The Montessori Math Progression
I’ve been taking a bit of a deep dive in the Montessori Math Progression lately. In our homeschool preschool, we introduced the Montessori Math Materials this fall and my daughter has really dove into her number work here lately. I am also DIY’ing the complete beads work (golden beads and the full beads cabinet – I seriously need to get busy on this!) and planning for the next school year plus trying to get ahead so that I am not overwhelmed after French baby #3 arrives. In addition to all this, language is my happy place and I did not learn mathematics in a hands on way. The way I was taught math was predominately abstract. I initially found Montessori Math quite intimidating because of this. It was just so much new. I took my time though. I read and re-read Montessori’s works, the albums I have, plus I am writing out the process (which has helped me so much!). I also I have been in conversations with other Montessorians online and have done a few trainings on the sequence and special topics as they pop up and are in budget.
In our Montessori homeschool we have leaned heavily on Info Montessori for our scope and sequence in 3-6. I recently grabbed The Montessori Sequence Timeline from Mainly Montessori for our 6-12 album. Both the math and language sequences overlap with the 3-6 classroom (as they should!). IF a child is doing the 3-6 Primary classroom, they will likely move through the start of both of those sequences on Pilar’s timeline between 4.5 and 5 so I grabbed this for our final year of 3-6 (the Kindergarten year!) to help with planning early since I planned to get it for 1st grade and on. I love her approach to the layout of the timeline. In addition to this, I grabbed “Montessori Matters: A Mathematics Manual” by Sister Mary Motz. I have found both resources to be so helpful in addition to the album on Info Montessori in preparing and working through the mathematical sequence. I have also referenced “Basic Montessori” by David Gettman.
This sequence will likely start for most children around 4.5 and then take them through through lower elementary or so. It could start earlier and it could take the child longer to work through the progression or the child could even finish this before the end of their kindergarten year. That is the beauty of the Montessori Method. It takes the child to the next piece wherever they are, not based on an arbitrary thing such as their birthdate or grade level. It is about the child’s mastery. Their unique educational journey. That said, based on where we are today, we will most likely finish through memory work and perhaps even start moving into abstraction when we finish the 3-6 Primary Classroom at home. Although abstraction can be parallel with memory work, it varies by the child and may come after memory work. In that case, we would be working on this in year one of the 6-12 classroom. Much of the memory work will be in the 6-12 classroom as well.
Ok, enough preliminary thoughts… on to the sequence!
MONTESSORI EARLY MATHEMATICS SEQUENCE OUTLINE
- Pre-Math Material
- Organic counting/Experiencing Math Naturally
- Red Rods
- Numbers 1-10
- Numerical Rods
- Sandpaper Numbers
- Number Rods and Cards
- Spindle Box
- Concept of Zero
- Cards and Counters
- Memory Game
- The Decimal System
- Introduction to the Golden Beads (Place Value and Decimal System)
- Large and Small Number Cards (Number 11-99)
- Formation of Numbers (Golden Beads + Number Cards)
- Changing (Golden Beads Work)
- Search for ten snake game
- Four Operations
- Addition (Small Cards + Golden Beads)
- Stamp Game Addition
- Multiplication (Small Cards + Golden Beads)
- Subtraction (Small Cards + Golden Beads)
- The Dot Game
- Stamp Game Subtraction
- Stamp Game Multiplication
- Division (Small Cards + Golden Beads)
- Stamp Game Division
- Linear and Skip Counting
- Teens: Quantity (Colored Bead Stairs)
- Teens: Symbol (Teens and Tens Board)
- Tens: Association/Counting from 11-99 (Teen and Tens Boards or Cards and Nine Ten Bars and Ten Unit Beads -Golden Beads)
- Linear Counting/Chains of 100 & 1,000
- Skip Counting – 10’s
- Skip Counting – 2’s & 5’s
- Skip Counting 3’s & 4’s
- More Skip Counting (as interested?)
- Number Roll
- Writing Practice
- Organic when interested with Sandpaper Numbers
- Invitations to Learn
- Options on the Shelf:
- Memory Work
- Addition Snake Game
- Addition Strip Board
- *Addison Bingo Chart
- Addition Charts
- *Addition Special Cases/Combinations
- Teen Snake Game
- *Bead Bar Activities for Addition
- Subtraction Snake Game
- Subtraction Strip board
- *Subtraction Bingo Game
- Subtraction Charts
- *Subtraction Special Cases/Combinations
- *Subtraction Bead Bar Activities
- Negative Snake Game
- Multiplication Beads (Decanomial Bead Bars)
- Multiplication Board
- *Multiplication Bingo
- Multiplication Charts
- *Multiplication Special Cases/Combinations
- Multiplication Snake Game
- Multiplication Decanomial Beads Layout (Different from the Bead Bars?)
- The Substitute Game for Decanomial
- Unit Division Board
- Division Charts
- Division Bingo Game
- *Division Special Cases/Combinations
- Passage to Abstraction
- Small Bead Frame
- Hierarchies with Golden Beads
- Large Bead Frame
- Racks and Tubes
- Fractions/Coins/Time
- Fraction Circles (Sensorial)
- Coins Unit
- Exploring Measurements
- Clock Unit
- Fraction Names
- Fraction Writing
- Fraction Labeling
*Activities listed with a star are activities that are listed as occurring with the memory work in my Montessori Matters Album (many of which state they are common around age 8) and are sprinkled throughout my Mainly Montessori Timeline starting at 7. Because of this, I am leaning towards holding off on prepping for these and circle back around if it makes sense in lower elementary work. I also may prep for them but hold the lesson unless it seems like the child is headed that way – if it makes sense? Especially with the special cases. I spoke with other homeschooling moms and teachers and the concensus seems to be the same. Hold off on some of this work.
Update: I connected with Pilar Bewley since writing this (author of the Timeline and creator of Mainly Montessori). I love what she had to say so I am just going to quote her here: “For Primary, the memorization sequence is pretty strict (in the sense that you present it in a certain order). Basically it’s Strip Board (or multiplication/division board) with one or more booklets, followed by the loose slips with the finger charts and then the blank chart. That’s all that’s presented in Primary, with the view that the child’s Absorbent Mind will pick up the operations quite easily.
In my experience, it’s best to wait until they’re 6+ to do extra games and especially to do the Special Cases, since these require mental flexibility that most Primary children don’t yet have. However, doing additional bead games can be fun for an older Primary child, since they provide a variety of ways of seeing the same operations.”
ALL that to say… the starred items will be something I am holding off on in Primary 😉
Pre-Math Materials
Montessori starts with naturally experiencing math in the environment. Counting objects, asking for three of an item (bring me three spoons to set the table), counting in games (like hide and go seek), or experiencing numbers as they are in the environment as we encounter them. The emphasis is not placed on needing to know how to count or remembering what number represents what quantity or even numerical order prior the introduction of the Math Materials.
In the 3-6 classroom, the math materials are normally introduced around 4.5 during the sensitive period for math (which spans between 4 and 6 for most children). The only prerequisite is the red rods. Once these are mastered, then you move on to the first mathematical material: the number rods. The sensitive period itself can be noted by a fascination with numbers and quantities.
Our society puts a lot of emphasis on getting started with these things prior to school as if the child will be at a disadvantage if they do not know how to count or recognize numbers at this point. It is important to realize that the child is experiencing math sensorially from birth. The child abstracts concepts, qualities, and even quantities without needing to name them. Eventually the child begins to build an internal order and desire a name for that order. The first names are typically names of things and people. Eventually they begin to count and want to put a name to the quantity of items. Follow the child on this but do not lead them. Just answer their questions, count with them, and play with them.
According to Maria Montessori, it is typically around age four that the child is ready for the language of mathematics. They will observe others counting before this and join in. They will also begin to understand small quantities of objects. Neither is an indicator that it is time for the math materials by themselves.
Prior to age four in the Montessori classroom, the foundation for mathematical learning has been laid by both practical life and sensorial work. The child has can work orderly/follow a process, has refined movements, can focus on a single task of interest, and is accustomed to the work cycle. The child has also likely begun the language sequence and has used symbols. All of this is quite important in preparing the child for the mathematical sequence. Don’t rush into it because your child can use the red rods. Wait for the readiness of the child (which is sometimes before 4! Follow the child!)
Numbers 1-10
This portion of the mathematical sequence is quite sequential whereas after this section, there begins to be quite a bit of parallel work. The work in number 1-10 is a concrete exploration of the numbers the child is already experiencing organically. This is the first practical application of the concrete knowledge the child already has. The number rods, spindles, counters, and the sandpaper numbers are tactile, playful, and fun ways to explore numbers 1-10. This playful exploration allows the child to truly understand what a number is as opposed to just the numerical representation of the number. The full sequence also allows for practice that doesn’t feel repetitive like math drills. This doesn’t feel like work. It feels like play (because it is!) A child is much more likely to remember something during play than during a time of forced learning.
The Decimal System
At this point in the sequence we move from sequential work to parallel works. The decimal system, four operations, linear and skip counting and writing practice will all move parallel to one another after the introduction to decimal system lesson with the golden beads. Clearly some things are in order while others are parallel. Personally, I found it helpful to have the outline above but to group the works into units in which the parallel works will occur together. I will discuss each concept briefly and then dive into the flow. I anticipate each unit to be about 2-3 months? This is just an estimate. It is to give an idea of when we will be working on each activity, but the key is to follow the child. So this is an overarching view and we will look at six weeks ahead of where we are based on what the child is working on right now.
The decimal system introduces the child to the different categories of numbers aka the language for the decimal system (unit, ten, hundred, thousand). This is first explored concretely and then the numerical representation is taught (abstract representation). This is accomplished through the following materials: golden beads, small number cards, and large number cards. The goal is to aid in reading large numbers and understand place values.
At this stage, the materials are explored first, and then the language for the activity is given. This shifts with the second plane so if you are just starting the materials with a child 6+, you will want to give them the language right away so that they can understand and see the relevance of working with the materials. You will also want to check out the Great Lesson as a kickstart for each subject area explored in the second plane. These lessons give the information the second plane child craves and sets them up for further exploration in areas of interest.
Four Operations
This section is actually grouped under the decimal system work on Info Montessori but I found it helpful to break it into another section like my album from Mainly Montessori (The Timeline) and Montessori Matters does. This includes addition, subtraction, multiplication, and division with the golden beads, the stamp game, and the dot game. It is quite common for multiplication to follow addition because it involves the same process. The end result of both addition and multiplication is a larger quantity. Prior to addition, the child will have worked with the golden beads, the teens and tens boards, the search for ten snake game, and will begin linear and skip counting work (although the later is likely parallel). Some teachers continue on to subtraction unless the child discovers multiplication on their own but I find the argument for continuing with multiplication first quite reasonable due to their similarities and at this point plan to continue forward with multiplication first either way. The presentation is the same for static multiplication as it is for addition and I find it just makes sense to move forward to this presentation next. Neither way is “wrong” though.
Linear & Skip Counting
Once the child has had the introduction to golden beads and number cards presentation, you can move forward to the works in linear and skip counting. This starts with the color bead stairs, then the teen and tens boards (or in our case, “tens cards”), followed by the bead cabinet, and more golden bead work. This section of work leads the child to work with number 11-1,000. The hands on, playful repetition helps establish the process of counting and prepares the child for arithmetic.
Writing Practice
Some albums have this listed and some do not. It happens organically during the progression but I will be opting to have it as an option in Primary (3-6) as a work to select from the math shelves just like I have writing work available on the language shelves. I will not be encouraging her through the checklist (pre-work plan option that will be on the shelf in kindergarten in our home) to choose this as a main writing work, but as a math work option during our kindergarten year. I will also not be ensuring this is touched weekly or daily as her guide. She will get plenty of practice organically for now so it is not a big deal if she isn’t selecting the work. If it is something that needs work in her first year of 6-12 (1st grade in traditional schools), then I will include it in our work meetings to formulate her work plan.
Memory Work
Up until this point in the sequence, we have playfully worked through all the decimal system operations and all the stamp game operations. The child will “realize that he can go no further. In doing golden bead addition equations, he is slowed down by his need to count out quantity. This realization is a stimulus to encourage him to learn the facts” as opposed to requiring him to learn (Sister Mary Motz, Montessori Matters). Further, the parallel exercises above use the subconscious to begin memorizing data before the child intentionally begins to memorize data. The child will begin to use this information at will during these exercises and as he continues down the path of mathematical mastery/exploration.
Passage to Abstraction
You cannot teach true abstraction but you can bring the child through experiences that lead naturally to the process of abstraction and allow them to organically begin to abstract on their own. That’s the beauty of the Montessori Math Progression. It does this for you! The child will begin to “leave the material, [and] very easily come to love writing out the operation thus doing abstract mental work and acquiring a kind of natural and spontaneous leaning towards mental calculations” (Maria Montessori, “The Discovery of the Child”). When you see this “spontaneous leaning”, it is time to begin the works in this section.
Fractions/Coins/Time
I grouped these three concepts together. Fractions is included in Info Montessori’s math sequence, but not coins and time. These are typically done in the 6-12 classroom BUT they are already of interest to my 4.5 year old and I can see that they would be beneficial to explore before then. In the 6-12 classroom they typically come in to play as parallel works with the Intro to Decimal System (fractions), golden beads addition work (coins) and then once measurement has been explored, the clock intro lesson can come into play and is likely parallel to the initial division work.
Parallel Math Work “Units”
Note: These “units” will take the child between 3-6 weeks to move through depending on interest of the child and number of homeschool days a week.
Unit 1
- Intro to Golden Beads
- Number Cards
- Changing (Golden Beads Work)
- Teens: quantity/color bead stairs *Parallel to Golden Beads but after initial presentation of 1 & 2 in this list
- Teens: Symbol/Teens and tens board
- Search for Ten Snake Game *Parallel to Teens work but after number 4’s initial presentation
- Tens: Association/Counting from 11-99 (Teen and Tens Boards or Cards and Nine Ten Bars and Ten Unit Beads -Golden Beads)
- Linear Counting/Chains of 100 & 10,000
- Fractions (Sensorial)
Unit 2
- Golden Beads Addition
- Coins Unit (Can be parallel or can wait until after a couple more presentations in this unit)
- Stamp Game Addition
- Golden Beads Multiplication
- Golden Beads Subtraction
- The Dot Game
- Stamp Game Subtraction (Could be parallel to the Dot Game)
- Stamp Game Multiplication (Could precede or be parallel to the Dot Game)
Unit 3
- Golden Beads Division (after this, any of the words below can be parallel)
- Stamp Game Division
- Skip Counting by tens (and then as interested proceeding through different numbers)
- Exploring Measurements (Whole Numbers)
- Number Roll
- The Clock (Intro Lesson)
Note: Unit 3 is somewhat parallel to Unit 2 and could be grouped together more. Truly all three units could be somewhat parallel. This is just based on my reading and albums that I am using. Follow the child!
It is helpful for me to sit down, outline, and then dive into each section (by writing it out) in order to take my understanding of the material used to the next level. I hope that by sharing this, you find it helpful in applying the Montessori Method in your own homeschool or preschool/early elementary classroom. In my digital tracking tool/homeschool portfolio, I have plugged in the state standards for our current state under each subject. If you are curious to see how following the Montessori curriculum not only meets but exceeds state standards, feel free to check that out. You are also welcome to save the Google Sheet and make it yours to use as your own tracking tool or portfolio at home.
Happy Montessori “Math’ing” =)
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