One-to-one correspondence: counting without number words

What to assume they know

• Counting a small collection aloud, and the idea that two collections can be the same size
• Addition and subtraction within the whole numbers
• What a prime number is, and the ability to test small numbers for primality
• Doubling
• Set notation of the kind {1, 2, 3, …} — braces, commas, and an ellipsis meaning "and onwards"

What they should be able to do

• Explain how pairing one token to one object decides whether a collection is complete, without either quantity being counted or named
• Run the pebble-and-pot procedure on a stated herd and read the verdict off what is left in the pot
• State what the procedure establishes and what it does not — in particular that it never yields the size of the herd
• Write the set of natural numbers in the chapter's notation and say where the set begins
• Report the location, approximate age and notch count recorded for the Lebombo bone, and the two purposes it is thought to have served
• Describe the column structure of the Ishango bone and identify the arithmetic the groupings are read as carrying
• Continue the sequence 11, 13, 17, 19 and justify the continuation by the property the four share
• Decide whether the natural numbers are closed under subtraction and defend the decision with counterexamples
• Count on the joints of one hand as the chapter describes, and connect the result to a base other than ten

Where it usually goes wrong

• "Counting means saying the number names." The pebble method never names a number and still answers the question exactly. Reciting names is one way to record a pairing, not what counting is.
• "The pot tells the herder how many cows there are." It does not. It reports a difference between two acts of pairing. Students conflate "I know none are missing" with "I know how many there are" — the whole section separates these.
• "Tally marks are just primitive counting, with nothing mathematical in them." The Ishango grouping is the counter-example the chapter chooses: the notches are arranged by a property of the numbers, which is a statement about arithmetic, not a record of a quantity.
• "29 notches means someone counted to 29 and stopped." Under both readings offered, the 29 is not a total someone aimed at; it is where a cycle came round again. The bone records a period, not a sum.
• "11, 13, 17, 19 — they go up by 2 then 4 then 2, so the next is 21." 21 is 3 × 7. Odd numbers and primes part company at 21, and this is the standard trap in Q2.
• "Subtraction always works, we just have not learned the answer yet." Inside the natural numbers 3 − 7 genuinely has no answer, and admitting that is what forces the extension in §3.2 and §3.3.
• "Everyone has always counted in tens." The finger-joint count gives twelve on one hand, and the chapter points at base-12 systems built on exactly that habit. Ten is a choice with an anatomical reason, not a necessity.

Questions to check understanding

• Given a number out and a number back, state what the pot contains and what the herder concludes
• Describe one-to-one correspondence and give a non-numerical example of it
• Write ℕ in set notation and say which familiar numbers it excludes
• Report the notch count, location and age of the Lebombo bone
• Given 11, 13, 17, 19, extend the pattern and justify the extension
• Decide with counterexamples whether a stated set is closed under a stated operation — the competency form of Q3, and the pattern the chapter re-uses for the rationals in §3.4
• Explain why counting finger joints suggests grouping in twelves
• Short-answer history: where each of the two bones was found and what its notches are read as recording

Examples worth working on the board

Values marked verified are worked out here on the chapter's stated inputs; the chapter prints no answers anywhere in pp. 41–67.

• The herd and the pot (§3.1, p. 41). The chapter states the procedure and no numbers. The point to land: the herder learns that two are missing while never learning that the herd is 23.
• The Lebombo bone (§3.1.1, p. 41). Located in the Lebombo range, on the border of South Africa and Swaziland. Age given as approximately 35,000 years. Carries 29 notches, described as distinct, deliberately carved and uniformly sized. Offered as either a counter for the moon's phases or a menstrual calendar. Useful when explaining it: 29 is close to the length of a lunar month, which is what makes both readings plausible from the same 29 marks.
• The Ishango bone (§3.1.1, p. 42). Recovered close to where the Nile rises, in what the chapter names the Democratic Republic of Congo; dated to around 20,000 BCE. Three columns of asymmetrical notches. One column groups the tallies into 11, 13, 17, 19 — every prime between 10 and 20. A second column is read as demonstrating multiplication by 2.
• Fig. 3.1 (p. 42, artwork over a photograph of the bone). A single long bone lies horizontally, carrying four separate groups of red notches, each with its own numeral and no two of them paired on the same group. Read closely, they run staggered along the bone: 11 sits below, near the left end; 13 above, left of centre; 17 below, right of centre; 19 above, near the right end. The caption calls the figure a representation of how the tallies group by primes. Read from the printed page: the figure shows the four labelled groups only, not all three columns described in the prose, so it is an illustration of the prime column rather than a facsimile of the artefact.
• Continuing the prime column (Exercise Set 3.1, Q2, p. 43). Inputs: the four numbers 11, 13, 17, 19, and the instruction to give the next three of the same kind. Verified: the shared property is primality; continuing past 19 gives 23, 29, 31.
• Closure under subtraction (Exercise Set 3.1, Q3, p. 43). Given as established: adding any two naturals lands you back among the naturals. Asked: does subtraction behave the same way, and the student must supply two or three cases backing the verdict. Verified: 7 − 3 = 4 stays inside; 3 − 7 does not, and neither does 5 − 5, because the chapter's natural numbers begin at 1 and so contain neither zero nor any negative. This is the hinge the next two modules turn on — say so here.
• Counting on one hand (Exercise Set 3.1, Q4, starred, p. 43). Inputs: each finger carries 3 joints, and the thumb is the pointer that counts them. Verified: four fingers × 3 joints = 12 on one hand, which is exactly the grouping a base-12 system uses.
• Trade arithmetic (Exercise Set 3.1, Q1, p. 43) belongs to the next topic — it is set at Lothal and is about exchange rates, not about pairing.

Figures to have open

• Fig. 3.1 (p. 42) redrawn as a schematic: a bone outline with two notch clusters and the four numerals 11, 13, 17, 19 placed as the printed figure places them. The printed version sits over a photograph of the artefact and should not be reproduced; a line drawing carries the argument.
• A gate-and-pot movement for sections 2 and 3. Standard schematic; the chapter supplies no figure for it.
• A 29-notch bone aligned against a lunar cycle strip, so the coincidence in section 7 is visible rather than asserted. Standard schematic.
• A number-line or set strip for ℕ showing that it begins at 1, with the region to the left of 1 conspicuously empty. This empty region is what modules m02 will fill, so it should be drawn to be re-used.
• A hand with the joints marked, for the base-12 count. Standard schematic.
• No photograph is needed anywhere in this topic.

Where this sits in the book

• NCERT Ganita Manjari, Class 9 Mathematics, printed Chapter 3, titled "The World of Numbers". Its opening section, §3.1, whose printed heading begins "The Dawn of Mathematics" and then names counting as a human necessity, occupies p. 41.
• §3.1.1, printed heading "A History Written in Bone", pp. 41–42.
• Fig. 3.1 and its caption, p. 42.
• Exercise Set 3.1, p. 43 — questions 2, 3 and 4 belong to this topic; question 1 belongs to Why India needed names for powers of ten.
• Chapter Summary, p. 66, first bullet, restates the natural numbers and dates their emergence to many tens of thousands of years in the past.
• Forward pointer inside the chapter: the closure gap opened by Q3 is answered in §3.3, p. 45.