Expression tree math

[Ted Kosan]

Over the past month, Michel Paul from this group and myself have been
experimenting with exposing his high school students to 1) expression
trees and 2) the simplest of the “unwritten” techniques that
mathematicians use to solve equations. MathPiper was used as the
visualization tool for this experimenting, and the following website
contains some examples of the output it is capable of generating:

http://206.21.94.61/tkosan/expressiontreemath/


Michel asked his students to provide written feedback about what they
thought of these visualizations. Here are their slightly edited
responses (I did not include feedback that was related to bugs in the
software):

============ Student Responses

1) I think that it shows finding the value of 'x' quite well because
it takes us through the order of operations without having us to
memorize PEMDAS. Basically, it makes solving an equation more clear.

2) It all makes sense.

3) I think it is smart that it shows the steps and it all makes sense.
I think that this is a very effective way to solve equations. The tree
diagram clearly demonstrates the relationship between the equation and
illustrates a clear understanding of the steps in the process.

4) How do you do this?

5) This is really neat! I enjoy the fact that the various operations
are not only expressed visually with the tree, but also expressed in
words.

6) This shows step by steps how to create a tree. It has great
explanation and images that make it easier for one to understand the
concept. I found it very helpful and clever.

7) I actually think its helpful because its shows the equation mapped
out on the tree. Rather then it just being numbers, its a visual to
help you understand the equation.

8) I personally do not think this is a good way to show the
function.The problem here is that the + and c together makes people
think that you are adding C.

9) It was very cool, it shows that he has been actively working on
this project, keep up the good work.

10) Very cool!

11) It is a lot more self-explanatory now. Before, I needed your
guidance to really understand it, but now it is very clear.

12) I think this makes sense and i like what he did with this. I think
that it is beneficial because the tree is helpful. The tree shows the
relationship and details of the equations.

13) I think the way the expression trees are explained is just a bit
confusing. your moving in the right direction, it is getter better and
better each revision. I think that it is pretty cool that this will
actually show you what to do and how to get the answer instead of just
handing you the answer.

14) Automate the solve equation process if it's not so already. Still
great piece of software. I think this is interesting... Of course, I
still find it easiest to do in your head, But I like that this shows
the order in which to do things. It is very clear: start at the top of
the tree and move down.

============

If anyone would like more information about these experiments, I can
answer questions about the software, and Michel can provide more
information about how his students reacted to the visualizations.

A minimal version of MathPiper is available on the above website if
anyone is interested in playing with the expression tree visualizer
and the “unwritten” technique equation solver. The readme.txt file at
the bottom of the website contains instructions for installing and
running MathPiper.

Ted Kosan

[Christian Baune]

Hi,

welcome to flatland :-)

I am a bit astonished that you had such good feedback for the method.

Having an arborescent thinking scheme and being developer I am always confronted to trees.

What you suggest is also known as AST(Asbtract Syntax Tree) and is well visible in RPN(Reverse Polish Notation) or CPE (Completely Parenthesized Expression).

When writing 1+2 you have this AST :

OP : +

|NUMBER : 1
|NUMBER : 2

That CPE : ((1)+(2))

Or that RPN : 1,2,+

Use of tree go far beyond arithmetic.

They can be used to explore possible paths when solving a problem.

In fact, you idea is not new, I considered it years ago : 

https://docs.google.com/presentation/d/1bc1491b8Ro4PnNpTN01lPSp_E2wK-jhr4IvlX6gf_-U/edit?usp=sharing

https://docs.google.com/presentation/d/11K8gslQ6mGvKpQsE4lTM4IH8AYEDZ-AIJmMtX4tOWhI/edit?usp=sharing

As you can see, tree and tree again.

This subject/methodology could be easily understood by a child of 8 year or more.

Basicaly it says : "Don't fear to use something you don't know, use it then find it"

It can sound really dumb but the trick is that doing so recursively, you will end up with things you know and would be able to climb the tree!

In the two links above, you can see how successful you can be with it.

But people aren't ready to do it that way.

For now, "mathematics for students" is all about "match and apply". 

It has been proven with the well known "Captain age" problem which is still used to assess schooling

and student still fail by finding a number.

The fact is that by using such solving strategy, you concentrate more on the steps involved than in actually processing them.

Teacher train student to process more than to think.

That's not the teacher fault, they have limited time-frame and the number of tools greatly improve (formulas, concepts, algorithms).

Maybe it would be more useful to abandon some of the tools (Eg. Cramer, Integrals) and focus on making student build thinking strategies.

I never understood why math courses were mostly about numbers and not about problems. 

Why we never had to play Tower of Hanoi and tried to model it in mathematical terms to have a rigorous approach on solving it.

There is a lot more to learn by modeling and solving puzzles during a year than in 6 years of taught mathematics in High School!

I participated at ESE2002, an event by Amavet in Bratislava. 

I met a Russian teacher who displayed instructional material for 6-9Y children.

They had an unprecedented level in maths (that's why they were there) and there was only games on the table! No books, no pen.

These children could draw a cut in a cube, tell you minimum characteristics to define a rotated square in euclidean space.

I live in Belgium and I know at least one 30year old unable to calculate the area of trapezium from the length of // and the height...

These kids would have simply laugh or felt insulted.

This guy is not handicapped, he is an IT professional, like me.

So, the world is not ready to think.

We have to wait and introduce things slowly, slowly,...

After all, that can be on purpose : people being able to find out by themselves rely less on others and material.

They are less prone to follow blindly too and gain a greater awareness and insight in any situation.

If you own the torch in the night, people will follow you. 

If the sun rise, they could see you drive the near the abyss for your profit.

This is quite unavoidable unless teacher become independent and people able to choose.

Kind regards,

Christian

Ted Kosan

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[Maria Droujkova]

On Mon, Mar 11, 2013 at 4:08 AM, Ted Kosan <ted....@gmail.com> wrote:

Over the past month, Michel Paul from this group and myself have been
experimenting with exposing his high school students to 1) expression
trees and 2) the simplest of the “unwritten” techniques that
mathematicians use to solve equations. MathPiper was used as the
visualization tool for this experimenting, and the following website
contains some examples of the output it is capable of generating:

http://206.21.94.61/tkosan/expressiontreemath/

WOW!!!

This can be directly applicable to my cause, namely, early algebra. This technique is so visual it can be used by little kids. Moreover, the structure yields itself so well to storytelling - roads, climbing trees, etc. 

Are any parents of kids ages 3-6 reading this, and interested in doing some experiments with me? I would modify Ted and Michel's techniques (slightly) and you will test the activities and report. 

Cheers,
Dr. Maria Droujkova
919-388-1721

[David Wees]

My son is 6. I'd be interested in trying something out.

David

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