PBL and IBL in mathematics classrooms

Since the publication of a paper on ‘minimal guidance during instruction’ 20 years ago the definitions of student-centred pedagogies have become blurred. Just as the title of the paper lumped together a range of approaches that “do not work”, so today advocates of a ‘science of learning’ continue to gloss over their specific characteristics. Within mathematics education, a focus on minimal guidance risks conflating two pedagogical models: project-based learning (PBL) and inquiry-based learning (IBL). Even though the two share intertwined histories and similar aims, there are, nevertheless, differences that have implications for the learning of the subject.

Origins of PBL

Kilpatrick’s pamphlet The Project Method (1918) presents the first systematic treatment of PBL. The unifying idea of the project is “the conception of wholehearted purposeful activity proceeding in a social environment, or more briefly, in the unit element of such activity, the hearty purposeful act” (p. 4) An example of a purposeful act is a student planning and making an item of clothing in the classroom. By putting the purposeful act at the centre of the educative process, Kilpatrick seeks to link the school with an outside world in which the purposeful act is typical of a worthy life. Education has become life, he claims, because school students now have the opportunity to live a worthy life through purposeful acts.

In the pamphlet Kilpatrick categorises types of project. Perhaps the main type is the one in which students embody an idea or plan in an external product, such as the item of clothing in the example. Students would, Kilpatrick suggests, take the following steps as independently as possible: decide on the purpose, plan, execute, and judge the outcome. Another type of project, which is the most familiar type to teachers of mathematics, involves solving a problem. Yet another involves obtaining a discrete skill or acquiring a piece of knowledge. Kilpatrick contends that PBL subsumes problem-solving and one-off learning as special cases. They serve as means to complete an end – that is, they are components of the bigger project.

Purposeful act, inquiry, and the curriculum

That Kilpatrick studied under Dewey and was heavily influenced by his philosophy is evident in the continuity they both saw between the school and society and their common aim of educating a citizenry to defend democracy. However, they emphasise different educational factors. For Kilpatrick, the purposeful act, in and out of school, gives rise to self-reliant and active citizens who are alert to threats to democratic institutions. For Dewey, it is the systematic school-based inquiry of students’ daily experiences that safeguards democracy by engendering a state of doubt.

These differences come into sharper relief when we consider the curriculum. Although followers of Kilpatrick asserted that a curriculum could be built on subject-specific (as well as cross-curricular) projects, the reverse is not true. Not every content standard in the current mathematics curriculum is ‘project-worthy’. McHugh lists measurement, geometry, and statistics amongst others as aligning well with PBL but other standards, such as those related to number, cannot be the central focus of a project. The problem lies in finding an authentic application of the standard. The project must not only ensure students are “mimicking real-world work” (Fancher and Norfar, p. 44), but also lead to a product that provides evidence of learning. Whether the product is a report, design blueprint, presentation, or physical object, the creation of the product is “one of the main distinctions between ‘projects’ and project-based learning” (Fancher and Norfar, p. 35).

The link to a real-world context leads to another problem in relation to the curriculum. There is the danger that the concrete scenarios at the heart of PBL cannot give students access to abstract concepts: “It is inconceivable how the higher, abstract levels of mathematical thinking can be based on real-life situations” (Van Oers, p. 64). If students were to access the whole curriculum through PBL, it seems the purposeful act would have to be reconceived for decontextualised settings.

For Dewey the curriculum and inquiry have a more sophisticated, dialectical relationship. In the The Child and the Curriculum (1902), Dewey suggests that the child and the curriculum are “simply two limits that define a single process” (p. 11). The experiences that students bring to school already contain within them facts and truths of the sort in the curriculum. The role of the teacher is to map the steps that take the student from present experience to a mature understanding:

It is the continuous reconstruction, moving from the child’s present experience out into that represented by the organised bodies of truth that we call studies. (p. 11)

For example, students might bake a cake or construct a simple machine to learn about measures and ratio and proportion respectively. The emphasis this time is not on a product, but rather on creating an organic connection between experience and the curriculum through inquiry.

Dewey's logic and mathematical inquiry

In Dewey’s logic, inquiry is a hypothesis-testing process that never gets “wholly beyond the trial and error situation” (Dewey, p. 177). The ‘reconstruction’ of the curriculum is achieved through a generative process in which students reflect on experience in order to create, under the teacher’s guidance, new educative experiences that convert ideas and theories into ever more grounded curricular knowledge. It is the formulation of a proposition as a hypothesis, Dewey argues, that encourages students to collect more evidence (Dewey, p. 266). A larger pool of observations enables them to confirm or modify the proposition. The classroom becomes a place where knowledge is tentative and provisional, and where “active investigation, testing our guesses or theories, imagining possible results of this or that physical or social relation could be carried on” (Mayhew and Edwards, p. 271).

While Dewey’s generic model of inquiry is akin to the experimental method of science, it does not conform to the way knowledge is created and validated in mathematics. There are two sides to mathematical inquiry: the inductive side that has similarities with Dewey’s approach and the deductive side that is unique to the discipline. The inductive side involves exploring patterns and making and testing conjectures; the deductive side includes analysing mathematical structure and proving a generalisation. The former is a creative process of plausible reasoning and speculating; the latter is based on systematic reasoning that delivers a formal proof (Polya, p. 158). Although the traditional focus In schools on the deductive side – that is, formal methods and proven results – has given the subject an air of “authoritarian mysticism” (Lakatos, p. 163), the antidote is not to adopt Dewey’s one-sided hypothesis-testing method. IBL in the mathematics classroom should encompass both sides of the discipline.

The relationship between PBL and inquiry

In PBL, inquiry is described as both a component of the project and an overarching method used to complete the project. Although Fancher and Norfar define inquiry as “the process of learning a concept through myriad strategies that are driven by the learner” (p. 33) and sustained inquiry as students asking questions and learning new information throughout the project (p. 46), they also state that inquiry “follows a very structured process” (p. 56) and an inquiry-based task is a standalone, tightly-scripted condensed form of a PBL unit (chapter 7). These seemingly contradictory ideas are difficult to reconcile. An example of an inquiry-based task starts with the question: How can we determine the relationship between an amount of fruit and the expected amount of liquid produced by the fruit? The task is part of a project in which students have to make recommendations about the amount of supplies required for a charity breakfast. The task seems to be a process of structured discovery in which the teacher directs students towards an understanding of ratio in the context established by the project.

McHugh gives a similar example of a task that forms part of a project. The task requires students to redesign the inside of a Meals-on-Wheels van to maximise the number of meals that can be delivered. It is part of a project based on the driving question: How can we support older adults who face food insecurities? McHugh calls the task a performance task, rather than an inquiry-based task, which seems more in line with its true nature. There are multiple pathways to a solution, but the start of the task is closed and there will be one optimum solution. McHugh reserves the word ‘inquiry’ for the process through which the project is completed. She defines productive inquiry as comprising an iterative process of student questioning, exploring, collaborating and reflecting to complete a particular, situated task (pp. 86 and 87). However, the student-focussed definition is undone when McHugh states that PBL might look like “mini performance tasks expertly strung together” (p. 62). This formulation is reminiscent of Kilpatrick’s description of the purposeful act as a unit element of purposeful activity. In reality, the PBL classroom might move from a string of teacher-led ‘acts’ to a student-led ‘activity’ as the participants gain more experience.

That inquiry in PBL is linked to a real world situation and follows a cycle of task and reflection makes it is similar to Dewey’s concept of inquiry. However, Dewey envisaged an experiential inquiry that helped the child develop a more sophisticated understanding of curricular knowledge. In PBL, inquiry is a process of designing a product that responds to a real-world situation. For Dewey, mathematical knowledge came out of the intentional experience; in PBL, students meet mathematical tools that help to model the situation through ‘just-in-time’ tasks from the teacher. Neither, we note, unite the two sides of mathematics. Dewey’s method corresponds to the inductive side, while PBL’s practical focus leaves abstract deductive reasoning out of reach.

Agency and identity

Giving students agency and developing their identity as mathematicians are important aims in PBL and IBL classrooms. However, the type of agency and identity are related to the position of the subject in each approach. Through IBL based on mathematical inquiry, such as the Inquiry Maths model, students learn to identify patterns and properties while they explore a stimulus internal to the subject. From conjectures based on particular cases they form a generalisation, which they go on to prove deductively or amend if they encounter a counter-example. As students become more familiar with mathematical inquiry, they take on a greater responsibility for setting aims and directing the inquiry. At first they benefit from the support of the regulatory cards, but, ultimately, they will act independently. Students simultaneously learn to exercise their agency and embody the competencies and actions of mathematicians.

Within PBL, agency and identity are connected less to the subject and more to the context. Those contexts range from the immediate (How can we create the dream school?) to those that are linked to the wider community (How can we attract pollinators to our community?). Students take on the identity of mathematicians in the service of other professionals, such as architects or ecologists, although they might also draw on resources from other school subjects and funds of knowledge from outside the school. They express agency through establishing goals, setting the direction of the project, responding to critical reflections from their peers, and refining the design of the product while working collaboratively in small groups. That the decisions students make involve factors outside the subject means their agency has implications beyond lessons. Indeed, as McHugh explains, students can use mathematics to effect community change. A public product, such as a presentation about the dream school to a panel of designers and school administrators or a blueprint for an expert’s construction of a ‘bee hotel’, attempts to offer a solution to an authentic problem.

PBL can also tackle social injustices to a greater extent than mathematical inquiry. In IBL classrooms, all students have a voice to pose questions, propose ideas, and suggest aims and directions based on the knowledge they bring to the inquiry. Such equity is a big step forward from traditional classrooms in which the teacher’s voice dominates. However, students have the potential to counter marginalisation and intervene against discrimination outside the classroom through the link that PBL establishes with the community. McHugh gives an example of a project that connected mathematics to social inequities. One student focused on gangs, whose members, she discovered, are predominantly Black or African American and Hispanic youth. After finding out the annual amount the US spends on gang violence, she calculated that the amount would cover the cost of sending all the gang members to college for four years. The student presented her findings in an infographic, which she sent to law enforcement experts and displayed in local youth clubs. While IBL challenges the inequities of the traditional mathematics classroom, PBL offers the student an opportunity to become, in McHugh’s words, a “change agent.”

PBL and IBL

The reduction of student-centred pedagogies to the catch-all phrase ‘minimal guidance’ belies the differences between PBL and IBL. In drawing out those differences in terms of origins, classroom practice, and the relationship to mathematics, we have attempted to demonstrate that they constitute separate approaches to the learning of the subject. Even if they espouse similar philosophies and employ some of the same terminology, students should experience PBL and IBL in the classroom if they are to appreciate the nature and applications of mathematics fully.

December 2025