The Problem With How We Teach Math
We need both direct instruction and inquiry-based learning (just not at the same time)
In the past decade, we’ve seen a lot of hysteria about declining math skills and the widening achievement gap between students in the United States and the rest of the world. Clearly, there is something wrong with how we are teaching math to our kids.
Traditionally, math has been taught through an “I do, we do, you do” approach: the teacher provides explicit instruction and modeling on a problem, then works through the problem with the class, and finally—once students have achieved mastery with teacher scaffolding—withdraws support so they can complete problems independently before starting the process again with a harder mathematical concept.
In theory, this should work. But clearly something has been going wrong. Fewer and fewer kids are achieving mastery of the foundational concepts they need to accumulate and build knowledge toward higher level concepts. As a result, students are unable to progress through the math curriculum as we would like.
One important factor that contributes to declining math achievement is low engagement and negative attitudes toward math. This may seem obvious, but if students hate math, think they are bad at it, and avoid it, then their achievement suffers. Attitudes, expectancy beliefs, and engagement are important across all subject areas, but research suggests they are particularly relevant in mathematics.
In response, many schools have turned away from direct, explicit instructional approaches and toward inquiry-based learning. In this model, students explore mathematical concepts and discover patterns and relationships through investigation rather than direct teaching. They work on open-ended problems and are encouraged to develop their own solution strategies. Proponents of inquiry-based learning seem to be correct that the approach increases engagement and improves attitudes about math.
The problem is inquiry-based approaches do not actually improve math skills. Most kids cannot actually “discover” new mathematical knowledge on their own, and explicit instruction is necessary to teach novice mathematicians.
So if inquiry-based approaches are ineffective at teaching new concepts, and direct instructional approaches are disengaging, what are we supposed to do? The answer is to use both approaches strategically at different points in the learning process.
First, teach new concepts with direct, explicit instruction. When students are encountering material for the first time, they need clear teacher-led guidance.
Second, use objective assessment data (i.e. tests!) to inform movement from one level to the next. Make sure that students achieve 90-95% mastery with scaffolding before moving to independent practice. Then ensure students achieve 90-95% mastery independently before moving to a higher level concept.1
Third, once students achieve mastery, provide inquiry-based learning and problem-solving opportunities for enrichment. These activities should involve integrating and thinking differently about previously mastered concepts. Here, teachers can get creative. These activities can involve group work and peer-to-peer learning, as well as connecting math to other academic subject areas. They should be fun and engaging and should reward creativity in problem solving.
The debate between traditional and progressive math instruction has been framed as if we must choose a side. But the real question isn’t which approach is better; it’s when each approach works best. Students need explicit instruction to learn new concepts and inquiry-based enrichment to deepen their understanding of what they’ve already mastered. Stop treating these as competing philosophies, and start treating them as complementary tools.
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Not as good as the 2003 Northern Valley Demarest production.
This procedure may involve increasing the number of opportunities for practice (i.e. repetition and rehearsal; with and without scaffolding) for students who are relatively slow to achieve mastery. Maximize tier 2 RTI/MTSS interventions for these purposes. (For those who aren’t familiar with RTI/MTSS, I’ll explain what this means in an upcoming post.)