Build & Learn Mathematics
Seven Operations. Six Structures.
How Much Mathematics Could That Possibly Contain?
Most elementary math curricula spend years teaching number sense: math facts, algorithms, decomposition strategies, benchmark numbers, mental-math strategies, and particular ways to move from one number to another.
Build & Learn takes a different approach.
We teach children to master seven operations, six Cuisenaire rod structures, and the rules for manipulating mathematical symbols.
Then we leave number sense to the child.
Because there are infinitely many numbers and an inexhaustible number of relationships among them. We cannot teach children every path through that landscape.
We can teach them how to navigate.
New Here?
Before you enroll, join the free Learning Well at Home community, take the NWD Orientation, try a free course, and ask questions. I want you to understand how we learn mathematics before you decide whether Build & Learn is right for your family.
What Do Children Actually Learn?
Seven Operations & Six Structures
Seven Operations
Children learn the rules for manipulating:
Addition • Subtraction • Multiplication
Division • Roots • Exponents
Fractions as Operators
When they learn the rules, they learn what can be done with them.
They can reverse them, repeat them, combine them, nest them, substitute expressions for expressions, and transform mathematical relationships while preserving what is true.
The point is not to get answers, but to enter the landscape of mathematics.
On the other side of the equal sign lies not just an answer, but a world of possibilities.
Six Structures
Children encounter mathematics by repeatedly returning to only six Cuisenaire rod structures:
Trains • Mats/Rectangles • Staircases
Crosses • Towers • L's
The structures stay familiar while the mathematics children can find within them becomes increasingly sophisticated.
A train that a young child can build and talk about can become a place to investigate addition, difference, multiplication, division, fractions, equivalence, unknowns, substitution, and algebraic relationships.
The structures don't change. What the child can see and express has.
What If We Left Number Sense to the Child?
You can tell me that ten comes after nine. That it is 8 + 2. Or 2 × 5. Half of 20. √100. One ten in our base-ten place-value system.
All of those are true.
But every one tells us about a relationship in which ten participates.
There are infinitely many such relationships.
Most number-sense curricula select useful relationships for children to learn. They build highways, major roads, and country roads between numbers.
Build & Learn gives children the tools to build roads.
Build & Learn gives children the tools to build roads.
The numbers themselves don't express the relationships. The operations do.
Once children know how to use the operations and manipulate the symbols, they can generate relationships nobody thought to teach them.
They can go somewhere mathematically because they noticed there was somewhere to go.
"Mind the symbols and the numbers will take care of themselves."
Caleb Gattegno
What Does the Work Look Like?
Mostly Like Play
Children build.
They compare.
They play games.
They make predictions.
They try something.
They talk about what happened.
They change something and try again.
They write relationships.
They substitute.
They hunt for everything they can find in a structure.
Through Notice–Wonder–Discover, we give children
something worth attending to, something to do,
and something to talk about.
Over time, they become remarkably fluent with
mathematical structures, operations, and symbols.
This is Normal
Composition for 10: First Grade
This work looks complicated at first glance, but the mathematics underneath it is remarkably simple. The student is composing 10 again and again while using the operations and symbols he has learned to make transformations and substitutions.
In the first line, the student begins with a true statement about ten: 5 + 5. He rewrites one 5 using a fraction and its inverse relationship. He then breaks apart the other 5 into 3 + 2, expressing the 3 and 2 as fractions of 6.
In the next line, he breaks 5 apart differently, this time into 4 + 1, and rewrites 5, 4, and 1 using fractions.
Then he continues playing with those same three numbers, 5, 4, and 1, using fractions of fractions while preserving the value of the expression.
On the next line, he rewrites ten as 9 + 1, expresses 9 as a square, and then transforms the 1 into a fractional expression.
The student continues in this way down the page, breaking numbers apart, substituting equivalent expressions, and making simple transformations while preserving what is true.
The complexity comes from the number of possibilities, not from complicated procedures. Once children master the symbols and learn what they can do with them, the other side of the equal sign becomes a place to explore.
Thee of One Number
In this game, the student writes true statements for relationships in which three of one number are equivalent to another.
If a = 9 then c must = 3. One of those 3s must be ⅓ of 9.
The same relationship holds for 1 and 3, 2 and 6, 4 and 12, and so on.
Once the student sees that relationship, everything else necessarily follows. The numbers change, but the relationship remains true.
How Do They Get There
Children begin with the rods, mathematical language, and the operations. They build relationships and learn to talk about what they are doing.
The progression moves back and forth between the concrete, spoken language, and mathematical symbols:
Build and read → Build from oral language → Translate into symbols → Read symbols → Write from oral language → Translate back to rods
Students write mathematical symbols when they have something to say.
First, they need experiences worth expressing. They build, compare, combine, separate, repeat, measure, and transform. Through this work, they learn how the operations behave and what they can do with them.
Once the operations are familiar, the symbols give the student a way to express relationships they already understand. Written mathematics becomes a language for communicating what they can see and do.
And once children can use that language, they can do much more than record what they built. They can manipulate the symbols themselves, substitute one equivalent expression for another, make transformations, and follow relationships into new mathematical territory.
Everyone Begins With Cuisenaire Rod Structures
Build & Learn: Cuisenaire Rod Structures is the foundation of the entire curriculum. Everyone starts here.
Children learn the six foundational structures while developing fluency with mathematical language, operations, equivalence, unknowns, substitution, and symbolic manipulation.
Games and investigations give them repeated opportunities to use what they know in new situations.
Build & Learn: Cuisenaire Rod Structures
This course is the foundation. Everyone starts here. Even if your child is older, has used other curricula, and knows about numbers. Even if...
Children learn the six foundational structures while developing fluency with mathematical language, operations, equivalence, unknowns, substitution, and symbolic manipulation.
Games and investigations give them repeated opportunities to use what they know in new situations.
Build & Learn: Numbers 1–100
After Structures, children can continue into Numbers 1–100.
And here's where we need to be very careful with the language.
This isn't “now we develop number sense.”
This is where children take the tools they've acquired and go exploring the landscape of numbers 1–100.
That's a much more compelling description of the second course.
What You'll Receive in Build & Learn: Structures
80 Short Video Lessons
Once you take the free Orientation, you can open the course and begin without hours of preparation.
Parent Guides & Task Cards
Enough guidance for you to understand both the math and the investigation so that you can support your child while letting them discover the necessary for themselves.
Games and Investigations
The work children actually do.
Skills Checklist
A way to see what your child can do without reducing progress to pages completed.
Family Access
You receive lifetime access to the course. This includes all updates and office hours where you can ask questions.
Community Support
Ask questions, share student work, see what other families discover, and get help when you need it.
Important: Start with Notice. Wonder. Discover.
Build & Learn is part of a larger way of thinking about education: how we attend to what is in front of us, how we make sense of what we see, and how what we discover changes the way we understand the world.
Notice–Wonder–Discover is the framework that holds that work together.
The free Learning Well at Home community is a place to begin exploring it. Take the NWD Orientation. Try a free course. Look through student work. Ask questions. Join conversations with other families who are learning to see education this way.
There is plenty of room to look around before choosing a curriculum.
Choose Your Build and Learn Curriculum
Build & Learn: Structures
Everyone starts here. Everyone.
Begin with the structures, relationships, operations, and symbols that make mathematics navigable. Through play, games, transformations, and rod investigations, students explore trains, mats, staircases, and rectangles while learning addition, difference, equality, multiplication, fractions, composition, decomposition, and equivalence.
Students move from building and talking about mathematical relationships to reading, writing, and manipulating those relationships with symbols.
Build & Learn: Numbers 1 to 100
Everyone starts here. Everyone.
Begin with the structures, relationships, operations, and symbols that make mathematics navigable. Through play, games, transformations, and rod investigations, students explore trains, mats, staircases, and rectangles while learning addition, difference, equality, multiplication, fractions, composition, decomposition, and equivalence.
Students move from building and talking about mathematical relationships to reading, writing, and manipulating those relationships with symbols.
Build & Learn: Numbers 1-100 is available as an add-on when you enroll in Build and Learn Structures.