[OLPC-GSoC] Math Tutor

Mon Apr 7 16:48:40 EDT 2008

On Mon, Apr 7, 2008 at 9:54 AM, Nidhi Gupta <coolnidhi13 at gmail.com> wrote:
> Math Tutor
>
> Idea
> The idea is to develop an application that would make the child good in
> maths through creative learning and developing problem solving abilities.
> The application would also use speech-synthesis to make user experience more
> lively.

An excellent idea.

Having read through your proposal below, I have a suggestion. Please
get a set of Cuisenaire rods (available from almost any school supply
outlet) and read the little book by Caleb Gattegno that comes with
them. Work all of the exercises yourself, and then try some of the
early ones out on children. Then think about how we can translate his
methods of teaching the ideas behind arithmetic to the XO. I am
suggesting that we teach mathematics visually and physically before we
teach notation and calculation. This is the natural sequence for
children, one that schools often ignore, at our peril.

Let me give you another example, which I call Kindergarten Calculus.
(See the Laptop Wiki page of the same name.) There are two essential
questions in calculus: to determine the direction of a curve, and to
determine the area inside a curve.

For the first, take any curved object and put a Cuisenaire rod or
other straightedge up to it. The straight line of the rod is the
tangent to the curve, and shows the direction of the curve at one
point. Physics gives us this tangent visually, with no notation and no
calculation. Then we can demonstrate that the top and bottom of the
curve are points where the tangent is level. These are two of the
fundamental ideas of differential calculus. Defining the numeric slope
and the operation of creating a derivative function have to wait a bit
until we introduce rather more geometry and some arithmetical
calculation skills.

Secondly, we want to demonstrate integrals. One bit of preparation is
needed. Take a standard grade of paper, draw a square on it, cut it
out and weigh it on a sensitive balance. Then calculate the size of a
square that will weigh one gram. Cut out some one-gram squares. Weigh
at least one to check your work. (For schools that cannot afford
paper, lay out the figure in clay, fill it with water to a standard
depth, such as the side of a Cuisenaire rod, and measure the amount of
water that the figure holds. Instructions for making a quite sensitive
balance out of local materials with no special tools can be found on
the Net.)

Now, on the same grade of paper draw the standard x and y axes and any
curve that is always positive. Draw vertical lines at the ends of the
curve to make a closed figure. Cut out the figure and weigh it on a
sensitive balance. The weight in grams is the area in your standard
squares. This gives an accurate value for the definite integral.
Negative values must wait.

Now take that same cutout and fill it with Cuisenaire rods aligned
vertically. In the same way, we can weigh the rods that approximate an
area, and weigh the smallest rod, and get an approximate area for our
figure. A few more steps (easier to do on a computer than physically)
will bring us to the idea of the Riemann integral.

We can do indefinite integrals by adding strips (single Cuisenaire
rods) to our figure in sequence, and then we can demonstrate the idea
behind the Fundamental Theorem of Calculus, that the derivative of an
indefinite integral is the original function, because the derivative
at each point of the integral is approximately the area of the last
strip. Constants of integration will not be difficult to do visually.

But we have demonstrated all of the core ideas of calculus in a way
that children can understand entirely. Then we can consider
introducing bits of deductive and analytic geometry, vectors, and
precalculus in the later grades, sincluding limits, trig functions,
and exponentials, at whatever age works for children, with a similar
progression from the visual and tactile to the formal and numeric.

Now comes the question. I hypothesize that children growing up knowing
these concepts since kindergarten will have far less difficulty with
calculus calculations and proofs when the time comes than those who
come to it cold in high school. It will take time to do the
experiment, but the stakes are great enough to justify putting
considerable resources into it. Starting, of course, with an
implementation of all of these ideas and more in either Smalltalk or
Python/NumPy.

What do you think, Sirs?

> Importance
>
> There is a need to make the children learn the basic mathematics, operations
> and to some extent problem solving also. Children needs to learn maths in a
> very interactive manner. This application proposed to develop an activity
> that will teach children maths using speech-synthesis and images so that it
> become fun for them.

They need to understand the ideas behind the math, not just do the calculations.

> Use case example.
>
> A person starts the application. A person can be a user or a guest. A normal
> user starts from the first level but has a capability to save the current
> level but as a guest one can enter any level at any time with no saving.
>  When the person starts this application, he has a choice which profile to
> start.
> Based on the profile, he learns a level in the most creative and easy
> manner. After every level, a test based on the current level is performed.
>  He goes in the next level only by clearing the test by a minimum criteria
> in a stipulated time.
> Thus expertizing in that particular set of basic mathematics.

Tests should be voluntary. Their usefulness to children is greatly
overrated. Children have a much sharper sense of what they do and do
not know or understand than any adult around them.

> To make the application interactive, at various levels, colorful and luring
> (creams, balls, fruits etc) figures will be used.

Cuisenaire rods, please. No distractions. Real math is compelling, and
should not be confused with entertainment.

> Levels
> Based on difficulty, the application has the following levels:
>
>     *beginner: This level will include counting and number formation.
>            eg  children will be taught "1", "2" up to 100 by means of
> suitable figures.
>                  this will include derivation of numbers above 100.eg
> 123=one hundred and 23(which is  already taught)

I rate this as stage 3. Stage 1 is tactile and physical, with
Cuisenaire rods. Stage 2 is still visual, but on the computer.
Notation, including number names, is a highly derivative, abstract
concept, and should not be the starting point.

>    *medium: This level will teach tables 2-10   followed by a dogging table
> test.
>
>     *intermediate: This will include basic operations +,-,*,/ in single
> digit
>        eg 3*7,4-2
>
>     *higher: This will include basic operations +,-,*,/ in double digits
>        eg 35*14
>
>      *expert:This will include basic problems on operations in figures. Hint
> can be given regarding the operation to be performed.
>
>    Speech synthesis would be used in all level to make the experience more
> lively. e.g. "two multiplied by three equals six"
>
> Proposed Features
>     -In the first level, learning counting will include a sequentially
> generated numbers along with a randomly selected figure from a repository of
> figures to represent that number. eg :2 can be represented as 2 candies.
>      -Hint will be pronounced and the child needs to write it down by
> pressing key(if input is keyboard).This will make him relate sound with
> number.
>     -The questions in test will be chosen from a set of randomly generated
> questions from a repository and corresponding hint will be displayed
> depending upon person's option to display it.
>
>     -The test will have to be passed through a minimum criteria in a
> stipulated time to proceed to the next level.
>     -User can save their previous levels and continue the next time.
>
> Future Scope
> 1.Increase the operations to roots etc
>  2.Put problems in statements and not figures.
> 3.Number Game can be thought of in this direction.
> 4.Input method can be further be varied and speaking of number can be
> incorporated.
>
>
>
>  --
> REGARDS
> Nidhi Gupta
> Information Technology
> Netaji Subhas Institute of Technology
> New Delhi
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>

-- 
Edward Cherlin
End Poverty at a Profit by teaching children business
http://www.EarthTreasury.org/
"The best way to predict the future is to invent it."--Alan Kay