The Hungarian Ten Frame: How Pattern Recognition and Manipulatives Transform Math Learning
Table of Contents
The Role of Manipulatives in Math Learning
One of the most fascinating takeaways from our conversation was Hungary’s unique approach to early math education. Unlike the U.S. or the U.K., Hungarian children do not begin formal schooling until they are nearly seven years old. Before that, their preschool experience is carefully structured to build foundational skills—without ever introducing symbols like numbers or letters.
Dr. Back explains that Hungarian preschools immerse children in mathematical thinking through spoken number names, counting games, number-based rhymes, and play-based learning. Instead of writing their names, children use a character stamp on their work, focusing instead on fine and gross motor development, color recognition, and set identification.
The Story of Six: A Deep Dive into Number Sense
Dr. Back shared an insightful example from her research in Hungary—a first-grade lesson on the number six. Unlike traditional math instruction, where numbers are often introduced quickly with minimal exploration, Hungarian educators take a much deeper approach.
She observed a highly skilled teacher guiding students through an entire week dedicated to understanding each number from zero to five before progressing to six. In one lesson alone, students were exposed to ten different representations of the number six. This allowed them to:
✔️ Recognize the “six-ness” of six in various contexts
✔️ Generalize their understanding across multiple representations
✔️ Build on prior knowledge of five and extend their thinking
By using familiar manipulatives and hands-on learning, students weren’t just memorizing numbers—they were deeply internalizing their meanings and relationships.
The Hungarian Ten Frame: A Powerful Visual Tool
One of the standout manipulatives used in Hungarian math education is the Hungarian Ten Frame, also called a “number picture” in Hungary. This differs from the traditional Ten frame (which consists of five boxes on top and five on the bottom) by using a domino-style structure with open slots, allowing students to physically place counters into each space.
Every child has access to these double-five frames, often using double-sided counters (blue on one side, red on the other). During a lesson on six, for example, students would fill five slots and then see the extra one, reinforcing that six is one more than five. By flipping counters, they could visually grasp multiple compositions of six:
✔️ 5 + 1
✔️ 4 + 2
✔️ 3 + 3
✔️ 2 + 4
✔️ 1 + 5
This structured approach allows students to recognize number relationships more intuitively. Unlike traditional Ten Frames, which often lead students to count one by one, the Hungarian Ten Frame promotes pattern recognition and subitizing, making it easier for students to grasp numerical structures at a glance.
Cuisenaire Rods: Building Proportional Reasoning
Another powerful tool Dr. Back observed in Hungary was Cuisenaire Rods—a set of proportional rods in lengths from 1 to 10 centimeters, each represented by a different color. Originally developed by Belgian educator Georges Cuisenaire and popularized by Caleb Gattegno, these rods are incredibly versatile in teaching number relationships, proportions, and algebraic thinking.
Cuisenaire rods emphasize proportionality. For example, two white rods (1 cm each) equal the length of one red rod (2 cm), while three white rods equal the length of a lime-green rod (3 cm). This structure naturally leads students to explore fractions, addition, and multiplication visually.
Dr. Back highlighted how Hungarian educators introduced Cuisenaire rods alongside other representations, ensuring students could make connections between different visual models. One of the most exciting aspects of these rods is their potential for tactile learning—advanced learners can even add fractions by feeling and comparing rod lengths behind their backs.
Hungarian Ten Frames vs. Cuisenaire Rods: When to Use Each
✔️ Cuisenaire Rods help students explore proportionality and relationships between numbers. Instead of focusing on counting, students see how different numbers relate to one another—how a six-rod can be made of two threes or three twos, reinforcing multiplicative thinking.
✔️ Hungarian Ten Frames highlight number structure and proximity to five and ten, which builds strong foundations for place value and arithmetic strategies. By consistently seeing numbers in relation to five, students develop efficient mental math strategies for addition and subtraction.
As Dr. Back points out, it’s important to help students see the connections between these different tools. A six-rod in Cuisenaire blocks represents the same number as six counters on a Hungarian Ten Frame, but each manipulative provides a different lens through which to explore number relationships.
The Importance of Generalizing and Number Lines
One common pitfall in teaching with manipulatives is over-reliance on a single tool. Heather Brand warns that students sometimes struggle to transfer or generalize their understanding beyond a specific manipulative. For example, children who only learn multiplication with Cuisenaire rods might believe that multiplication only works with rods, rather than seeing it as a universal concept.
To build true mathematical understanding, educators must encourage students to:
✔️ Work with multiple representations to generalize concepts
✔️ Explicitly discuss connections between different tools
✔️ Transition gradually from concrete manipulatives to abstract reasoning
Another critical visual tool for mathematical thinking is the number line. Unlike manipulatives that emphasize units (like counters or blocks), the number line helps students understand the continuity of numbers. Dr. Back points out that numbers are not just separate objects but part of a continuous scale, which is essential for understanding measurement, fractions, and graphing.
At Made for Math, we help students build number line understanding using bead strings—a simple yet effective way to help children see numbers as points on a continuous scale. When students construct their own number lines using counters or beads, they internalize that numbers represent quantities rather than arbitrary symbols.
The key is blending step-by-step instruction with meaningful, rich tasks that keep students excited about learning—a balance that Made for Math strives to achieve.
Want to Connect with Dr. Jenni Back?
Dr. Jenni Back has had an extensive career in mathematics education, developing resources, leading research projects, and training educators. Here are some of her key contributions:
📌 Published Books
Hooked on Mathematics: A Programme of Study Based on Free Quality Resources – Developed over a six-year project with a school in Brighton, incorporating NRICH materials and Hungarian methodologies. Purchase Here
Making Numbers: Using Manipulatives to Teach Arithmetic. Purchase Here.
Making Fractions: Practical Approaches to Fractions and Decimals Purchase Here.
📌 NRICH Project – Developed primary math resources for NRICH, a project dedicated to enriching students’ mathematical experiences. Explore her work: NRICH Website
📌 National Centre for Excellence in Teaching Mathematics (NCETM) – Led research projects on effective professional development for math teachers, including:
- RECME Project (2007-2009): Research on effective CPD in mathematics – Final Report
- Host Schools Project (2011-2013): Supporting schools in developing best practices – Evaluation Report
📌 Higher Education & Teacher Training – Delivered undergraduate and postgraduate training in mathematics education at Middlesex University (2004-2007) and Plymouth University (2009-2012).
📌 Nuffield Foundation Research – Led two projects that resulted in practical teacher handbooks:
Using Manipulatives in Arithmetic (2016): Read More
Teaching Fractions & Decimals (2023): Read MoreDr. Back’s extensive work bridges hands-on learning with research-backed strategies, making math more accessible to educators and students alike.
Other Resources Mentioned in this Video:
📌Learning Trajectories Resource for early childhood research. (Unlocking Dyscalculia episode with Clements coming soon!) Learning Trajectories
This post contains affiliate links. (We would never recommend something we don’t love and think is worth purchasing ourselves!)
💡 Is multisensory math right for you and your family?
If you are looking for professionals trained to work with kids with dyslexia, dyscalculia, ADHD, and more–you’re in the right place. See if multisensory math is right for you and your family!
MFM Authors
Jennie Miller
Marketing Assistant
is our Marketing Assistant and content creator here at Made for Math. Jennie loves being part of a company that is working to make mathematics accessible to children with dyscalculia.