Response to Joanne Jacobs and John Dewey...

Imagine the following activity, Joanne. You guess the grade where this was actually demonstrated. Also, how would you characterize this lesson? Traditional? Reform? Constructivist? Discovery-based? Active learning? Ah, labels!

Children working in pairs draw, with a ruler: a triangle, quadrilateral, pentagon and hexagon.
The instructor tells each group to take their scissors and cut the three corners of the triangle and rearrange the pieces to form a straight angle. The teacher is demonstrating this on the overhead with color transparency film or with an opaque projector. This activity is repeated for the other figures, except the teacher doesn't tell them how many straight angles can be formed. She tells them to make as many as possible with the pieces, but pieces cannot be used more than once. The children are told to record their findings in a data table, which is also demonstrated by the teacher.
# of sides # of vertices (corners) # of straight angles formed
[Note: This will not appear spaced correctly]
3 3 1
4 4 ??
5 5 ??
6 6 ??

The teacher is circulating, assessing, guiding, asking many questions... She asks the students to predict how many straight angles could be formed from cutting the vertices of a decagon. She doesn't tell them what a decagon is -- someone in the class will make an educated guess and she'll guide them by relating the prefix deca- to common words.
She will then ask each group to formulate a rule in words for the relationship between the numbers of sides and the number of straight angles. After 10-15 minutes, groups volunteer to discuss and display their findings. The teacher is asking many questions of varied levels of taxonomy.

Later she has the children combine pairs of straight angles to form 'circles' and begins to formulate the numerical version of this important rule.

So, what grade level? Could it be introduced as early as 4th or 5th? Should all children have had similar experiences BEFORE taking high school geometry? Is this lesson far too time-consuming just to get at a simple algebraic formula 180(n - 2)? Personally, I have seen lessons like these at the middle school level, but, in other countries, younger children (4th grade) have these kinds of experiences. My Korean students told me so! If you would like to read further views like this, pls visit my blog (http://www.MathNotations.blogspot.com). I will repeat this comment there and expand on it...

Ok, I'm expanding! Just to invite any reader to offer useful lesson plans like this and make them accessible to many. Yes, there are many web sites for lesson plans, study guides, submitting lessons (and getting paid for them), etc. For math, we need to have common standards for each grade for ALL our students and to support these goals with effective lessons that develop conceptual thinking, communication and get kids excited about learning. Discovery-based lessons like the one described above are NOT the goal five days a week. That's obvious. But why we do we see fewer and fewer of these kinds of lessons in higher grades. Is it because most educators and administrators would view this as appropriate only for younger learners? Do they continually echo the refrain, "In the upper grades, there's just SOOOO much more material to cover -- theres' no time for 'fun and games'... " Now what did the TIMSS study reveal and suggest about math lessons in our country?? Could he have been suggesting that 'LESS IS MORE'!?! Will anyone out there ever get it? National Math Panel, ARE YOU LISTENING?? By the way, does everyone remember that I am not a reformist, not a traditionalist, just an iconoclast, who expects to be ignored in my own land... I detest labels! HAPPY HOLIDAYS - PEACE IN OUR TIME...

Joanne, I enjoy reading your excellent edublog as much as anyone's. You cut through the b******* and make sense. I hope you will direct your many readers here as well.
Dave Marain