Wednesday, September 25, 2013

Montessori Math: Fitting It All In, Elementary

You are going to either love this post or hate this post depending on if you think like me.  Just a warning.  You'll probably know pretty quickly which category you fall into.  I will forgive you if you abandon me part way through and come back tomorrow when I'm speaking your language again.

If you follow me on Facebook, you already know that I spent two hours at the bookstore this weekend getting the Montessori elementary math sequence(s) organized in my head.  I want to share some of my work today, but I want to come out and say right from the start that what I accomplished was not rocket surgery.

It's not rocket surgery.  It's rather superficial work when it is all said and done.  However, it is helpful for determining what albums I needed to purchase and which works I needed to start in the next few weeks.  Another thing it does is help me absorb the big picture of the full six-year Montessori elementary math "curriculum."  The understanding now stored in my brain is far more valuable than what wound up on paper at the end of the day.

If you read yesterday's post about the primary math sequence(s), you read about how I like to organize a big category like "math" into multiple threads.  Instead of one road, "math," that you are puttering down with a big "oversize load" sign on your backend, you have several roads that you are chugging along on simultaneously.  You don't necessarily start your journey down every road at the same time, some roads are longer than others, and you don't travel them all at the same speed.  I don't know if you've seen an AMI Elementary math album.  It's HUGE.  Thinking that you start on page one and work your way through one day at a time until you get to page 443 could be disastrous because the albums aren't meant to be used that way.  However, when you realize how many threads of work there are (13 plus "threads within a thread")  it can be intimidating if you think you are going to be working in all threads at the same time.  Fortunately, you won't be.  I needed to figure out which 5 or 6 threads DO start in year one.

At first glance, it seems like Montessori elementary math will look like this:

Instead it looks like this:





squaring and cubingsquare root and cube root
*word problemsword problems

decimal fractions


powers of numbers

non-decimal bases

ratio and proportion


  • If you want to pin this chart, the chart at the TOP of the post is pinnable. The non-pinnable charts look better than the pinnable ones so I kept this one as is.  The reason it won't pin is because it is a spreadsheet, not an image. 

  • I have "word problems" in two different colors starting in two different places. As you will see below, some of the albums have a category called "Word Problems" in which the word problems are all about Distance, Velocity, Time, Principal, Interest, Rate and Percentages.  This is different than the idea that the child should be doing "word problems" using appropriate skills (such as the four operations, measurement, etc.,) throughout the entire sequence (including primary, start it about when they start doing the memorization work).  I didn't want to forget this so "appropriate skills" work problems are in light grey from year 1-2.  I changed the color to dark grey for the actual category several albums call "Word Problems" (DVTPRIRP, LOL!) that might begin in year three.

Another thing you might have discovered in yesterday's post is that I use multiple album brands like a total rockstar   crackpot    enthusiast.  My main, go-to, album is Keys of the Universe.  However, I also really like Cultivating Dharma.  I tend to have them both open, side-by-side, on my iPad because they are extremely similar but I sometimes will prefer the pictures or script in one over the other.  Both of them are great at delivering the big picture.  Both have great scripts and illustrations.  What AMI albums tend to do with each new skill is have you take your child through a couple of examples and then let them continue working with the apparatus from day to day inventing their own equations "making sure they cover x, y, and/or z."  And philosophically this is the best way to go.  Psychologically, for me, my obsessive tendencies kick in and I want to make sure that they have covered x, y and/or z.  So, I like to make equation slips instead.  Really I'm exchanging one kind of work (careful observation) with another (making materials).  But, I know my strengths.  The Montessori Research and Development albums (and KHT Montessori albums, primary only.  Both are AMS) tend to give a strict, broken-down list of the types of equations you need for each work and in many cases provide a list.  I like the lists. I don't like having to generate equations.

The work this creates for myself is that if I'm going to be sometimes using three different albums, I need to figure out what is where in each album.  This can be tricky because different albums call things by different names.  Plus, I hadn't bought the necessary MRD albums yet because I didn't know which ones I needed for year one and didn't want to buy extra albums this fall. I need to spread out the expense.  They spread out what is in the KotU/CD albums into FOUR albums that are each about two inches thick.  That's not counting geometry, fractions, and decimal fractions which each have their own albums (Fractions has TWO; geometry has THREE.). So, I needed to figure out which ones I needed to buy.  

This is where I wind up at the bookstore for two hours with the tables of contents printed out from three albums PLUS the handy-dandy, super-helpful scope-and-sequence from KotU, complete with "approximate" start and ending years for each topic.  I went through them all and decided how many threads there were and what they should be called.  I also marked each table of contents with the page numbers the corresponding lesson is on in the other album (for KotU and MRD.  CD is organized the same way as KotU so it wasn't necessary to do that.).  

The Cultivating Dharma Math 1 album (There are two math albums and one for geometry.  KotU has one big math album and a separate for geometry as well.) says right in it that there are "13 threads and they are:  a, b, c..,"  I still had to check and make sure that was the same as the other albums (it wasn't, the other albums had MORE, but they were really subsets of a larger category.  Here's the chart of the threads by album, what they called everything, how many categories I decided to have and what I decided to call it (because these categories will be in my record books):

Notes:  On this chart "Word Problems" refers to the DVTPRIRP topics I mentioned earlier.  The chart will also get bigger if you click on it.

I put the (caveat, caveat, caveat) "approximate" start year for each thread (from the KotU scope and sequence support pages) on the chart while I was at it. I much prefer the "year  numbers" to "ages" but you must always follow the child.  There will be "threads" within threads of course.  For example, in "numeration" we might be on one road for a long time, but we will eventually get to "multiples" which will carry on alone for some time but eventually split into, "multiples" and "factors" and run concurrently.  The "multiples" thread starts before "factors."  Also, remember all the threads don't start at the same time.  You might start the elementary "numeration" thread while your child is still completing multiplication and division in his or her primary "memorization" thread.  You eventually run out of large bead frame work under "numeration" which clues you in to start the "multiplication" thread where the large bead frame work continues.   The albums have clues for you throughout as to how to coordinate across threads, but you have to read your whole album to find it.  I'm still working on sorting that all out.   

It is easy to find all the work and categories in the two AMI albums.  I am NOT providing a chart of which volume of MRD math all of the work is in.  It's crazy.   I determined that I need to buy MRD volumes 2 and 3 for this year by-the-way.  Some of the Elementary Year One work (mostly numeration) is actually in MRD volume we will be working across FOUR MRD albums this year, NOT counting Geometry, using Fractions vol. 1 (and maybe vol. 2) and parts Math vols. 1, 2, and 3.

If you are trying to decide what materials you need to buy for a particular year (again, year numbers are approximate, follow the child!) you can look at Jessica's scope and sequences on the Keys of the Universe website on the albums page.  All the lessons are listed with the start and end year afterward.  If you need to, then you can just buy what you need for a particular year.  I did.  I sometimes will buy a little ahead if something is on sale and it helps me meet a free shipping threshold.

Related Post:
If you are looking for similar information but at the primary level you might like...
Montessori Math: Fitting It All In, Primary

Montessori Monday


  1. Replies
    1. Amy, thank you for always commenting! I make very little money on the blog (which all goes back into supplies) and it's really comments that keep me going. I really appreciate it!

  2. MBT,

    I have had math on the brain, too! I was doing the same exact thing this weekend, and feel so much better after the process. I felt so overwhelmed looking at the scope and sequence of KofU, but when I started breaking everything down it didn't seem so scary! Thanks for the post... good to know that I am not alone. You didn't lose me. I read it from start to finish :-)

    1. Cristina, THANK YOU for letting me know you stuck with it and commenting. I rely on comments to keep motivated here, and it gets quiet sometimes. Thank you for lifting my spirits :) Isn't it crazy how that organizing process really just makes you feel so much better about that album? I'm glad I'm not alone too :)

    2. Absolutely, the deeper I go into it... the more I am reminded of why I love the Montessori Method. I was really drawn to the educational philosophy for many reasons, but one of the main reason was the approach to mathematics. I got my degree in statistics, and the way math is taught is genius!

    3. LOL! I have my degrees in I tend to count to four and start over . So, I on the other hand am just amazed at the level of math that is in there! And thanks to the Montessori method, I understand it.

  3. I'll go back to a past post and repeat: you ARE amazing. I have looked at so many math albums over the years, and I just ditched them ALL because I couldn't combine them in my mind/organization at all. The other subjects, I have kept pieces here and there to use as supplements if interest arises or if it is a local requirement (meaning, I am the one requiring the topic ;) ). But math - nope. So I am amazed by the fact you got them all aligned!

    (now we do add in Life of Fred, other "living math" - but specifically as supplement and usually just for fun, on the side)


    1. Well, when you have such a strong handle on the one album and there is so much in there in the first place there probably isn't much motivation to complicate things. The other albums are definitely easier.

      I on the other hand did NOT have a strong handle on this. I think my reading of multiple albums is a way of making up for the lack of formal training. I understand everything much better after I've read it three different ways. In training, of course, the albums are the final product of lecture and practice and putting that all together. Like everyone likes to remind us untrained folk...nothing can take the place of making your own albums. BUT that's something you don't have time to do when you are already in the trenches. I feel like reading multiple ones and gaining extra info each time is something I CAN do. :)

      You are TAUNTING me with the Life of Fred. I don't have those yet. You make me want pick them up and start adding columns to my charts. Meanie.

    2. You know you want to.... ;)

      You make a good point - one that has been at the edges of my mind, but you finally put the words to it - exploring all the options does take the place of "training" in the sense that it becomes your own work to sort out what will work for your own children and situation... I remember being there and doing that before going to training. Yes, training was definitely easier in that sense - Baptism by FIRE in my case ;) I'm still applying burn cream - and loving every second of it. I am just happy I get to share with people like you and others who are just as enthusiastic about Montessori :)

    3. Nice visual on the burn cream :)

      I reread my previous comment and that sentence "The other albums are definitely easier" makes no sense at all. What I meant was that the AMI albums made more sense (were easier) than the AMS (Montessori R&D) so why bother learning the other albums (especially MRD) when you already have such a strong handle on your own album. TOTALLY lost in translation and I wanted to make sure it didn't sound like I was saying that all the other albums are easier than your album because that is NOT the case. Phew.

      Must go google Life of Fred contents...

    4. I wondered what you meant, but I guess I read between the lines ;)

      I have to admit - I didn't like Fred at first - he gets exploited or otherwise maltreated in several of the books, and the references to him not eating so he doesn't grow kind of get to me. But Legoboy (as most children) has a different focus in life, and it all provides great discussion fodder.

  4. Love, love, love it. Savoured everything you were saying. Thanks so much.

    Talking about Maths and Montessori, I totally agree. Just love, love, love it! Going to be fun making or buying the cubing material.

    But one thing missing, I feel, is alternatives/natural strategies for doing math. Chunking etc? I don't know if this is covered in elementary? But where you teach stategies for working out math quicker - basically what a lot of people do, naturally in our minds, like 7 + 5 - thinking goes ... getting to the nearest 7 + 3, then adding 2. Where is the covered in Montessori? Or chunking 54 + 23 ... thinking 50+20 and 4+3

    This is one supplement I think is needed.

    An excellent book on how maths is now taught in (UK/AUS) school is Maths for Mums and Dads - I think is a good supplement to understand just the extra strategies and are you covering them, whatever program you are using? Also the interesting little math facts (like the reason the cirlce is 360 deg) and games in the book are really good.

    One supplement I like, here is Australia, we have Natural Maths - it's taught from age 5 and kindergarten. Their parents guide book is excellent and they have a download now on their site.

    And they have just released a new series for at home called Back to Basics. Here is a sample. and only $8!!!

    MBT, any, any chance we can have your excel spreadsheet to adjust to our own needs, instead of starting from scratch, please???? :)

    thanks again for excellent ideas about the threads and breakdown. Hadn't thought about it, and excellent ideas.


    1. Tracey - the math concepts you mention are naturally picked up throughout the material as the children are encouraged to find patterns in their work. The specific chunking you mention is a huge part of the experience of manipulating the golden beads; then later with the colored beads, the cards, then the pegboard and other material, and finally into abstraction. All the manipulation leads them directly to those patterns and conclusions.

    2. Tracey,

      I will look into putting up the excel spreadsheets. Might take a few days because one is in Open Office. I have so much trouble with all the free spreadsheet programs. Nothings works as well as a big, paid-for Office suite...but I have a Mac.

      As to the natural math strategies, my husband uses those all the time and I rarely do. The ones I have used since youth I used as a "cheat" and never understood the math behind them. So, my strategy has been to try to keep my husband from teaching them "cheats" until after they understand the concept. They aren't really "cheats" of course, but rather valuable ways of understanding the math in general. For example, when my husband rounds up to and then subtracts the difference to find an answer to an equation more quickly.

      It's only one example, but when I read the 7+5 example you cited I immediately thought about the snake games. The boys already do that kind of chunking because the snake game requires that you replace the 7 and 5 bar with a ten and two bar as you work. When the boys see 54+23 they think "get seven units then seven tens" instead of an on paper equation.

      When the boys haven't seen what I might call a "cheat" right now I used to be tempted to tell them what it was so they could find the answers faster. Over time I discovered that if they spent enough time with the material to understand the basic operation first they eventually would discover the shortcuts on their own in order to do the work faster. ("hey mom! Did you know whenever you add a single-digit number to nine the sum is 'that number' minus one and then you say 'teen'.") I decided it the discovery was ultimately better made on their own.

      That said, it is early days and we are not doing very advanced math yet. Because I don't personally tend to do the types of things you asked about I might not be familiar with techniques we truly want eventually, AND the boys may not continue to figure them all out on their own. So, it is something to keep in mind and I'm definitely bookmarking that resource you mentioned.

  5. Thanks so much Jessica and Andrea. Yes, you what you say is so true, I hadn't thought about it, but these concepts are taught quite naturally, through the manipulatives. And maybe our kids will naturally pick them up.

    And many of the concepts aren't needed to be taught, like doubles (knowing what all the doubles up to 10 equal to, e.g 6+6=12) - because these are done through memorization. But it is so interesting how these are natural strategies that MANY people just work out, like your husband and his cheats. :) I'm sure he would love to read the booklet and see how many he naturally knows, and finally someone is on his side. He's say, 'see its a strategy, not a cheat. I was right all along'. Ooops. :)

    In discussions with my best friend, she has learnt these natural thinkings herself, and didn't understand how people didn't just know them. And then there is me, who just can't even think about how to mentally add units, then pass the extra 10 over and add those - as we learn in simple math. Now that I know some strategies, I don't feel so daunted by trying to add mental math. It used to leave me stumped and too much mental capacity thinking overdrive!

    What I like about teaching these strategies, is then I know I've opened this door of thinking to the kids and they can use it as they wish, instead of it being hoped for through the montessori process. And it's fun.

    I haven't really taught my son the strategies yet, except for one. And I was interested to see that he is natually picking up these strategies. Last week, he said, 'Mum 6+7 = 13'. I asked him how he worked it out. He said, 'well 6+6=12, then it's equal to 13'.

    Andrea, if anything, one thing you would like is a lot of their work is math stories. Scenarios where kids have to work out the answer. Once I have the books, I'll copy some of the pages for you, to have a sneak peak.

    1. I hope you don't think we were picking on you! Horrors!

      I was reading one of the Montessori R&D math albums this weekend and I kept thinking of you as I kept seeing these little concepts hidden everywhere.

      I guess what I was trying to say is that maybe the reason we didn't learn those strategies accidentally like your friend is because we had traditional textbook math backgrounds and that the Montessori way of manipulating materials makes it easier to come by them naturally. Some people, like my husband and your friend, are talented and will come by those strategies regardless of the teaching method.

      I like what you said about opening the door of thinking for your kids.

      Would love to sneak peek those pages when you have time!

  6. Now that I've been presented up to the squaring and cubing, I'm taking a look at the sequence in depth. I'm kind of confused by the squaring and cubing part in the spreadsheet. From Jessica's scope and sequence it looks like you do the intro to squares and cubes from 1st to 3rd, but squaring and cubing starts in 2nd and you don't really get into the bulk of it till 4th grade really.

    I have an AMI style album and my problem is seeing what kind of follow up work you could potentially be doing after the presentation. It's all kind of glossed over. Do you use the MRD albums for that purpose? How do you decide on what to work on within the first year for example? My confusing comes in figuring out the sequence within threads that are covered around the same time. I tried reading through my album, different presentations within each "thread" are pre-requisites for other threads.

    1. Jessica's scope and sequence gives both a "start year" and an "end year" for each presentation or sometimes a group of presentations. So your refer to ""intro to squares and cubes" is a specific presentation that takes ten minutes. Her scope and sequence is telling you that you can present this as early as 1st year or as late as third year. You won't be working "on that presentation" for three years, LOL. Her squaring and cubing work is divided into two sections. Some is in the "squares and cubes of numbers" section. The rest is in the "squaring and cubing" section. During "year one" I present the work she has listed as beginning in "year one." In year two I present the work she has listed as "year two" and so on.

      Of the 23 presentations or groups of presentations in this category only five of them have a "start" year of four or higher so I don't know where you would get the impression that most of the work is done in year four. I'd say most is done in year three. Four presenations/presentations categories are "year one", five are "year two", and nine are "year three." Three are "year four" two are "year five."

      I think maybe you are trying to find follow up work where there is none. A lot of the presentations are just presentations. The "follow up" is to move on to the next presentation. The ones that need follow up are somewhat obvious. I think the first year presentations don't require a lot of follow-up. Each subsequent presentation practices what you learned in the previous so the follow up is built in. It is not until you start doing sums that you might have specific follow up work. The follow up is simple for that. They need to invent equations or go through a set of equations.

      This is easier to describe in the "Multiples" section. The scope and sequence just says "multiples." This is an example of a "group" being listed instead of a specific presentation. Within that group some presentations were JUST a presentation and the follow up was to do the NEXT presentation. But then, one of the presentations is counting bead chains. Follow up would be to count OTHER bead chains and this would continue independently until they were able to count them all well. Next presentation is labelling two bead chains side by side and playing games. Follow up would be to do this with other pairs until finished. Later on they are circling multiples on paper 100 boards for one number at a time (i.e. 9's). They would do OTHER numbers as follow up until finished. Usually the follow up is "more numbers to do" or "equations." or "story problems" if you have them.

    2. You never say what album you have. It might help me help you if I knew.

      The free cultivating dharma albums are set up only slightly differently than Jessica's. He makes a "follow up" work list at the end of most presentations. Maybe reading through the albums he has available will help you. I find that his "follow up works" (which are just a list) are actually fleshed out presentations in Jessica's albums so I just stick with her albums.

      I am worried for you about your training course. In all my years "training myself" I always daydreamed that if I took "real training" I wouldn't have these questions, but you seem to have all the same questions I used to have.

      The purpose of the chart I made is to show which threads are happening at the same time. It tells me exactly how many threads I have to be poking around with at once. I know I can completely ignore "integers" for example during year one. I know I can ignore squaring and cubing in year one. Among the threads I AM working in I have to read the year one presentations completely and I highlight any prerequisites as I find them and then write the cross-reference on the partner presentation if it isn't already indicated. If AS I'M WORKING with the child I find I missed one I don't panic. It just creates a "natural pause" in one thread while we move back to the OTHER thread and work to where we can move on again. These "pauses" are what keep you from being in SIX math threads every week. You are in six threads across three years or even across one year, but not daily or weekly. Sometimes not even monthly.

      FYI the reason I split "squaring and cubing" into also "square root and cube root" is because all the albums but Jessica's did it that way. The items in the "square root and cube root" section of those other albums is in the regular "squaring and cubing" section of Jessica's albums and are the same works that she has in her scope and sequence as year three or higher.

      I don't use my MRD albums much. I use their fractions albums and sentence analysis albums because they match my materials. Otherwise I stick with Jessica's. READING many different albums is part of my personal training process. And, BECAUSE I've read them sometimes I choose to do another album's presentation for a particular lesson occasionally if I happen to like it better. I dislike generating equations. The child should be generating their own equations, but I do it differently. Because I don't like generating equations but WANTED sets of equations to use I bought the MRD fractions album because it had lists of all the equations. I also buy the Nienhuis activity sets and the like in order to use their equations. You don't need those things. I don't use MRD for follow up work, but it may seem that way because they have longer sequences for some things and I will sometimes choose to do their additional activities. The biggest example of this was with the memorization boards but that is technically primary not elementary. Mostly my MRD albums sit on the shelf after I read them. I use the Cultivating Dharma more frequently. The busier I am, the fewer albums I use and my favorite is KotU (Jessica's). If I don't understand her presentation due to some mental block of my own, I will read the same work in one or more other albums to see if their approach helps me understand.

    3. Thank you for the detailed reply. I basically have Cultivating Dharma's album. I also have 2 other AMS style albums, one is NAMTA. What confused me was that follow up work sounded so random, rather than a sequence of activities you do so the child understands the material. In checking my AMS album, i see that much more and it helps to have the details flushed out. One AMS album I have also actually lists the presentations from different threads in the order you would present them. So having all of these helps. But having to reconcile between albums is hard work.

      I feel like in my training they leave these questions to be magically answered during your internship year, where you naturally will see how a curriculum is planned and the concrete work that will be done. The internship is also itself a learning class. But I don't have this luxury to wait till an internship when I'm teaching now. So yes, I'm worried about my training as well and am actually contemplating using Jessica's album, if she provides those kind of details on scope and sequence? What would help me the most is actually someone listing follow up work in detail after each presentation so I know what materials are involved and just seeing one child's work plan for a year.

      My training actually uses the same scope and sequence in math as Jessica's. I wonder if they just borrowed from her?!

      I wonder if Jessica's training would answer my questions. I know the type of learner I am. I do better with a lot of information all laid out for me. It helps me figure out the big picture, the big idea and I then can use that big idea to guide me in making decisions when variables and new info comes out. So if I saw someone's curriculum planning for a whole year, or what a child does for a whole year, then it helps me understand the idea behind it. I won't follow it verbatim, but I would see why it's done that way.

      It's similar to that question I had about how many "work" a child typically does in a day. I realized after the long discussions with Jessica that really what I was asking for was what a normalized child behavior looks like. Before I took my training, just reading from Montessori books, I had this idea that a child kind of goes from one activity to another, quietly, concentrating, for a few hours at a time. After my observation class, I realized that is NOT typical child behavior, and I could see the range of normalization for primary kids. But I didn't observe elementary so it threw me off when I started homeschooling. Knowing the general idea and the whys really help me when I encounter things like this.

      For your Nienhuis and MRD equations, I was told similarly to just have the kids generate equations because the point is the process and not the product. And I've heard of the magical things children do when they are learning in that kind of environment. However, having those equations helps me see all the concepts that a child probably needs to have covered before they really understand that concept. Often it's having 0s in equations or introducing digit multipliers in sequence. But my album (and I assume Jessica's album) doesn't tell me this for many presentations. I don't know how I can assess otherwise if a child has mastered a concept otherwise.

      As I have the Cultivating Dharma album, do you find that you use Jessica's as the main one, and his as the backup? Just wondering if I should shell out the money and take that leap to use Jessica's album and forgo my training.

      Your explanation of how it actually works definitely helps me. I will see about doing something similar, make sense of this scope and sequence once and for all.