Lesson Plans for Mathematics.

[Alexander]

That's for mathematics as opposed to merely "math"

I'm putting this up on the board to remind me that smeone kindly asked for this, and it is within the "Understanding the Information Era" spectrum.

Madly busy. Must dash.

a

[Alexander]

This Thread aims to describe some lesson plans for enabling children to get a feel for Maths (Mathematics, not "math" See Real Mathematics. (Not "Math") and Preparation for the 21st C and the Information Era. What our kids SHOULD be learning! for related discussions), to help them build robust internal models and to clarify, if possible the role of the "teacher".

As can be seen from my sig. at the end of all my posts, I have considerable respect for children's natural ability to make sense of the world around them, and little patience for the teachers and parents who try to "project" or "inject" knowledge, morals or common sense onto children. I hope that some ideas that are posted will inspire others to trust children more than they currently do, and to think carefully about "teaching".

a

[Alexander]

I take my inspiration, from a story told to me describing how Seven Oaks School introduces Pi to it's children. I have no doubt that the story is true, (and even if it were no, it is still a good lesson for us all)! Hearing the story was a revelation, as it enabled me to finally understand how my physics teacher had been so successful with me in physics.

At Seven Oaks School, there is a strict rule that NO child or adult, neither teacher, parent nor private tutor, may ever explain what Pi is. This is to keep young minds fresh, and un-confused.

Then, one week, there is a "special" maths lesson, which lasts all day. Part of the children's assignment is to bring in from home, 2 or three "round things", all sizes, shapes. Bottles, lamp-shades, tins, anything they can get their hands on. The children are broken up into teams, provided some string and rulers, and asked to find and note, all measurements, and to check the relationship between all things measured, (you know, make a list with measurements and little pictures).

The teacher moves round the class, asking each team to report on their findings, but not "indicating" at all that they are near what he "wants" them to find. Through the discussion with each team, he is able to monitor how close they are to noting any patterns. Pretty soon, the room is a-buzz with excitement as team after team "discover" something important. That there is indeed a relationship between the roundy bit and the length across the middle of the roundy bit. Not only this, but some students will even suggest that it is independent of length of the object!

This is discussed in the class, and if not challenged by another child, then is by the teacher. And so, every team becomes involved in re-measuring every item, to prove that the relationship is independent of length, (which turns out to be true in every case!), and by lunch-time, this relationship has been shown to be just over 3. As the kids go to lunch, the question asked is, "Is this 'just over 3 number' exactly the same for all round things? Discuss."

Not only that, but the challenge is issued. Which team can measure to the highest accuracy? By early afternoon, all teams have become convinced that the number is the same for all sizes of circle, but it is not clear what the number is. The largest circles have been measured, but the number is not settled. This is the point that the teacher suggests using the athletics tape-measures and the cricket ropes on the cricket field. By the end of the afternoon, the class has become one big team, co-operating to get this number down.

Very hands on. Very real. Not a child has missed what they have been doing. Not single child has a fear of this 3 'n a bit number.

a

[Alexander]

The rectangle game.

I'm not going to tell you what this lesson plan is designed to teach. See if you can guess. But even if you can not guess the "official" name, it is perhaps less important than noticing the patterns.


If you do work out the "subject", or think you have, please do not post the answer. Just PM me. The reason for this is that once you "see" the answer, the exercise becomes useless.

This is another exercise to encourage the investigation of the natural world.

O.K. As the teacher, you have some preparation to do. First the ingredients.


103 marble set. (assuming the marbles are 1cm in diameter)

Minimum requirement:
***2 cloth bags with a pull chord on the top...
***1 bag, say 7x7 cm, the other say, 10x10 cm
***103 marbles, all the same size. (they can be stored in the bags)
***1 large piece of thick card (the board) 54 cm x 54 cm (54 diameters)
***Pin holes should be drilled every 1 cm (or the diameter of the marbles)
***thick card strips (triple or more thickness) 2 x 54 cm and 2 x 27 cm
***(Pin holes should be drilled every 1 cm (or the diameter of the marbles))
***blunt pins that fit in the holes, so the strips can be held on the board.
***A printout with all the numbers from 1 to 103 down the side.

For a flashy set, you can have a wooden board and wooden strips, more bags, more marbles.

The game.

Rules.

On the first turn, children may choose any number from 22 to 66. (Subsequently, any number is OK). They put that number of marbles into a bag, and take the bag and card strips to the board (physically separated from the large bag of marbles.)

They must create as many rectangles as possible with these marbles. (eg 48 marbles = 12x4 and 6x8 etc)

They must make rectangles, not lines. Anything 1 x something is not allowed.

No spaces in the rectangle are allowed, and no left over marbles either.

They must note down on their print-out against the numbers in question, the rectangles they find.

Which numbers have the most rectangles?

Are there any numbers with no rectangles?

©Alexander Streater 2001

[truly_sarah]

Oh, I wish I were in those kind of math classes. Do you think a PhD program could figure out how to teach statistics without the lectures? Then I might do it!

Here is a new exhibit at the Smithsonian, for everybody who's been following this and the related threads:

Just go to the http://www.americanhistory.si.edu Smithsonian Web Site and then click on the exhibit link for Math Education - it provides a brief history of American Math Education and at the very end has loads of references, some of them look like the type of education that Alexander is propsing...might be some good information for homeschoolers?

Thanks, Alexander for starting these discussions. I'm really busy with my work right now so am not blowing anyone off, just that my computer is tied up.

Sarah

[Alexander]

Thanks Sarah!

What a sweetie.

This link looks good.

Does anyone have a subject they would like "planned"?

Stats. ??? Mmmm, I'll think about that one. Probably easy though.

a

[Alexander]

In Sarah's link, it mentioned "new Math". This is kind of sad, because the origional proponents were trying to get the idea across that understanding the mathematical concept was all important. The difficuly was:

1) It was ahead of it's time, (at the peak of the Industrial Age)

2) conceived by mathematicians, implemented by beaurocrats (sorry, "educators").

3) Mathematics is difficult, so the success rate of children appeared to be much lower, so the system was watered down to "math".

Oh well. A new generation. perhaps we will see it succeed again, though under a possibly new name.

a

[truly_sarah]

Oh yes, I noticed all that...but I thought it did give a nice history of math education and intent. How we arrived to where we are today...I was impressed that in the resource links they did have more progressive materials!

My son seems happy with his Singapore Math. It will be interesting to see if he spends much time trying to 'proof' the trick of dividing fractions by flipping around the second term. Even at almost 38 years I still have to convince myself of how this works. Somehow it never really quite sank in. Hoping with #2 when we use manipulatives I can 'get it' with him.

Sarah

[Alexander]

A good way to teach about fractions is to use pizza. Real pizza. Nothing like hands on practice!

a

[Dot.mom]

I like Alexander's examples because it demonstrates children learning on at least two different levels. First they learn the concrete facts (Pi, or whatever stage is set). Secondly (and I believe more importantly) is they are empowered by their own discoveries. They learn they are capable of finding out universal truths for themselves and are not dependent on others to "teach" them.

[Piglet]

mmmm Pizza and fractions, maybe if my teacher had used that I would find them easier! I use liqorice allsorts for counting with 4 and 5 year olds and we eat them all when we are finished.
I remember arguing with my math teacher and saying " I can not think of a time in my life when I will ever need long division so why should I learn it?" I was right I never have, and I resent the hours I had to sit through mind numbing maths lessons, if only it could have been a bit interesting and a lot more fun.

[Alexander]

Originally posted by Piglet
if only it could have been a bit interesting and a lot more fun.

Or relevant!

a

[graken]

Fractions are like number different based systems like binary.
Bead boards are used to teach decimal.
Bead boards could be used to teach other number systems such as trinary or hexadecimal and conversion between them.
Bead boards could also be used for fractions where one bead is equivalent to the denominator number on the board above.

Charles