Adrianne opens the conversation with a simple goal:
to clearly explain what the science of learning is and how it shapes effective math instruction.

The term gets tossed around often, but it isn’t a strict checklist or a single method. Instead, the science of learning represents everything we currently know—and everything we will discover—about how people learn and how the brain builds knowledge. That means the field is always growing, but there are key principles that consistently show up in the research.

Table of Contents

Discovery Learning vs. Explicit, Systematic Instruction

To illustrate one of these core principles, Heather references an iconic M.C. Escher artwork—the one filled with staircases twisting in every imaginable direction. People on the stairs wander up, down, sideways, and even appear stuck with nowhere to go.

She contrasts this with a second image: one simple staircase with a single clear path upward.
This represents explicit, systematic instruction.

Heather explains why this matters so much in math:

Algorithms and procedures weren’t invented randomly—they evolved over centuries because they work and they scale.

When students invent their own methods without guidance, those methods often break down in higher-level math. What works in 3rd grade can become a barrier in algebra, geometry, or beyond.

And as Heather points out, elementary teachers understandably may not see how an improvised algorithm will fail later on. That’s why systematic and explicit instruction is essential—not just helpful—for long-term success.

What the Research Tells Us

Research consistently supports systematic and explicit instruction as one of the most effective practices in teaching math.

It benefits all learners but is especially powerful for students who struggle with working memory, executive functioning, or language processing—common traits for many students with dyscalculia.

Why Background Knowledge Matters in Math

Another key aspect of the science of learning is background knowledge—not just what students know about math, but what they know about the world around them.

Heather explains that while linking new ideas to what students already understand is powerful, sometimes students simply don’t have the background knowledge they need. And in those cases, teachers must intentionally build it first.

She shares an example from her daughter’s high school coursework: a math problem involving the cross-section of a gutter.

For many adults, those words seem straightforward—but for students?
Her daughter didn’t know what a cross-section was. She also didn’t know what a gutter was. She doesn’t own a home. She’s never had a reason to think about gutters, and many students haven’t, especially those who live in apartments or have limited exposure to certain types of real-world structures.

This is a reminder for educators and parents:
We can’t assume students have the vocabulary or life experience required to make sense of every math problem.

Terms that seem simple to us—like “yard,” “volume,” “altitude,” “quarter turn,” or “gutter”—may be brand-new to some children.

That’s why intentional teaching of vocabulary and real-world reference points is an essential part of effective math instruction. Before students can connect new ideas to prior knowledge, we must make sure that prior knowledge exists. Otherwise, we risk building a staircase with missing steps.

The Role of Math Facts in Background Knowledge

Another form of background knowledge that often gets overlooked: math facts stored in long-term memory.

When students know their math facts—addition, subtraction, multiplication, and division—they free up their working memory during problem-solving. Instead of juggling a high cognitive load of many pieces all at once, they can focus on the concept being taught.

Without that foundational knowledge, students must constantly pause to compute simple facts, which increases frustration and slows learning. For neurodivergent students or those with dyscalculia, that mental load can become overwhelming.

Math facts are a deep topic on their own (and Heather and Adrianne tease upcoming podcast episodes on that very subject!), but the takeaway is clear:

Memorized math facts reduce cognitive load so students can focus on learning new content.

Working Memory: The Hidden Challenge Beneath Many Math Struggles

Another core principle within the science of learning is working memory—the brain’s ability to temporarily hold and manipulate information. It’s one of the biggest factors influencing whether a student can follow multi-step math procedures, remember directions, or stay engaged when concepts become complex.

To illustrate this, Heather gives Adrianne a quick working memory challenge. She shows a picture of ten animals—rhino, fox, crab, bear, elephant, squirrel, buffalo, turtle, giraffe, and stingray—and asks Adrianne to recall as many as possible after just a few seconds.

Adrianne does better than average, remembering six. But as Heather explains, that’s typical for adults—especially educators—who often use mental strategies without realizing it. The average person can hold three to four pieces of new information in working memory at a time. Adults may stretch that to six or seven, but for students, especially those who struggle, it may only be one or two.

This has massive implications for math instruction.

If we teach a new algorithm with six or seven steps, most students simply won’t be able to hold all the steps in working memory long enough to internalize them. That’s why children forget steps, mix up the order, or freeze during multi-step tasks—their working memory is overloaded.

This is especially true for neurodivergent learners or students with dyscalculia, who may have reduced working memory capacity.

Adrianne adds a relatable example: playing Bunco. If you’ve ever tried to chat with friends while mentally adding dice rolls at the same time, you know exactly how difficult it is to juggle multiple cognitive processes. Some people can do it effortlessly; others need silence and focus. The difference? Working memory capacity.

Moving Information From Working Memory Into Long-Term Memory

The next goal of instruction is to help new learning leave working memory and move into long-term memory. One of the most effective ways to do this is by activating prior knowledge.

When new information is connected to something familiar, the brain knows where to “file” it. Heather shares an example from Marilyn Zecher:

• When teaching division of whole numbers, we ask:
  “How many groups of ___ can I make out of ___?”
• Later, when teaching division of fractions, we use the same structure:
  “How many groups of ¼ can I make out of ¾?”That consistent language helps students recognize the relationship between concepts, making them easier to store and retrieve.

Chunking: A Proven Tool for Reducing Cognitive Load

Returning to the animal example, Heather describes how teachers can support working memory through chunking—grouping information into meaningful sets.

Instead of asking students to remember all ten animals, she could have chunked them like this:

• 🦀Sea animals: stingray, crab, turtle
• 🦒African animals: rhino, elephant, buffalo, giraffe
• 🐻Forest animals: bear, fox, squirrel

Chunking reduces the load on working memory by organizing information into familiar or logical groups. And in math, this can look like:

• teaching steps of an algorithm in small segments
• grouping facts by patterns
• breaking a complex task into manageable parts
   
  These strategies help students focus, remember, and ultimately learn more effectively.

  Retrieval Practice: Helping Students Remember What Matters

  Building on working memory, Heather introduces another powerful tool from the science of learning: retrieval practice. Essentially, retrieval practice is low-stakes quizzing—asking students to recall information they’ve previously learned. This process strengthens memory and signals to the brain that the information is important.

  Heather demonstrates by revisiting the animal exercise from earlier. Adrianne tries to recall the animals again, this time using the categories Heather suggested (forest, sea, African). She remembers eight of the ten—an improvement from the first attempt. This simple repetition illustrates a key principle: each act of retrieval strengthens the memory trace.

  Heather emphasizes that forgetting is normal and expected. Initial difficulty recalling information doesn’t mean the student didn’t learn it or that the teacher hasn’t taught it well. Instead, these “misses” are part of the process, helping the brain recognize that this information will be needed later. With repeated retrieval, students become more confident in their ability to remember. Over time, this practice can reduce test anxiety, because the brain has already had multiple opportunities to recall the information in a safe, low-pressure context.

  Supporting Students Who Struggle With Retrieval

  For students who find retrieval practice challenging, Heather recommends additional strategies:

• More chunking: Breaking information into smaller, manageable pieces
• Checking for gaps in prior knowledge: Ensuring foundational concepts are understood
• Adding retrieval cues: Using mnemonics, jingles, acronyms, images, or rhymes
   
  Treat it like marketing for memory: catchy, memorable cues help the brain remember what matters. For example, a logo, a song, or a visual can serve as a retrieval cue that makes recall easier.

Dual Coding: Using Multiple Channels to Strengthen Learning

Heather also introduces dual coding, a principle covered in a previous episode with Dr. Nidhi Sachdeva. Dual coding recognizes that the brain can process two streams of information simultaneously, such as verbal and visual input, if they are related.

This has clear implications for classroom instruction and teaching materials:

• Avoid slides or worksheets overloaded with text while speaking. Students cannot process multiple verbal inputs at once.
• Align images and words carefully, and provide time for students to engage with one modality before adding the other.
  Just as we saw with Bunco and mental math, trying to juggle too many cognitive demands at once overwhelms working memory. By using dual coding thoughtfully, teachers can reduce cognitive load and help students retain information more effectively. 

  The Impact of Visuals in the Classroom

  Dual coding also highlights the importance of how visuals are used in learning environments. Heather points out that classroom visuals—posters, decorations, and colorful displays—can sometimes be more distracting than helpful.

  While we often assume that bright, fun decorations engage young learners, research shows that students actually learn better in environments that are visually minimal and focused. Overly busy walls or worksheets with unnecessary characters and decorations can pull attention away from the content and increase cognitive load.

  When using visuals, the key is intentionality:

• Relevant visuals: Only include images or colors that directly support the math concept. For example, color-coding an equation to highlight important parts can be helpful.
• Avoid unnecessary distractions: Characters or “fun” elements that don’t relate to the math problem can hinder learning rather than enhance it.
   
  In short, visuals should support understanding—not compete for attention. By aligning the classroom environment with dual coding principles, teachers can help students process both verbal and visual information more effectively.

  Read more about dual coding with Nidhi here.

Subitizing and Worked Examples: Tools to Strengthen Number Sense and Memory

Subitizing is the ability to instantly recognize the number of items in a small set without counting. For parents and teachers, this skill can be transformative for helping students understand numbers quickly and confidently.

One practical tool to build subitizing skills is subitizing playing cards. These cards allow children to see numbers instantly, making math more visual, fast, and intuitive. Using subitizing cards regularly helps students build a solid foundation in number sense, which supports more complex mathematical thinking later.

Heather adds that we can further support students’ working memory by using worked examples or task analyses. These are step-by-step demonstrations of how to solve a problem, often accompanied by visual cues or small images that guide students through each stage. For example, a worked example might show addition with regrouping using pictures or icons to track each step.

These strategies help students:

• Reduce cognitive load: By having the steps visually laid out, students don’t have to hold everything in working memory at once.
• Internalize procedures: Students can refer back to the examples, practice with retrieval, and gradually learn to perform tasks independently.
• Identify tricky steps: Worked examples make it clear which parts of a problem students find difficult, allowing teachers to provide targeted support. Over time, as students practice and retrieve steps from memory, prompts can be gradually faded, strengthening both memory and understanding. This combination of subitizing practice and step-by-step visual guidance provides a research-backed approach to helping students succeed in math.

Interleaving: Strengthening Understanding Through Variation

Another powerful principle from the science of learning is interleaving. Interleaving involves mixing similar but distinct topics or problem types rather than practicing only one type repeatedly. This helps students learn to differentiate between concepts and apply the right strategies in the right context.

Heather uses the example of the animal activity from earlier:

• Students can first compare all the gray animals, discussing their environments and counting each type.
• Then they examine the colorful animals, noting differences and patterns.
• By alternating between the two sets, students learn to notice subtle distinctions and apply analytical thinking.
  This same principle applies directly to math instruction. For example, if students only practice part-part-whole addition problems repeatedly, they may eventually stop paying attention to the story context and just add the numbers mechanically. Interleaving different problem types forces students to pay attention to critical details, such as the problem’s structure or the signs in an equation.

   

  Heather notes that interleaving is also essential for practicing math facts. If children practice all addition facts first and all subtraction facts later, they may make “careless mistakes” when switching between operations—not because they don’t understand, but because they haven’t practiced mixing the facts. Intentional interleaving helps students fine-tune their focus and develop more flexible problem-solving skills.

  In short, interleaving ensures students learn to notice what matters and apply knowledge correctly across different contexts.

  The Learning Hierarchy: From Acquisition to Adaptation

  Another core concept from the science of learning is the learning hierarchy, which outlines how students progress through acquiring and mastering a skill. According to resources like Intervention Central, the hierarchy has four key stages:

  1️⃣Acquisition – This is when students are first learning a skill. Accuracy is inconsistent, and adult support is often required. For example, a child learning addition for the first time may need guidance with every problem. Many educational challenges occur here when instruction moves too quickly to the next topic before mastery is solidified.

  2️⃣Fluency – At this stage, students are accurate but may still be slow. Fluency means the skill can be retrieved quickly and reliably without heavy cognitive effort. For instance, a student may now solve addition problems correctly but needs more practice to answer instantly, like recalling math facts from memory.

  3️⃣Generalization – Here, students apply their skill across different contexts. Interleaving practice is critical at this stage. For example, a student might know their math facts in warm-up exercises but struggle to apply them in double-digit multiplication unless they’ve practiced transferring knowledge across contexts.

  4️⃣Adaptation – The final stage involves using skills flexibly in new or real-world situations. A student may know the algorithm but may need to adapt to knowing when it is more efficient to solve without the algorithm. Adaptation is ongoing as students adapt and apply their knowledge to higher-level skills.

  Many learning struggles arise when students are pushed to a stage they’re not ready for. Ensuring students progress appropriately through each stage is critical for long-term retention and confidence.

  Spaced Practice: Helping Skills Stick

  Spaced practice is another essential strategy. It involves reviewing material after intervals of time rather than all at once. For example, after learning about animals or math facts:

• Check recall at the end of the lesson.
• Review the next day or a week later.
• Revisit after a month, then after two months. This repeated retrieval signals to the brain that the information is important and needs to be stored in long-term memory. Forgetting occasionally is part of the process; it helps the brain strengthen recall. Spacing practice also reduces test anxiety because students gain confidence through repeated, low-stakes retrieval.

Applying Science of Learning Beyond the Classroom

Adrianne points out that these principles—acquisition, fluency, generalization, adaptation, and spaced practice—aren’t just for math or children. They apply to learning any new skill, from sports habits to professional skills. The hierarchy helps learners of all ages understand progression and build mastery systematically.

The science of learning isn’t just theory—it’s actionable, practical, and transformative. By:
🔴Building background knowledge,

🟠Supporting working memory,

🟡Using retrieval practice and spaced repetition,

🟢Leveraging dual coding and interleaving,

🔵Following the learning hierarchy, and

🟣Using Explicit Instruction

Educators and parents can help students not just succeed in math but build skills that last a lifetime.

Whether you’re teaching addition or life skills, these strategies provide a roadmap for learning that works for anyone, at any age.

The CRA Club: Putting Science of Learning into Practice

For educators and parents ready to apply these strategies for their struggling learners, The CRA Club provides a wealth of resources:
🟣Step-by-step materials aligned with research-based strategies for students with dyscalculia, dyslexia, and SLD.

🔵Coaching calls with Heather and Adrianne for personalized guidance.

🟢Tools to support working memory, retrieval practice, interleaving, and more.

At $47/month, the CRA Club is designed to be affordable while giving members access to everything our team uses in their sessions. It’s about more than just resources—it’s about creating a movement for math education where students thrive and teachers feel supported.

Download our FREE Science of Learning Printable Bookmark

Looking for a simple, research-aligned way to introduce the Science of Learning to teachers, parents, or colleagues?

Check out our FREE printable bookmark highlighting the Science of Learning.

Curriculum Alone Can’t Rescue

a Student Struggling With Math.

We have the tools and strategies

that help students with complex learning needs

finally understand it. Let us show you how.

MFM Authors