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From Linguist to Educator: Dr. Swain’s Journey Through Japanese

Dr. Nathaniel Swain’s path to education began with a fascination for language, especially Japanese. His early studies in linguistics were fueled by a desire to master the complex writing and structure of Japanese. But over time, it became clear that what truly interested him wasn’t just Japanese—it was the nature of language itself.

The Intersection of Language and Mathematics

As their conversation deepened, Adrianne asked an important question: how does language play a role in learning mathematics? Dr. Swain offered a compelling perspective—math, in many ways, is a language. It has its own structure, vocabulary, and system for communicating complex ideas. For educators, this means being intentional about how mathematical language is taught, rather than assuming students will absorb it naturally.

He highlighted the work of researcher Dr. Sarah Powell, who explores the overlap between language and math learning. According to Swain, mathematical terminology isn’t something students intuitively grasp; it needs to be taught explicitly and linked to both concrete experiences and abstract reasoning. When teachers skip this step, students may end up memorizing procedures without truly understanding the concepts behind them.
Dr. Swain illustrated this point by describing a challenge his young son was facing in first grade. His son was working on basic subtraction problems involving 100, but without a strong understanding of place value, it became an exercise in memorization rather than comprehension. Swain explained how breaking the number 100 into “ten tens” helped clarify the concept. This kind of language-based scaffolding—where students move from tangible representations to abstract ideas—is key in building foundational math knowledge.

Learn more about the CRA Method here.

Adrianne added that instructional language matters deeply, and that even small shifts can make a big difference. She shared a classroom strategy that uses bundled craft sticks to help children physically grasp quantities like 10, 100, and 1,000. The tactile weight of the bundles helps students internalize the idea of magnitude and structure in a hands-on way, supporting both conceptual understanding and language development.

Supporting Students with Language-Based Learning Difficulties in Math

When asked how to support students with language-based learning challenges in mathematics, Dr. Swain emphasized the importance of consistent review and intentional vocabulary instruction. He explained that students first need to understand core mathematical terms during the initial learning phase—this includes identifying examples and non-examples, and connecting terminology to operations and place value concepts.

Beyond this initial stage, fluency becomes critical. Dr. Swain stressed the value of retrieval practice—regular opportunities for students to recall and use key math language and concepts. One effective approach is a daily math review at the start of class. Even just five to ten minutes of revisiting foundational ideas can help reinforce understanding and build automaticity, especially for students who need repeated exposure.

He pointed out that students with language-based difficulties often try to memorize math facts the way they would memorize words, without fully grasping the underlying structure. Helping them move from rote memorization to conceptual understanding takes time, targeted practice, and exposure to error-free learning opportunities. The goal is to develop fluency not only in vocabulary, but in the thinking and manipulation of math concepts themselves.

Making Learning Stick: Interleaving and Memory in Math Instruction

Adrianne highlighted one of the key insights from Dr. Swain’s book: while our working memory is limited, there’s no proven cap on what we can store in long-term memory. The challenge lies in how to move information from short-term to long-term storage—and keep it there. One effective strategy Dr. Swain discussed is interleaving, which plays a crucial role in helping students retain and apply mathematical knowledge.

In many classrooms, math is taught in isolated blocks—several weeks on addition, followed by subtraction, and so on. This approach may feel productive in the short term, but doesn’t support long-term retention or the ability to distinguish between similar types of problems.

Interleaving, by contrast, involves mixing different types of math content more frequently. For example, a week of lessons might include a day on addition, another on measurement, and another on geometry. This regular switching helps students practice retrieving information from memory, which strengthens their understanding over time.

Dr. Swain likened this to how we teach language arts. We wouldn’t spend four weeks only on reading, then ignore it while focusing solely on spelling or writing. Just as language skills develop best when integrated, so too does mathematical understanding benefit from varied and repeated exposure. Interleaving helps students make deeper connections and reinforces learning through retrieval, not just repetition.

What “On Track” Math Instruction Looks Like

In his book, Dr. Swain outlines clear indicators for when math instruction aligns with the science of learning. These “on track” practices help students develop both fluency and deeper understanding—and support all learners, regardless of ability.

One key marker is the integration of conceptual understanding and procedural fluency. Rather than treating them as separate skills, effective instruction weaves them together. For example, teaching an algorithm shouldn’t come at the expense of understanding why it works; instead, both should reinforce each other.

Dr. Swain also emphasizes the power of visual tools like number lines and bar models. These help students break down and solve problems in ways that are concrete and accessible, particularly for those struggling to grasp abstract concepts.

Daily review is another crucial component. Spending just 5 to 15 minutes revisiting previous content—from the day before, the previous week, or even last term—helps students retain knowledge and build long-term mastery. This regular retrieval practice is especially important for students who need repeated exposure to mathematical language and ideas.

Finally, high-quality instruction should be available to all students. That means providing explicit instruction and guided practice not just for those who are struggling, but also for students who appear to be advanced. Every learner benefits from clear explanations and structured opportunities to practice.

Too often, students who find math difficult are allowed—or even encouraged—to opt out. Dr. Swain shared his concern that in some classrooms, students are told to read a book instead of participating in math lessons, effectively cutting them off from opportunities to grow in this essential area. For him, this isn’t just a misguided choice—it’s an educational injustice.

What “Off Track” Math Instruction Looks Like

While well-intentioned, certain popular approaches to teaching math don’t align with what the research shows about how students actually learn best. In the book, Dr. Swain highlights a few of these “off track” practices as gentle red flags—signs it might be time to reevaluate.

One of the biggest is the push for invented strategies over formal algorithms. Some educators are taught to avoid traditional methods because they’re seen as too rigid or boring. Instead, students are encouraged to invent their own ways of solving problems. But the research suggests that formal algorithms—when paired with good conceptual instruction—are essential for building fluency and understanding.

Another common approach that may be off track is launching open-ended tasks without any prior explicit teaching. This model often promotes “productive struggle,” where students are left to grapple with unfamiliar problems in the hope that they’ll eventually arrive at understanding. While it sounds empowering, this method can actually overload students’ working memory and lead to confusion rather than clarity—especially for those with learning difficulties.

As Dr. Swain points out, struggling students are often labeled as “not math people” when in reality, they’ve simply experienced ineffective instruction. Good teaching—including clear explanations, scaffolded practice, and well-sequenced content—can significantly reduce frustration and build confidence.
One of the most inspiring examples Dr. Nathaniel Swain shares comes from Bentley West Primary School in southeast Melbourne, Australia. At this school, more than a third of students have diagnosed specific learning disabilities (SLDs)—yet every single one of them is performing at or above grade level.

How? It’s not magic. It’s great teaching.

With high-quality instruction and targeted intervention, Bentley West ensures that all students—regardless of their learning profile—build functional literacy and numeracy skills. And for students without SLDs, the impact is just as remarkable: they’re working an average of 18 months ahead of the standard curriculum.

As Dr. Swain puts it, the school actually has to teach 18 months beyond the expected level because otherwise the content is simply too easy.

This is a powerful reminder: when we raise the bar on instruction—when we embrace explicit teaching, daily review, visual models, and structured problem solving—we lift all learners. Struggling students aren’t left behind, and advanced learners aren’t left waiting.

What Keeps Math Learning Off Track?

Let’s unpack several common but problematic practices in math instruction that, while well-intentioned, can actually hold students back.

Here are a few key “off-track” indicators:

❌ 1. Pre-test/Post-test Teaching Cycles
Teaching a topic once and then not revisiting it for months is a recipe for forgetting. This “mass practice” model ignores what we know about memory: students need spaced repetition and interleaving to retain and build fluency.

“You might teach geometry in the first part of the year, and then not return to it for six months. Of course they forget—there’s no opportunity for spaced review.”

❌ 2. Tracking Students Into Rigid Ability Groups
Streaming students into “high, medium, and low” math groups may feel logical, but in practice, it often worsens achievement gaps. The low group goes slow, the high group goes fast, and the distance between them grows.

A better alternative? A multi-tiered system of support:

✅Strong Tier 1 instruction for everyone

✅Small group or 1:1 interventions as needed

✅Opportunities for extension and enrichment

“When you allow students of all levels to access challenging material, and support those who need more help within that setting, everyone rises.”

Why Cognitive Load Theory Should Shape Every Math Lesson

Dr. Nathaniel Swain offers a powerful lens through which to rethink how we teach: Cognitive Load Theory (CLT). At its core, CLT helps explain why “figuring it out” isn’t always the most effective way to learn, especially in math.

“We might feel like students learn more if they’re just thrown into the deep end. But what cognitive load theory tells us is that their working memory is too limited to thrive in that kind of situation.”

So what is Cognitive Load Theory?

Cognitive Load Theory is built on a simple premise:
🧠 Working memory is limited — often to just 2–4 elements at once.
🗃️ Long-term memory is unlimited — and that’s where fluency and flexible thinking come from.

CLT helps teachers decide when to use explicit instruction, when to model and guide, and when students can truly benefit from independent problem-solving.

The Two Loads to Manage:

Connect with Dr. Nathaniel Swain

If you’re ready to go deeper into this work, we highly recommend picking up Dr. Swain’s book:

📘 Harnessing the Science of Learning
Purchase the book here!

This resource is an accessible, compelling guide for any educator ready to align their practice with the science of how students learn best.

Visit Dr. Swain’s website here!

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