Speak Maths

Maths Language Course

Learn to read and write mathematics as a language: A1 to C2, in four skills. Reading, saying, writing and explaining, with no calculating needed.

Drafts: the levels, texts and checks are not final, and nothing here is an official document. The levels are CEFR-inspired. They are not CEFR levels and nobody has checked yet that the steps are equal.

Drafted with AI and checked by an independent AI. Not yet reviewed by a person.

Every unit is one narrated lecture: watch the diagram, listen to the voice, and do the practice exercises at the end of the chapter.

1. A1 Numerals, the four operations, = and the order signs, brackets, letters, short statements.

   1. 01 Numerals and Place ValueI can read and say a number of up to four digits, and tell what each digit stands for.
   2. 02 The Four Operations and How We Say ThemI can say + − × ÷ in words, in the right order, and write them from words.
   3. 03 The Equals SignI can say what = tells me and find it in a short statement.
   4. 04 The Order SignsI can read and write < > ≤ ≥ and chains like a < b < c.
   5. 05 BracketsI can say what a pair of brackets holds together and read the group as one quantity.
   6. 06 Boxes and LettersI can say that a box or a letter stands for a number, and read 'Let … stand for …'.
   7. 07 Units and QuantitiesI can read and say a number with its unit as one quantity, and write one.
   8. 08 Reading a Number SentenceI can read a whole number sentence aloud, in order, and choose the words that match it.
   9. 09 Writing a Number SentenceI can write a spoken number sentence in signs, with the right sign for "more than", "less than", "is equal to", "is not equal to", "at least" and "at most".
   10. 10 Fractions as NotationI can name the top and bottom of a fraction, say it aloud, and tell a fraction bar from a division sign.
   11. 11 Short Statements and What They ClaimI can tell a claim from a question and say whether a short statement is true or false.
   12. 12 Reader: Five Short TalesI can read a short tale that mixes words and signs, and say what its statements claim.

2. A2 Parsing, powers, letter roles, formulas as sentences, for all and there is in words.

   1. 01 Precedence: What Attaches to WhatI can bracket an expression to show what attaches to what, and tap its main sign: × and ÷ attach before + and −, brackets come first, and signs of the same rank go from the left.
   2. 02 Hidden MultiplicationI can tell when things written side by side are multiplied, such as two letters or a number before a letter or a bracket, and when they are not, such as digits next to digits; and I can write a product with no times sign.
   3. 03 Powers and RootsI can say a power and a square root, show what an exponent reaches, and tell apart an exponent on one symbol from an exponent on a whole bracket, a minus before a power from a minus inside it, and a root over one term from a root over two.
   4. 04 Letter RolesI can tell whether a letter is an unknown, any number, a label or a parameter, using the words around it.
   5. 05 Formulas as SentencesI can read a formula as a sentence, name the role of each letter, say it aloud, and write a formula from a description of what depends on what.
   6. 06 The Fraction Bar and DivisionI can tell what a fraction bar groups, read a one-line division and see when it is unclear, and write a division with a bar or with brackets.
   7. 07 Percent, Ratio and RateI can read % as "per hundred", a:b as a ratio and a slash between units as "per", and write each one from a sentence.
   8. 08 Absolute Value and the BarsI can read |x| as the distance from zero and |x−3| as the distance between x and 3, tell bars from brackets, and write them from words.
   9. 09 Points, Pairs and IntervalsI can tell when (a,b) is a point, an ordered pair, an interval or a greatest common divisor, and write a point or an interval from words.
   10. 10 “For All” and “There Is”I can read and write “for every” and “there is” in words and with the signs ∀ and ∃, and I know that one counter-example breaks a “for every” claim but one example does not prove it.
   11. 11 Labels in Geometry: A, B, CI can read the names of points, segments, angles and triangles in a figure (A, AB, ∠ABC, △ABC), also with Persian labels, and I know that the letters are names, not numbers to multiply.
   12. 12 Digits and DirectionI can change between the digits 0–9 and ۰–۹, write the decimal mark in both, and read a formula inside Persian or English text from its left, and I can spot a formula that has been turned around.
   13. 13 Saying It: Formulas in English and in PersianI can say x²+1, (x+1)², ½, a/b, √x, |x| and 2(x+3) in words in English and in Persian, in a short and a long way, and write them from dictation.
   14. 14 Reader: Five Short Texts with FormulasI can read a short text that mixes words and formulas, and say what its formulas, ratios and labels claim.

3. B1 Functions, sets, intervals, quantifiers, definitions, short proofs.

   1. 01 Expressions as Objects, and EquivalenceI can tell an expression (a name for a number that depends on a letter) from an equation (a sentence that claims something), and I know that two differently written expressions can name the same number for every value of the letter.
   2. 02 Functions: f, f(x), f(2), f: A → BI can read f(x) as “the output of the rule f at the input x”, tell f from f(x) and f(2), and read f: A → B and x ↦ x² as statements about where inputs and outputs live.
   3. 03 f², f⁻¹, f(f(x)) and the Order of CompositionI can read f(g(x)) and f ∘ g with the right rule acting first, read f⁻¹ as the inverse rule and not as one over f, and know that f² has two conventions and the text must say which.
   4. 04 Sequences and Subscripts: Index or LabelI can read a_n and a_{n+1} as terms in given positions of a sequence, read formulas such as a_n = 2n and a_{n+1} = a_n + 2, and tell an index from a mere label such as P_A.
   5. 05 Sets and Set-Builder NotationI can read and write sets in braces and with {x ∈ A : P(x)}, and say what ∈, ⊂, ∪, ∩ and ∅ claim.
   6. 06 Intervals and Their BracketsI can read and write intervals such as [a,b], [a,b), ]a,b[ and (a,b), know which ends are included, and tell an interval from a point.
   7. 07 Inequalities and Solution SetsI can read an inequality as a claim about x, say which numbers its solution set holds, and write that set with braces or as an interval.
   8. 08 Logic Words: And, Or, Not, If … Then, IffI can read and write statements joined by and, or, not, if … then and “if and only if”, in words and with ∧, ∨, ¬, ⇒, ⇔, and I know what each one claims, including that mathematical “or” allows both.
   9. 09 Quantifiers in Statements: ∀∃ and ∃∀I can read and write statements with two quantifiers, I know that ∀x ∃y and ∃y ∀x say different things, and I can tell how far a quantifier reaches and which collection it is about.
   10. 10 Implication: Direction, Converse, Contrapositive, NegationI can tell the direction of an if-then, write its converse and contrapositive, and negate statements with ∀, ∃ and if-then, and I know that a claim and its contrapositive say the same while its converse does not.
   11. 11 Definitions and What They LicenseI can read a definition such as “n is even if n = 2k for some integer k”, find the new word and its condition, know that “if” means “if and only if” there, and say what the definition lets me write in each direction.
   12. 12 Probability Notation: P(A), P(A | B), E[X]I can read P(A), P(A | B) and E[X] as sentences, say them aloud, write them from a sentence, and explain what they claim, without computing any probability.
   13. 13 Statistics Notation: x̄, σ, ΣI can read x̄, σ and Σ as words, read a sum with its start and stop as “add up over”, and tell small σ from capital Σ, without computing any statistic.
   14. 14 Vectors and Matrices as NotationI can read bold and arrowed letters as vectors, say A_ij as “the entry in row i and column j”, tell A transpose from a power, and write these from a description, without doing any matrix arithmetic.
   15. 15 Statement TypesI can tell whether a statement says that something exists, gives a bound, is an identity or is an equivalence, say it, and write one of each from a plain description.
   16. 16 Reading a Short ProofI can find the hypothesis and the conclusion of a claim, say which fact each line of a 3 to 5 line proof rests on, and say in a few words what the proof shows.
   17. 17 Writing a Definition and a ClaimI can write a definition of a new word in words and signs, write a claim with its hypothesis and conclusion, and tell the two apart.
   18. 18 Reader: Five Short Texts at B1I can read a short text that mixes words and formulas — a definition, a claim, a proof, a sheet of formulas, a sequence — and say what it claims and what it uses.

4. B2 Sums, limits, integrals as sentences, proof moves, overloaded signs.

   1. 01 Σ and Π as SentencesI can read a sum or a product as a sentence (index, start, stop, body), say it aloud, and write one from a plain description, and I know which letters are placeholders.
   2. 02 Limits as SentencesI can read lim notation and the ε–δ sentence as a sentence and as a game, say who chooses what, and write the pieces, without evaluating any limit.
   3. 03 Derivative Notation: f′, dy/dx, ∂f/∂x, ḟI can read the four common ways of writing a derivative, say them aloud, write each from a description, and tell where they conflict with other uses of the same marks.
   4. 04 Integral Notation: ∫, ∬, ∭ as “Add Up Over a Region”I can read ∫, ∬ and ∭ as “add up over a range or a region”, say what the bounds and the dx name, and write an integral from a plain description, without evaluating it.
   5. 05 Nested Quantifiers and Their NegationI can read statements with several quantifiers in the right order, say who depends on whom, and write the negation by flipping each quantifier.
   6. 06 Proof Moves: Assume, Suppose, WLOG, It SufficesI can say what a proof is doing when it says “suppose”, “without loss of generality” or “it suffices to show”, and I know which of them claims something, which changes the goal and which hides a symmetric case.
   7. 07 Proof by Cases and by ContradictionI can read a proof by cases and a proof by contradiction, say what each one assumes and what it must reach, and check that the cases cover everything.
   8. 08 Induction as a Text PatternI can find the base case, the induction hypothesis, the inductive step and the conclusion in a proof by induction, and say what each one claims and what it does not claim.
   9. 09 Overloaded Signs: |·|, ′, ∼, ⊂, Δ, Hats and Bars, e, iI can say which meaning a many-meaning sign has in a given text, by looking at the object beside it and at the definitions, and I do not read every sign with the first meaning I learned.
   10. 10 Convention Clash: the Same Idea in Two Fields and Two CountriesI can recognise when two texts write the same idea in different ways (intervals, ℕ, log, decimal mark, i and j), say that neither is wrong, and find out which convention a text follows.
   11. 11 Linear Algebra NotationI can read and say Aᵀ, A⁻¹, ‖x‖ and ⟨u,v⟩, and I know what kind of object each one names: a matrix or a number.
   12. 12 Probability Notation at B2I can read and say P(X ≤ x), X ∼ N(μ,σ²), E and Var, and I know which letters are random and which are fixed.
   13. 13 Asymptotics: O, o, Θ, ∼I can read and say O, o, Θ and ∼ in statements about large n, and I know what each one claims and what it does not.
   14. 14 Modular Arithmetic and Number Theory NotationI can read and say n ∣ a, a ≡ b (mod n) and a mod n, and I know which of them are statements and which are numbers.
   15. 15 Chained Equalities in a Derivation: What Changed at Each StepI can read a derivation line by line, say what changed and what stayed the same at each step, and say what the whole chain claims.
   16. 16 Reading the Words Around a FormulaI can read the words that connect formulas (since, hence, where, e.g., i.e., in particular, clearly), say what each one licenses, and use them to write a short justified step.
   17. 17 Reading an Applied-Science Paper’s Methods SectionI can read the formulas and the words of a short methods paragraph and say what was studied, which letters are parameters or estimates, what is assumed and what is compared.
   18. 18 Writing a Theorem Statement; Its NegationI can write a theorem statement with its letters, hypothesis and conclusion, and write the negation of a statement with quantifiers or with “if … then”.
   19. 19 Mediation: Formula and Plain Words, Both WaysI can say a formula in plain words without changing what it claims, and write plain words as a formula, and I can spot a paraphrase that loses or adds something.
   20. 20 Reader: Five Longer Texts at B2I can read a longer text that mixes words and formulas — a derivation, a proof by induction or by cases, a limit definition, a methods paragraph — and say what it claims, what it assumes and what each line uses.

5. C1 Local definitions, ambiguity, papers outside your field, writing a note.

   1. 01 The Symbol-Table SkillI can read an unfamiliar sign from its local definition, record it in a symbol table with its meaning and where it holds, and see when the same letter changes meaning.
   2. 02 Ambiguity and ScopeI can see when a notation or sentence has two readings, write both out, decide from the text, and state the convention I use.
   3. 03 Notation-Introduction SentencesI can read “Let …”, “Fix …”, “we write …” and “denote by …”, say which name is new, what the letter may be used for, and that these sentences claim nothing.
   4. 04 Reading an Abstract, a Notation Section and a Main TheoremI can read the abstract, notation section and main theorem of a short paper, and say what is claimed, what each sign means, and what is not claimed.
   5. 05 Summary and Critique of a ProofI can say what a proof starts from, what its key idea is and where it ends, say what each line rests on, and name a step that is not justified or a hypothesis that is not used.
   6. 06 Writing a Self-Contained NoteI can write, and check, a short note whose definitions, claim and proof sketch use only words and letters it has introduced.
   7. 07 Translating Between ConventionsI can carry a statement from one book’s conventions to another’s (intervals, ℕ, decimal mark, logarithm, ⊂) so that it keeps its meaning, and I can say what I changed and why.
   8. 08 Formal Restatement of a Prose ClaimI can restate a prose claim in symbols by fixing the domain, the quantifiers and their order, and I can say the result back in words to check that it is the same claim.
   9. 09 Detecting a Wrongly-Scoped or Ambiguous SentenceI can find a sentence whose not, quantifier, and/or, sum sign or pronoun reaches further or less far than it should, write out its readings, and state which one I use.
   10. 10 Sign-Language of Other ErasI can read a passage written in an older notation by using the key and the date of the text, say what it claims, and put it into today’s notation.
   11. 11 Reading a Paper Outside One’s Field (guided)I can read a short extract from a field I do not know by finding where its signs are introduced, sorting what is defined, standard or to look up, and saying what is claimed.
   12. 12 Capstone: Read, Summarise, Formalise, CritiqueI can read a short mathematical text, summarise exactly what it claims, write a prose claim in signs, and critique a proof or a sentence for scope and ambiguity.

6. C2 Long and complex proofs in any field: their structure, their gaps, what each lemma is for.

   1. 01 Architecture of a Long ProofI can sort the blocks of a long proof (lemma, claim, remark, corollary), draw its dependency map in words, and say what loses its proof if one part fails.
   2. 02 Hidden Steps and What They LicenseI can find the steps a proof hides behind “clearly” and “it is easy to see”, say what each word promises, and test a hidden step against the hypotheses instead of accepting or rejecting it.
   3. 03 Reductions: WLOG, It Suffices, DensityI can say what a reduction replaces and what it must be licensed by: symmetry for “without loss of generality”, an implication for “it suffices”, and continuity for a density argument.
   4. 04 Existence: Constructive and Non-ConstructiveI can tell whether a proof of “there is” names the object or only shows it exists, and I can read the words choice, Zorn’s lemma, compactness and pigeonhole as steps with a stated licence and a stated limit.
   5. 05 Long Analytic Proofs: the Order of ChoosingI can read a long ε–δ or ε–N proof, say which constants are chosen in which order, what each depends on, and what “uniform” changes.
   6. 06 Induction and Its RelativesI can read a proof by strong, structural or minimal-counterexample induction or by infinite descent, say what is assumed, how many cases it must check and why it ends.
   7. 07 Algebraic ProofsI can read an algebraic proof with quotients, an isomorphism theorem and commutative diagrams, say aloud what each arrow and equation claims, and say why each check is needed.
   8. 08 Counting and Probabilistic ArgumentsI can read a double-counting, expectation, union-bound or coupling argument, say what is counted or averaged, which facts it needs, and what it does and does not prove.
   9. 09 Abstract Language: “Canonical”, “Natural”, “Up to Isomorphism”, “Well-Defined”I can say what each of these four words promises the reader, what it does not promise, and what check or choice-free rule stands behind it.
   10. 10 Proofs Outside Your Field: Tracing Definitions and DependenciesI can read a proof in a field I do not know by tracing every symbol to where it was defined, every “by Lemma n” to the result it points to, and every result to what it rests on.
   11. 11 Critique: the Gap, the Unused Hypothesis, the Circular Step, the Scope ErrorI can read a proof as a critic: say which kind of fault a step has (missing warrant, unused hypothesis, circle, scope), where it is, and what is and is not shown by a proof that has one.
   12. 12 Capstone: Read a Long Proof, Map It, Summarise It, Say What Each Lemma is ForI can read a long proof in a field I know the words for, map how its parts depend on each other, say what each lemma is for and which step uses it, and summarise it in a few sentences.