Trial and error

One more childhood story.

MN:

My father is a mathematician. I was interested quite early in numbers
and mathematics. My father answered all my questions counting on the
fact that I would just stop listening when I get bored.

I would estimate that when I was about 11 or 12 years old I had the
opportunity to take part in a voluntary afternoon class about
mathematics in school, where we were exposed to actual mathematical
problems. These problems were of course quite simple but of a flavour
that it was not just applying some previously learned trick or
technique but rather one needed an actual mathematical idea.

This is where my story starts. I remember that I was in the beginning
quite desperate because it was totally unclear to me how one would
find such new ideas or tricks. What was the standard procedure to get
to them? I could easily follow such tricks once they were presented to
me, but how should I find them myself???

Of course, I asked my father. And to my big surprise he simply
answered something along the lines of: “You have to try more ore
less random ideas and just see which of them work!”
And indeed, my problems were gone and I knew what to do and I could
relatively quickly solve “proper” mathematical problems myself.

I could imagine that lots of students in school who have difficulties
in maths never had this kind of experience with “problem solving”.
In my experience most of them implicitly assume that mathematics is
all about learning some techniques and then simply applying them
verbatim, without any trial and error. And therefore they get stuck as
soon as a “non-standard” problem comes up.

Sheer madness: testing at 4

From NY Magazine: The Junior Meritocracy: Should a child’s fate be sealed by an exam he takes at the age of 4?

“Shadows of the Truth”: first criticism

First critical words about my draft book Shadows of the Truth (where I analyse childhood stories sent to me by readers of my blog). They came from good colleagues of mine, real experts in mathematical education.

Your material may be like using Olympiads to analyse mathematical development: those who perform at this level in their teenage years certainly exist, and one feels should be nurtured by the mathematical community to keep them fresh, so that their mathematical ability grows naturally into whatever they may do in adult life, leaving them free (if they choose) to graduate from adolescent problem solving to more serious mathematics.

[At present too many of those who feature in olympiads, move away
from mathematics for bad reasons! And too many of those who move
from Olympiads to more serious mathematics never really mature as
they should.]

But 95+% of mathematicians do not feature in Olympiads during adolescence – reflecting the greater importance of other “affects” (stubbornness, persistence, delight in global features – rather than local problems, …).

So we suggest that the “healthy norm” for mathematical development may be to reach 16 or 17 or 18 without noticing anything in particular, other than a sufficient affinity for the mathematics one has met to want to continue.

[...]

It seems bizarre to suggest that it is “normal” for kids to philosophise consciously about what goes on in the classroom. Healthy schools are simply not like this: the classroom is the least important part of what goes on in school (the communal, the physical, the food, contests – good and bad – in the the playground or on the sports pitch, the sheer rhythm of the timetable, etc. are all so much more important at the time). Most kids, including those who might subsequently be seen as culturally / mathematically important, simply need a cocoon which allows them to emerge as stable young adults, who are also literate, moderately knowledgeable, and mathematically competent.

In particular, why should any kid be surprised that some things do not make sense at the time? For most kids that is normal. So your precocious examples

(i) may be unusual in coming from backgrounds that encourage (either actively or incidentally) this kind of precocious reflection;

(ii) may be extreme (See Chapter 17!!!) in experiencing this kind of puzzlement *so rarely* that it leaves almost a scar!

If so, then you are missing out

(a) those who *never* experience this in a domain that matters to them,

(b) those who accept this feature and learn to handle it quietly while getting on with the things that seem to matter, without ever becoming fully conscious of what they are doing.

In short, those who are precocious in their inner reflections strike us as being a rather small minority – though [like those who stand out in olympiads] you are quite right to explore what their memories and experiences have to tell us, as long as we do not make the mistake of thinking that such self-conscious reflection is to be viewed as “good”, or “normal”.

Dear Reader, if you have your opinion about my book, I will be happy to hear it.

Abstract Thoughts? The Body Takes Them Literally

A nice article in the New York Times, based on Moving Through Time, by L. K. Miles et al., and Weight as an Embodiment of Importance, by N. B. Jostmann et al. From an abstract for the second paper:

Four studies show that the abstract concept of importance is grounded in bodily experiences of weight. Participants provided judgments of importance while they held either a heavy or a light clipboard. Holding a heavy clipboard increased judgments of monetary value (Study 1) and made participants consider fair decision-making procedures to be more important (Study 2). It also caused more elaborate thinking, as indicated by higher consistency between related judgments (Study 3) and by greater polarization of agreement ratings for strong versus weak arguments (Study 4). In line with an embodied perspective on cognition, these findings suggest that, much as weight makes people invest more physical effort in dealing with concrete objects, it also makes people invest more cognitive effort in dealing with abstract issues.

One more research cited is even more interesting: Children Learn When Their Teacher’s Gestures and Speech Differ, by Melissa A. Singer and Susan Goldin-Meadow. Abstract:

Teachers gesture when they teach, and those gestures do not always convey the same information as their speech. Gesture thus offers learners a second message. To determine whether learners take advantage of this offer, we gave 160 children in the third and fourth grades instruction in mathematical equivalence. Children were taught either one or two problem-solving strategies in speech accompanied by no gesture, gesture conveying the same strategy, or gesture conveying a different strategy. The children were likely to profit from instruction with gesture, but only when it conveyed a different strategy than speech did. Moreover, two strategies were effective in promoting learning only when the second strategy was taught in gesture, not speech. Gesture thus has an active hand in learning.

[with thanks to muriel]

Google has sense of humor

Just make Google search for “recursion”.

math stories – Nobel laureate Wole Soyinka

From http://www.indianexpress.com/news/The-Lone-Ranger/571235, courtesy of Dan MacKinnon.

Nobel laureate Wole Soyinka is a man of few words, but such beautiful words they are.

As a school student, Nigerian author Wole Soyinka loathed mathematics. “When my final exams for school were over, I remember taking all my mathematic books and making a bonfire and burning them. I even did a little jig on the ashes,” says Soyinka to a devoted audience who braved the scorching mid-day sun to listen to him at the Jaipur Literature Festival.

Little did the Nobel Laureate know at the time, that mathematics would save him from losing his mind, during his imprisonment in 1997. “When I was imprisoned, I was thrown into solitary confinement. I had been placed under trial but it was a barren existence. I invented games in my head. I began doing mathematics again. I’d scratch on the floor of the cell with a stone, working out permutations and combinations, using different formulae. Hours would pass but it nearly drove me crazy too,” says Soyinka, who later managed to befriend the jailor and smuggle in a book to read. He later began to make ink out of smuggled in coffee and wrote on scraps of paper. “I had to create an interior life to survive,” says the 75-year-old.

From fish to infinity

A brave attempt of popularisation of mathematics, in a New York Times blog.

First Proper Mathematics

A new childhood story,  from RSR.  Dear readers,  please send more!

On a large sheet of paper I made a triangle of numbers and addition signs as below.   Down the right side I made a list of the results of the additions.   It was clear that this process could continue as long as I wanted and my attention went to the vertical sequence on the right.

.                                                                                            0

1                                                                                           1

2 + 2                                                                                  4

3 + 3 + 3                                                                          9

4 + 4 + 4 + 4                                                                16

5 + 5 + 5 + 5 + 5                                                        25

6 + 6 + 6 + 6 + 6 + 6                                                 36

7 + 7 + 7 + 7 + 7 + 7 + 7                                       49

…….                                                         ..

It was clear that the numbers in the sequence increased more rapidly as you went down so I formed the sequence of first differences.   Of course I obtained the odd numbers.   So I thought, “Is this true in general?  Does the sequence continue always to generate the odd numbers no matter how far we go?”   I also thought, “Can I prove it?” and asked my father, who had a PhD in chemistry.   He confirmed that the odd numbers were indeed correct and mentioned algebra.   I wondered how can he know and can I prove it?   I think I thought in terms of a proof based on counters; I did not know my addition tables and certainly not my multiplication tables, and performed the additions by counting, mainly in my head but possibly also using my fingers.   I did not properly formulate a proof based on counters until grown up, as I later had algebra that made the result obvious anyway.   A proof based on counters is quite easy and possibly I got near to it at the time.

Perhaps I did not continue thinking about the matter to the point of constructing a proof because I became aware of the question, “Even if I get a proof, how will I know the proof is correct?”   This question bothered me.   I think I was aged four at the time, coming up to five, just after the Second World War was ended.

The point at the top of the triangle denoted zero zeroes added together.   The symbol “0” would not have been correct and I had a little difficulty deciding what I should put at the top.

Evidently I understood zero.   At some point, probably earlier than the research, I had discovered that you can continue counting forever, using the usual representation of numbers if one ran out of names.

My mother tongue is English and the above mathematics was all in English.

Islands and continents

A new childhood story,  from NA.  Dear readers,  please send more!

I do not whether the short story I am going to tell you fits the requirements of a story about math education, since it takes place in the family. I was about eight-nine years old (Italian third-fourth grade) and I was learning about continents. “Is Australia a continent or an island?” I asked my father. He answered it was BOTH a continent and an island; an answer I found deeply unsatisfactory. I thought for a while about islands and what makes them different from continents, until -weeks later- I reached the conclusion that, by stretching and contracting, Eurasia could be an island of the Oceans as well as an island of the Como lake (“all its shores are on the Como lake”).

A sunny day right after rain I was walking with my mother, I pointed to a puddle and I said: “we are on the island of that puddle”.

She shrugged and replied “why do you always say such stupid things”. (Only many years afterwards I learned that was part of something called topology).

Please send me more stories like that!

Escher and the Droste Effect

A wonderful site of an intriguing project.