MƒA knows that many teachers have an interest in learning about current and foundational education research in their content area – and who better to provide this insight than a practicing Master Teacher.

Over the past school year, Doug Shuman, two-time MƒA Master Teacher and mathematics teacher at Brooklyn Technical High School, has summarized key mathematics pedagogy research articles. Read his final submission below and previous pieces here and here. We hope this has served as a valuable resource for both our teacher community and the greater education community. 

Article of Focus:

Kirschner, Paul, Sweller, John, and Clark, Richard, Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching, Educational Psychologist (2006), Vol.41, No. 2, Pg. 75-86.


Ultimately how much instructional guidance is best will be determined by the goals of our education system. If the goal is to maximize short-term content mastery, the authors of this paper make a compelling argument that direct guidance is superior to minimally guided instruction: minimally guided learning consumes much of the brain’s resources and strains its architecture. But if another goal is for students to think analytically and critically, they will need to practice this skill, inefficiently assembling their knowledge while learning how to assemble their knowledge, and quite possibly sacrificing a precious point or two on the next standardized test. As educators, most of us see this as a small price to pay so that students may learn to teach themselves. Unfortunately, policy makers, and these authors, aren’t there yet.

In their 2006 article, Paul Kirschner, John Sweller, and Richard Clark took on the long-running debate between minimal instructional guidance and “strongly guided instruction” for novice learners. Despite what sometimes feels like a series of condescending swipes at inquiry learning, the authors ultimately make an important point for mediated instruction. The argument constructed against minimally guided learning finds its basis in current theories of human cognitive architecture. Kirschner et al. relish pointing out hyperbolically that “minimally guided instruction appears to proceed with no reference to the characteristics of working memory, long-term memory, or the intricate relations between them.”

According to the authors, minimally guided instruction places a double burden on working memory. Novice learners first need to engage in “problem-based searching” for “problem-relevant information” before new content can be isolated and sorted through in working memory. Furthermore, they claim that during this process long-term memory cannot be accessed for learning. The result is that the “cognitive load” placed on working memory exceeds the brain’s ability to efficiently and effectively alter long-term memory.

The authors suggest two interventions to increase retention of new content in long-term memory. The first, “Worked Examples,” involves students studying already solved problems, rather than having them struggle to solve unfamiliar problems. A second intervention is called “Process Worksheets.” This approach evidently includes the graphic organizers that many of us have seen and used. Like most of you, my reaction to these being the centerpiece interventions of this article was immediate and unpleasant.

The authors discount minimally guided instruction without addressing its benefits beyond the boring interventions they offer. Namely, that students who learn in an inquiry environment also learn how to inquire, which is both useful and engaging. In the authors’ view, there is no role for inquiry because learning how to inquire gets in the way of learning new material. This conclusion stems from the authors definition of success: content mastery as displayed on standard assessments given shortly after instruction. In her seminal 1998 article, “Open and Closed Mathematics, Student Experiences and Understandings,” Jo Boaler presented the alternative, and more important, result that students who learned how to approach math problems with an inquiry mindset were more willing and better able to interrogate and solve unfamiliar problems after a significant period of non-exposure than students taught in a traditional setting.

In my Algebra I classroom, inquiry and direct instruction are just two tools in my pedagogical toolbox. Cognitive load theory warns us that the capacity of working memory is severely limited. Yet by designing investigative tasks that are appropriately scaffolded and interesting, students can construct their own knowledge. Day 1 of my quadratic functions unit is an investigation into what causes a parabola to form a ‘U’.  Do students struggle? Absolutely. Do they become familiar with the grace and symmetry of the parabola without acquiring any skills that will directly improve their score on the New York State Algebra I Regents Exam? Guilty. A week later I teach them exactly how to complete the square to find the roots of a quadratic function. Do students efficiently learn an important procedure, ever-present on the Regents Exam? Absolutely. Does their previously acquired understanding of the symmetry of the parabola contribute to their quickly grasping the presence of the newly introduced symbol, ‘±’? You bet.

Focusing exclusively on the cognitive load that inquiry learning places on working memory ignores another important ingredient in successful learning. We learn more effectively when we give meaning to what we’re learning. Meaning can come from many places, not just the strict interpretation that meaning is equivalent to metacognition. I subscribe to the notion that allowing students to arrive at their own conclusions gives an emotional and psychological punch to their learning that is absent from direct instruction. As Boaler demonstrated, when students control the direction of their learning, and learn how to approach unfamiliar problems, the learning is more “usable”, versus the more “inert” learning inherent in direct instruction.

To prepare our students in the weeks before the New York State Algebra I Regents Exam most of us employ efficient techniques these authors would applaud. We review model solutions and rehearse procedures. But to their credit, the creators of the exam sometimes manage to write one or two constructed response questions that are well-crafted, imaginative and unfamiliar to students. It takes experience and practice to navigate this terrain. In the end, this paper is a warning that relying solely on pure inquiry-based learning is an inefficient way to learn content. But measuring educational efficiency solely by the amount of pure content learned for a single assessment is like measuring how much a person can eat in a lifetime by the amount of fish you’ve given them today.