https://www.cognitivecardgames.com/

By Michael Leppert

To begin this article, it is important to remind the reader of recent findings about the value of learning with games and/or play, vs. rote memorization. Playing bean-bag catch regularly, while reciting multiplication tables is an effective, powerful mode to enable a kinesthetic learner to master the tables for life. Cognitive Card Games recognizes these findings into the brain’s nature vis a vis learning.

Cognitive Card Games is an online platform and physical card deck, dedicated to educational card games built around math and cognitive skill development — primarily through a flagship patented product called X-Squared. According to the site, the company’s mission is to “Advance the state of mathematics comprehension throughout the world” by creating tools that help players think more deeply about math, rather than simply memorizing facts.

At its core, the site aims to blend education and game design, transforming traditional math learning into interactive card-based challenges that can be played individually or cooperatively. It also suggests future expansions beyond math.

Company Mission & Educational Philosophy

The Cognitive Card Games mission page is clear about its educational intent:

• • The founders want to “transform how the fundamentals of mathematics are learned, understood and practiced.”
  • They emphasize learning by doing, asking questions, and thinking critically, reflecting quotes attributed to Einstein and Feynman about learning beyond rote memorization.

Mathematics is often taught as a set of rules and procedures rather than as a playful problem-solving process. If a card game can build intuition for order of operations, arithmetic fluency, and algebraic thinking, that seems to offer tangible benefits for learners who struggle with abstract instruction.

Flagship Product: X-Squared Math Card Deck (patent 12525142)

Core Concept

At the heart of the website is X-Squared, billed as a math brain game built around PEMDAS/order of operations (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). You use Integer (whole number) cards, Operation cards, and other functional cards to craft equations that match a given Solution card.

X-Squared is the first publicly available Order of Operations game that includes two critical elements: 1) Complete and 2) Simple.  These attributes make this product universally applicable.

It is complete in that it physically contains all elements of PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition and Subtraction.  It enables game play that allows players to fully express their understanding of these concepts.  There is another game on the market, called PEMDice, that includes each of the elements mentioned, but it is contrived.  The play is limited.

It is simple in that the instructions can be easily understood and followed.  There is one other product on the market that is structurally complete, Equations.  That is a brilliant game that was created in 1962 and is still used in math competitions today for gifted students.  While complete, Equations is not simple.  There are numerous complex rules that constrain its accessibility.

In more detail:

• • A basic game challenge involves using specific integer and operator cards to create an equation equal to a target number.
  • Parentheses and exponent cards can optionally be used to expand possibilities.
  • The deck supports unlimited game possibilities, including formats akin to puzzle challenges or even “game show”-style play.

This format resembles classic educational card games like Krypto — a math card game where players create valid expressions to reach a target number — but Cognitive has more structured rules and the Cognitive custom cards, rather than standard playing cards.

Deck Contents & Structure

X-Squared decks contain a large set of cards:

• • Integer cards (whole numbers)
  • Operation cards (+, −, ×, ÷)
  • Parentheses cards
  • Exponents and solution cards

A companion product (X-Cubed), to be released in 2027, includes additional mathematical functions like factorials.

This structure gives the game a modular and customizable quality — users can adapt difficulty or introduce optional rules, adding to its long-range value.

Gameplay & App Integrations

The site also offers digital versions and tools to support gameplay:

• • A Daily Challenge section presents math challenges based on the X-Squared logic, which players can solve against archived problems.
  • A digital App called X-Squared Challenge aims to let players practice these puzzles on web or mobile. A Beta version is currently available to the public for free at:

https://x-squared-challenge-basic-classic.base44.app

Games for All Ages?

The site states that the games are “for all ages” — from young learners, getting comfortable with arithmetic, to adults looking for brain training.

This wide range is appealing. Math puzzles can be intellectually stimulating and fun, especially when presented clearly. It is up to the parent or teacher to determine the level of math familiarity of his/her child.

Pricing & Purchasing

Cognitive Card Games currently sells two products:

• • X-Squared Math Card Deck — ~$18.18
  • X-Squared Challenge Digital App— ~$11.11
    App available for download through the App Store and Google Play on April 1, 2026.

These prices are competitive with many specialty card games and educational tools.

Similar products associated with the X-Cubed Math Card Deck will be available in 2027.

Strengths

1. Educational Focus

There’s real value in games that encourage conceptual understanding over memorization — especially in subjects like math, where many students struggle with abstraction. X-Squared’s mechanics, building expressions to match targets, likely do reinforce operational thinking and fluency with arithmetic concepts.

Card games and puzzles have been linked to cognitive benefits such as improved working memory, pattern recognition, and executive functioning.

2. Flexible Gameplay

Because the deck supports unlimited possibilities and multiple game modes (basic play, challenges, game show style), the user experience potentially remains fresh longer than with rigid board or puzzle games.

3. Cross-Platform Support

Offering both physical decks and digital challenges expands accessibility for different types of learners. Physical decks are appropriate in a classroom setting, apps more for individual practice.

3. Broad Accessibility

Claiming “all ages” suggests broad accessibility, but math card games that require PEMDAS mastery may be challenging for younger children unless significant scaffolding is provided. The product is accessible to young learners simply through its content. While the embedded structure enables a full exploration of PEMDAS, it requires no knowledge of PEMDAS to use.  Children between ages 3-7 can simply lay out the cards in front of them and manipulate them in a free form interaction.  For instance, they can count using the integer cards 0-9. They can practice basic operations by putting an Addition card between two integer cards. The cards themselves may be used in any manner imaginable. This has been demonstrated already, by setting an X-Squared deck in front of a group a young children. They responded by naturally placing cards in various configurations to practice counting and basic math operations.

Games with Cognitive Value

Card games have long been used to build cognitive skills:

• • Classic memory games help focus and recall.
  • Number puzzles like Krypto help arithmetic fluency.
  • Strategic card games stimulate planning and decision-making.

In educational contexts, card decks are often used to reduce anxiety around learning and make abstract concepts more tangible.

Who Cognitive Card Games Is Best For

This product is best suited for:

• • Math enthusiasts who enjoy puzzles and numerical challenges.
  • Educators looking for creative classroom tools.
  • Parents seeking a tactile, non-screen alternative to apps for practicing arithmetic.
  • Older students or adults who want playful drill on order of operations.

Finally

Cognitive Card Games — particularly through X-Squared — presents an innovative and appealing concept: A card deck that turns mathematical operations into interactive challenges. The research-supported notion that engaging with puzzles builds cognitive skills aligns with broader findings on the benefits of card games.

Overall, Cognitive Card Games is a clever concept with real potential, especially for those who enjoy math and card-based challenges. Considering its value and price, this is a game worth keeping and using for many years at home or in the classroom! Ω

https://umathx.com/

“Students come to see mistakes not as failures, but as essential steps in learning.”

Assessed by Homeschool Magazine

UMathX, created by Neufeld Learning Systems, is a powerful mathematics-learning environment that spans K through Algebra 1. It emphasizes deep conceptual understanding, rather than rote procedures, offering more than 3,500 lessons with a blended model that includes on-computer, off-computer, and hands-on work.

Within this larger math-learning system is Code ‘n Play, an introductory coding-to-learn component designed to combine mathematical concepts with playful, tangible activities. On the UMathX website, Code ‘n Play is described as an experience where “kids scan coding cards to program a KaiBot¹ for screen-free fun,” then test, explore, and eventually move to a virtual coding environment.

This fusion of robotics, coding, and mathematics makes Code ‘n Play a particularly promising tool for building not only arithmetic and algebraic understanding, but also computational thinking. Below we review its structure, strengths, potential challenges, and its broader educational importance.

Structure and Design

Code ‘n Play unfolds in three stages (as indicated on the UMathX site). UMathX While UMathX doesn’t fully detail each stage on the homepage, the design is clearly scaffolded:

1. Screen-free coding with physical cards and a KaiBot robot: Students manipulate physical “coding cards” which, when scanned, guide the KaiBot. This tactile interaction is more concrete than abstract code and supports learners who benefit from hands-on experience.

2. Predict, test, and explore: After programming the robot, students make predictions, test their code, and observe what happens. This cycle encourages hypothesis testing, debugging, and iterative thinking – problem solving through stages of observation and refinement.

3. Shift to virtual coding: Once students are familiar with the logic through physical interactions, they progress to coding on the computer screen in a virtual environment, retaining the same step-by-step coding logic.

The design aligns with UMathX’s broader pedagogical philosophy: Building from concrete to pictorial to abstract, giving students multiple representations, and encouraging “productive struggle” to deepen understanding.

There is also an associated “First Steps in… Coding to Learn” eBook, to further ground coding in mathematical contexts. The eBook provides structured examples and student-led exercises, making it easier for learners to navigate coding in a math-rich way.

Strengths and Educational Benefits

1. Hands-On Learning
The initial use of coding cards and the KaiBot, offers profound benefits, especially for younger learners. Physical artifacts (cards, robots) make abstract logic tangible. This multisensory approach, touching, scanning, seeing results, supports learners who struggle with traditional abstractions.

2. Scaffolding and Progression
By starting with concrete manipulation and gradually moving to virtual environments, Code ‘n Play respects the developmental journey of students. As kids gain confidence, they can transition to more abstract representations without losing understanding.

3. Integration of Math and Coding
Rather than teaching coding in isolation, Code ‘n Play weaves in mathematical thinking. Programming the robot isn’t just about control logic; it’s deeply tied to math concepts. That helps students see coding not as a separate domain but as a way to explore and reinforce mathematical ideas.

4. Promoting Growth Mindset and Productive Struggle
UMathX is built around encouraging students to wrestle with ideas, make mistakes, and reflect. Code ‘n Play contributes to that by requiring prediction, testing, debugging, and iteration. When students try a sequence of cards, see what happens, and revise, they are effectively practicing the fundamental mindset of “learning from failure.”

5. Accessible for Varied Learning Environments
UMathX is designed to be used in many contexts: Whole-class, small groups, individual learning, virtual or in-class. Code ‘n Play’s flexibility fits well within this model. Whether in a school with robotics resources or at home with minimal devices, students can engage with the coding curriculum.

6. Teacher Professional Development
Neufeld Learning Systems supports educators with training. Their professional learning plan teaches a “teach, don’t tell” philosophy, focusing on unpacking concepts, scaffolding appropriately, and using a multisensory, three-part lesson model. This ensures that teachers can confidently facilitate coding-math integrations, even if they lack prior coding experience.

Potential Challenges and Considerations

While Code ‘n Play is rich with potential, there are some challenges and limitations to keep in mind:

1. Resource Requirements
To use the physical card-and-KaiBot phase, schools or families will need to obtain the KaiBot robot and coding cards. This may have a cost barrier or logistical hurdle for some.

2. Learning Curve for Educators
Although UMathX offers professional development, implementing a blended math-coding curriculum still requires time, planning, and comfort with technology.

3. Transition to Virtual Coding
The shift from physical to virtual coding must be carefully managed. Some students may resist moving to the screen if they prefer hands-on activities, or conversely, may struggle with the more abstract virtual interface.

Importance for Students Today

The importance of Code ‘n Play in today’s educational landscape cannot be overstated. Here are several reasons why this kind of coding-integrated math learning is particularly valuable:

1. Bridging Math & Computational Thinking
In a world where computational literacy is increasingly important, teaching students to code through mathematics builds both domains in tandem. This approach demystifies code and shows its relevance to numeracy, patterns, logic, and problem solving.

2. Aligning with 21st-Century Skills
Skills like resilience, iterative design, debugging, and logical reasoning are central to modern educational and work environments. Code ‘n Play gives students early, structured exposure to these practices within a safe, scaffolded space.

3. Supporting Diverse Learners
The blended model, physical cards, robot, virtual interface, makes coding accessible to different learning styles. Kinesthetic learners, visual learners, and those who struggle with abstract thinking all benefit from the multi-representational nature of the program.

5. Encouraging Growth Mindset
By requiring prediction, failure, and revision, Code ‘n Play reflects one of UMathX’s core values: Understanding through struggle. Students come to see mistakes not as failures, but as essential steps in learning.

6. Making Math Engaging and Relevant
For many students, math can feel disconnected from real-world coding or robotics. Code ‘n Play bridges that gap, showing how math is not just theoretical but instrumental in building behaviors, controlling robots, and programming systems.

7. Preparing for Future Learning
Early exposure to coding in a mathematical context can lower barriers to more advanced computer science education later on. It builds both confidence and a conceptual foundation, helping students feel prepared for more complex STEM learning.

Code ‘n Play within the UMathX system is a thoughtfully-designed, rich resource that brings together mathematics and introductory programming in a highly accessible way. Its multi-stage structure, from scanning physical coding cards to programming a robot, to transitioning into virtual environments, aligns beautifully with UMathX’s philosophy of moving from concrete to abstract understanding. Through playful experimentation, prediction, testing, and reflection, students not only learn math concepts more deeply but also develop foundational computational thinking skills.

Moreover, the integration of professional learning for teachers ensures that the program doesn’t just benefit students: Educators are supported in learning how to guide, scaffold, and facilitate this blended coding-and-math pedagogy. While there are practical considerations, such as acquiring hardware like the KaiBot ($210) or ensuring smooth transitions between physical and digital phases—the potential gains in engagement, conceptual understanding, and 21st-century skills are significant.

In today’s education environment, where both mathematical literacy and coding ability are increasingly valuable, Code ‘n Play represents an important bridge. It helps students understand why math works the way it does, rather than just remember how to perform procedures. It gives them a playful, hands-on way to explore logic, sequences, and abstraction. And it sets them up with resilience, curiosity, and a growth mindset—all crucial for success in modern STEM learning and beyond. Ω

• A KaiBot is a hybrid coding robot that teaches STEM, various academics and coding to children ages 5 to 15.

By David Chandler, Math Without Borders

pi is an important number in mathematics. It is also a difficult concept for students when it is first introduced.

pi = 3.14… approximately, but that is not quite right.  pi = 3.14159… approximately, but that is not quite right either.  pi = 22/7 but that is an approximation too.  pi has been computed out to millions of digits, but the result is still an approximation.  pi is mysterious to most students at first because it is the first irrational number they have encountered in school.

To be an irrational number means it cannot be written down as a fraction or as a decimal number with a finite number of digits, or even as an infinitely repeating decimal number like 0.33333….  pi can theoretically be expressed, in decimal form, as an infinite, non-repeating decimal, but such a number cannot be written down or even fully computed. How can we use a number we cannot write down or even fully compute? This is a lot for a middle school student to try to comprehend.

pi has a very simple, clear, easy-to-comprehend definition:  pi is how many times farther it is to go around a circle than to go straight across it. In more mathematical terminology, it is the ratio of the circumference to the diameter of any circle. What is that ratio? It doesn’t work out exact in digits, but simple measurement shows that it is approximately 3.14.

Position a bicycle wheel with the tire stem at the bottom. Mark the starting point with chalk on the pavement.  Then roll the wheel until the tire stem comes to the bottom again and mark the ending point. That is a measurement of the circumference of the wheel. Measure the diameter of the wheel and divide. The result should come out very close to 3.14. Do the same with a soda can, a jar lid, or a circle of any other size. The ratio of circumference to diameter will always come out the same. That number is the same for any circle of any size.  We may not know all the digits of that number, but it is a definite number so we give it a name. It is called Pi, the Greek equivalent for the letter “p”.

I have created a series of three videos explaining pi with animated drawings, posted on YouTube with a playlist for all three, posted on my web site, www.mathwithoutborders.com. Go to the site, then go to the tab labeled Math Explorations, then look down the page to the Pi Movie.

The first video in the series presents the most important fundamental concept for understanding pi, the idea that there is a fixed ratio of circumference to diameter for every circle, no matter the size.  I do this by starting with a square, and showing that the distance around any square is exactly 4 times the distance across through the center.  I call this “the magic number” for all squares. If you trace a circle in each square, the circle cuts the corners off, so the “magic number” for circles must be a little less than 4.

I then show that the distance around a regular hexagon is always 3 times the distance across, measuring from opposite corners through the center. Therefore the “magic number” for regular hexagons is exactly 3. A circle traced around the outside of a hexagon would make it bulge outward a bit, so the “magic number” for a circle must be a little greater than 3.

The conclusion of the first video is that there is a fixed ratio of circumference to diameter, the “magic number,” for any circle, and that that number is somewhere between 3 and 4.  Even though we have only found one digit of the value of  pi, we have illustrated the meaning of pi.

Archimedes did something like this in the 3rd century, B.C. He started with two hexagons, one inscribed inside a circle and one circumscribed outside a circle. He then found a way to compute the new perimeter when the number of sides is doubled. From the perimeter of a 6-sided figure he was able to deduce the perimeter of a 12-sided figure and hence a closer approximation of pi. Applying the same method repeatedly, he was able to double the sides repeatedly, finding the perimeters of 24, then 48, then 96-sided figures. Based on his computations for 96-sided figures, he estimated the value of pi to a value equivalent to 3.14 in modern decimal notation.

Keep in mind that Archimedes did not have modern mathematical tools or even modern mathematical notation to work with. He did not even have our system of base 10 decimal fractions. He carried out all of his calculations by hand, including finding square roots, using ordinary fractions.

If I were presenting pi to a middle-school class, I would stop after the first video in the playlist and tell the story of Archimedes along the lines of what I have done above. By clarifying the meaning of pi and giving an indication of how one could approach the problem of finding its value by successive approximation, pi is demystified. That is really the goal at this stage.  Once pi is demystified, students can continue with the usual practice, finding circumferences from diameters, and vice versa.

The second and third videos in my playlist are more appropriate for a high school audience. They require familiarity with the Pythagorean Theorem, which also assumes familiarity with finding square roots of numbers.

The second and third videos illustrate how a spreadsheet can be used like a computer program to run repeated calculations. The details of the method shown here are somewhat simpler than the way Archimedes originally reasoned it out, but the general idea is the same. The method boils down to repeated application of the Pythagorean Theorem. Whereas Archimedes, working by hand was able to find the lengths of the sides of a 96-sided figure, we are able to set up a repetitive process on a computer spreadsheet which in just a few seconds can be extended to polygons with thousands of sides. Archimedes was able to derive pi to an accuracy equivalent to 3.14. We can apply our method, using a spreadsheet program on a computer, to get a value accurate to 15 digits!  This is far more than enough accuracy for most practical applications.  For most purposes pi is only needed to a few digits of accuracy.

We are not learning to calculate pi because we will need to know how to compute pi in practical applications.  Scientific calculators usually have a key that gives pi accurate to 8 or 10 digits with a single click. We are seeing how pi could be calculated to become familiar with this special number, in particular, and with the new category of irrational numbers in general. This “magic number” is not magic at all. It is a number we do not know how to compute exactly, so we compute it approximately, but the degree of approximation can be made as close as you like.

The most important things to be learned from this lesson are:

1. that the ratio of the circumference to the diameter of any circle is the same,
2. we cannot compute that ratio exactly, but we can compute it very accurately by successive approximation,
3. since we cannot write the complete number in digits, we usually refer to the number by its name, pi, there are methods, understandable even to a high school student, to approximate the value of pi as closely as you like. Ω

By Kathleen Cotter Lawler, https://rightstartmath.com/

When we think of playing games, we think of good times, laughter, and family fun. When we think of math, rarely do these same thoughts come to mind. What a shame! Math should be good times, laughter, and family fun. So how can we make that happen? Read on and let me share some ideas.

First, if we are going to blend math and card games, we need to make sure the games are not glorified flashcards or disguised worksheets. The math games need to entice laughter and joy.

Then, the games need to help the child learn, not just review. If there isn’t a learning component involved, a child who doesn’t know their facts can’t play the game. Yes, those who know the facts will review and solidify their skills. But we want to make sure both elements, learning and review, are involved in the game.

So let’s look at two math card games; one for addition facts for 10 and another for comparing and understanding fractions. These are games that are found in Math Card Games, written by Dr. Joan A. Cotter.

The first game, Go to the Dump, is similar to the popular children’s Go Fish game. Instead of having the pairs be matching numbers, we’re going to have the pairs be two cards that make 10. So 8 and 2 will be a pair, 4 and 6 will be a pair, and so on.

To help a young child learn or verify the facts of ten, give them 10 counters; 5 of one color and 5 of another color. This grouping in fives helps with recognizing, or subitizing, the quantities. Line up the counters as shown here.

XXXXX OOOOO

Then, when a card, let’s say 3, is being considered for its match, slide that number of counters to the left. See how the quantity of 7, the match for 3, is quickly recognizable?

XXX XXXOOOO

Here’s what 9 would look like. This helps the child learn the facts of 10!

XXXXX OOOO O

Allow the child to use these counters as long as needed. After repeated use, the child will begin to visualize the quantities and not need the actual counters.

So let’s play the game! It’s best if three to four people play, although more or less can certainly play. Use cards with the numbers 1 through 9, six of each card. Each player starts with five cards. Place the remaining cards face down in the center. This is the “dump.”

Have the players check over their cards for pairs that equal ten. Use the counters to help determine what is a match. If any pairs are found, place those two cards face up, one card on each of two stacks in front of the player. This allows pairs to be easily verified and, more importantly, this allows the child to have a visual image of two numbers being a pair to make 10. Additionally, it makes shuffling unnecessary for the next game.

The first player asks the player to his left for a number that he needs to make 10. Let’s say he has a 4 in hand. He would ask “Do you have a 6?” If the second player has the card, she must give it to him. The first player gets another turn. If she does not have the requested card, she says, “Go to the dump!” The first player picks up the top card from the dump, the stack in the center. Even if the card picked up pairs with one in his hand, his turn is over.

The second player takes her turn by asking the player on her left and so on. If a player runs out of cards, he takes five more cards, but his turn is ended. When the dump is gone, players can ask any player for a card. At the end of the game, all the cards will be paired.

There is a variation of this game available for your iPhone, iPad, or Android. Search the appropriate store for “Go to Ten,” download, and enjoy!

Now, let’s look at a fraction game, Fraction War. Start by drawing a linear fraction chart shown here and let the child use it as they play the game. This provides assistance for those just learning and helps everyone visualize fractions and understand the relationship between one, halves, fourths, and eighths. It also helps children to learn to read rulers.

To play this two-person game, you will need the following cards:

3 each of 3/4, 3/8, 5/8, 7/8
5 each of 1, 1/4
4 of 1/8
8 of 1/2
With only 34 cards, this game goes relatively fast.

Remember how to play the traditional war game? Each person lays down two cards and whoever has the higher card takes both cards. You try to capture all the cards from your opponent.

Shuffle the deck then split the cards evenly between the two players. Keeping the cards face down, each player takes the top card from her stack and lays it face up in the middle of the table. Use the fraction chart to help figure out whose card is greater, and that person takes both cards. Have the children alternate deciding whose card is higher so that you don’t get one child drifting along on the coattails of the other.

When the same cards are played, it’s a “war!” Both players lay one card face down, then play a card face up. Looking at the two new cards, the player who has the higher card takes all six cards. This is a game that children will play for hours — to the point that you will cheat to lose to get out of there!

This game is also available for your iPhone, iPad, or Android. Search for “Fraction War.” Then, if you would like additional math card games that teach as well as review, create good times, laughter, and family fun, check out the Math Card Games kit from RightStart™ Math —

https://rightstartmath.com/