Sometimes mathematical empowerment is achieved by individuals of a group and sometimes not. Dr. Edgell first cites an example of a fifth grade student achieving mathematical empowerment while involved in an academic year, formative, field research project studying the accessibility of applications of algebraic modulo structures by fifth and sixth grade students. Secondly, Dr. Edgell cites an example of first grade students not achieving mathematical empowerment, collectively or individually with respect to large number estimation ability while involved in an academic year, formative, field research project studying conservation of one-to-one correspondence, application to understanding counting numbers and the operations of addition and multiplication.

When preparing pre-service teachers of mathematics I generally try to get students to cite examples which have characteristics of an idea as well as to cite examples which do not have such characteristics. Sometimes I might suggest examples such that some have the defining attributes and others do not have the attributes and challenge the students to separate the examples according to the criteria of the idea. The essential message is that if one is to claim understanding of an idea one should be able to cite those which are versus those which are not examples of an idea. It is not enough to simply be able to cite a couple of examples which have the properties of an idea and feel that one has the idea. It seems to me to be appropriate, for the purposes of initiating this discussion on Developing Children’s Mathematical Power , that an example of a child exhibiting mathematical power and an example of children not exhibiting mathematical power be discussed.

Caitlyn, a fifth grade student, had volunteered with other fifth and sixth grade students of the Sunset After School Program located at Hernandez Intermediate School, San Marcos CISD, San Marcos Texas, to be involved in a formative, field research project which I was directing and conducting. I was interested in studying the issue of accessing applications of algebraic modulo structures for the academic year 1999-2000. The group met on seventeen Friday afternoons for an hour or so each session. I suspected that Caitlyn was using a guessing strategy and a diagnostic tool indicated that occasionally she was guessing. I discussed the results of the diagnostic with Caitlyn and made the issue that guessing was not an acceptable strategy clear to her and indicated that she needed to make a decision about continuing her participation with the study (or not). It was clearly her wishes to continue participation and she would work hard on her strategies and I conditionally included her with the group. The others members of the group were all female sixth graders.

The goals of the project were a continuation of the goals of the previous Spring ‘99 semester, to access the cast out structure for numbers , counting, integral, rational, expressed in base ten and the checking of arithmetic, addition, subtraction, multiplication, division of such numbers. There were some modifications of these goals in that we would only consider addition and multiplication and also consider these same goals applied to these categories of numbers and arithmetic of these numbers expressed in other bases.

We started, as indicated, with base ten. The sixth graders were already familiar with casting out nines with base ten numerals and checking arithmetic of such and readily recalled their experiences of the previous semester, in fact they had become more sophisticated with respect to their skills. Caitlyn was much slower in applications and perhaps because of our previous interaction she appeared to be very methodical with respect to using her strategy. She seemed very appreciative of the use of the initial graphics and manipulatives, base ten blocks, in determining the cast out of a number. Initially when she was introduced to ruler models for fundamentally understanding the operations of addition and multiplication she seemed somewhat less than appreciative. When we encountered arithmetic of integers she began to become more interested in the fundamental ruler models, sort of accepted the graphics of the cast outs and really became dependent upon the base ten color coded blocks both for expanding upon the fundamental ruler models and for determining the cast outs. It appeared that we were on the edge of her arithmetic experience. When we expanded to the rational numbers expressed in the digit-point format we were somewhat beyond her range of school experiences and so the fundamental ruler models and the manipulative base ten (re-ordering the value assignments to the blocks) models really became her basis for determining sums, products., cast outs, and applying cast outs. She gradually began to expand her acquired school arithmetic skills and in the process her thinking in terms of base ten was expanding beyond her previous understandings. The sixth graders were encouraging and patient, although from time to time they would like to show off their relative expertise.

The day finally arrived when we shifted to expressing numbers in base five and casting out fours. This was different for the sixth graders as well as Caitlyn. The formative ruler models took on a new role of importance for the sixth graders, but for Caitlyn these had become familiar. I encouraged everyone to think in terms of base five rather than thinking in terms of base ten and translating (or, translating from base five, thinking in base ten, and translating back to base five, as it was becoming readily evident that Caitlyn was engaging). With some practice with addition and multiplication of counting numbers the students began to get free of the fundamental ruler models, but most still seemed to be thinking (notes, talking themselves through a problem, describing their results, body language, etc.) in terms of base ten. It was clear that Caitlyn was translating back and forth. The casting out of fours seemed to come naturally to the sixth graders, particularly after some base five block manipulative experiences. Caitlyn remained very methodical in this process. Applications of the cast out of fours to checking addition and multiplication seemed rather easy for the sixth graders, but tedious for Caitlyn, although she gradually became rather proficient. The sixth graders capitalized upon the fundamental ruler models in understanding the arithmetic of integers and, after some experiences with the color coded base five blocks, readily checked such arithmetic. Again, we were on the edge of Caitlyn’s experiences, but she plod forward. Sometimes she expressed frustrations but generally she continued progress, however tedious. Expanding to the rational numbers seemed natural to the sixth graders who seemed to be almost thinking in terms of fives by this time. Their explanations were gradually more and more in terms of fives and their demonstrations were clearly in terms of fives. Initially Caitlyn was becoming desperate, but began to see some patterns in the arithmetic of rational numbers which she seemed to have insight upon and became increasingly skillful. She was beginning to catch up to the sixth graders in skills even though it seemed to take her a considerable while to do so. Essentially it was clear that she was still translating.

There were only a couple of research sessions left, Caitlyn was skillful (and not guessing) and had managed to stay with the group. We made a transition to base three, their choice. We quickly brushed through the fundamental ruler models, the familiar cast out graphics, and the familiar base three block manipulatives. We were adding and checking addition of counting numbers with the cast out of twos, and, somewhat because of the simplicity, everyone was making good progress, even Caitlyn who was still translating. Then it finally happened! I had challenged Caitlyn to the following problem.

The work looks quite ordinary, but the manner in which she communicated what she was doing clearly indicated that she was not translating. She was thinking in threes! She easily and flexibly expressed herself with confidence, a clear expression of mathematical power. Caitlyn breezed through the arithmetic of integers and rationals and the checking of such. She was in command. I asked her if she would assist me the next year if I choose to study similar issues but start with other bases and she confidently indicated that she would be excited to be my assistant.

This next example came about as a result of noticing some rather sophisticated native small number , less than one hundred, estimation skills during an academic year, 1999-2000, formative, field research study with a class of first grade students, taught by one of my previous students, Mrs. Benner, at Goodwin Primary, Comal ISD, New Braunfels, Texas. The initial goals of the study included studying accessibility of conservation of one-to-one correspondence, application to understanding counting numbers and arithmetic, addition and multiplication, of such. There were thirty-one sessions, which generally met on Fridays for about an hour just before lunch.

Based upon the observation of these small number estimation skills I wondered about their large number, several thousand, estimation skills. So, I designed a sequence of seven episodes to investigate their native skills and to see if such skill might be modified to become more sophisticated. Initially I filled a large onion sack with native pecans and covered the sack with a black garbage bag and left the bag, labeled "mystery bag", in front of Mrs. Benner’s desk for a few days.

Interview format: Interviews always began with a promise of a reward at the awards ceremony for the best estimate of each session. Quantitative information determined during a session was always listed on an adjacent board to the interview station and referred to during the interview. Students were asked leading/expanding questions - "Do you think that there are more than 0,10,100,1000, ..., pecans?" With a response of "no" , for example at 1000, then the same leading/expanding questions would start with the last power of ten to which there was agreement , such as - "Do you think that there are more than 200, 300, ..., pecans?" Until another response of "no", for example at 600. Then "..., more than 510 pecans?", etc.. Students were never informed as to how close their estimate was to the number of pecans in the "mystery bag". It was always indicated that the interviewer was appreciative of the participation of the student in the session. Generally interview sessions would be interpreted as friendly sessions.

First Session, the "Gut Level Estimate": I was able to set up a station outside and behind the room to initial reveal the contents of the bag to individual students and to conduct the initial interviews. This "Gut Level Estimate" was clearly a "gut level guess".

Second Session, Group Interaction Estimate: Mrs. Benner had the uncovered mystery sack in her room in view to all the students. She encouraged the group to openly discuss various opinions as to the number of pecans for an extended period and then during the day she conducted individual interviews with the same results, guesses.

Third Session, Weight Model: We weighed the bag of pecans to be about 50 lbs. Then we carefully counted 400 pecans which weighed the to be just a little under, 5 lbs. Individual interviews were conducted with the same result, guesses.

Fourth Session, Volume Model: Mrs. Benner choose a plastic tub which when filled contained 480 pecans and the onion sack of pecans almost filled 9 tubs. Mrs. Benner interviewed each individual following the same format and with the same results, guesses.

Fifth Session, Area Model: I brought two large unfolded cardboard boxes upon which was marked several congruent squares 0.5 m by 0.5 m. The students carefully arranged the pecans on the squares which almost covered ten squares and then carefully counted the number of pecans on one of the squares, 420 pecans. I then conducted individual interviews with the same results, guesses.

Sixth Session, Linear Model: I brought corrugated iron roofing sheets each of which has valleys upon which the students carefully laid the pecans end-to-end. The students carefully counted the number of pecans to one valley to be 90 pecans and there were 44 valleys of pecans. I again conducted individual interviews with the same results, guesses.

Seventh Session, One-to-one Correspondence, Base Ten Model: Students formed teams, two teams to carefully put ten pecans into each plastic sandwich bag, two teams to carefully put ten sandwich bags of pecans in a larger plastic bag (each containing 100 pecans), two teams to carefully put ten of the larger plastic bags of pecans in a large paper shopping sack (l000 pecans per shopping bag). The students then determined the number of pecans to be three shopping bags, nine large plastic sacks, three sandwich bags of pecans or 3930 pecans. All of the students were excited to learn and understand the number of pecans, finally. But analysis of their estimates and professional observations indicates that none of the students, individually or collectively had attained mathematical power in estimating a large number of objects.