Math

The Importance of Learning Progressions in Math Education

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As an instructional coach in a district that recently began Standards-Based Learning in math, I hear similar concerns across the district, especially this time of year. The pressure to have mastery of all the Priority Standards before the year ends. (It’s important to note that all the standards are being taught. There are Priority Standards and supporting standards. The supporting standards do just that: support. They are the prerequisites, if you will, to the Priority Standards.)

But there is a hidden “Instructional Debt” that makes these standards feel like an uphill battle. If we want our students to succeed at high-level problem solving, we have to talk about the one thing that has become a bit of a “taboo” word in modern math: Memorization. Okay, the act of memorization isn’t taboo; some of the old methods are no longer supported by current research. It’s a frustration for all K-12 math teachers. So let’s talk about it and how we can help students master facts using current research.

The “Cognitive RAM” Problem

Every student has a finite amount of mental energy (let’s call it “Cognitive RAM”). I can hear you all now, “So do the teachers!” When we ask a student to solve a multi-step word problem, that task requires a massive amount of RAM for reading comprehension, translation, planning, and strategic persistence.

If that student hasn’t memorized their basic addition, subtraction, or multiplication facts, they are forced to use their limited RAM for “manual labor”: counting on fingers, drawing tally marks, drawing, modeling, or skip-counting. By the time they get to the actual logic of the problem, they’ve run out of “mental memory.” The whole task seems insurmountable.

The truth is: Memorization is Creative Freedom. When the facts are automatic, the brain is finally free to be creative in the approach to solving the problem. It breaks down a barrier. Think about it. If you are trying to solve a problem and realize you need to multiply 376 by 48, but you don’t have your facts memorized, this task just became a slow, muddy drudge. However, if you know you will need to multiply 376 by 48 AND you know your facts, the hard part is behind you once you know what to do. Suddenly, things don’t feel so unattainable.

The Progression is Non-Negotiable

To be clear: I am not advocating for “rote memorization” without understanding. Memorization is the final step of this Learning Progression. It only works if it is built on a solid foundation:

  1. Concrete: Manipulating base ten blocks and counters.
  2. Representational: Drawing tape diagrams, number paths, and arrays.
  3. Abstract (The Goal): Automaticity, mental fluency, and algorithms.

If we jump straight to memorization, we build a house of cards. But if we stay in the “Representational” phase forever, allowing students to rely on skip-counting patterns or finger-counting, we are capping their growth. We are asking them to do “back-breaking” math every single day. We do need to nudge them to move beyond the Representational model and help them see/understand that they are ready for the abstract and that the abstract is, in fact, your friend.

The Three Gaps Holding Students Back

When I listen to teachers discuss where students are “stuck” on a standard, they usually find that there is a gap in one of these three essential progressions:

1. The Missing Floor (Addition & Subtraction Facts)
If a fourth grader is still “counting on” to solve 14 + 6, they aren’t just slow, they are overloaded. Mental math strategies like “Make a 10” are the building blocks for every standard that follows.

2. The Fluency Wall (Multiplication & Division)
Skip-counting (7, 14, 21, 28…) and arrays are beautiful ways to learn multiplication, but it’s a weight around a student’s neck during long division. We have to move them across the bridge to automaticity.

3. The Magnitude Gap (Flexible Thinking)
When a student looks at 1/4 and 5/6, do they see numbers to crunch or magnitudes to visualize? Flexible thinking means knowing that 1/4 is “a little bit” and 5/6 is “almost a whole.” If they can’t visualize this, they aren’t ready for the standard of comparing fractions.

Bridging the Gap with MathReps

This is exactly why the MathReps framework exists. We don’t just “hope” kids learn their facts or develop flexible thinking. We build consistent, high-frequency opportunities to practice these skills alongside the priority standards.

A MathRep ensures that students touch the concrete and/or representational models every single day until those skills settle into the abstract. It allows us to pay off the “Instructional Debt” in small, daily installments so that when students are expected to solve two-step word problems with multiple operations, our students have the mental capital to win.

The Bottom Line: Don’t be afraid to slow down and build the floor. You aren’t “behind” on the pacing if you are busy building the progressions that make those standards possible.

Moving from “Turn It In” to “Explain It”: The Power of Student Voice

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“I have never heard them talk so much all year.”

Every teacher has that one student – the one who is mathematically brilliant or deeply creative but remains quiet during whole-class instruction. When a teacher finally hears that student’s voice through a tool like Snorkl, it’s a powerful moment of validation.

It’s validation that the tech we are using is meaningful and having a measurable impact. It’s the proof that the student is absorbing the information and is, in fact, brilliant. Most importantly, it gives us the data to back up our suspicions that the student is capable of much more than they show in a group setting.

The “Autopsy” Model: Why the Whole-Class Check Isn’t Enough

For years, the gold standard of education has been the “Turn It In” culture. A student completes a worksheet, the teacher collects it, and it is eventually graded (usually 24–48 hours later).

Even when we try to speed this up by “checking together as a class,” we are often still stuck in the Autopsy Model.

While a whole-class check provides “immediate” answers, it isn’t personalized. A student might mark an answer wrong, but the “why” remains a mystery. The teacher is still acting as a coroner, documenting a collective result after the thinking has stopped, rather than coaching the individual student through their unique misconception.

The “Biopsy” Model: Catching Learning While It’s Alive

We need to move toward a Biopsy Model, where feedback is instant, alive, and happens while the student is still in the “messy middle” of thinking.

Looking at this image, which leads to better understanding and engagement? The left shows an autopsy; the right shows a biopsy.

In my district, I’m seeing teachers achieve this through a powerful combination of analog and digital tools: Wipebooks + MathReps + Snorkl. And soon, a Kami component will be used for assessment.

  1. The Low-Stakes Rehearsal: Students start on a Wipebook. Because it’s erasable, they aren’t afraid to make a mistake. They can “draft” their thinking, erase, and refine until they are ready to share.
  2. Cognitive Unloading: Students snap a photo and record their explanation. For our ELD learners, this is a game-changer. Snorkl allows them to record in their primary language. By removing the “Brain Tax” of translation, students can use 100% of their mental energy on the math logic itself.
  3. The Immediate Pivot: Because Snorkl provides instant, personalized AI feedback, the teacher’s role shifts. You aren’t just reading answers off a key to a silent room; you are intervening based on real-time data.

Moving with Data-Driven Intent

The most exciting part of this shift isn’t just the tech; it’s the Strategy.

Editor’s Note: The “Insight” Advantage Don’t just look at individual scores. Use the Insights feature to see patterns across your class. Snorkl analyzes the results and groups students with common mistakes together. This allows you to pull a small group immediately and address the issue while the interest is still ongoing.

A Rallying Cry for Success

When we see a quiet student step into the arena, using their own voice, their own language, and their own logic, it validates that our instructional choices are working. It proves that the student can do the work, and now we have the data to back it up.

I want to hear from you: When was the last time a “quiet” student shocked you with their brilliance? How did you uncover it? Share your success stories below. Let’s celebrate the moments where the “biopsy” saved the day!

Related post

The 10-Minute Mastery Loop: Why Daily Repetition Beats Weekly Review

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It’s the same story every year. You leave school on Friday, thinking the lesson was solid. The exit tickets were great, the students “got it,” and you have the data to prove it.

Then Monday morning happens. You know where this is going, right?

You pose a warm-up question, and instead of hands in the air, you get a sea of blank stares. It’s frustrating, defeating, and it feels like you’re constantly playing catch-up. Reteaching isn’t fun for you, and it certainly isn’t for them. But why does this happen? I’m so glad you asked.

The Science of “Monday Amnesia”

The culprit isn’t your teaching; it’s a psychological phenomenon called the Ebbinghaus Forgetting Curve.

In the late 1800s, psychologist Hermann Ebbinghaus discovered that human memory declines exponentially. Without active reinforcement, we lose 50-70% of new information within the first 24 hours – yikes! By the end of a week? Up to 90% can be gone.

When we wait until “Friday Review” to revisit a concept, we aren’t reviewing; we are re-learning. We are trying to fill a bucket that has already leaked dry. This is where our frustration begins.

How to Flatten the Curve

Ebbinghaus also found a solution: Spaced Repetition. And if you follow John Hattie’s work, he has found that spaced repetition has an effect size of 0.62 to 0.65. Every time you revisit information, the forgetting curve flattens. The memory becomes more durable, and the rate of decay slows down.

The “sweet spot” for that first repetition is within the first 24 hours. This is where MathReps comes in. See, I got you!

Why MathReps is the “Escape Hatch”

MathReps is built on the 10-Minute Mastery Loop. It works because it solves two major classroom problems at once:

  • The 24-Hour Reset: Instead of waiting for a weekly quiz, MathReps hits the same standards/skills daily. By practicing on Tuesday what was learned on Monday, you “reset” the forgetting curve before the “blank stares” have a chance to set in.
  • The “Brain Tax” Refund: Usually, reteaching involves a new layout or activity. This forces students to pay a “Brain Tax”, using their cognitive energy just to understand and navigate the instructions and tasks. With MathReps, the format stays the same for 6-8 weeks. Because the structure is predictable, the “Brain Tax” is zero. Students spend 100% of their mental energy on the math. And this is exactly what we want them to do.

Build Mastery, Not Just Distant Memories

When you use the same MathRep for 6-8 weeks, the skill moves from working memory into immediate recall. You don’t need a hundred different activities; you only need 5-7 core MathReps to cover the entire year’s essentials. Right there is what every teacher wants to hear. No digging through sites to find another activity, no extra work for you, just 5-7 MathReps that you work with during the year.

Check out this 30-second breakdown of how the protocol works:

Ready to start your first loop?

Don’t waste another Monday morning playing catch-up. Use our searchable database to find the perfect daily rep for your current unit or one that will help with your yearly review for those state tests.

🔎 Search the MathReps Database Here

💡 Pro-Tip: Start your first 8-week cycle with Place Value. It is the foundation for almost every other standard, and building automaticity here will pay dividends all year long.

Every MathRep You Need, Just One Search Away

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Well, I’m back. It’s been a hot minute since I’ve written here. I was in the February slump that somehow dragged into March – you know how that goes. But during that quiet time, I was busy building the tool you’ve been waiting for!

As many of you are aware, I am the creator and curator of MathReps. For a long time, I have wanted a searchable database where you can look up MathReps by standard or keyword (s). I am pleased to say that I have created such a system! Yes, that’s right, you can now search through the entire K-12 MathReps library and find what you need. The system is seriously slick; it will take you to the exact slide you need. No more searching through stacks of slides. No more wasted prep time. We’ve all been there. We know there is the perfect MathReps for 4th-grade fractions, but you’re clicking through a few different slide decks to find it. I decided that HAD to end.

Here’s a quick 55-second walkthrough to see it in action.

What You’ll See:
What this new database does for you

  • Search by standard: type in “4.NF.1” and go
  • Search by keyword: need “Decimals” or Array”? Just type it in
  • Direct Access: It doesn’t just take you to the slide deck; it takes you to the exact slide

This is just the beginning. I have a few ideas for making it even better. Stay tuned. I’d love for you to go try it out at mathreps.com/search-all-mathreps and tell me: What’s the first standard you’re going to search for?

Empower 4th Graders with Decimal Mastery

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Why This MathRep Matters

For 4th-grade educators who are guided by the CCSS.Math.Content.4.NF.C.6 and CCSS.Math.Content.4.NF.C.7 standards (part of the Number & Operations—Fractions domain), this MathRep is a game-changer.

  • CCSS.Math.Content.4.NF.C.6
    • Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.
  • CCSS.Math.Content.4.NF.C.7
    • Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.

Why Teachers Love It

  • Low floor, high ceiling: Students can start with simple conversions and move toward rich reasoning and comparisons.
  • Multiple entry points: Some may begin with fraction-to-decimal conversion, while others may focus on comparing decimals; yet, both pathways are supported.
  • Discussion built in: The MathRep encourages students to explain their thinking (“I know 0.59 is less than 0.6 because …”), which deepens understanding. Using this MathRep in Snorkl can further support student reasoning.
  • Standards-aligned and ready to use: Especially helpful when you need a targeted resource for 4.NF.C.6 and 4.NF.C.7.

Ready to Get Started

Download or open the accompanying MathRep (see video) and begin your lessons with this ready-to-go template. Embed the video in your class expectation or homework link to give students a chance to revisit the concept later. Doing it on paper? Why not print out a blank template and a completed template on the back and insert it into a plastic sleeve? Students then have a reference if they get stuck.

Visit MathReps.com for free templates and more resources.

Final Takeaway

This MathRep is a powerful, standards-aligned tool for supporting 4th-graders in mastering decimal notation and comparison. By anchoring learning in discussions, visual models, and student reasoning, it simplifies complex content into manageable and engaging experiences. Add this to your toolkit and watch your students build confidence with decimals.

Let me know how it goes in your classroom – I’d love to hear your success stories and any tweaks you make!

Customize Your MathReps in Snorkl Easily

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This past summer, I’ve been sharing insights about the MathReps collection available in Snorkl. In my recent post, New MathReps Available in Snorkl Library, I provided a quick guide on how to easily navigate the platform to discover your favorite MathReps. Don’t miss the latest video that demonstrates how to seamlessly add these resources to your library and customize them to suit your specific needs.

Math Dash Chats: Boost Classroom Discourse

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Earlier this year, a simple idea sparked a solution to a common challenge in many classrooms: how to review math concepts and encourage student conversation when time is short. This led to the creation of Math Dash Chats.

Our district, like many others, was grappling with a noticeable gap in our curriculum—a lack of dedicated time for math discourse. We know that talking about math helps students solidify their understanding, but with so many standards to cover, where do you fit it in? I created Math Dash Chats for 3rd Grade, as an instructional coach who works closely with 3rd-grade teams, it felt like the perfect place to start. Since then, I’ve created sets for grades 2-6 and am excited to expand to grades 7 and 8 soon.

So, what exactly are Math Dash Chats, and how can they help your students? I’m so glad you asked!

What are Math Dash Chats?

Math Dash Chats are 36 prepared slides for your grade level (currently grades 2-6). The activity is designed to be a quick, five-minute daily review that gets students talking.

The slides are divided into six sections, five of which are based on Common Core domains like Geometry and Measurement, and the sixth is a directions section. Problems are hidden behind colorful “doors” [01:05], which you can view beforehand. Then, simply drag the questions over for a fun and engaging reveal.

How Do They Work?

The idea is simple: choose one “door” a day to discuss for about five minutes. This brief, focused discussion ensures a consistent review without taking up valuable class time. The topics covered are not just standard procedures; they encourage students to explore reasoning, number sense, and even domains like geometry or measurement that are often rushed through or left for the end of the year.

The video provides an example from the “Convince me that” category, where students are asked to prove that “4 tens is the same as 3 tens and 10 ones” [01:53]. This type of question promotes collaboration, and you might find that students want to use personal whiteboards or manipulatives to work through some of the problems together.

The Result

The response from teachers has been overwhelmingly positive. They love the ease of a no-prep, ready-to-go resource that gets students talking about math. Who doesn’t love a well-thought-out, free resource that is proven to work?

If you’re looking for a quick, impactful way to review math concepts and get your students engaged in meaningful math conversations, Math Dash Chats are for you!

Math Dash Chats Folder: Please make a copy of the desired slide deck for yourself by selecting ‘file’ > ‘make a copy’. If you receive a message that says ‘Access Denied’, it may be an issue with your district account. I’ve encountered this recently. If this happens, I suggest trying your personal account and sharing it with your district account. If that doesn’t work, contact me and we can try a few other options.

Let’s Talk Math

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My current position, Tech TOSA/Instructional Coach, affords me the opportunity to go into teachers’ classrooms and share the amazing things that are happening. So here I am sharing what this talented first-grade teacher in my district is doing.

Earlier in the week, I was in my Tech TOSA role. I went into this first-grade classroom to teach a lesson on coding using Beebots. I’m sure you’ve figured out this is not what I’m here to discuss. When I walked in, I noticed this chart on her board. What struck me about it was its simplicity. As you can see, it has some basic concepts and images to accompany it. What also drew my attention was the title: Let’s Talk Math. This implies that Math is something that should be discussed. It’s not something we do in isolation or keep to ourselves. While I did not have time to discuss with the teacher what she does with this chart, I know her well enough to say that she references it consistently.

I think it’s worth noticing that this is not flashy, cutesy, or Pinteresty in any way. Too often, we teachers ‘do too much,’ as the kids would say. That isn’t to say that those who make their room match or aesthetically pleasing “do too much.” It’s just to say that if you’re not that kind of teacher—like me—that’s okay. Having something as simple as a chart on your whiteboard works just as well. The important thing here is accessibilty – both in terms of understanding and placement – is most important. If it’s in a place where students can’t see it, like being too high up, or teachers don’t reference it, it’s no good to anyone.

What is my point in all of this? I wanted to celebrate the awesomeness of what this teacher is doing and highlight its simplicity. What are some simple ways you keep your students engaged while encouraging discussions?

NOTE: You may have noticed that I use hyphens in my writing. This has been a thing for me for many years: you can find them in previous posts dating back to pre-AI. I also know that AI uses them often, and it is one of the tell-tale signs that something has been written with AI. I felt the need to point out that while I use them, they are not a result of AI, just of my own knowledge of how to use them.

Effective Place Value Techniques for Teaching Addition

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Are you looking for a way to help students truly see what’s happening when they add multi-digit numbers? One powerful approach uses place value-based strategies that build from representational thinking toward more efficient, abstract methods.

Start with Expanded Form

In the first part of the video, I model a place value strategy using expanded form. Students break apart each number into hundreds, tens, and ones, add those values separately, and then combine their sums.

This representational method supports flexible thinking and strengthens their understanding of how numbers work. It also lays a strong foundation for future strategies that depend on place value fluency.

Scaffold Toward the Algorithm

Next, I introduce a slightly more advanced approach that continues to use the Hundreds Chart as a scaffold. This visual support helps students begin to internalize regrouping and transitions them toward the traditional addition algorithm, a 4th-grade standard.

This shift is intentional. By gradually moving from expanded form to a structure that supports the algorithm, students develop a deeper understanding of why the algorithm works—not just how to use it.

Support with Consistent Structure

The real power of MathReps lies in their consistency. Each template reinforces key math skills in a familiar format, allowing students to focus on developing strategies and precision rather than navigating new instructions each time.

Whether students are practicing during warm-ups, small groups, or independent work, MathReps create a rhythm of reflection and growth with immediate feedback.

Grab the Free Templates

The MathReps template shown in the video is available for free at MathReps.com. And if you’re looking for a reusable option, check out the dry-erase Wipebook versions—perfect for centers, partner work, or teacher modeling.

One Rep at a Time

With MathReps, you’re not just assigning practice—you’re building confidence, one rep at a time.

Why Manipulatives Matter Beyond Early Grades

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In the early grades, it’s not uncommon to see math manipulatives used daily in the classroom. Teachers pull out connecting cubes, counters, base ten blocks, and more—tools that help students build a concrete understanding of math concepts. Whether they’re exploring different ways to make 10 or practicing addition, these hands-on tools support their thinking in meaningful ways.

This follows the CRA model—Concrete, Representational, Abstract. We start with the concrete, like manipulatives. Once students have a solid grasp of the concept, we move into the representational, such as drawings or visual models. Finally, we introduce the abstract, using numbers and symbols alone. For example:

  • Concrete: Use connecting cubes to model 5 + 3
  • Representational: Draw 5 circles and 3 more
  • Abstract: Solve 5 + 3 = 8

But here’s what I’ve been thinking about lately: why do we often abandon manipulatives once students move into the upper grades?

Sure, by fourth or fifth grade, many students no longer need cubes for basic addition and subtraction. At that point, they’ve likely mastered those foundational skills and can work abstractly. But what about when we introduce new, more complex concepts—like volume in upper elementary or integers in middle school?

Volume is a great example. It’s a tough concept to grasp without something physical to hold or build. Yet so often, we hand students a formula and expect them to just “get it.” What if we instead gave them time to build with cubes, experiment, and see what volume means before jumping into the numbers?

The same goes for concepts like negative numbers. A number line or clothesline math activity can help students visualize and understand the relationships between positive and negative values. Why skip that step?

All this has me wondering: Is it a time issue? A training issue? Have we simply forgotten how powerful manipulatives can be for older students, too?

I’d love to hear your thoughts.
Do you use manipulatives in your classroom? What grade do you teach? What concepts do you use them for?

Let’s keep the conversation going—and keep math meaningful at every grade level.