Author: mwiernicki

I am a husband and father of two quickly growing children. I have taught elementary, middle and high school math and a math course for preservice teachers at Mercer University. I am currently a math teacher on special assignment and a consultant in Henry County, GA. In my spare time, I study (and perform a little) magic, swim, and love to spend time with my wife and kids!

Transparent head silhouette with interlocking gold and silver gears inside representing brain function

Connecting Numeracy and CT: The Structure was Always There – Part 3
Transparent head silhouette with interlocking gold and silver gears inside representing brain function

In the last two posts (Part 1 and Part 2), we looked at what fluency actually is and the three structures that sit underneath it. In this post, I want to make a connection that might surprise you.

Those same three structures — place value, properties of operations, and equivalence — are also at the heart of computational thinking.

I know. Stay with me.

What is Computational Thinking, Actually

Computational thinking isn’t about coding. It’s about a particular way of approaching problems — one that computer scientists and mathematicians have always shared. It has four moves:

  • Decomposition — breaking a complex problem into smaller, manageable parts
  • Pattern recognition — noticing what stays the same across different contexts
  • Abstraction — stripping away surface detail to see the underlying structure
  • Logical sequencing — ordering steps where each one depends on the one before

Sound familiar? It should. When a student solves 27 × 4 by breaking it into

20×420\times 4

and

7×47 \times 4

they are decomposing. When they notice that the same move works with 27 × ¼, they are recognizing a pattern across abstractions. When they express that as a(b + c) = ab + ac, they have abstracted. These aren’t separate skills layered on top of mathematics. They are mathematics — the structural kind.

Where Numeracy Fits

Numeracy is what happens when a student can take those structures into the world and use them confidently. Not just in a math class, on a familiar problem type, with a procedure they’ve practiced. But in a grocery store, a spreadsheet, a situation they’ve never seen before.

That flexibility — the ability to deploy structure in unfamiliar contexts — is what separates a student who has memorized math from a student who understands it.

So fluency, numeracy, and computational thinking aren’t three different goals pulling in different directions. They’re three expressions of the same underlying capacity: the ability to reason from structure.

What this Means for Teaching

If the structures are the through-line, then the question worth asking about any lesson, any task, any resource is: does this make the structure visible?

Not just — is it a good activity? Not just — will students enjoy it? But: will students walk away having experienced something about how numbers behave that they can carry forward? Will they be able to use that understanding the next time they encounter something unfamiliar?

That’s the goal. Not fluency for its own sake. Not computational thinking as a trendy add-on. But students who see the structure — in numbers, in patterns, in problems they’ve never met before — and know what to do with it.

And when those students finish a challenging task and they tell you they want to do more — because somewhere along the way, they started asking “What if…?”, run with it. They’re posing rich questions and ideas to explore because they’re curious. Many of their ideas will likely involve some mathematical modeling. Another huge win.

So, the end of the lesson is not, necessarily, the end of a lesson. But it is the beginning of a mathematician.

Connecting Numeracy and CT: The Structure was Always There – Part 1

When students compute, are they executing steps or reasoning from relationships? Most of us were taught the former. But the latter is where mathematical power lives.

Feel the Structure

We all want our students to become fluent with facts. There are many ways we can try to get them there. Some focus on strategy development and building mathematical relationships. Others focus on speed and memorization. One way builds confidence and connections. The other creates anxiety.

To build fluency, we need to understand what it is.

If fluency is represented by this walnut.

When students are able to apply strategies to multiple contexts to solve problems, they are fluent. They are also working their way toward numeracy: having the confidence to use basic maths at work and in everyday life.

Let’s look at some ways to solve 27 x 4:

What’s the underlying structure of each? They all break the factors into friendlier numbers and multiply, then put them back together. Decomposing and Composing. All with the underlying structure of the Distributive Property. One uses the Place Value structure as well (top left). Another also uses Equivalence as a structure (bottom left). But all use the structure of distributive property. Naming the property isn’t something students in grades 3 -5 need to focus on. They just need to understand what’s happing and apply it.

Sometimes teachers go with timed tests rather than focusing on strategies because they think that the strategies are replaced by fluency. It’s not really the strategies, though. It’s the underlying structure of these strategies. And fluency doesn’t replace the structures, it compresses them.

In my next post, I’ll discuss the three big structures that encompass all of K-5 mathematics.

Connecting Numeracy and CT: The Structure was Always There – Part 2

Name It and Connect It

Part 1 of this series was all about feeling the structure of mathematics. We looked at what fluency actually is — and what it compresses.

In this post, I’ll be discussing the three big structures behind all of the strategies for K-5, and how they connect to fluency and numeracy.

Across all of K–5 mathematics, three big structures do most of the heavy lifting. They don’t belong to any single grade level or content area. They run through multiplication, fractions, place value, and early algebraic thinking like a spine. Once you see them, you can’t unsee them.

Place Value 

Our number system is built on a simple but powerful idea: position determines value, and each position is ten times the one to its right. That structure is what makes 27 × 4 manageable — we can split 27 into 20 and 7 precisely because of how our number system is organized. It’s also what makes the multi-digit addition and subtraction with whole numbers and decimals work. An algorithm isn’t the structure. It is the structure, compressed.

Properties of Operations

This is the one students use constantly without knowing it has a name. When a student solves 27 × 4 by doing 25 × 4 + 2 × 4, they’re using the distributive property. When they flip (3 × 8) x 5 to 3 x (8 x 5) because it’s easier to think about, they’re using associativity. These properties aren’t rules to memorize — they’re descriptions of how numbers actually behave. And they work the same way whether the numbers are whole numbers, fractions, or decimals.

(3 x 8) x 5

3 x (8 x 5)

Equivalence 

This one tends to get underestimated. Equivalence isn’t just “same value, different form.” It’s the engine behind some of the most important mathematical moves students make. Finding a common denominator before adding fractions? That’s equivalence. Rewriting 28 as 20 + 8 before multiplying? That’s equivalence. Recognizing that

12\frac{1}{2}

and

24\frac{2}{4}

describe the same relationship? Also equivalence. Students who own this idea deeply have a tool that works across every content area they’ll encounter.

Here’s the thing about these three structures: they don’t stay in their lanes. Place value only does its full work because of equivalence. The distributive property only works with fractions because equivalence lets us rewrite them. Early algebraic thinking is what happens when students start to see all three as a unified system of structures rather than separate topics.

That’s not a coincidence. It’s the structure.

Fluency compresses the structures — place value, properties of operations, equivalence. Those structures are the shared foundation of both numeracy and computational thinking.

In my next post, I’ll connect these three structures to computational thinking — and why the connection has been there all along.

Open Patterns

When we teach students about patterns, we’re really teaching them how to begin to think algebraically. This is especially true when we use Growing Patterns or Visual Patterns from Fawn Nguyen. When we use these patterns, we usually give the first three like you see in the image below.

We often ask students to build the next 2 or 3 stages which opens the door to describing how it grows which leads to a discussion about what is staying the same and what is changing (algebra).

But, what if we wanted to make this a more creative exploration to ignite curiosity and engage students more. What if we gave them this, instead:

What might your students do with this? How might their pattern grow? These are questions I asked myself before I did this exact thing with some fifth grade students. The results were amazing. A few students created the same pattern as above, but most created patterns that were much more interesting. And because they created themselves, they had a real “need” to know how to describe it. Some of their patterns can be seen in the images below:

Students shared how their patterns grew each had a chance to share how they saw what was changing and what was staying the same as their pattern grew.

The greatest thing about this is that instead of looking for another pattern to investigate more deeply, i.e., come up with a rule to find the number of red and yellow hexagons in any stage, we could just choose one of the students’ patterns. And we could do this for as many days as we had patterns.

This teacher wanted to use this as a springboard to plotting points on the coordinate grid, so I created this simple Amplify Patterns Task for them to complete the next day – and I added some their creative patterns into the task.

If you like, here is recording sheet that the students used. When I print these, I print two pages per sheet.

If you try this idea, I’d love to hear how it goes.

Curiosity, Wonder, Creativity, and What if…?

Check out Extending Curiosity and Wonder with “What if…?“ , published in the NCTM journal MTLT in December, 2025.

You can also download my Free Resource: 10 “What if…?” Prompts to Spark Curiosity & Discourse in Math Class.

If you’re here because of you attended a session about What if…? at a conference, and would like to go straight to the conversation, click here to jump to the comments and type away! If you got here by some other means, please read on and feel free to join in the conversation as well.

Engaging students in rich tasks engages students, allows for multiple strategies, and brings everyone to the problem-solving table. But, often, at the end of the lesson, students stop thinking about math. Math class is over, so why should they keep thinking about it? What if there was a way to keep students curious? to keep students wondering? to give them permission to be creative? and keep them thinking about mathematics?

There is… It’s What if…?

Below are some resources that can be used to help incorporate What if…? into your mathematics teaching routine. You can start with one of these rich tasks. I’ve listed some other great resources for these types of problems at the end of this post.

After students have solved a rich task, we ask students to think about the problem they just solved and about what they’re still curious about. This might fall into one of three categories:

  • A change in the context of the original problem
  • A challenge of an assumption in the original problem
  • A new idea/alternate possibility added to the original

Students should only focus on one of these categories, initially. Trying to think in multiple directions will make this a frustrating experience, rather than a creative one. Most students tend to focus on adding a new idea/alternate possibility to the original problem or change one part of the context. Below you will find some structures to use to support students as they begin to think in terms of What if…?

Helpful tips:

  • Provide students with time to think privately about one of the questions in the template.
  • Provide students with time to talk with a partner/group about their What if…? ideas.
  • When it’s time to share as a class, be patient. It may take a minute to get sharing started – especially if this is the first experience with What if…?

Other resources for rich tasks:

Share your experiences, questions, and comments about using What if…?. Keep the conversation going. Leave a comment, below.

Resources for ‘What if…?’

What if you wanted to learn more about incorporating What if…? into your mathematics teaching routine. You’re in the right place. Below, you will find some resources to help you get started. Click here to share your thoughts or ask a question.

You’ll need to start with a rich task. I’ve listed some other great resources for these types of problems at the end of this post. Below you will find some structures to use to support students as they begin to think in terms of What if…?

Helpful tips:

  • Provide students with time to think privately about one of the questions in the template.
  • Provide students with time to talk with a partner/group about their What if…? ideas.
  • When it’s time to share as a class, be patient. It may take a minute to get sharing started – especially if this is the first experience with What if…?

Other resources for rich tasks:

Share your experiences, questions, and comments about using What if…?. Keep the conversation going.

The Slinky Task – FAQs

The Slinky Task was dropped a couple of weeks ago and I’ve spoken to a number of people who had a lot of questions about it, so I’ve added some interesting facts (FAQs) about this task below. Enjoy.

  • How did you come up with this idea for the task? I was at a nearby school last fall working with one of the teachers there and a friend of mine, who also works there, mentioned that some students were eating lunch in her room and one of the boys was playing with a slinky that she had. He actually said, I wonder how far this slinky will stretch. She said she immediately thought of me. And then I showed up the next day. It did take me a couple of months to figure out how I wanted to do Act 1.
  • How long did it take to straighten the slinky? The initial straightening (what you see in the beginning of the Act 3 video) took about 45 minutes and the slinky still had a lot of fairly sharp bends in it (see below) . It also gave me several blisters on my thumbs and a couple of fingers. Lesson learned: Wear gloves.
img_2651
  • So, how long to straighten the slinky after that? It took about 18 hours. I worked at it for about 2 hours a day, when I could. There was a huge learning curve involved. I made several mistakes that added to the time needed to get this done. It was tedious, but with music playing in the background, it was fine.
  • What tools did you use to get the slinky straightened? As I mentioned, there was a huge learning curve. Some of the suggestions I got from Google searches actually prolonged the work, so I eventually just clamped the slinky in a vise and used pliers and vise-grips to bend it a couple of inches at a time.
img_2652
  • Any surprises? Yes. The slinky actually snapped three times as I straightened it. This was not good, but I did come up with a way to hide this in the Act 3 video. I clamped the slinky to the measuring tape using vise grips in multiple places to keep it aligned. Those vise grips are strategically placed to hold (and hide) where the slinky broke.
  • Did anything else not work out the way you thought or hoped? Yes. After I straightened the slinky, I asked my son to help me get it measured. We got it clamped and everything was set, but the measurement wasn’t even close – I think it was about 2 feet off from what I had computed the length should be. I kept thinking that I didn’t straighten the slinky enough, so I kept trying to get more of the bends out. I wasn’t going to even share this task – even after all the time I spent on it – because the numbers just weren’t even close to the real world measurements. Then it occurred to me that maybe my reasoning and computation was wrong. I was so close to this. Maybe I wasn’t seeing something that I needed to see. So, I sent the unfinished task and my mathematical thinking to some math friends and colleagues. I thought some fresh eyes (and minds) might be able to see what I couldn’t. The next morning, I got an email back from @KCwetna, with just the right amount of wondering that helped me reason through my own thinking to find the mistake. And, this mistake she provided me with is now included as a part of the task. It introduces another level of thought for students as they engage in the task – all based on the wonders she shared. Brilliant! So, essentially, my mistake ended up making this whole task better with some help from my friends.
  • What’s next? When the idea for this task was shared with me, I initially thought about doing this for slinky jr and the giant slinky. As of right now, those are on the back burner, but would be great sequels to explore and they are suggested at the end of The Slinky Task. I do have the slinkies for these tasks, but haven’t started the sequels yet.

Hubbub

A partner game for developing student flexibility with number combinations to five and ten (or even 20) that has connections to the first Thanksgiving.

If you teach Kindergarten, this post (and game) is for you! Ok, 1st Grade teachers can use this too, in the beginning of the year, or with more ten frames.

I can’t believe I haven’t written a post about this. I learned about Hubbub about 17 years ago, when I was just starting as a math coach in my district. I heard in mentioned in a video that some Kindergarten students were watching to learn about the first Thanksgiving. Apparently, children played Hubbub at the first Thanksgiving. I jotted down the rules and added a double ten-frame as a score board.

Materials needed:

  • 5 two-color counters per pair of students
  • Double ten-frames (one per pair) (one per student, if working on composing numbers to 20)
  • Cubes (10 per student) (20 per student if each child has a double ten-frame)

Rules:

  1. The object of the game is to score 10 points (or 20 points if you use a double ten-frame for each student).
  2. Players alternate shaking and spilling the 5 two-color counters. A player continues their turn (shaking and spilling counters to earn points) until they shake and spill 4 counters of one color and 1 counter of another color.
  3. Points are scored according to how the two-color counters land (see scoring sheet linked below):
    • If the counters show 3 of one color and 2 of another color, the player earns 1 point.
    • If all five counters show the same color, the player earns 2 points.
    • If the counters show 4 of one color and 1 of another color, the counters get passed to your partner.
  4. Players keep score by placing cubes on a ten-frame for each point earned. Players never lose the points they earn.

Teaching the Game

This is a partner game. I always model this game with the teacher on the carpet and we talk about the rules of the game, how to toss the counters, and good sportsmanship.

I sometimes use same-different images as an opener to this game. The slides for these can be found below, along with my lesson notes (standards included).

I introduce the two-color counters and then I shake them in my hands and gently drop them on the carpet. The counters will show one of six combinations:

As you can see, the possibilities are grouped by combinations to five (3 yellow and 2 red or 3 red and 2 yellow, etc.)

These combinations have scores/ consequences associated with them.

As we model the game, each time a new combination shows up, we discuss what happens. The scores and consequences for each pair of combinations is shown below.

A player’s turn only ends if they roll 4 of one color and 1 of another color. This is important, because most games we play with students, players each take one turn after each roll. We continue to model playing the game discussing good sportsmanship and asking questions, like “How many more do I need to get to 10?” “How many points do you have?” “Who has more points?”.

When someone wins – almost always not me – we ask the students if they are ready to play. The teachers pair the students with their partners, and we give them their materials. Students then begin to play and we monitor, asking students questions about combinations to five and 10 as they play. Students get very excited and, since we model it, we often hear students who do not “win” say, “That was a good game, would you like to play again?”

Hubbub Materials

If you use this game, please share your experiences. I’d love to hear about your students’ experiences and how you may have changed the game to suit your students’ needs.

Full disclosure here – I never won this game (17 years of playing) until last week when I went to my wife’s school to teach this to each of the Kindergarten classes before Thanksgiving break. They took a picture because it was the first time I ever won!

What is Mathematics?

In my previous post, I shared my personal experience with playing with mathematics as a child. That reflection prompted some digging about when we as humans begin to think mathematically, and I found something fascinating:

Experimental research shows that infants as young as 6 months have the ability to:

  • recognize the approximate difference between two numbers
  • keep precise track of small numbers, and
  • do simple subtraction and addition problems.

And when babies are mathematizing like this, they activate the same parts of the brain that are associated with mathematical thinking and reasoning in adults (I told you this was fascinating). So, before we can even speak, we have the ability to quantify. Our inherent curiosity and ability to think and even reason mathematically is on display here.

The idea that teaching and learning mathematics isn’t just computation, that it involves sense-making through reasoning, is research-based and builds on the natural curiosity, and the mathematical ideas and abilities we possess at a very young age. 

Mathematics helps us make sense of and explain the world around us. It is the science that deals with the logic of shape, quantity, and patterns. Mathematics is a subject created based on the need to solve problems and, in my opinion, should be taught that way. It’s a beautiful, creative, and fascinating subject with applications in every field: teaching, economics, engineering, biology, chemistry, physics, entertainment, shipping, food service, geography, geology, technology, real-estate, and politics, to name just a few.

The common myth is that mathematics = computation. While computation is embedded within mathematics, it is really a very small part of a greater whole. The strong, flexible core of mathematics is all about reasoning and sense-making. The “computation part” of mathematics can be taught with this strong, flexible core in order to make sure that the computation students learn makes sense so that it can be applied to solve problems in the real world. 

Ultimately, mathematics is about sense-making. The mathematics we use today to solve problems was developed by creative thinkers who asked questions like. “What if…?” “Maybe we could try…?” and “I wonder what would happen…?”  This creative thinking is still happening today to solve problems like coastal erosion from tropical storms. You, your students, or your children can be one of these creative thinkers that uses mathematics and mathematical modeling to solve some of the world’s biggest problems. Let’s keep students thinking about mathematics as much as possible!

Learning to Play with Math

This is the first in a series of posts about learning to think like a mathematician. This is my first memory of playing with numbers, questioning my own thinking, and making sense of new ideas. As my brother Peter said, when I shot the video clip below, “and this is how it all began!”

When I was around 5 years, my older sister showed me an adding machine that was in my grandmother’s closet. It was large (to a fiver year old) and very heavy. It was completely mechanical and had 81 numbered hexagonal keys – 9 rows of 9 keys (see image, above). Each column of keys was numbered with the digits 1-9. Pressing a digit in a given column would display that digit in a corresponding window along the bottom row. There was also a lever on the right side that could be pulled to reset all of the column windows to zero.

My first experience with this incredible machine, thanks to my sister, Susie, was to press the “1” in the lower right corner (the ones place) continuously.  The tenth press felt different and the display at the bottom returned to zero, but the place just to the left turned to display a “1.” Pressing the same “1” button felt the same again for 9 presses, then on the tenth press it felt different again, and again the window below turned to a “0.” And the window to the left turned to a 2. I could make the second window count to ten, too? By only pressing the one button? This was fascinating! After a long time, and lots of presses of the same key, when all of the bottom windows displayed nines, I would press that same “1” button and it would “feel a little different” and something amazing and extremely satisfying happened. Watch the video below to experience it for yourself: 

All of the 9s would alternately flip to 0s, like a row of dominoes flipping over. It was a sensory explosion! I could hear the dials flipping. I could see them flipping, and I could feel when it should happen. It seemed magical at the time. 

I (we) were told not to play with this because it was an antique and I (we) might break it. I was a pretty good rule follower, but the allure of this machine was too much. I would sneak into my grandmother’s closet and lug that heavy machine out to play with it – a lot – even when I was a little older, just to get to see, feel, and hear that domino-like effect of numbers flipping over. Even now, it makes me smile to think about it. 

I would spend a lot of time pressing that “1” key until it got to 9, then press it one more time and a “1” would pop up in the place to the left and a zero was in the spot below the key I was pressing. It never got old. I kept pressing that key, not just to see what I had come to know would happen, but to figure it out. I began to make predictions and ask myself questions, like when the display read “249” one more press and the digit in the place to the left of my key changed to a 5. “I bet it changes to a six next time.” When it did, I was hooked. This continued and every time the next column got to a 9, I’d quickly press my “1” key one more time to get it to click over. What I noticed, though, is that it took a lot longer to get each row to 9, but that wasn’t enough. I wanted to know how much longer. I kept going because the more nines I had, the cooler it sounded when they all flipped over! Eventually, I figured out that it took ten presses of that “1” key to make the next column change, and that column had to change 10 times to make the next column change. I discovered a pattern of tens. 

What I didn’t realize, initially, was that I didn’t have to keep adding ones to get the full row of nines (I was still only about 5). I could just press each of the nine “9” keys 1 time each, then add 1 more by pressing the far right “1” key. Then, satisfaction and amazement came much sooner! Once I figured this out, it was a much quicker experience but, frankly, a little less satisfying. Watch the video below to see and hear what I loved so much.

This was one of my first experiences with playing with mathematics and the effects it can have in the sense-making and building deep understandings of mathematical ideas. I believe it is one of my earliest mathematics learning experiences, and I believe it had a huge impact in how I think, mathematically – especially about place-value.

My hope is that this and some future posts may cause you to reflect on some of your own, similar, experiences. If so, please share. I’d love to hear your experiences. Stories like this, I think, have the potential to bring to light just how beautiful mathematics can be and the connections that can be made by studying this amazing subject! 

Side note: I really wanted to take this amazing contraption apart to see how it worked. My parents are thankful that I never did, but I still wonder what the inner workings of this adding machine look like. Unfortunately, I never got to find out, but I am still very curious. Perhaps there’s a video out there that I can watch so I don’t ruin this antique with my tinkering.