Today I learned about a couple theories and some interesting research on numerical cognition in children. And I thoroughly enjoyed debunking Piaget!

Transcript | References | Media Credits

## Featured Books

My First Book of Time (DK Games)

The Number Sense: How the Mind Creates Mathematics

What Counts: How Every Brain is Hardwired for Math

## More about New Math

- What exactly was the "new math"? on The Straight Dope
- Why Guessing Is Undervalued by Annie Murphey Paul, TIME Ideas
- An A-Maze-ing Approach to Math by Barry Garelick, EducationNext
- Whatever Happened to the New Math? by Ralph A. Raimi

## Transcript

So I was taking a little stroll down memory lane, and I found the book that saved my life. Because learning to tell time was probably the most difficult thing I have *ever* learned, to this day, and it's still a bit of a struggle for me. But that's for another video. The reason that I dug up this book is because I was trying to figure out at what age I first understood what a number is. And I didn't have any luck, unfortunately.

But there's this pretty famous guy named Jean Piaget, who did a lot of research on this well before my time. In the 1950s he did some famous experiments on children -- and that doesn't sound right... But here's one of them. You present a child with two rows of objects, and you ask them, "Does this row have more, does this row have more, or do they have the same?" After the child agrees that both rows have the same number, you stretch out one row and ask again, "Are they still the same amount? Which one has more?" *(Video clips showing children picking the row that looks longer.)*

Kids are so adorably stupid, aren't they? I love kids. Especially making fun of them. So Jean Piaget thought that children are born as blank slates, with absolutely no understanding of number whatsoever. As he says in his own words,

"...mathematical knowledge is constructed, not pre-formed. If this knowledge was pre-formed, [...] it would have to exist implicitly in babies and even in animals." — Jean Piaget in Piaget on Piaget: The Epistemology of Jean Piaget (1977) by Yale University Films (Watch clip on YouTube)

Now, the mathematician Tobias Dantzig, who I mentioned coined the term Number Sense, believed the exact opposite. As he says,

"Man, even in the lower stages of development, possesses a faculty which, for want of a better name, I shall call Number Sense." — Tobias Dantzig, Number: The Language of Science, page 1

So, who do you think is right? The psychologist, or the mathematician? Are we born a blank slate, or are we born with number sense?

Well, Jean Piaget's theory dominated psychology in the 1950s and had a *huge* impact on education.

To give a really brief and one-sided history lesson, in the 60s and 70s there was a popular reform in math education known as New Math. You might've heard of it from the popular song by Tom Lehrer:

The very funny, very confusing New Math movement was heavily influenced by Piaget's theories. People thought Piaget had proven that children can't understand numbers until they first have a logical foundation for them, so teachers thought it made perfect sense to focus on really formal, abstract ideas like set theory. And so math education was stripped of all its intuitiveness, because Piaget thought children have no intuition!

But hold on one sec... When Piaget's famous experiments were redone slightly differently, the results told a very different story.

For example, in one experiment they show the children two rows of candy. And instead of asking them which row has more objects, they simply ask, "Which row do you want to eat?" And in this case, regardless of which row looked longer, the children picked the row with the most candy, even if they were as young as two years old.

And here's something interesting: when the experiment is redone but with younger children, they get the answer correct even *without* candy. For some reason, there's a gap in understanding. The youngest children get the answer correct, and so do the oldest children, but for some reason, the children in the middle get confused. So some other researches set out to discover why this is happening.

The researcher warns the child that there is a naughty teddy bear puppet. (Something like this. I know it' not a teddy bear, but it's the best I got.) A naughty teddy bear puppet likes to come in and ruin the game for us, because he's evil. Evil teddy bear. The teddy bear comes in, does his evil stuff (oh no, he's ruining the game, oh no!), and after the tedd bear rearranges things, the researcher asks, "OK, well, now which row has more objects?" And in this experiment, the children got the answer correct, indicating that yes, they *do* understand numbers. They just get confused when you ask them the same question two times in a row. And I would probably be confused, too.

So take that, Piaget! Or should I say, *vous sont incorrectes, Monsieur Piaget*. And more recent research has found that infants as young as *four days old* have a basic sense of number.

And in another experiment, if you show infants two screens showing a different number of objects, and you play a certain number of drum beats, the infants will look much longer at the screen in which the number of objects matches the number of drum beats they heard.

And infants can even understand basic addition and subtraction. Dr. Karen Wynn discovered this with a little magic show. One object is put on the stage, and then a screen hides it from view. Then another object is clearly put behind the screen. When the screen lowers, if it reveals only one object, the infants looked much longer than if it reveals two.

And even more impressive is that this still works with larger numbers, too. So if you show them 5 plus 5 equals 5, they'll look much longer than if they see 5 plus 5 equals 10.

So, it seems that we probably are born with this innate number sense! And it should come as no surprise that Jean Piaget is considered completely wrong about some things. But you know what's really discouraging? When I was doing research on YouTube, I found a bunch of videos on Piaget's experiments, but not a single video *disproving* Piaget. Not one. So I want to fix that, and I want to get my hands on some children! :D I mean... Does anyone want to lend me their kids for a little while? For science?

Actually, if you have access to young children, you can do the experiments yourself! I would *love* to see the results! And I think we need to show the future generations of educators and psychologists (who will undoubtedly learn everything they know from YouTube) that children aren't actually as stupid as we used to think.

"Not bad for the first day. Hoo-ray for New Math, new hoo hoo math. It won't do you a bit of good to review math. It's so simple, so very simple, that only a child can do it!" — "New Math" by Tom Lehrer

## References

- Number: The Language of Science by Tobias Dantzig
- The Number Sense: How the Mind Creates Mathematics by Stanislas Dehaene
- What Counts: How Every Brain is Hardwired for Math by Brian Butterworth
- The Math Gene: How Mathematical Thinking Evolved And Why Numbers Are Like Gossip by Keith Devlin
- Numbers Guy: Are our brains wired for math? by Jim Holt, The New Yorker
- A Brief History of American K-12 Mathematics Education in the 20th Century by David Klein
- What exactly was the "new math"? on The Straight Dope
- Why Guessing Is Undervalued by Annie Murphey Paul, TIME Ideas
- An A-Maze-ing Approach to Math by Barry Garelick, EducationNext
- Whatever Happened to the New Math? by Ralph A. Raimi
- Child's Conception Of Number by Jean Piaget
- Mehler, J., & T. G. Bever. (1967). Cognitive Capacity of Very Young Children. Science, 158, 141-142. http://www.sciencemag.org/content/158/3797/141.short
- McGarrigle, J., & M. Donaldson. (1974). Conservation Accidents. Cognition, 3, 341-350. http://www.sciencedirect.com/science/article/pii/0010027774900031
- Bijeljic-Babic, R., J. Bertonicic, & J. Mehler. (1991). How Do 4-Day-Old Infants Categorize Multisyllabic Utterances? Developmental Psychology, 29, 711-721. http://www.sciencedirect.com/science/article/pii/S001216490200846X (Read PDF)
- Starkey, P., E. S. Spelke, & R. Gelman. (1983). Detection of Intermodal Numerical Correspondences by Human Infants. Science, 222, 179-181. http://www.sciencemag.org/content/222/4620/179.short (Read PDF)
- Wynn, K. (1992). Addition and Subtraction by Human Infants. Nature, 358, 749-750. http://www.nature.com/nature/journal/v358/n6389/abs/358749a0.html
- McCrink, K. & Wynn, K. (2004). Large-Number Addition and Subtraction by 9-Month-Old Infants. Psychological Science, 15, 776-781. http://bc.barnard.columbia.edu/~kmccrink/Koleen_McCrink/publications_files/McCrink%20%26%20Wynn.pdf (PDF)
- Wynn, K. (1995). Origins of Numerical Knowledge. Mathematical Cognition, 1, 35-60.

## Media Credits

- Conservation task by jenningh
- Conservation of number task by Shena Kiper
- Preoperational Conservation of Number (5 yrs old) by angelface1030
- Piaget on Piaget: The Epistemology of Jean Piaget (1977) by Yale University Films (Watch clip on YouTube)
- "New Math" by Tom Lehrer (On YouTube)
- Graph for candy experiment - McGarrigle, J., & M. Donaldson. (1974). Conservation Accidents. Cognition, 3, 341-350
- The Infant and Child Cognition Lab, Boston College by Boston College
- Magic show experiment diagram - Wynn, K. (1992). Addition and Subtraction by Human Infants. Nature, 358, 749-750
- Stimuli videos courtesy of Dr. Koleen McCrink

And public domain media:

- String Quartet No. 3 in E flat major Op. 51 (B. 92) by Antonín Leopold Dvorák, from Musopen.org
- Top hat Silhoette Profile by studio_hades on OpenClipArt.org
- Spider web by nayrhcrel on OpenClipArt.org
- Silhouette of a brain by laobc on OpenClipArt.org