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- You thought they were lost. Turns out, they were thinking.
You thought they were lost. Turns out, they were thinking.
What looks like zoning out might actually be learning in progress.


MAKING IT EASIER TO BE A BETTER TEACHER
Ever watched a student finally “get it”… five minutes after you almost gave up?
This week’s issue is all about those moments, when the process finally clicks, especially in our math classes.
Our Tech Tool find helps students see how math works instead of just hunting the answer, while in Brainy Bit this week, we look at a study that reveals why those blank-stare pauses might actually mean understanding is loading.
Sometimes progress just needs space - and the right explainer.
Here’s how you’re about to become an even better teacher in the next 6 minutes:
Tech Tool: An actually useful math app ➗
Brainy Bit: How to know they actually ‘get it’ 🤔
🚀 Noteworthy News
👉️ Thought-provoking: Can AI keep students motivated—or does it do the opposite?
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And now back to making you an even better teacher 👇️
TECH TOOL
A math app that actually explains itself
Remember “turning to the back of the book” for answers?
This week we’re looking at a math app that doesn’t just give the final answer but shows exactly how to get there. Used right, it’s less about cheating the process and more about finally understanding it.
The Solution: Cymath
Cymath acts like a patient, digital tutor that breaks down math problems one step at a time. Students can enter equations by typing or snapping a photo, and it’ll explain how each step works—perfect for after-school help, late-night study sessions, or when you’re away and the substitute’s allergic to math.
It’s also surprisingly teacher-friendly. You can project Cymath’s explanations to model problem-solving strategies or assign it as a self-check tool for independent learners. The free version is more than enough for most classrooms, while the optional ad-free upgrade adds extra examples and clarity.
Still, Cymath’s strength is also its biggest risk: students can skip the “thinking” part entirely if it’s not introduced with the right guardrails.
Is This For YOUR Classroom?
If your students value the how, yes.
But if they’re answer-hungry and allergic to process, you’ll need to frame it carefully. Cymath only works when curiosity does—without it, it’s just an online answer key with better lighting.
Strategies That Work:
Step-by-Step Practice: Have students solve a problem first, then compare their reasoning to Cymath’s steps.
“Explain the Explainer”: Ask students to describe why Cymath took each step—it’s great for reflection.
Emergency Math Sub Days: Let Cymath lead the lesson when your substitute’s more Shakespeare than statistics.
Cymath isn’t about shortcuts - it’s about second chances.
When framed right, it helps students fall back in love with the process, not just the product.
“The only way to learn mathematics is to do mathematics.”
Brainy Bit

When the Lightbulb Flickers On
TLDR: A new study of professional mathematicians found that right before a breakthrough (the “aha!” moment) there’s a measurable shift in how they think and write. Their work becomes more structured, less erratic, and their pauses grow longer. In other words, insight looks surprisingly calm.
Even though this study focused on elite mathematicians, the findings hit close to home for us K–12 teachers: those quiet moments when students pause, stare off, or frown at a problem might not mean they’re lost, they might be right on the edge of understanding.
The Study: Analyzing the Eureka Moment
Researchers from Princeton and Oxford analyzed thousands of chalkboard images from mathematicians at work. Using computer vision and information theory, they measured how unpredictable each stroke of chalk was—what scientists call “information entropy.”
They discovered that as mathematicians approached a new idea, their writing became more predictable and organized.
That shift—less noise, more order—was a signal that insight was near. The quiet before the lightbulb wasn’t wasted time; it was the brain making sense of the chaos.
The Results:
Breakthroughs didn’t appear as random sparks - they built up like slow, steady pressure before a storm.
Thinking became structured, not scattered.
Pauses grew longer before major insight.
The “aha!” moment was the result of struggle, not luck.
So those blank stares in your classroom? They might not be disengagement—they might be neurons reorganizing.
In YOUR Classroom:
Students’ hesitation often means their brains are doing the heavy lifting—just beneath the surface.
Here’s how this study might improve your math class this week:
Strategies That Work:
Normalize the pause. Give thinking time purpose; don’t rush it.
Model your own aha moments. Show how mistakes lead to clarity.
Spot the build-up. Look for students who are “almost there”—their struggle is the spark.
Learning doesn’t always look like motion.
Sometimes, the quietest moments are the loudest signals that understanding is on its way.
WHAT DO YOU THINK?
We would LOVE to hear from you!
Reply to this email, or send us a message on Instagram! We’re here to walk with you in these crazy times!
Part of what makes The PEN Weekly community so special is the fact that our readers are teachers from around the world! We’re not going to lie, we think that’s pretty darn cool!
We’ll see you again on Monday 🍎
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References
Today’s newsletter adapts information from the following sources:
Tech Tool:
Cymath. (2025). Join millions of users in problem solving. Retrieved from https://www.cymath.com/
Brainy Bit:
S. Tabatabaeian, A. O’bi, D. Landy, & T. Marghetis, An information-theoretic foreshadowing of mathematicians’ sudden insights, Proc. Natl. Acad. Sci. U.S.A. 122 (35) e2502791122, https://doi.org/10.1073/pnas.2502791122 (2025).

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