Math Symbol Hierarchy
The Unicode Standard 7.0 has 2311 math symbols not including the ASCII letters and the standard combining marks like tilde, which are also used in math zones. Such a large number of symbols can be confusing if not intimidating to people who don’t use mathematics professionally. Even those who use math a lot don’t use most of the advanced symbols. The reason so many symbols are in Unicode is to fulfill the promise that all math symbols in modern print be defined, thereby allowing the Unicode math symbol set to be used as a basis for technical publications. This is a very worthy goal, but we don’t want it to get into the way of students learning math or more advanced folks that only need to use a fraction of the symbols. Accordingly it’s desirable to define some subsets that work for people of varying levels of mathematical expertise.
Three general levels seem pertinent:
1) Basic (arithmetic, fractions, i.e., pre algebra):
+/÷×*±∓([{}])=≠≈~≅≃∝<>≤≥≪≫←↑↓→↔
2) Intermediate (basic + algebra, calculus, set theory)
Alphabetics 
a..z A..Z ℂℕℙℚℝℤℵ Γ∆∇ΘΛΞΠΣΦΨΩ αβδγεϵζηθϑκλμνξπρστυϕφχψω 
Math alphabetic styles 
Italic, bold, bold italic, Fraktur, openface, script 
Operators 
•∘⊗′†!∴∵⇐⇒⇓⇑⇔∞≡∀∃∄∂∅∠∡∢∟∧∨¬∩∪∊∍∈∋⊂⊃⊆⊇≶⟨⟩∥⊥ 
Nary operators 
∫∮∏∐⋀⋁⋂⋃ 
Accents 
′ ̀ ́ ̂ ̃ ̅ ̆ ̇ ̈ ⃗ 
Not symbols 
Many symbols can be negated by overwriting them with a /, e.g., = → ≠ 
3) Unicode (all Unicode math symbols)
see Unicode Technical Report #25 Unicode Support for Mathematics and associated data files
The basic set is useful generally and should suffice for K6 students, who have no need of integrals, summations, products and the like. The intermediate set should suffice for high school and college students, who have no need for most of the rarer mathematical symbols. In our user interfaces we can have options to expose only basic symbols, intermediate, or on demand, the full set. For any level, a list of the most recently used symbols is helpful since people almost always use a subset of the Unicode math symbols.
Within the sets, other classifications can be used to show symbols that have substantial similarity to one another. For typographical and syntactic purposes, math operators can be classified into the following general categories:
Class 
Meaning 
Example 
N 
Normal  includes all digits and symbols requiring only one form 
0..9 § 
A 
Alphabetic 
a..z 
B 
Binary 
× ÷ 
C 
Closing  usually paired with opening delimiter 
)]} 
D 
Diacritic 
̂ ̃ 
F 
Fence  unpaired delimiter (often used as opening or closing) 
 
G 
Glyph_Part  piece of large operator 
⎩ 
L 
Large  nary or large operator, often takes limits 
∫ 
O 
Opening  usually paired with closing delimiter 
([{ 
P 
Punctuation 
; : 
R 
Relation  includes arrows 
= → 
S 
Space 

U 
Unary  operators that are only unary 
∀ ∃ 
V 
Vary  operators that can be unary or binary depending on context 
+ − 
In addition, these classes can be subclassed, e.g., grouping all the symbols that have equality in them like =≠≈~≅≃≤≥. There are often multiple ways to group symbols, e.g., ≮ would be included in a <like symbols group like <≮≤≪∈∉∊⊂⊄⊆ as well as in a notlike symbols group like ≮≠≯∄∉∌≁≄≇≉≭≢≰≱≸≹⋡⊄⊅.
You can see a variety of math symbol collections by searching the Internet for mathematical symbols. For example, in addition to Unicode Technical Report #25 and its associated data files, there’s the article Mathematical operators and symbols in Unicode. That article displays the Unicode code charts for all the math symbol blocks in Unicode along with math symbols in other blocks. Scanning through it provides a quick way to view the Unicode math symbols. Other articles like Math Symbols list many math symbols and include descriptions of how the symbols are used.
Subclasses of math symbols are helpful for organizing symbol galleries such as in the Office math ribbon and in math touch screen keyboards. The math ribbon symbol gallery offers a superset of the intermediate class above broken into seven subclasses: Basic Math, Greek Letters, LetterLike Symbols, Operators, Arrows, Negated Relations, Scripts, and Geometry. The ribbon’s Basic Math class is more extensive than the basic level given here. Touchscreen keyboards can include both symbol galleries as done for emoji collections as well as surround menus that appear when you hold down a key like <. The possibilities that such keyboards offer for efficient math input are very intriguing. A keyboard based on the basic level set defined here would be ideal for many scenarios in addition to K6 math. Keyboard galleries could be used for more advanced symbols. In principle the entire Unicode math symbol set could be accessed in a way that’s not intimidating to K6 students.