Body parts and tally marks: counting before numerals

What to assume they know

• Why any number system needs a fixed, ordered sequence of symbols — what a standard sequence is, and the one-to-one pairing that a count consists of
• That a system can be unending yet inconvenient, or convenient yet finite
• Reading a labelled diagram in which the numbers are printed inside the artwork
• Familiarity with tally marks from primary school, which the chapter assumes

What they should be able to do

• Trace the printed body-part sequence from 1 to 27 and identify the point of symmetry at 14
• Explain why a body-part sequence qualifies as a standard sequence under the chapter's own definition
• State the ceiling of the drawn body-part system and say what a user must do to count beyond it
• Describe how a tally differs from the stick method of §3.1, and what the difference buys
• Report the two bones the chapter names, with their find-sites, their notch evidence and their stated ages
• Explain how a tally of days becomes a calendar, and why 29 notches is a suggestive number
• Say precisely what a tally cannot do, and connect that failure to the grouping idea that follows

Where it usually goes wrong

• "Body counting is just holding up fingers." Fingers give ten. This system gives 27 because the sequence walks over wrist, forearm, shoulder, ear and eye as well — every station is a named place, and the naming is what makes it a sequence rather than a gesture.
• "Everyone would count the body in a different order, so it cannot work." The order is fixed by the body's own layout and by the community's convention, and the mirror symmetry makes it easy to hold. Two people who share the convention will always stop at the same place.
• "A tally is not a number system." Under the chapter's definition it is one: a sequence of marks with a fixed order, paired one for one with the objects. What it lacks is convenience, not legitimacy.
• "The bones prove ancient people did arithmetic." The chapter says the marks are thought to represent numbers and that the calendrical reading is a possibility. Keep the hedge; it is in the text.
• "Tally marks are the same as sticks." Almost. The difference is permanence: the marks survive the counter and can be read next season, which is precisely what a calendar needs.
• "Grouping tally marks in fives is how tallies have always worked." At Part I p.57 the chapter replaces each group of five marks with a new single symbol, pointing at the Roman V of its Table 1 as the example; it never draws a crossed group of five, and neither should the explanation. Here the marks are ungrouped, and that is the state of affairs the next topic repairs.

Questions to check understanding

• Given the body-part figure, name the station for a stated number, and the number for a stated station
• State the largest number the drawn system reaches, and describe one way its users could go further
• Explain, in the chapter's own terms, why a tally is a number system
• Given the two bones, report which is older and which carries the stated notch count
• Compare the stick method and the tally method and state the one property that separates them
• Explain why an ungrouped tally is slow to read even though it is quick to write

Examples worth working on the board

Items marked counted were read off the printed page image; the numbers in this figure are printed inside the artwork and do not appear in the extracted text at all.

• The body-part figure (Part I p.55, top artwork, belonging to §3.2 I on Part I p.54). A standing figure with dashed leaders running from printed numbers to points on the body, and two long curved arrows showing the direction of travel. Counted: the numbers run 1 to 27 with no gaps and no repeats. The walk starts at the digits of one hand — 1 to 5 across the fingers — continues up that arm through wrist, forearm and shoulder, crosses the head by way of ear and eye to 14 at the head's midline, and then descends the other side in mirror image, ending at 27 on the far hand. The structure to show is 13 + 1 + 13: thirteen stations on one side, the midline station alone, thirteen matching stations on the other. That midline station, number 14, is the nose — the walk crosses the face eye → nose → eye. No leader on the figure ends at the crown.
• The stated home of the practice (Part I p.54): a Papua New Guinea community, which the chapter says counts on hands and body parts and does so still, in the present tense as well as the past.
• The tally definition (Part I p.55): a mark made for every object counted, so that the finished collection of marks is the count. The chapter's own comparison is with Method 1 of §3.1 — the only difference is that a mark is cut rather than a stick added.
• The Ishango bone (Part I p.55): a find from the Democratic Republic of the Congo, dated 20,000 to 35,000 years, its notches set out in columns, with a possible calendrical reading. Photographed on the page beside the other bone.
• The Lebombo bone (Part I p.55): a South African find carrying 29 notches, estimated at around 44,000 years old, called one of the oldest surviving mathematical objects and read as either a tally stick or a lunar calendar. The page shows it as three photographed fragments to the left of the Ishango bone.
• The general age claim (Part I p.55): marked bones older than 20,000 years have been found that appear to carry tallies. The two named bones are the oldest known such finds.
• A number worth showing. 29 is close to the length of a lunar month, which is why a 29-notch bone invites a calendar reading. State the coincidence; the chapter does not compute anything from it, and neither should the explanation.

Figures to have open

• The body-part sequence with all 27 stations numbered and the travel path drawn. This is the chapter's own figure (Part I p.55) and section 2 cannot be taught without it. Redraw it as a clean schematic: the numbers must be legible and the symmetry about 14 must be visible, which the printed art achieves only loosely.
• A folded version of the same figure for section 3, showing the two halves matching. This is an added construction and should be labelled as such.
• A long unbroken run of notches for section 9, deliberately too long to read. Standard schematic.
• Photographs of the two bones are printed on Part I p.55 but are not required; a drawn bone carrying a countable 29 notches serves section 8 better, because the printed photograph does not let a student count them.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part I, printed Chapter 3, "A Story of Numbers", §3.2 "Some Early Number Systems", subsections I. Use of Body Parts (Part I p.54, its figure carrying over to Part I p.55) and II. Tally Marks on Bones and Other Surfaces (Part I p.55).
• Back-reference: Method 1, the stick pairing, is at Part I p.51, and §3.2 II names it explicitly.
• Forward pointer: the replacement of every group of five marks by one new sign is argued at Part I p.57 and belongs to Counting in twos, and what number-names reveal about a culture's base.
• The chapter's SUMMARY (Part I p.81) does not mention tallies or body parts; they support the first bullet rather than appearing in it.