Inventing a myth 

Analysis paper

Our reply

Myth

The myth remains that inquiry-based learning should be the primary method used to teach mathematics. 

The authors provide neither evidence for the existence of a myth, nor an explanation of from where and when it 'remains'.

To back their claim, they cite just one meta-review about the experiences of disabled students. The review is not even about inquiry as such, but rather the dehumanising aspects of rote learning in mathematics education. The review calls for increased inclusion of pan-disabled students in research focused on inquiry and meaning making - a call that is both reasoned and timely, but in no way proof of the existence of a myth.

The first issue with this myth is the assumption that inquiry-based instruction should be the standard method of learning. 

As the academics cannot present any evidence, we do not know who assumes that 'inquiry-based instruction' - a confusing term in itself - should be the  standard method.

Data from international assessments (e.g., PISA) suggests that many students experience challenges learning mathematics. In addition, educators are going to see the impact of recent and unprecedented interruptions to instruction on students’ mathematics performance for years to come. A majority of students will benefit from more structured initial instruction. 

This is an example of a logical fallacy. Both statements are true, but they do not justify the conclusion. 

In fact, we would argue that the opposite might be true. Remote lessons during lockdown necessarily focussed on discrete procedures and provided less opportunity for discussion and reasoning. Our emphasis back in the classroom should be on inquiry processes.

Next, inquiry-based learning is suggested to increase achievement, curiosity, and interest in mathematics, as well as promote motivation to solve new and novel problems, yet we do not have sufficient data to support these claims.

Teachers who use inquiry know from experience that it leads to an increase in all three. We have research to show that to be the case. Perhaps we do not have enough to convince the academics because research funding is currently controlled by politically-appointed guardians of cognitive science.

Truth

Students have difficulty with learning when instruction is misaligned with student learning needs and readiness. While some students may thrive with true inquiry-based learning, their success is an exception rather than the standard outcome. 

The danger of misalignment is greater with forms of direct and explicit instruction. In the inquiry classroom, students' questions and conjectures help the teacher assess their learning needs and adapt the course of the inquiry accordingly. All students benefit from such tailored provision.

In fact, decades of research evaluating effects of inquiry-based learning and guidance demonstrated that more specific supports and guidance have been more effective than inquiry without supports in a wide range of contexts. De Jong and Van Joolingen reported that the forms of inquiry that were most beneficial were those that also included access to relevant information, in addition to support to structure inquiry and monitor progress - all elements that align with explicit instruction.

Just because support, guidance, monitoring progress and access to relevant information 'align' with explicit instruction does not mean that they cannot be elements of inquiry as well. Indeed, they would be features of most pedagogical models.

The paper by De Jong and Van Joolingham refers to computer simulations that aid discovery learning in science. In conflating computer simulations with classroom activity, science with mathematics, and discovery with inquiry, the academics reveal their confusion.

Essentially inquiry-based learning is reactive rather than proactive to student readiness. Lazonder and Harmsen noted that many of the supporting scaffolds reported in studies in their meta-analysis were added ‘ad hoc’. 

The implication here is that inquiry teachers do not plan. Instead, we are both proactive and reactive, preparing support and guidance before the lesson and adapting during the lesson. (See Planning and unplanning mathematical inquiry).

The authors of the analysis paper, however, have misunderstood the point. Lazonder and Harmsen do not say that classroom scaffolds in inquiry classrooms are ad hoc; rather, their point is that studies of inquiry learning classify types of guidance ad hoc (p. 684). If researchers had an a priori classification based on a theoretical framework, Lazonder and Harmsen argue, we might have more fruitful findings about the types of guidance that are most effective.

Rather than assuming inquiry-based learning is the best approach to introduce and teach mathematics, it is more appropriate to design instruction based on content, existing evidence of effectiveness, and likelihood of success considering student strengths and learning readiness.

Overall, Lazonder and Harmsen say that their meta-analysis provides "convincing evidence that inquiry-based methods can be more effective than other, more expository methods of instruction." That the authors of the analysis paper omit this conclusion and falsify another part of the meta-analysis suggests they are reading research from a particular ideological perspective. 

To do this, consider student readiness to engage in a learning activity. Novice learners, for whom the concepts and procedures are new or not-yet understood benefit from explicit instruction more than inquiry-based learning. With explicit instruction, the educators consider the scope and sequence of mathematics, building and connecting new concepts with previously learned concepts, and guide the learning process by modeling the skill, providing scaffolded practice, and finally encouraging independent practice and application of skills. 

The argument that 'novices' require explicit instruction is now used to consign students to a meagre diet of transmission and practice.

Students new to inquiry require more support and guidance, but that does not mean they should be denied the opportunity to be part of classroom inquiry. All students have the right to experience the disciplinary processes unique to mathematics. 

Metaphorically, a builder wants to build a birdhouse. They need the appropriate tools and materials to complete the project. If the builder has the materials, but not the tools, the builder will likely not be successful in the process or may build something that takes more effort and is less effective.

The metaphor is ill-conceived. Teaching is a far more complex process than building a birdhouse. Even within its own parameters, however, the metaphor reveals the limitations of explicit instruction. The builder might have the materials and the tools, but not the self-awareness and abilities to regulate and monitor the building process. It is precisely through inquiry that students learn to become effective self-regulating learners.

In Maths Teaching

To provide all students with high-quality mathematics instruction, consider:

(1) Teach with evidence-based practices first. One evidence-based practice with a large corpus of evidence includes explicit instruction. Explicit instruction is a combination of modeling, practice, and feedback. Explicitly teaching skills students are initially acquiring affords students to be successful applying skills later to new and novel concepts provided through inquiry.

'Evidence-based' has become code for practices backed by cognitive science. The reliable and robust research behind inquiry is accorded no recognition.

However, it seems from the confusing last sentence that inquiry is appropriate to apply skills to new concepts once explicit teaching has occurred. The authors are treading dangerously close to the myth they set out to refute. Inquiry might, after all, be the best approach for part of the learning process.

(2) Rather than offering true inquiry, provide modified inquiry learning with built-in scaffolds and support for student success. Explicit instruction is important when the discovery process may be inaccurate, incomplete, inefficient, or inadequate, which is often for many students. Anticipating student challenges during inquiry and proactively providing scaffolds and supports will help students make connections. Incorporate evidence-based practices including: guided notes to support learners progression through the task, self-regulated strategy development (SRSD) to support self-regulation in multi-step problem solving tasks, and incorporate teaching how to use concrete manipulatives or the concrete-representational-abstract instructional framework to develop conceptual understanding.

It turns out that structured and guided inquiry are permissible, as long as they incorporate explicit instruction to avoid misconceptions. What is not permissible, apparently, is inquiry as discovery learning. 

It might have helped their case if the authors had mentioned discovery in the myth they invented. 

They do not present evidence that the myth exists and, then, in a surprising twist, accept that their own myth might be true in certain circumstances.