With emphasis on mathematical relationships, these CGI classrooms were fertile grounds for challenge to take place. However, challenge is relative to the individual, and the
relationships must be at a level permitting each individual to experience Ascending Intellectual Demand. For instance, Geraldo, the top tier fifth grader, reported to have understood the concept of common denominators since third grade and did not find the discussion of equivalent fractions challenging. However, when the class digressed one day to explore how multiplication of
on his interest and gave him a related challenge problem the next day, after he had finished the regularly-assigned problem. The challenge problem was designed to discover the fraction division algorithm. Geraldo was temporarily stumped but, with a small amount of teacher’s scaffolding, could see the relationship between dividing by a number and multiplying by its reciprocal. The teacher had placed him in his zone of proximal development, thus increasing the intellectual demand. Knowing when a gifted student needs intellectual challenge requires skill and attentiveness on the part of the teacher, recognizing even brief signs of student intrigue and making instant decisions to facilitate that student in a new direction. Reminding that Ascending Intellectual Demand is an “escalating match between learner and curriculum” (Tomlinson et al., 2009, p. 11) it guides teachers as they consider how to challenge their students.
Knowing the extent to which students are challenged is related to ways in which they were challenged. The next sections will focus on other significant ways that the CGI teachers pressed the intellectual demand for their students beyond what might be considered a more average classroom experience. Limitations of these ways are also discussed to further describe the extent of the challenge.
Problemization of simple problems. The students’ self-reports of perceived challenge can approximate the degree of challenge they experienced as they solved the assigned problems. Their average perceived challenge level of the assigned problems was low (1.4 for top tier advanced students and 1.7 for second tier advanced students, on a scale of 1 to 5 with 1 meaning ‘not challenged’). The top tier advanced students finished the problems within 5 minutes (out of an average of 22 minutes allotted to the class for problem solving) 57% of the time, which suggests the problems were relatively easy for them. The teachers did, however, frequently use strategies to extend their students’ thinking during this problem solving phase.
The students perceived the challenge of the entire lesson as higher than their rating of the assigned problem alone (2.1 for the top tier students and 3.1 for the second tier students).
Hiebert (as cited in Empson, 2003) pointed out that the cognitive level of a problem could be low but the problemization of it in the hands of a skilled teacher and an active group of participating students could make it a high cognitive level experience. Problemization of a math problem refers to the extensive elaboration of the problem that extends the mathematical thinking beyond a simple solution. In the cases of the two CGI classrooms in my study, at each observation I witnessed problemization as the teachers used the somewhat-simple word problem assigned that day as a springboard for a deeper mathematical discussion. Teachers led discussions in which students compared different strategies for solving the problem and looked for connections between them. They delved into more depth at the math concepts behind the solutions as different number choices for the same root problem were discussed. Teachers’ resourcefulness for problemization was evident as they took advantage of opportunities for students to discover new related concepts and to express the numerous mathematical relationships that were
discovered by using mathematical notation of equations. These rich discussions almost always led to exposure to above grade level topics and lasted an average of 42 minutes, further
supporting the idea that although the individual problems may have been easy for some to solve, the problemization raised considerably the level of thinking.
Number choices for solving problems. The most common strategy of differentiated instruction in these CGI classrooms was the use of more challenging number choices for the same root problem. Although it is a practice that efficiently allows students in a mainstream classroom to self-differentiate, its limitations are apparent when the results show top tier students completing all number choices in a fraction of the time that other students take. It works for a
while with a new topic, such as when the third grade students were transitioning from adding thirds and halves with a common unit of sixths, to being challenged to finding a common unit for thirds and fourths.
With mathematically gifted students possessing a high ability for processing
mathematical information and keen mathematical memory (Krutetskii, 1976), these students catch on quickly to both procedures and concepts and retain the knowledge to apply at a
subsequent time. Once the concept is mastered, a mathematically gifted student may see adding eighths and twelfths as no more difficult than adding halves and thirds. The teacher must be attentive to the moment when these students have conceptually mastered a topic and are ready for a greater challenge. This may be evidenced by a quick finish time with complete explanation of the process.
The selection of number choices for the root problems observed in this study were often designed to elicit relational thinking from one number choice to the next, encouraging students to look for relationships between sets of numbers. Problems that could be solved using relational thinking between number choices often led students to quick solutions particularly when the advanced students recognized the relationship between number sets meant there would be a relationship between answers. Although the third graders were just beginning to see such relationships, the fourth and fifth graders were getting accustomed to looking for them and getting good at applying them to produce quick answers to subsequent problems. However, the students did not always take advantage of the relationships between number choices to solve the problems, perhaps finding their chosen methods to be just as easy. However, these relationships between the number choices did offer another interesting way to solve problems that ordinarily may have been easy for some students, and serves as another way for the students to look at the
problem once they have finished with their initial strategies. The teachers must recognize that when relational thinking in this manner becomes easy for these students, they should be prepared to challenge the student in new ways.
Emphasis on notation of mathematical relationships. Both teachers emphasized expressing mathematical relationships using equations and making connections to mathematical properties. This elevated the level of instruction beyond any specific math topic, promoting a deeper algebraic understanding of numbers. The prevalence of this notation in the students’ work and in class discussion is a reflection of the teachers’ efforts to push this higher level understanding.
Intellectual Peer Groups. Placing the advanced students together provided increased opportunities for discussion among these intellectual peers that prolonged their engagement with the problem. This also presented students with the opportunity to lead their own mathematical discussions, increasing autonomy, a kind of challenge in itself (Diezmann & Watters, 2002). In the group investigations, the students were challenged to develop generalizations beyond those expected of the other students.
Higher order thinking skills emphasized in teacher extensions. The teachers interacted with the advanced students on a regular basis with the intent of extending their
thinking to higher levels. This attention facilitated students in seeing connections between ideas, notating their work, making conjectures, and justifying their solutions. Other types of higher order thinking were prevalent, too, as categorized in Tables 6 – 9, thus increasing the intellectual demand.