As its name indicates, Algebra: Themes, Tools, Concepts (ATTC, by Anita Wah and myself, Henri Picciotto) is an algebra textbook. However, we included a strong geometric strand within it. Why we did this is summarized briefly in Algebra for all? How we did it is outlined in detail in A New Algebra.
The geometric strand of ATTC includes work with assorted manipulatives, not because of any magical belief in their effectiveness, but because they are a tool to promote discourse, and to help make connections between algebra and geometry. For my views on manipulatives in general, see that section of my article Early Mathematics: A Proposal, or else the front matter of several of my books, particularly ATTC and my Lab Gear books. (where to get them).
One of the geometric tools I used in ATTC (and the subsequent Geometry Labs) is the geoboard, because work on a lattice is a good preparation for continuous work in the Cartesian plane. On the one hand, the geoboard provides a good environment for studying similarity, equivalent fractions, and slope. On the other hand, work on geoboard area leads to a range of worthwhile problems, including some that lead to a preview of the Pythagorean Theorem. The latter use was criticized on the Web by one of my heroes, Don Chakerian.
Note that the controversial activity I pioneered in the 1990's has gained broad acceptance today. Search the Web for "tilted square" to see how many curriculums now incorporate this approach.
October 16, 1997
Dr. G. D. Chakerian
University of California
Dear Dr. Chakerian,
You don't know me, but your Geometry: A Guided Inquiry had a profound impact on my teaching. I never miss your talks at Asilomar, which are always the high point of the weekend for me. As a result, I was terribly upset to read your attack on my approach to previewing the Pythagorean Theorem (as found in the book Algebra: Themes, Tools, Concepts).
Let me answer your attack point by point, and describe the result of using this approach with actual students.
- Here is the way geoboard area is developed in the book:
- Find areas of right triangles in the "easy" orientation on the geoboard
- Use addition and subtraction of such right triangles to obtain the area of arbitrary geoboard shapes, including especially "tilted" squares (it is true that those squares are eyeballed --I'll come back to this point below-- but areas found this way are absolutely correct, and Pick's Theorem is not needed. In fact we are using the most basic thing about area, mathematically and pedagogically, when we obtain new areas by adding and subtracting known areas.)
- Find the distance between any two lattice points by building a square on that side, finding its area, and taking the square root -- this is a distance algorithm that is absolutely correct, and which is completely understood by a much broader cross-section of students, at a younger age, than the distance formula.
- Generalize this technique to the distance between (a,0) and (0,b) where a and b need not be whole numbers. This is of course the well-known dissection proof of the Pythagorean Theorem, which is much easier to understand after having done this work in a number of special cases.
- Far from objecting to that proof on ideological grounds as you claim, laying the groundwork for that proof is what this is all about! This work helps students see what's in front of their face geometrically in the classic figure, and the work with specific numbers is valuable in preparing them for the same process with variables.
- Finally the lesson you refer to, where the sum of the areas of the squares on sides of right triangles (and also of acute and obtuse triangles for comparison) are investigated.
- I would certainly trust a carpenter to eyeball a right angle on an 11 by 11 lattice, in any orientation, and so would you. An eighth grader, on the other hand, may need help. The best way to build a right angle on a lattice is to make use of the negative reciprocal relationship of the slopes of the sides. This is precisely the approach I use to help students -- without using the terminology.
By the way, eyeballing a right angle on a small lattice and finding areas by using logic is considerably less cumbersome and more accurate than using a ruler, as you ask students to do in your Geometry book, in preparation for the formal proof of the Pythagorean Theorem.
- Remember that we are talking about kids who will see a formal introduction to the Pythagorean Theorem, to the distance formula, and to the negative reciprocal relationship in Geometry and Algebra 2. The idea here is to help prepare them for that formal work. Whether this is constructivism does not matter to me: it's good teaching. Just because it wasn't done this way before does not invalidate it. Comparing this treatment (with middle school students) to the traditional deductive one (in high school) is meaningless. Each one has a place : one helps prepare for the other.
Your claim that they are being led into a "chaotic jungle" is just not what actually happens! As the results of the independent study comparing both groups in Davis show, the algebra skills of students who use my book are not harmed, on the contrary: my approach narrows the gap between the top and bottom third of the class, without hurting the top. This is a result worth celebrating. (More information on this is available from the school district, or from Dr. Alfred Manaster in the Math Dept at UCSD.)
In addition, through the geoboard, the students who use my book get the special bonus of a rich geometric model for the work with radicals, which follows the lessons outlined above: looking at the dissection of the square with area 20 into four squares with area 5 helps throw light on why sqrt(20) = 2 sqrt(5). And a final project of using the Pythagorean Theorem to tabulate the distances of all geoboard vertices from the origin, first as square roots and then in simple radical form reveals many patterns involving slope and radicals, as well as providing a useful drill.
All I can say as a teacher is that this approach has made the Pythagorean Theorem and the now unfashionable simplification of radicals more meaningful to my students, not less. The strategy outlined above is far from haphazard. It has been honed over many years of teaching (starting when I was using your textbook!) It is not a substitute for the proof of the Pythagorean Theorem, of course, but a preparation for it. Certainly, in the hands of "unskilled guides", this approach will not work, but neither will a deductive geometry curriculum in grades 7-9!
If you want to pursue this discussion in person, please call me. (I realize this is less important to you than it is to me, so I won't be sitting by the phone.) I have no illusions that my Algebra book is perfect. It certainly will never have the significance of your Geometry. But I do think that your attack on it was not deserved. Yet the attack sits forever in a public site on the World Wide Web ("2+2=4"), where I found it today.
I hope you are not offended that on page 309 of my book there is a mention of a Sherman's Department Store, where two of the clerks are named G.D. and Cal! Even if you don't like my book and misunderstand my views, I owe you, Stein, and Crabill a lot, and will always remain a fan.
University of California, Davis
Department of Mathematics
Nov. 8 1997
Dear Mr. Picciotto,
I appreciate your letter of October 16, and I really am sorry for the impression that item on "Mathematically Correct" left with you. Kreith and I should certainly have worded this in a way making it absolutely clear that our "attack" is not on your excellent book but on the rapidly spreading philosophy of constructivism.[...]
The success of a constructivist approach, or any approach, depends in the first place on the teacher, and your book is certainly an excellent tool in the hands of a competent teacher. We simply hoped to point out the dangers of an uncontrolled "constructivism" in the hands of those "unskilled guides" who are prone to neglect foundational logical structure in mathematics. It is not your book that might lead students into a "chaotic jungle", but an unskilled teacher. Of course, this could happen with any book.
The points you make in your letter are extremely reasonable and well-taken. Also, as another point in favor of your book, I am aware that it has been a success here in Davis -- that certainly is a result worth celebrating.
I certainly thank you for your praise of my geometry book and the impact it had on your teaching. Again, I have seen my book used to good effect in the hands of good teachers, and I have also seen it fail in the hands of unheeding and disinterested teachers.
I apologize to you, and I thank you for your very thoughtful letter and your many kind words. I was unaware of Sherman's Department Store and its two clerks G.D. and Cal I am proud that you let me work in the store, and I shall never again have a harsh word for a customer!
Very best regards,