Representation & communication
Move between concrete/pictorial/symbolic; use precise vocabulary; link multiple representations.
Representation is the many different ways we can show the same mathematical idea; communication is saying and writing it so clearly that another person understands exactly what you mean.
What it means
A single idea in maths can wear many outfits. Take the number six. You can show it as six real counters you can touch (that is concrete), as a drawing of six dots or a filled bar (that is pictorial), or as the symbol 6 or 4 + 2 (that is symbolic). All three are the same amount — only the costume changes.
Being good at maths partly means being able to move between these three forms and to see that they agree. If your blocks say six but your written answer says five, one of them is wrong, and comparing the representations helps you catch it.
Communication is the other half. It means using precise vocabulary — calling a corner a vertex, a flat surface a face, the answer to an addition a sum — so that words carry exact meaning instead of vague gestures. “The pointy bit” might mean several things; “the vertex” means one. Good communication also means linking representations out loud: “This tower of ten cubes is the same as writing 10, which is one ten and zero ones.”
None of this is decoration. When you can translate an idea into a picture, a symbol and a sentence, and show they all match, you understand it far more deeply than if you only memorised one form.
Worked examples
1 — One idea, three costumes. The fact “three plus two makes five”:
| Form | How it looks |
|---|---|
| Concrete | ●●● put next to ●● |
| Pictorial | a bar split into a 3-part and a 2-part |
| Symbolic | 3 + 2 = 5 |
2 — Translate a symbol back to a picture. Given 2 × 4, draw it as an array: 2 rows of 4 dots.
● ● ● ●
● ● ● ●Count them: eight. The picture is the multiplication, not just a hint about it.
3 — Say it precisely. Point at a triangle. Loose: “it has three pointy bits and three lines.” Precise: “it has three vertices and three sides (edges).” Same shape, sharper words.
The generative-art connection
Generative art is representation you can watch being built. In the dot-multiplier tool, you start with one dot and a number — say eight — and the tool multiplies that dot into a symmetric starburst of eight. The picture on screen is the number: you can literally count the arms, so the symbolic “8” and the concrete cluster of dots become the same object in front of your eyes. Change the number and the art changes with it, so the link between symbol and picture is never abstract — it is the thing you are making.
Tools like Polypad and the Math Learning Center apps do the same across other ideas: drag concrete tiles into a group, and the matching number updates alongside, keeping all three representations visibly in step.
Common misconceptions
- “The symbol is the ‘real’ maths and the picture is just for babies.” Backwards. The picture and the symbol are equally valid representations of the idea; experts move fluently between them, and the picture often reveals why an answer is true.
- “Any words will do.” Vague words hide misunderstandings. Precise vocabulary (sum, difference, vertex, equal) is what lets two people be sure they mean the same thing.
What a learner should be able to solve
The benchmark for mastering this concept — the problems a child at this stage is expected to be able to work through.
-
Pick any number from `1` to `10`. Show it in THREE different ways: as a group of real things you could touch or draw (dots, fingers, counters), as a picture (a filled bar or a row of shapes), and as a written symbol. Then say one sentence explaining why all three show the same amount.
Answer
Any consistent choice is correct. Example for `7`: seven counters; a bar cut into 7 equal parts; the symbol `7`. Sentence: "Each one has seven, so they all mean the same number." Success: all three forms match the same quantity AND the child links them in words.
Art hook Canvas: type a number 1-10; the sketch draws that many counters scattered, that many bar-segments in a row, and prints the big numeral — three panels updating live so the child watches one number wear three costumes.
-
Someone shows you `4 + 3 = 7` written down. Draw or describe a picture that shows this exact fact, and use precise words to say what the `4`, the `3`, the `+` and the `7` each mean.
Answer
Picture: 4 dots next to 3 dots making 7 dots (or a bar of 4 joined to a bar of 3). Words: "`4` and `3` are the parts I am joining, `+` means put together (add), `7` is the total, also called the sum." Success: picture matches the symbols and the vocabulary (part, add, total/sum) is correct.
Art hook Canvas: draw two dot-clusters side by side (sizes from two number boxes); a glowing bracket sweeps under them and prints the sum. Change either number and the clusters and total redraw.
-
Look at this loose sentence about a cube: "it has some flat bits, some lines and some corners." Rewrite it using precise maths words, and give the exact number of each.
Answer
"A cube has `6` faces (flat surfaces), `12` edges (lines) and `8` vertices (corners)." Success: correct vocabulary (face, edge, vertex) paired with the correct counts 6/12/8. (Check: faces - edges + vertices = 6 - 12 + 8 = 2, Euler's rule.)
Art hook Canvas: a slowly rotating wireframe cube where hovering highlights a face gold, an edge blue, a vertex red, and prints the word plus a running tally (6 faces, 12 edges, 8 vertices).
-
Your friend and you both work out `20 - 8`. You get `12`; your friend gets `13`. Instead of just arguing, describe TWO different representations you could each build to check who is right, and say what the correct answer is.
Answer
Correct answer: `12` (your friend's `13` is wrong). Checks: (1) a number line — start at 20 and hop back 8, which lands on 12; (2) counters — take 20, remove 8, count what is left = 12; (or use the inverse fact `12 + 8 = 20`). Success: names at least two valid representations and identifies 12.
Art hook Canvas: a horizontal number line 0-20 with a dot that hops backward 8 steps on a button press, each hop drawn as an arc, landing lit up on the answer.
Training exercises
Practice problems, easier first, that build toward the bar above. Each doubles as a seed for an interactive artwork.
-
Here is the symbol `5`. Show the same amount by drawing dots. How many dots did you draw?
Answer
Five dots. The symbol `5` and five dots are the same number in two forms.
Art hook Canvas: type a numeral, sketch draws exactly that many dots in a neat row — the simplest symbol-to-picture translator.
-
Here are four dots in a row: `● ● ● ●`. Count them, write the number symbol that matches, and say the number word out loud.
Answer
`4`, said "four." This goes picture -> symbol -> word, the reverse direction of counting out dots from a symbol.
Art hook Canvas: dots appear one at a time with a soft click; when they stop, the matching numeral fades in beneath them.
-
Match each picture to its symbol. Picture A: three squares in a row. Picture B: a bar split into two equal halves with one half shaded. Symbols to choose from: `3` and `1/2`.
Answer
A -> `3`; B -> `1/2`. A shows three whole things; B shows one of two equal parts shaded.
Art hook Canvas: two draggable picture cards and two symbol targets; a card snaps and glows green when dropped on its matching symbol.
-
Show the number `six` in as many different ways as you can think of (aim for at least three). For example, one way is six fingers.
Answer
Any three valid forms, e.g. six fingers, six dots, the symbol `6`, `4 + 2`, `3 + 3`, a 2-by-3 array, six tally marks. All represent six.
Art hook Canvas: a grid of mini-panels each rendering six a different way (dots, array, tally, dice-face, bar, numeral) so the child sees many costumes of one number at once.
-
A picture shows two rows of five dots. Write a multiplication symbol that matches it, then a repeated-addition symbol that matches it too.
Answer
`2 × 5` and `5 + 5` (both equal `10`). The same array can be read as multiplication or as repeated addition.
Art hook Canvas: a 2-by-5 dot grid; one button labels it `2 × 5`, another sweeps the two rows lighting them and prints `5 + 5` — same grid, two symbols.
-
Spot the mistake. Ben says: "My blocks make a tower of nine, so I'll write the symbol `6` because a nine looks like an upside-down six." What went wrong, and what should he write?
Answer
The symbol must match the AMOUNT, not the look of the digit. Nine blocks means he should write `9`. Representations have to agree on quantity.
Art hook Canvas: a tower of nine blocks beside an editable numeral; if the typed digit doesn't equal the block count the numeral shakes red, and turns green on 9.
-
Read this sentence about a triangle-based shape and turn it into precise maths words: "the pointy solid has a flat bottom, some slanted flat parts, and a point at the top where the lines meet."
Answer
"A pyramid has a base (a face), triangular faces meeting at an apex, which is a vertex where the edges meet." Key precise words: face, edge, vertex, base, apex.
Art hook Canvas: a rotating pyramid; tapping the tip labels 'apex/vertex', tapping a slanted side labels 'face', tapping a line labels 'edge'.
-
A number line is drawn from `0` to `10`. Explain, in a clear sentence, exactly where you would put a dot for the number `7` and why.
Answer
Put the dot 7 steps to the right of `0` (the same as 3 steps left of `10`). It sits between the `6` mark and the `8` mark, right on the `7` mark, because 7 is 7 units from zero. Success: locates 7 correctly and justifies it by distance from 0.
Art hook Canvas: a number line 0-10; click anywhere and a dot snaps to the nearest whole mark and prints its value — turn it into a 'hit the target number' mini-game.
-
The symbol is `3 + 4`. Build it as a picture AND write it as a different-looking symbol that gives the same total. Do they agree?
Answer
Picture: 3 dots joined to 4 dots = 7 dots. Another symbol: `4 + 3` or `7`. Yes, all equal `7`; addition can be shown as a picture and written in more than one way.
Art hook Canvas: two dot piles you resize; a bracket prints the sum, and a toggle flips the order to show `3 + 4` and `4 + 3` give the same picture and total.
-
Reasoning challenge. Your teammate wrote `12` but you can't tell if they mean twelve counters, the number twelve on a line, or one-ten-and-two-ones. Describe two DIFFERENT representations of `12` and explain how they help make sure everyone means the same thing.
Answer
E.g. (1) a group of 12 counters (concrete/quantity); (2) 12 on a number line, 12 steps from 0 (position); or (3) one ten-rod and two ones (place value). Linking them removes doubt: all point to the same value 12. Success: two distinct valid representations plus a clear reason.
Art hook Canvas: three synced panels for 12 — a heap of counters, a marked number line, and a ten-rod-plus-2-ones — all lighting together when you press play.
-
Open-ended maker task. Choose your favourite number from `1` to `12`. Design a tiny 'number portrait': show it as dots arranged in a pattern, as a symbol, and describe the pattern using precise words (row, column, array, equal, total).
Answer
Any well-chosen design is correct. Example for `12`: a 3-by-4 array of dots ("3 rows, 4 columns, 12 dots in total, an array") plus the symbol `12` and maybe `3 × 4`. Success: picture, symbol and precise words all agree on the chosen number.
Art hook Canvas: pick a number; the sketch offers rectangle-array layouts for it (e.g. 12 -> 1×12, 2×6, 3×4) and prints the matching `rows × columns` symbol under each — a number-portrait generator.