Dr. Edgell, over several years, has conducted formative, field research programs with students at grades K-S at the public school levels. Many of these programs have focused upon several aspects of accessing geometry. Such research has impacted the preparation of pre-service teachers of mathematics and in-service teacher training programs. This program, of discussion, occurred during the 1997-98 academic year with a class of kindergarten students. The program consisted of twenty-six research sessions, each between forty-five minutes and an hour and fifteen minutes. Sessions focused upon formative ideas which relate to understanding geometry from a measurement, tessellation, cube, simplex, construct perspective. This particular paper is focused upon a couple of episodes within a session.

First Episode - Jeremy

It finally had happened! It actually really happened! Jeremy had made a volunteering motion and the class was spell-bound. In all of the previous twenty-five research sessions, each of about an hour duration (Mrs. Green and I had informally agreed before the program began that about twenty minutes was probably as long as I could expect the students to stay focused and on task. But, the students were so excited (and we were similarly excited) that before we realized it we had easily stayed focused and on task for over forty-five minutes, somewhat to our surprise when we realized the passage of time. So, we revised our game plan time allotment to be somewhat flexible. Some of the sessions extended to over an hour, just before lunch.), Jeremy, an unusually timid, kind soul, had not volunteered to resolve a problem, demonstrate a manipulative or take a leading, independent role in activities of a session. Every one, Mrs. Green, the twenty or so classmates and myself were almost instantly tuned to the event of Jeremy volunteering. During virtually every previous session Jeremy had opportunities, just like every one else, due to a random cloths-pin technique Mrs. Green had shared, to personally take responsibility for some aspect of the session activities. Students usually were expected to become the center of attention in addressing some phase of the session activities and to be personally responsible for such. Sometimes these responsibilities revolved around resolving a geometric related problem solving episode, leading a discussion about a geometric object or idea, manipulating geometric related materials in explaining an idea, etc. Students could decline or accept the role and from time to time students declined for various reasons and a decline was never allowed to reflect negatively upon the student. Similarly, students who accepted the role and were unsuccessful were not treated in negative fashion in that usually such events were verbally rewarded for the effort to make a contribution. Generally, students were excited about the activities of the sessions and often just couldn’t wait to be called and tried to volunteer. Occasionally I looked for volunteers and selected various candidates for several reasons. There had been very few attempts by Jeremy, when chosen by the random technique, which virtually gave him an opportunity during each session, but never successfully. Usually such attempts were aborted very early, even with lots of peer encouragement, through his overwhelming shyness he would simply withdraw. So, having volunteered, the attention of the whole class was focused upon him. I immediately called upon Jeremy to be responsible for the demonstration and we waited with baited breath for his inspired insight which had led him to volunteer. We had selected the small tangram square as a unit of measure to discuss the measures of other tangram pieces or combinations of pieces. Area of a figure was described as the number of small tangram squares that would tessellate the figure. We initially constructed a figure with two small squares sharing a common edge. The students were asked about the number of square units that would tessellate this quadragon with zero points of concavity. Then we constructed a figure with three small squares such that each shared an edge with a square and all three had a common vertex. The students were asked about how many square units that would tessellate this hexagon with one point of concavity. The next activity was constructing an isosceles, right trigon by taking the two small congruent isosceles, right trigon pieces of tangram sets and having the two share a common leg. Then I asked for the area of the larger constructed isosceles, right trigon. That’s when Jeremy volunteered. Actually, when I first constructed the figure and everyone anticipated the question, a lull had occurred while they thought about this newly constructed figure and how the area might be explained. But, Jeremy had inspiration and volunteered during the lull and now he had everyone’s focused, hopeful attention. Jeremy advanced to the center of the circled students, grasped a couple of trigons congruent to the construct pieces, showed that he understood the tessellation of the figure in terms of the two pieces he held and then did a two-dimensional flip with one of the trigons and placed them on top of a small square piece thus demonstrating the tessellation of the square and declared that the measure of the larger constructed trigon was one square unit. The room exploded with applause. Jeremy had this self glow about himself and returned to his place on the carpet while Mrs. Green and myself tried to retain our composure (Later, I am sure, we both gave way to emotion. Speaking for myself, when I recounted the event to others I simply cried).

Lots of times during previous sessions there had been multitudes of occasions when individual students demonstrated real insights that clearly demonstrated an idea or a real growth in geometric sophistication. The idea of measuring beyond counting vertices or counting of categories of geometric objects was not an initial goal that I had anticipated. About mid-way through the sessions we were constructing geometric shapes with steel balls as vertices and congruent magnetic rods as parts of edges of the seven types of trigons when we sort of naturally started describing the lengths of edges in terms of the number of congruent magnetic rods and laying the edges end-to-end to represent the perimeter of such trigons (before, when constructing trigons, quadragons, pentagons, etc., with perhaps straws and strings, we would untie the beginning and end point to stretch out the straws on the string to represent the perimeter without counting the congruent straws , if any, which tessellated the edges) and count the number of congruent rods. I then decided that measuring in units of other cubes was a real possibility. So, we from time to time counted various dimensional cubes that tessellated figures, getting a feel for which cubes didn’t lap or gap in the tessellation. Also, we had been involved in constructing various dimensional figures with appropriate dimensional simplexes. And then, later we were involved with tessellating figures with such simplexes, usually using the vertices of the figures.

Second Episode - Sami

This is a continuation of the previous episode, which was to use the small cube of the tangram pieces as a unit cube and to determine the measures of tangram pieces or combinations of tangram pieces from various sets. After getting over Jeremy’s event we continued by considering one of the mid-sized isosceles, right trigonal pieces with the same inquiry about the measure. This piece wasn’t constructed by two congruent isosceles, right trigonal pieces and it took a moment or so before someone realized that the two pieces that Jeremy had selected also tessellated the larger trigon and demonstrated such while declaring that the area also was one square unit. This demonstration resulted in arranging the two trigons to tessellate the larger trigon and then transferring such to tessellate the same unit square (conserving area in the process). The next problem was to determine the area of the parallelogram piece. It appeared that this might stump the group and there was a couple of futile efforts. Jeremy then easily demonstrated that the same two trigons selected earlier could also be arranged on the quadragon with zero points of concavity to tessellate such and that the pieces could again be transferred to the unit cube, again demonstrating the measure to be one square unit. Everyone was sort of amazed, was Jeremy emerging? The next problem was posed by the largest of the tangram trigons. Someone demonstrated the larger trigon could be tessellated by two of the mid-sized trigons which in turn were tessellated by a pair of the smallest trigons for a total of four congruent trigons which tessellated two of the unit square, thus the area was two square units. Wow! We constructed a large square with two congruent large isosceles, right trigons sharing a common hypotenuse. Students demonstrated the measure of this large square two ways. One of the students followed through by using the previous demonstration for both of the tessellating large trigons. Another student merely picked up four of the smaller squares and tessellated the larger square (double the edges and the area is multiplied by four) with the four unit squares having the same measure of four square units. We then constructed a quadragon with zero points of concavity (a right trapezoid) by having a leg of the largest trigon share an edge of the large square. One of the students demonstrated that the figure could be tessellated with three of the large trigons, each large trigon had been shown to measure two square units and so two plus two plus two accounted for six square units. We next constructed the last problem of the last session and the air was tense with excitement because they knew that my routine for classes was to pose increasingly difficult problems and they could hardly wait to meet the challenge. This problem didn’t initially appear to be too difficult. We constructed another figure similar to the last figure with three of the smaller trigons. Two trigons shared a common hypotenuse and then one of the trigons also shared a common leg and all three trigons shared a common vertex (a right trapezoid similar to the preceding figure). A little experimentation by students showed that one unit square would not tessellate the figure, a gap, and two unit squares would not tessellate either, a lap. After some trials and errors it became Sami’s turn. Sami liked center stage, liked to volunteer, and usually did really well with demonstrations and explanations. She started with a little bluster and then began to wilt, but not give up. It became rather obvious from her trials and efforts, body language, etc., that she probably knew the answer but couldn’t find expression for what she knew, she simply couldn’t express the number and neither could anyone else. She pondered, squirmed and appealed but to no avail. My usual problem solving modality was to wait awhile before offering a hint. Mrs. Green couldn’t stand it any longer and offered a hint. "Sami what do you have when you have fifty cents?" With open arms and rolling eyes she responded, "fifty cents?" Mrs. Green continuted, "Think about it Sami, how many dollars do you have when you have fifty cents?" Then the obvious became evident to her, but not to any of the others, and she blurted out, "Half of a dollar, this extra piece is half of a square unit!" Everyone applauded. Then she felt obligated to finish up the expression for the measure of the figure by amending her observation with "The figure measures one and a half square units."

Over the many years of teaching, conducting formative, field research with public school age students from kindergarten to the senior level , working with pre-service teachers, working with in-service teachers and noticing educated adults, I have noticed some commonalties with difficulties with geometric measuring. Geometric measuring is important. Geometric measuring is quite often when/where we really gain measuring experiences which extend to understanding one’s environment, recognizing and forming conjectures from patterns of phenomenon, and measuring is at the heart of probability and statistics. Measurement geometry not only has a rich cultural history but also predicated formal geometries. It seems that the idea of geometric measuring, the counting of the non-negative real number of unit cubes that tessellate the region of the space bounded by the figure, has not been synthesized during the formal educational process. People do not seem to understand why it is that cubes, unit cubes, were selected to communicate a measure. Apparently they have not constructed patterns in tessellating spaces with cubes to realize that for any space, cubes countably tessellate that space. Consequently, a non-negative number of cubes tessellate sub-regions of spaces. There doesn’t seem to be any real insight into constructing cubes of one dimension from cubes of another dimension or in reversing the situation or in discussing the facets, sub-facets, etc., of cubes. People do not seem to realize that the choice of cubes to express the measure is important (getting beyond the tradition vs international unit verbiage) for some cubes will always result in a measure of zero unit cubes and other cubes will always result in uncountably many unit cubes. Although cubes seem to have intrinsic values and a level of universal communication, the bottom line is that educated people simply do not really seem to know much about cubes, applications and uses in measuring. The same comments apply to the use of simplexes with measuring. The traditional educational process exposes children to a seemingly myriad of formulas in terms of seemingly obscurely related references such as "base" and "height" which in effect only formulate measures of a few of the quadragons. Usually students may have a few token hands-on measuring experiences then the routine becomes a pre-algebra substitution, grind out the measure (make arithmetic mistakes) situation. Students don’t usually come out of the experience being independent in geometric measuring by any stretch of the imagination. Essentially the central idea of tessellating figures with cubes, one idea, has been splintered in the process. Years ago, realizing some of the issues associated with sharing the ability to measure with children, I decided to write a simplex measure formula which can be simply formulated onto a programmable calculator so that a user friendly screen simply asks for the edges of tessellating simplexes and that calculator can almost immediately provide the measure which can be summed with the measures of other tessellating simplexes to determine the measure of a figure. The following formula is an expression of that simplex measure formula: , where represents the measure of a n-dimensional simplex, , where represents the numerical part of the measure of the distance between vertex i and vertex j, vertex 0 is arbitrarily chosen and serves as a reference vertex of the simplex and represents a n- dimensional unit cube. From observing the use of rulers, distance or angular (I prefer both to be in international units), I would recommend that students not simply be handed a commercial ruler (many of which have serious flaws), but instead be encouraged to construct rulers, discover the applied use of congruent linear or angular units in tessellating rulers. Students then discover why on a linear ruler one looks for a ruler where zero is not an end point and why, at least initially, just one unit of measure is chosen and then discover later the need for smaller units on both distance and angular rulers. Why do we seem focused primarily upon two-dimensional models? The world that we live in is somewhat closely modeled with three-dimensional models! Moving between dimensions may encourage young minds to think in terms of the next dimension, regular octatopes or tesseracts and pentatopes the cubes and simplexes of the fourth dimension. There is much to be learned about sharing important geometric ideas at the pre-primary school levels and we need a lot more effort in formative, field research, similar to the work of E. Glenadine Gibb and others, so that we can get rich ideas for the comparative analysis experiments to follow.