Montessori Mathematics is Incomplete

Or so the story goes ;)
(just at elementary)

So is the language album - and history - and geography....

It is true that the traditional Montessori elementary level mathematics album (and the others listed) does not contain some information that is required by every state in the US and probably by most standard school requirements around the world's first-world countries.

But let's think about cosmic education.

Mathematics is not a stand-alone subject by itself.

Mathematical principles are embedded in the very life around us; the very things that support that life.

Mathematics as a human language has come down to us through history as a gift from our ancestors. We have built upon it, used it, expanded it, created a supranature with it -- we live it every day and we pass it on to those who come after us.

We humans by nature have a mathematical mind (bear with me on this one - future Montessori Nugget!).

So let's think about this.

What is "missing" from the mathematics elementary album? 
--graphing: look in geography (economic geography with production and consumption); also see botany and zoology studies; it is also included in math, but not as a stand-alone album page; be sure you have an album with ALL the exercises. Specifically designated coordinate planes are not included.
--estimation: used in practical situations all over the place, most notably in geography (economics), fractions, long division
--statistics and probability: really get into it with economic geography; also covered in history studies
--comparisons (greater, equal): exercises within math presentations
--measurement: may be introduced in math, but heavily utilized in every other area of the room in practical life situations --- check Geometry ;)
--money: practical life (Goings Out, literal practical life); extension of various decimal fraction album pages and experience is assumed when children are working on divisibility (early math)

What is "missing" in  language?
--Research skills: no formal album page exists, but exercises, presentations and almost all follow-up work requires it. The entire elementary sequence focuses on research - show the children how to use tools as they come across them; formally these things are introduced in economic geography (upper elementary), but the children are very likely to know it all by then, if they've truly been exploring their own interests already.
--telling time: see the history album
--5 paragraph essays: they're not necessary.

"Missing" in history: 
--local history - timelines and charts: the children build these themselves according to the local educational requirements and their own interests (no album can cover every single "local history" requirement - so guidelines are given to create your own)
--the album just seems too short: because the history album provides the framework and the highlights - the children's interests and the local educational requirements dictate the details; encouraging the use of many different sources for information, rather than depending on just one source that may or may not be accurate or non-bias, or even cover everything.

"Missing" in geography: 
--everything that should have been done in primary, can be taught quickly in elementary with similar materials - so elementary geography "officially" speaking does not include functional geography. This is a research area for the children - as they study cultures, climatic zones, peoples of various places and times, they will be learning political geography at the same time.
--basic map skills: Goings Out - maps of museums, bus routes, maps of the city, etc.

The main thing to remember is that, since elementary Montessori is based on cosmic education, just picking and choosing one or two subjects won't cover all the concepts needed in that subject (ie mathematics - you'll need biology, geography, history and language to "complete" the math cycle).

This is where it is crucial to utilize the local educational requirements ("public school standards") in the 3rd and 6th years of elementary. These are a tool for the children to compare where they are with their locally schooled peers, check in on language usage, etc. It is a tool that builds personal responsibility as they do the comparing themselves, look for suitable tools to fill in, or in a few rare occasions the adult will provide the needed tool. 

Please reply back with anything that seems to still be missing. I'd like to make this post as comprehensive as possible, so we can honestly say which public school requirements are truly missing. 


  1. I find the areas that are missing from the maths curriculum that you have listed but are included in other areas of the curriculum still need to be flagged up and taught discretely. This is less because they are not well taught but because when the children move on to senior school at 11 there is no option for non-mainstream education in the area so they have to understand and use the same language as the children who come from the mainstream.

    I also find that unless there have been discrete lessons that are specifically about a subject, for instance graphing, then the children often do not get the point of the lesson. It needs to be introduced and then amalgamated into other subjects.

    Missing from the list you have included is symmetry; rounding numbers to the nearest 10, 100 and 1000.

  2. Thank you for your insights!

    I agree it is important to make the points clear when teaching about these connected areas, so that the child can utilize the same language as other children.

    Rounding is included within the economic geography section, though as you point out: not discretely. It is utilized within the sections that track produce, import/export, population, and the like. Perhaps this is something that could be added to a command card set somewhere - it is pretty basic to find the rules (and the variations on those rules) and have some practice with them, without drilling and killing as most of us did in public school :)

    Symmetry should be part of the geometry album, though again, not discretely (I can't even find it right off-hand!). I recall in training doing the work with a string over the geometric cabinet shapes; then tracing them and drawing or cutting on the lines of symmetry. This is certainly something that could be included with the geometry command cards and/or the geometry booklets. Since the children have had so much work with the insets in various forms and so much other experience in geometry, they've had it all sensorially, it is just the language that needs to be added. Symmetry command cards would be a fun extension, and the children could be invited to write their own extensions off of those. Hmm. I'm feeling inspired ;)

    It does need to be emphasized for everyone just coming to Montessori that when a child learns a skill in one area, it should not remain there. It should be applied to other areas, through their researching and projects - not just the work itself, but the presentation of that work, whether public or private; so utilizing graphs in their research for example.

    Too, the children should be checking over those local education standards as they phase out of the Montessori educational environment - 3rd and 6th grade should be posted in the appropriate environment, so that they are taking responsibility for filing in any gaps between what is developmentally appropriate and what is expected.

    You make some good points! Please do let me know if you find any other areas that seem to be missing! :)

  3. I have been perusing our state's standards - I've not looked at them since 2009 and they were updated in 2010. They are not much different, but I can say with certainty that not even most public school children actually have an understanding of everything in the standards. Exposure, perhaps, if they weren't absent that day.

    I did find other math concept:

    --mean, median, mode (good for command cards, or for the children to see the standards and find their own way of learning it) - part of statistics within mathematics
    --actually plotting ordered points on a grid (part of geometry)

    Oddly enough, some of the concepts discussed above as "missing" aren't even on my state's standards for mathematics; however, they are still important concepts. Also, Montessori math definitely covers far more experience, understanding and depth of knowledge than any state's standards I've seen in the last 5 years. So even with the "holes", there is a more thorough foundation.

    I'm interested in how others' state standards compare?

  4. In England and Wales we have the National Curriculum Levels. They cover 4-18 years and maths is divided into different areas; Shape, Space and Measure, Using and Applying Mathematics, Number and Algebra, Statistics. Parents inevitably ask about what level their child is at, even at our school. I have therefore had to try to align the national curriculum to the Montessori materials and sequence. It was SO hard! For instance, the NC values and pushes children into working in the abstract as soon as possible. We know that many children need materials far beyond the age of 7 but in the mainstream these children are regarded as "failing". Therefore, when I match up the two differing systems I have to give lower NC levels to our children because they are doing fractions using the materials, rather than working from worksheets and regurgitating the facts they have been taught. We have found that the children in the 3-6 class zoom ahead in understanding and skills as compared with children in the mainstream. When they come into my 6-9 class, there is a period where they are learning using materials and their progress through the levels slows down. When they reach the stage of not needing the materials they suddenly jump ahead by many levels because we can tick off skills that they have internalised and can use in the abstract. We have to ask the parents to trust us for a couple of years while their children build their understanding.

    Overall, I think that the montessori materials allow children to build such a strong understanding that they can transfer those concepts to new skills and theories with confidence.

  5. Annicles,

    It is interesting timing that you say this! My son has just started doing long division without paper, but he does like to go back to the beads to check once in a while - not every problem and not every step (he'll just set up the step that he's on).

    Yesterday, a friend (who is very non-Montessori) commented that if my son "really" knew long division, he wouldn't need the beads at all. I politely reminded him that my son is getting a thorough foundation so that when he is ready for abstraction, it will be solid - and the age that happens matters less than the strong foundation. He JUST turned 8, my friend's daughters' curriculum does not introduce long division until level 4 (about 1/3 into the school year, approximately age 10) and immediately delves into 5 steps for it. His girls are very intelligent girls, yet I'm not sure they entirely understand what long division DOES.

    I can understand your frustration trying to align the national standards with the albums and trying to provide the children developmentally appropriate skills. I am blessed to live somewhere I don't *have* to do that, but I have been working with some individuals that do. It is not an easy task!


  6. If you actually take the AMI training at an AMI center as a posed to buying a copy from someone on the Internet and try to use it without any of the theory and practice you would know that there is NOTHING missing. It is a disservice to the children to try and "teach" something you have not experienced and practiced.

    1. Anonymous -

      Please note the beginning line of the post "or so the story goes" - meaning, the statement in the title simply isn't true, at least in the sense that most people would see it.

      I also chose that subject line for search engine purposes - it's a popular one.

      As applied to what a child actually *needs* and what is developmentally appropriate for the child, YES - you are absolutely correct that the mathematics albums (and all areas of elementary Montessori) are absolutely complete.

      However, this post was not titled to imply that the math (or any other) is NOT complete - only to cite what I hear so many people saying to me when *I* say, "Oh, Montessori is SO complete and wonderful for children!"

      I say that to people based on having both primary and elementary training through AMI at highly reputable training centers, with 15 years experience in AMI primary and elementary environments.

      I am also on a campaign to get people to actually read, study and apply the theory aspects of Montessori - not just pick up an album and follow the album pages (closely or otherwise). The theory is SO foundational.

      But I digress from my second point I'd like to make - and that is that not every school or homeschool has the luxury of "ignoring" their state standards. AMI training is *quite clear* in stating that there *will* be items that the teacher needs to make or needs to encourage the child to create to cover those state standards.

      Hence the posting of the local educational standards is discussed several times throughout the AMI elementary theory album. Not to use as a checklist, but as a way to develop responsibility in the children.

      So, even AMI itself (at least in the USA) recognizes situations where something will need to be "added in". The point to this particular post to emphasize that instead of immediately jumping to making something new to add in, consider that it is likely covered in another subject area (such as charts and graphs in research).

      Thank you for sharing your insights!

      And for everyone reading: THEORY is *needed* if you're going to do anything with Montessori!

  7. Adding this comment after this post was listed on Facebook ;)

    I have found the following things help to "fill in" even those discrete lessons, at least for math:

    1) Life of Fred Math series
    2) the adolescent algebra album available through NAMTA