This month’s topics:
Crows can count … to four.
Diana Liao and her colleagues in Tübingen were able to train crows to caw a specific number of times in response to visual and auditory cues. Their article in Science was picked up by Audubon on May 28th and by Smithsonian Magazine on the 29th.
Three male crows, borrowed from the University aviary, participated in this study. They were of the species Corvus corone (Carrion crow), common in Western Europe.
The crows were trained over a year to respond to a cue by cawing one, two, three or four times. They were then evaluated over a period of ten days. In a typical trial, in visual mode, a crow would be shown a colored number: purple 1, orange 2, blue-green 3 or pink 4. (Presumably in this context the Indo-Arabic numerals just served as four different shapes). It then had 10 seconds to caw a certain number of times and peck at an “enter key” to signal it was done. If the number of caws matched the cue, the crow was rewarded with a meal worm or a birdseed pellet. Otherwise, time out.
The distribution of responses is compatible with a real engagement of the subjects with the underlying number concepts. For example, as the authors observe, the errors were usually off by plus or minus one unit and tended to be larger when the target number increased. Another significant observation the authors report is that the reaction time—the delay between the stimulus and the first vocalization—was systematically longer for larger target numbers; as they remark, this “suggests that crows plan the entire number of impending vocalizations before motor production.” Furthermore, as they report, the acoustic quality of the first “caw” allows fairly accurate prediction of what the total number of vocalizations will be.
This item was covered by Ari Daniel for NPR on July 18. The four-minute broadcast includes the four auditory cues and a recording of one of the auditory-cue trials, and ends with a quote from Chris Templeton, a biologist unaffiliated with this project: “Animals are smart in a whole bunch of different ways, and those may or may not be the same things that we do.” To which Daniel adds: “Meaning that if crows were to give us an intelligence test, we may not pass.”
“When algebra makes you smile.”
Two Tufts University professors, Barbara Brizuela and Susanne Strachota, published an article in the journal Learning and Instruction (May 17, 2024) with the title “When algebra makes you smile: Playful engagement with early algebraic practices.”
Brizuela and Strachota followed the mathematics instruction of 69 children (four classrooms) for two and a half years. The level is early algebra, grades 2, 3 and 4, with students aged 7-10. The authors wanted to track “the joy that students can experience when engaging with mathematics practices.” As they explain, most studies about students’ emotional responses to mathematics instruction have focused on the negative side. Even studies on how best to support self-driven, “playful” learning were aimed at preventing boredom and “math phobia.” No one considered the possibility that children could actually enjoy math, or experience what the authors call “positive epistemic affect” (essentially, the pleasure of understanding).
The authors set out to characterize instances in which students demonstrated positive epistemic affect in response to the four practices the late James Kaput identified as characteristic of students at this stage:
- generalizing mathematical structures;
- representing,
- reasoning with and
- justifying these generalizations.
The classes were recorded, and Brizuela and Strachota identified observable behaviors that could be correlated with students’ enjoyment: overt utterances, body postures and movement, facial expressions (e.g., smiles or smirks) and gaze. As they put it: “we identify markers of joy and describe how those co-occur with students’ engagement with the specific mathematics practices within early algebra.” In this paper they single out three cases involving different mathematical tasks, analyzing how they were taught and how the students engaged with the work.
One nice example comes from a class on the number line. The teacher-researcher had asked an open question about how far the number line would extend. The authors reproduce the transcript of the ensuing conversation, annotated to record the practices represented and the markers of affect manifested. At one point, one of the students, Talik, mentions “infinity.” The teacher asks him, “What’s that, Talik?” He explains with a smile: “The number line just goes on and on.” A bit later the teacher repeats what Talik said: “It should go on and on and on,” and asks: “When would we ever stop?” An unidentified student says: “It’s infinity. We would never stop.” The teacher notices Felipe wants to speak and calls on him. He volunteers: “We would stop at infinity and beyond.” The teacher repeats what he said and asks, “What does that mean?” Felipe answers, in statement form and without smiling: “Buzz Lightyear.”
The authors note that Buzz Lightyear, from Toy Story 2, counts as “relevant and familiar context,” part of the criteria for “playful stances to learning.” (Lightyear’s catch-phrase, “To infinity … and beyond!” was once voted the best film quote of all time).
The meaning of the equal sign.
An ArXiV posting by Kevin Buzzard (May 16) was brought to the wider public on June 16 by Clare Watson, on Science Alert: “Mathematician Reveals ‘Equals’ Has More Than One Meaning in Math.” The title may call to mind the Monty Python sketch where Peter Cook asks John Cleese: “Are you using ‘yes’ in the affirmative sense?” or Bill Clinton’s Grand Jury testimony where he states: “It depends on what the meaning of the word ‘is’ is.” But this is serious. The problem arises when we attempt to enlist computers into processing mathematical arguments, for example checking if a proof is correct or not. It turns out that our use of “equals” follows human conventions, often tacit, which lead to statements that are indigestible to a machine.
Buzzard’s posting focuses on fairly arcane examples from the work of the celebrated algebraic geometer Alexander Grothendieck, but there are more elementary ones. The group of symmetries of a polygon is the set of rigid motions (rotations and reflections) taking that polygon to itself. For example, if the polygon is an equilateral triangle, a 120$^{\circ}$ rotation clockwise or counterclockwise brings it back onto itself. This is a symmetry. Likewise, reflecting the triangle across any one of its side bisectors brings it back onto itself. This gives three more symmetries. Calling a set a group means that there is a multiplication combining any two elements to give a third. For symmetries of a polygon the “multiplication” is composition: performing the first motion, then the second. The top table below shows how this works for an equilateral triangle. For example, rotating 120$^{\circ}$ and then rotating 240$^{\circ}$ takes us back to where we started. (The symmetry that leaves everything fixed is labeled ${\bf I}$, the identity).
Googling “group of symmetries of equilateral triangle” leads to this page where you can read that “the dihedral group ${\bf D_3}$ is the symmetry group of an equilateral triangle” and “in mathematics, ${\bf D_3}$ … equals the symmetric group ${\bf S_3}$.” On another page you will find that ${\bf S_3}$ is the group of permutations that can be performed on 3 symbols. What does it mean to say that these two groups are “equal”?
When we say that two sets are equal, we mean that they have the same elements. The groups ${\bf D_3}$ and ${\bf S_3}$ obviously don’t have the same elements, but a comparison of the two multiplication tables shows that the correspondence that matches the 120$^{\circ}$ rotation with the permutation $\lt \mathsf{abc}\gt$, the 240$^{\circ}$ rotation with the permutation $\lt \mathsf{acb}\gt$, etc., going down the two lists, takes products to products and shows that the groups are really “the same” up to notation. Could this be the meaning of “${\bf D_3}$ equals ${\bf S_3}$”? The problem is that a machine would need to know which element goes with which, and the identification given here is not automatic.
In fact, our matching amounted to labeling the vertices of the triangle with the symbols $\mathsf{a, b, c}$, reading counterclockwise. But we could just as well have labeled them clockwise; then the 120$^{\circ}$ rotation would correspond to the permutation $\lt\mathsf{acb}\gt$, etc.; the two group structures would still match perfectly, but in a different way.
Buzzard explains that this imprecision in talking about structures and “equality” has not led to any errors in mathematics, but that success in formalizing mathematics to the point where computers can do it usefully depends on clearing it up.
—Tony Phillips, Stony Brook University