Tony’s Take December 2025

This month’s topics:

• Numbers in prehistoric mathematics?
• The dark side of algorithms

Numbers in prehistoric mathematics?

How long have people been using numbers—words like two, three, four that represent quantities? We know that the ancient Sumerians had an advanced numerical culture: This 4600-year-old Sumerian tablet shows computations for the areas of six rectangles, and their sum. These rectangles are extremely elongated (aspect ratio $1:60$), so the tablet was most likely used in a school. The tablet’s use of format (spreadsheet-like), notation and units, the very idea of a computation school, suggest that by the time it was written, numbers had already been in use for many, many years.

An article published on December 5, 2025 in the Journal of World Prehistory presents evidence that the use of numbers might go back to the Halafians, who inhabited Mesopotamia (map here) between approximately 6200 and 5500 BCE. (That’s some three millennia earlier than the Sumerian tablet, and long before the emergence of written language.) The authors, Yosef Garfinkel and Sara Krulwich of Hebrew University, analyzed pottery remnants dug up from 29 Halafian sites. The pottery is of very high quality and often finely decorated, with a sophisticated use of symmetry.

Garfinkel and Krulwich noticed an unusual organizational pattern in Halafian pottery decoration: In depictions of flowers, the number of petals is almost always 4, 8, 16 or 32, i.e. almost always a power of 2. Among the 375 specimens of floral decoration, only a handful use other numbers (6, 7, 12 and 13); the authors describe these as “isolated cases that are probably the result of less skilled craftsmanship.”

So what does this tell us about the Halafians and numbers? Very little, unfortunately. The authors write: “The topic of prehistoric mathematics seems at first glance to be beyond the borders of knowledgeability.” We are still outside those borders.

In fact, it may be that the Halafians did not even have numbers as we know them. For us that is almost inconceivable, especially for a society which had developed craftsmanship and art to the level shown in these shards. But we do not know how they thought, and what kind of systematic and possibly non-numerical methods they may have used to organize their lives and their creativity.

The dark side of algorithms.

In the United States, boys and girls start kindergarten with similar mathematical skills, but by third grade the boys are already performing better and reporting better confidence. Why? A recent study about how teenagers of different genders perform mathematical tasks like addition, subtraction and multiplication attempts to explain. A December 5 headline from The Conversation sums up the findings: “Girls and boys solve math problems differently — with similar short-term results but different long-term outcomes.”

The authors of the study, Martha Makowski (University of Alabama) and collaborators, report that boys are more likely to use strategies like decomposition. For example, they might reframe $38 + 27$ as $30 + 20 + 8 + 7$. By contrast, girls are more likely to use the standard algorithm for adding multi-digit numbers: Add the units and then the tens, remembering to “carry” the $1$. Both methods work. But on more difficult problems, the students who used the algorithm tend to perform significantly worse than those who did not.

The study tested 213 students at a U.S. high school, 116 female and 97 male. The 50-minute test included:

• Three multiple-choice computations:
  • $25 \times 9$,
  • $600 – 498$,
  • $19 + 47 + 31$.
• Questions about how the students had solved each of the items. Regarding the last one, for example, they were given four options:
  • I used a written algorithm, setting it up like this$$\begin{array}{r}19\\47\\+31\\\hline\end{array}$$
  • I added $19+31$ in my head, and then added $50+47$.
  • I guessed.
  • Other: (please explain)
• A set of eight mathematics problems “shown to have gender differences in national samples.” One example:
  

  

In their discussion, the authors report “a persistent pattern” of female students using algorithms more than males, and that more frequent algorithm use “persistently predicted” lower performance in problem-solving.

Not every country suffers from a U.S.-style gender gap in mathematics. A 2016 study by Chen Shen et al. in the Journal of Experimental Child Psychology involved first graders from the U.S., Russia and Taiwan. Among American and Russian students, the authors reported the same pattern of gender differences that is documented above. In Taiwan, on the other hand, they reported no gender differences in strategies or in accuracy. They suggest educational context should be examined when investigating the differences between boys’ and girls’ performance in mathematics.

It is clear from the Shen et al. report that what they call “educational context” begins to impact gender differences in children’s mathematical development as soon as they start school. Additional evidence is provided in the report of a 4-year longitudinal study of all French first and second graders, over 2.5 million children. “Rapid emergence of a maths gender gap in first grade,”, by Pauline Martinot, Stanislas Dehaene and collaborators, appeared in Nature last June.

—Tony Phillips, Stony Brook University