics’ in mathematics lessons. Tasks that promote active mathematical thinking by encouraging learners to use their own informal methods before memorising procedures are necessary for sense-making.

Some features of mathematical modelling as they relate to learner choice and sense-making are presented in this section. The first feature of modelling is that problems are devolved. Model-eliciting problems typically include a messy real-world situation where students make assumptions and limit the information they use based on these assumptions. Handing over problems and the responsibility for solving them implies that these decisions and choices are also handed over to students. Blomhoj and Jensen (

The second feature of modelling that relates to learner decision-making is that the starting and ending point for problems is reality. Cirillo, Pelesko, Felton-Koestler and Rubel (

The third feature of modelling where learner choice is embedded is that learners

Gann et al. (

A fourth feature of model-eliciting problems that encourages decision-making and supports sense-making is that learners work in groups. Competencies of the group are likely to be greater than those of individuals (Hatano,

A final feature of modelling is that learners make use of their own ideas. Authors such as English and Watters (

The features of modelling that contribute to learner choice and enhance sense-making can be summarised as: problems in their entire complexity are devolved (handed over) to learners, problems are set in authentic realistic contexts (where learners make their own assumptions), learners produce a model and learners use, create or decide on the tools they need. Modelling involves more competencies than only being able to follow set methods and procedures. Furthermore, learners work collaboratively in groups in what is largely a decision-making process.

A conditional analysis for the context conditions for which sense-making may be enhanced is now considered. In mathematics classrooms, the context conditions discussed in this section are the type of problems learners solve, as well as the classroom environment. Mazur (

Non-authentic problems.

The given problem has an unknown answer that can be found using a known procedure. Learners simply recall the correct formula and apply it to the problem once they have removed the words in the problem. It may not necessarily be ‘the answer’ per se that is unknown, but also not knowing that the answer is justified or suitable. This is evident when learners give senseless answers to problem contexts (e.g. giving an answer of 7½ people).

When Mazur defined

Authentic problems.

For Kramarski, Mevarech and Arami (

According to Mazur (

In terms of a classroom environment that supports sense-making, Schoenfeld (

Treffers uses the term ‘interactivity’ (

explicit negotiation, intervention, discussion, cooperation, and evaluation are essential elements in a constructive learning process in which the student’s informal methods are used as a lever to attain the formal ones. (p. 451)

The constructive learning process may come about through group collaboration, since it promotes ‘students’ mathematical understanding by creating opportunities for students to reexamine the validity of their reasoning’ (Francisco,

Boaler (

In summary, problems where the solution procedure is not always known and explicitly followed may enhance sense-making in mathematics classrooms. Problems that are so tightly structured that only one possible method can be used may also limit sense-making. The role of group processes may encourage sense-making since learner choice needs to be negotiated and validated by members of the group.

Bliss and Libertini (

One way of transforming a mathematics problem into a modelling problem.

In this additive process, the problem is being opened up to allow for learning, interaction and sense-making through adding interpretations, meanings and labels. An example of this process may be that the mathematics problem is to calculate rate, for example ‘Simplify 10 ℓ:100 km’. Adding labels would lead to a word problem, for example ‘A car uses 10 ℓ of fuel for every 100 km; at what rate is fuel consumed?’ If some meaning were added: ‘Dan is choosing between two cars and wants to buy the car with better fuel consumption. Car A uses 11 ℓ per 120 km and car B uses 12 ℓ per 130 km. Which car offers better fuel consumption?’ A modelling problem may involve asking if crossing a border to the next country 20 km away for cheaper fuel is worth the effort.

A possible explanatory framework that extends their diagram and that can help us think about sense-making through learner choice is presented in

Supporting sense-making through learner choice.

The NCTM (

Boaler (

The article set out to conceptually describe sense-making in mathematics classrooms and to understand some of its features. The description or highlighted features are not an all-encompassing account. Wilson (

Encouraging learner choice and freedom to get involved in mathematical discussions with peers may allow for greater levels of sense-making. Learner choice may come about in methods or representation or simply in engaging in alternative procedures or finding connections between ideas and procedures. Featuring more learner choice in lessons may involve re-negotiating the didactical contract that exists in classrooms. This necessitates that a teacher ‘lets go’ of doing most of the mathematical work in the classroom.

The author declares that she has no financial or personal relationships that may have inappropriately influenced her in writing this article.