specialty math manipulatives

Recently, I came across and advertisement for a particular Montessori manipulative: the Pink Tower, and it got me thinking about math manipulatives in general but especially the costly specialty manipulatives of this kind:

The great thing about the pink tower is that each cube is exactly one

centimeter smaller in each direction than the prior cube -- with the smallest
cube being a perfect cubic centimeter. When the tower is aligned on one corner,
the smallest cube can "walk" up (or down) the tower and can slide around two
sides of each of the other cubes -- the perfect alignment is a point of interest
for the child.

Those kinds of sales pitches sound so appealing, don't they? But in terms of evaluating their actual value, it can help a lot to step back and look at the underlying principles. The Montessori-Method "Pink Tower" communicates, with proper instruction, a set of mathematical principles, but those same principles can be discovered naturally by a child playing with any set of unit blocks - and sets of blocks have a far, far wider range of applications than the Pink Tower, which is used for... what a dozen exercises at most? The graduated blocks of the Pink Tower can't be used for much beyond building the tower and a few variations because there's only one of each, but a set of blocks - ah! Blocks can be used to build a million different things *and* used to represent other things. The
Pink Tower is only ever the Pink Tower.

The best math manipulatives in the world are unit blocks and some of the best unit blocks ever made are, frankly, Legos. Yup, a big commericial toy wins the day, I'm afraid. With legos you could build your own Pink Tower... but why bother? Kids will learn exactly the same concepts playing with the blocks in their own way - or with other types of unit blocks. Not every child loves Legos.

The significant aspect to consider, when it comes to blocks, is that they are *unit* blocks - there are regular units of increase between one block size and the next. Most sets of plain wooden blocks are unit blocks because it's utterly logical to make them that way. (And wooden blocks are great for toddlers and younger children, a muuuuuuuch better investment than a Pink Tower, and probably a tenth of the cost, even if you buy a really nice set!)

the light reflecting differently off the different flat planes of the tower
when the child walks around it to admire his work is a point of interest.

This is a point I find, frankly, amusing. For a theoretical system that shies away from plastic, it's a little ironic that one of the selling points of the a toy is that it's smooth and shiny.

The tower has ten cubes because our number system is based on ten, and the
knobbed and knobless cylinders, pink tower, brown stair, and red rods are all
meant to reinforce that.

Okay, my math button has been pushed! Since base 10 is everywhere in the real world, why would you need to teach it? This is a case of Montessori toys being made to turn a sterile classroom into a substitute for a rich home environment. A noble cause, but not particularly applicable to home educators. There's no real need, if you live in any developed nation, to "reinforce" base ten. Kids figure it out (arguably they derive it) without specific lessons in the subject.

Besides, in the real world we use multiple bases every day, not just base ten. Consider: How many days in a week? How many seconds in a minute? And if you live in a country like the US which still uses inches and ounces....? Reinforcing base ten, under those sorts of circumstances, is counter-productive, and lessons with reinforce base ten have the side effect of then requiring additional lessons to give kids the tools they need to work with other bases. Montessori does this better and sooner than mainstream education, it is true, but at the cost of needing yet more, highly specific teaching tools. The Pink Tower is a one trick pony.

Now Legos, by contrast, do something that's far better than reinforcing base ten, they give kids a format to construct mathematical concepts in multiple bases simulateously and in the process give kids tools with which to derive the principles of mutiplication and division/fractions. How do you build a wall when you have a mix of 2x2, 2x3, and 2x4 blocks? That's the most basic Lego problem, one with which every Lego builder is familiar, since those are the most common building blocks - and it involves the principles of converting from one base to another... the same principles which are needed to add fractions.

Also, a very important part of this material is its baric and sterognostic
value -- "Montessori-speak" for the way the weight and feel of each piece varies
in a precise order -- the largest cube is 1000 times the weight of the smallest

Once again, I'm amused. The above is going to be true of any set of wooden unit blocks where the pieces are solid.

One of the problems with placing too much value on this idea, though, is that in real life objects do not, in fact, increase predictably in weight based on size - composition and construction matter. An empty 55gal drum weighs substantially less than a fairly small farrier's anvil. A 2x4 made of oak weighs more than one made of pine or poplar. A laundry basket full of laundry weighs and carries differently than a laundry basket with a cat in it.

So like base ten, this particular Montessori lesson is one that needs a hundred other lessons to modify and give kids a picture of the real world - lessons in empty vs full, lessons in pouring, lessons in lifting chairs and pulling drawers - because all the specific materials can do is give simple lessons. They're all one trick ponies, which is a big, big percentage of the cost - it's not just a matter of shelling out for one great toy but of getting a hundred items that you won't need a year

young children are acutely attuned to order; the sensorial materials provide a
very orderly environment in which they can notice very small differences
(something that most toys and the natural world don't often provide).

While the first part of that statement has some validity, the last, parenthetical statement is false - it sounds good, but it's not true at all. The natural world is full of order and small differences - heck there's a Montessori material which compares leaf shapes and sizes! Show me a bouquet or roses or a litter of kittens and I'll show you order and small differences.

And since commercial toys are made to appeal to children (unlike many "educational" toys which are made to appeal to parents) they include a lot of order and small differences. As much as parents deride this aspect, it's part of what makes some toys successful. Some examples of popular commercial toys designed around those concepts: My Little Pony, Littlest Pets, Barbie, Hot Wheels, American Girl dolls, Legos... and then there are toys and themes with more substantial differences like Pokemon, a wildly successful series of games, toys and shows because it so very much appeals to kids' interest in order and classification.

The basic principles of Montessori are excellent: learning is intrinsic for all people; learning flows best when it is chosen by the learner; observe the child to discover his or her needs and priorities; see behavior as a way of communicating needs; set the child up for success; when something isn't working, change the environment and recognize that you, the adult are part of that environment; punishments and rewards get in the way of ethical development, so avoid them. All of those principles have been re-discovered and derived again and again in the century and more since Maria Montessori first articulated them - not merely by wild eyed dreamers, but by serious scientific research into education and learning.

All the techniques and materials special to Montessori are just tools to help teachers and classrooms mimic the benefits of a rich, responsive home environment. It can be useful to read about the techniques and materials as a way of learning more about how children develop and learn - I've learned an enormous amount by reading the "whys" of various apparatus. The trick is not to get bogged down in the specifics but to discover the principles. The Pink Tower is pretty, but the principles are everywhere and human beings are wired to understand them - otherwise human kind would have never invented mathematics, much less the Pink Tower, to begin with.

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