Category Archives: Math

A Color Sequence for Representing Number Order

Match each number a specific color, consistently – not a bad idea when making colorful learning materials for early numeracy.

1 2 3 4 5 6 7 8 9 10

1=brown, 2=red, 3=orange, 4=yellow, 5=green, 6=blue, 7=purple, 8=gray, 9=white. 10 I represent with the color for 0=black, and 12 with red, and 16 with blue.

I propose that educators use the Electronic Color Code standard and adapt it as needed. It reflects Nature as in the rainbow/visible electromagnetic spectrum, and is also the internationally accepted convention and standard to represent numbers using colors.

That is all you need to know regarding this matter :-) especially now that you can buy Real Life color-coded numeracy materials hefractionsre

If you want a deeper look at the problem of saying numbers using colors and other attempted solutions, keep reading

Schooling and Real World can be friends

In Math as with everything else STEM and education, I am of the opinion that “schooling”, defined here as the intentional process to help a student achieve learning and knowledge, should be as much as possible connected with the Real World. Those of us who do not subscribe to the “because I said so” school of teaching already agree that learning is not necessarily connected with schooling. Thus, for schooling to be have some chance to become valid, relevant, useful to learning, it has to be part of the real world, beyond its pretty red building. “School stuff” if by itself,  bad, but when connected to “Life stuff”, can be good.

A quick look at “good try”

Most illustrators of children books will make some attempt to use color. Understatement. When dealing with numeracy, this is overwhelming and quite random. Random is not actually too bad, IMHO. If it’s going to be done wrong, it better be random that follow a weak or made-up standard.

ArcolorCuisenairee there standards? well, not quite, but a valiant attempt especially fashionable from the late 60s until the 80s were the Cuisenaire rods and derivatives. The fact that they were protected with copyrights and patents (since then lapsed or lost exclusive right, at least in Canada) might have influenced their lack of adoption by others.

There is an attempt of logic at the Cuisenaire color choices, well portrayed here, but still far far short of the simple logic of the rainbow sequence. I do like some advanced relationship work that can be done, like with 12. Yet, it remains closed into itself in a school-y way, not much Real World.

The Montessori colors are as follows: 1 = red, 2 = green, 3 = pink, 4 = yellow, 5 = light blue, 6 = light purple, 7 = white, 8 = brown, 9 = dark blue, 10 = golden
The Montessori colors are as follows: 1 = red, 2 = green, 3 = pink, 4 = yellow, 5 = light blue, 6 = light purple, 7 = white, 8 = brown, 9 = dark blue, 10 = golden

Then there is the Montessori Beads. The chosen color sequence appears to be random and its minor impact in Public Schools has made for even less of a “standard”, except of course among true devotees of said tradition.

quoting an honest view from

If you are teaching in a Montessori classroom, I do recommend that you follow this color scheme because your materials to be consistent with the entire Montessori curriculum. If you are making the beads at home, the color choices are completely up to you – just make sure you are always consistent yourself.
The basis for Good Choices in schooling content

While admitting that in evidence-based schooling there is no single one-size-fits-all solution, certain principles can help us make less-bad choices. For one thing, do reference Nature in your teaching – if a good organic, biologic-dynamic solution exists, that might be better than a totally artificial alternative. Yet, an established standard or convention that is commonly understood to many people is an important thing to have. Say, a person learning a totally made-up language or script might be cute, that endeavor probably will not help much with employment and other life needs, not as much as learning a language used by many will.

When Nature deals with colors as a sequence, there’s the rainbow. As a representation of the visible fragment of the electromagnetic spectrum it is quite connected with notions of quantity – thus its use for numeracy is sort of obvious and a Good Thing. Visible Red corresponds to a smaller frequency than visible Blue.

trick question: in Nature, is blue a “cold” color? See the answer later.

colorECodeAs to established conventions, the only example that I know of a color sequence reflecting the decimal number order is the Electronic Color Code. I like it that 0 is black – i.e., the absence of color=absence of quantity. Oh, BTW, the conventional order is the geek one for digits: 0 to 9. Thus here on right, where we go 1 to 10, black stands for the later.

Of course there is also the Hex RGB code, but even you, dear polymath reader, will agree that trying to use that as a basis for pre-school numeracy is a bit too much.

Limitations disclosure

Sincere educators admit that teaching, and especially schooling, demands simplification of concepts, a (hopefully) careful choices that do not alter facts. IMHO that is necessary, which trumps “perfect” any day. Thus there is no goal – during the basics of numeracy – for the whole of multiple-band Electronic Code values. Right now I just focus on the “unit” digit for a count of parts, as on the denominator in fractions.

One request: if you are aware of some color sequence used in school materials that makes any sense, please contact me? I am not afraid to be a pioneer with a new idea, but it does worry me some that this matter has not had much attention paid to it – and we are talking Sagan numbers here (“billions and billions”) of school items that are maybe not “wrong”, but certainly not “right”, wasting the opportunity to be Real Life relevant and useful because they use some random color scheme, when they could do better… Maybe there has been another initiative out there that deserves being THE color sequence for numeracy purposes?

Color my learning!

My first product was the Fraction Circles.

whole 1/1 -> brown, 2/2 red, 3/3 red, etc. Available for purchase here

12 Fraction Circles set – available for purchase in ATXinventor’s Etsy shop

I will soon be making numeracy rods

As to blue, it is of course the color of the hottest stars. See the Hertzsprung-Russell diagram. Red happens to be the colder star color…

(the comments system in my WordPress appears to be borked right now. Please use the “contact me” link on top of the page instead)

Introducing the STEM PK Angle Tools

constructing01very important:
Every child is unique – the path to learning
is not the same for all.
Please revise these guidelines as needed
to suit them to your individual child.
Thank you!

General Guidelines

Only introduce vocabulary as new concepts become intuitively understood. Better not to rush with words: you  want to avoid memorization without understanding. One of these days I will attempt to go through the whole process with some child, using no words at all – remains to be seen…

On that same sense, do not worry about a complete immediate understanding of each concept – this is a “construction of knowledge” + “agricultural” approach – provide good soil, open sun, water, and the structure to pull into, and they will amaze you as they grow and flourish!

For your information we will give you, as end notes, some background of the many things that the child is learning that are NOT part of the “official” Lesson. Good for you to know, but, again, they are not the focus – the paced and guided exploration and individual engagement is what you want, because that is what opens understanding, and, ultimately, learning.

Most importantly, never forget that “teaching” is not at all the same as “learning”. Make it possible and enjoyable to learn – you and your child will accomplish much more that way than by any amount of teaching you do…

Materials in your set

STEM PK Angles Tools
STEM PK Angles Tools
  • 3 single-angle black-orange angle vertices
  • 4 single-angle black-yellow angle vertices
  • 5 single-angle black-green angle vertices
  • 6 single-angle black-blue angle vertices
  • 6 four-angle black-4-color angle vertices
  • 4 color geometric shapes – triangle, square, pentagon, hexagon
  • 6 segments (that is our name for the straws – you receive extra, as these get easily damaged, but you seldom use more than 6 at a time)
  • felt working mat
  • presentation tray

To avoid loss of attention and focus, I suggest to keep all materials out of sight of the child until you use them.


Orange (three sides)

Prepare your materials on the tray:

  • the orange triangle
  • 3 segments
  • the 3 black-orange angle vertices
  • cover it all with the mat.

Place the tray in your designated work area, not between you and the child but to one side. Some people prefer to do “work” on a table, some do it on the floor, etc.  I have no suggestion except to set a routine and reasonably stick to it.

Set the work mat in front of the child.

Place the orange triangle  on the mat generally in the center. Holding it with one finger of one hand, follow the contour with one finger of the other. Carefully (sharp!) touch each vertex.

While still holding the triangle with one finger, motion or verbalize for the child to follow the contour and touch the corners. If necessary, very gently guide the child as he tries, though try to avoid intruding too much – better slowly and self-paced than fast empty success.

Release the triangle. If necessary help the child to hold it in place with one finger, and use the other hand to follow the sides and touch corners.

If the child is focused in the triangle, you may hold it from the mat and hand it to the child for further manipulation. Touching the flat surface from side to side is a good thing, but not necessary yet.

Ideally the child will imitate what you did, but do not be worried or concerned if he does not fully, as long as his attention is there to some extent. If the child pays no attention at all, as in not even touching the triangle by himself, he is not ready. Try again in a couple hours, or days. Same if his actions are clearly not conducive to a lesson, like biting the triangle, or using it to hit the work surface.

Now gently ask for the triangle and place it back on the mat, but to one side. Neatly place on the mat central area the angle vertices, then the segment. If and only if the child asks what they are or uses other words do use the words “segment” for the straws and “angle vertex” (or just “angle”, if it must, but better if you go all the way to “angle vertex”) for the hub connectors, or help the child get used to those words. Otherwise just move on without calling them anything. Just avoid naming them “straws” or “pieces” or anything such.

Pick up one of the angle vertices. Pick up one of the segments. Gently attach the segment to the black connector. It is somewhat important that you do use the black connector and not a color connector, reasons explained later. Place the assembly back on the mat. Motion or verbally encourage the child to proceed.

Now comes the most important part. Usually the child will pick up each of the remaining angle vertices and attach one segment to the black connector. Or maybe not. After each angle vertex has received one connector, there often is a pause. For me, this is the most marvelous moment of all – the child is reflecting, the first time in his life, about this curious task he is supposed to do, which he did, but  something else is possible – here are the other connectors with nothing, and the segments still have one unused ending… At some moment the child will have a glorious “aha!” moment, and will attach that segment end to one of the color connectors, and proceed to complete all of those, and to his surprise, notice that here is a triangle!

I strongly suggest not to try to rush this. If the child does not follow through for a few minutes, gently guide him to another, different activity and try again in a couple days maybe. You absolutely DO NOT want to intervene or cut through or do it for the child – If you do, you might feel you completed the lesson, and you will be right. YOU completed the lesson, but the learning process DID NOT HAPPEN.

Of course there are individual variations, for example the younger child who has seen an older sibling do it already. To avoid that, I respectfully would suggest to tackle this lesson with at least some minor amount of privacy, but, again, don’t make a big deal out of that.

The process of transferring mentally from one paradigm – one segment fits one angle vertex – to the other – each angle vertex has TWO segments – is too valuable to lose or rush. The third one –  vertices and segments form geometric shapes – is useful, but not as momentous as the previous one

Actually, there is is the “deep down” lesson: an angle is the figure formed by two segments meeting in one vertex (Wikipedia article). If you get that intuitively, angles will be your friends and you will have a much easier time with them from now on.

Let the child manipulate the triangle he has completed, then gently help him disconnect the parts, put them back on the tray, cover it with the work mat, put the whole away.

Later, let the child do the lesson by himself – do not worry at all that he seems to pay little attention to the orange plastic triangle – HE knows that there is something powerful happening regarding those segments and angle vertices! Maybe, if it hasn’t happened before, you can introduce those words, but you do not need to, actually I wouldn’t, unless the child does have a need to use them.

Yellow (four sides)

Now we will go for the four-sided shape, the square.

Same as above – just remove the Orange set from reach, and follow through with the Yellow set, of course this time with four segments

Green (five sides)

A variation to the previous two lessons: leave on the tray the plastic figures for Three and for Four. Make no comment about those, just follow through the lesson.

This time the lesson will probably go quite fast, the routine has been established. Good moment to work on other abstractions, like those fancy words. Angle, vertex, vertices, segment, side, face, triangle, square, pentagon… Some children enjoy enormously handling new words, and the strange plural form of “vertex” might be a source of enjoyment. Same as anything else. please  no forcing what doesn’t build in a reasonably healthy dynamic.Don’t forget it, “normal” people don’t do angles until 3rd Grade anyway… no need to rush!

(Author’s note: I will be adding to this later. need to catch up with my damaged freezer…)