Peter Liljedahl opens his wildly popular book on mathematics instruction, Building Thinking Classrooms, with a bold gambit. He tells the story of one teacher whose students do well on end-of-course exams and standardized tests and who receives high marks from parents, administrators, and students. Despite such success, Liljedahl thinks that everything she does has to change. If it ain’t broke, break it, I guess.
Liljedahl—who has come to these conclusions after visiting forty classrooms, he reminds us several times—believes these students aren’t thinking. There was plenty of activity, as students were “taking notes, answering questions, filling in worksheets, and starting on their homework.” But apparently none of this activity classifies as thinking. It’s only mimicry. Instead, to Liljedahl, thinking only occurs when students are fumbling to solve problems for which they receive little instruction or guidance. It’s only swimming if Uncle Joe pushes you into the deep end.
In place of traditional instruction and practice, Liljedahl envisions a classroom where the walls are whiteboards and the teacher sets students to solve increasingly complex tasks in small groups. In his ideal environment, schools would “shed the burden of curriculum” to instead busy students with various mind games, brain teasers, and even card tricks (he’s very excited about the card tricks). But alas, abolishing curriculum is a political nonstarter, so teachers must muddle through.
He implements such a scheme in a number of classrooms and declares success because students are thinking more, which he defines as working on problems at wall-mounted whiteboards. This shouldn’t surprise us. If you don’t teach students how to complete a problem, they’ll spend more time trying to figure it out. Whether they actually learn the content is apparently a secondary concern to engagement.
With school-level book clubs, fawning profiles in education media, and conference presentations dedicated to it, Liljedahl’s scheme is popular in itself, but also a piece of a broader movement in mathematics education away from direct, explicit instruction. Invariably called discovery, inquiry, experiential, or constructivist learning, this student-directed philosophy of mathematics instruction is common at the district level—and most influentially, forms the basis of California’s recently adopted mathematics framework.
The fundamental misconception to this approach is that mimicry, memorization, and structured practice somehow constitute lower-order thinking. Liljedahl includes a staircase diagram that proceeds from doing to justifying, explaining, teaching, and creating to demonstrate his point. Merely doing math problems isn’t thinking in his framework.
This hierarchy of thinking almost perfectly mimics (I thought mimicry was bad) the ubiquitous “Bloom’s Taxonomy,” which similarly ranks types of thinking from remembering and understanding up to evaluation and creation. But in the 1956 Handbook where he introduced the topic, education professor Benjamin Bloom created his taxonomy simply to help teachers discuss classroom activities “with greater precision,” not to rank or judge them.
In fact, he refers to knowledge acquisition as the “primary” objective in education. Knowledge and comprehension are the bases of far more complex thinking. If students are busy counting out basic math facts on their fingers, they can’t attend to more difficult math problems. If they’re busy sounding out words, they can’t attend to higher-order thinking, such as evaluating the text in hand or creating their own poems. Factual knowledge, memorization, and comprehension facilitate and allow robust analysis, synthesis, and creation.
Moreover, both mimicry and explanation are the primary means through which we learn. At the edges of human knowledge, we must rely on experimentation and discovery, but these processes are incredibly inefficient. Instead, for everything that society has already learned, we have language to communicate these ideas and models to follow. It took humanity thousands of years to discover algebra, calculus, and geometry. Why expect students to repeat that process when we can just teach it to them?
From these fundamental misconceptions, ineffective classroom practices develop. In a telling passage, Liljedhal lets slip the truth: “The lessons where direct instruction was used allowed more students to successfully complete the task at hand.” I’d encourage him to read that sentence slowly in the mirror and reflect on what he just said.
And decades-worth of research vindicate direct instruction as the most effective model for student learning. In a seminal article on cognitive science and education, three cognitive scientists summarize the research: “The past half-century of empirical research on this issue has provided overwhelming and unambiguous evidence that minimal guidance during instruction is significantly less effective and efficient than guidance specifically designed to support the cognitive processing necessary for learning.”
Look closely at one such study and the reasons grow apparent. Researchers split students struggling in math into either an inquiry or explicit instruction group. In the inquiry group, the teacher presented a problem and relied on group discussion to develop solutions. In the explicit instruction group, the teacher presented a problem, demonstrated how to solve it, and then had students practice numerous problems of a similar type. In the end, those who learned from a more formulaic lesson demonstrated greater proficiency. The researchers posited that the students in the inquiry group encountered both correct and incorrect solutions, and so these methods simply confuse students.
The What Works Clearing House, a research consortium housed within the federal Institute of Education Sciences, published a 2021 practice guide on the most effective, research-backed methods for math instruction. It includes systematic instruction, representation and models, and timed activities, all of which Liljedhal denigrates for no other reason than they don’t constitute “thinking”—which by the end of the book, in a textbook example of circular logic, he seems to only define as activities that he already approves of.
When put into other contexts, the silliness of this approach is self-apparent. Over at Education Next, Ryan Hooper asks us to apply this discovery-approach to other tasks:
What if I told you that lifeguards have a new method for teaching toddlers how to swim by throwing them in the deep end of a swimming pool without supervision, in hopes that they will learn from their productive struggle? Or that grandma’s cookbook would be thrown out because of its limiting step-by-step approach to baking a pie? Or that sixteen-year-olds should discover how to drive from their peers or, better still, on their own?
I call it silly, but it’s an unfortunately serious matter. In the previous half-decade, American education has had to reckon with the reality that generations of students struggled to read because pseudoscientific approaches to early literacy proliferated; teachers just didn’t vibe with the explicit nature of phonics instruction. We’re about to repeat those same mistakes in with math instruction, and the Pied Pipers leading us astray are applauded for it.
