According to the Professional Standards for Teaching Mathematics (NCTM 1991), a primary factor in teachers’ professional growth is the extent to which they “refl ect on learning and teaching individually and with colleagues”. Reflecting on their classroom experiences is a way to make teachers aware of how they teach. 

 

o   Tasks that ask students to perform a memorized procedure in a routine

manner lead to one type of opportunity for student thinking; tasks that require students to think conceptually and that stimulate students to make connections lead to a different set of opportunities for student thinking. Leads to the development of students’ implicit ideas about the nature of mathematics—about whether mathematics is something about which they can personally make sense and about how long and how hard they should have to work to do so

 

o    lower-level approaches to the task consist of memorizing the equivalent forms of specific fractional quantities, for example, 1/2 = 0.5 = 50%, or performing conversions of fractions to percents or decimals with standard conversion algorithms in the absence of additional context or meaning, for example, converting the fraction 3/8 to the decimal 0.375 by dividing the numerator by the denominator

 

o   When these lower-level approaches are used, students typically work many similar problems, twenty or more, within a given task.

o   A different approach to this same task—one that presents higher-level demands—might also use procedures, but in a way that builds connections to the mathematical meanings of fractions, decimals, and percents. One way to build such connections is to encourage students to grapple with the underlying concept of part-whole relationships by working with a 10 × 10 grid

o   Students might also be asked to record their results in a chart containing the decimal, fraction, percent, and pictorial representations, thereby allowing them to make connections among the various representations and to attach meaning to their work by referring to the pictorial representation of the quantity every step of the way.

 

 

Another high-level approach to the task—a doing mathematics approach—could entail asking students to explore the relationships among the various ways of representing fractional quantities. Students would not, at least initially, be given the conventional conversion procedures. They might once again use grids; but this time, grids of varying sizes, not just 10 × 10, would be used. F

When students use the visual diagram to solve this problem, they are challenged to apply their understandings of the fraction, decimal, and percent concepts in novel ways. For example, once a student has shaded the six squares, he or she must determine how the six squares relate to the total number of squares in the rectangle. In

“procedures with connections” or “doing mathematics” approaches are used, students typically perform far fewer problems, sometimes as few as two or three within a given tas

 

 

o   Tasks as they appear in curricular instructional materials should be set up by teachers then implemented by students which leads to student learning. Task in book not always identical to the task set up by teacher- differentiated learning. 

 

Theresa’s procedure

o   Students after being presented with a hard task- Their inclination—fortified by years of school experience—was to wait until someone, usually the teacher, showed them how to do it. Theresa was drawn unwittingly into this scenario because she was most comfortable with it

Ron’s practicing using task analysis 

o   For a short time, teacher Ron turned the question back to the students, telling them that it was their job to fi gure it out. As the students became increasingly anxious about their lack of progress, however, Ron began to tell them that they should try starting with the fraction fi rst. Most students had no difficulty figuring that six shaded squares would be 6/40. Then they found the decimal by dividing 6 by 40 to get 0.15 and then turned to the “tried and true” method of moving the decimal point two places to the right to convert from 0.15 to 15 percent. What had started as a completely intractable problem was solved in a matter of minutes! 

When Ron asked for feedback on the lesson, 

o   one of his colleagues noted that by moving through the problem in this way, the students had completely divorced their thinking from the diagram and consequently Designing and Enacting Rich Instructional Experiences 13 from the meanings of decimal, percent, and fraction.

o   Another teacher found it curious that the students showed no inclination to even check the plausibility of the answers that they came up with against the diagram.

o    After more discussion, the teachers agreed that by succumbing to the students’ requests of “how to do it,” Ron had reduced or eliminated the challenging, sense-making aspects of the task, thereby robbing students of the opportunity to develop thinking and reasoning skills and meaningful mathematical understandings. 

 

 

o   So next instead of giving in to students’ pleas for simplification, Ron suggested that they look carefully at the rectangle, noticing both the total number of squares and the ways in which the squares were organized into columns and rows. As he walked around the room, he noticed that those students who were making the most progress had observed that each column represented one-tenth of the rectangle and had shaded in six squares, almost as if they were “fi lling up” one and one-half columns. If a column was one tenth, or 10 percent, then a “column and a half,” they reasoned, would be 15 percent. The students who were having the most diffi culty were working with rectangles in which the shaded squares were not in columns but rather in some other configuration. He helped these students find other ways to figure the percent by asking questions that would allow them to build on the particular configuration that they had shaded.

o   Ron’s assistance encouraged the students to persist with figuring percent and, more important, made the students think about what percent meant in relation to this particular diagram.

 

o   Although it took nearly the entire class period to get through this one problem, Ron found spending the time to be worthwhile. By the end of the lesson, several students had presented alternative strategies at the overhead projector in the front of the classroom. Even Ron was surprised at the many different ways in which the students solved the problem!

 

 

Factors Associated with the Maintenance of High-Level Cognitive Demands 

1. Scaffolding of student thinking and reasoning is provided. 

2. Students are given the means to monitor their own progress. 

3. Teacher or capable students model high-level performance. 

4. Teacher presses for justifications, explanations, and meaning through questioning, comments, and feedback. 

5. Tasks build on students’ prior knowledge. 

6. Teacher draws frequent conceptual connections. 

7. Sufficient time is allowed for exploration—not too little, not too much.

 

Factors Associated with the Decline of High-Level Cognitive Demands 

1. Problematic aspects of the task become routinized (e.g., students press the teacher to reduce the complexity of the task by specifying explicit procedures or steps to perform; the teacher “takes over” the thinking and reasoning and tells students how to do the problem). 

2. The teacher shifts the emphasis from meaning, concepts, or understanding to the correctness or completeness of the answer. 

3. Not enough time is provided to wrestle with the demanding aspects of the task, or too much time is allowed and students drift into off-task behavior. 

4. Classroom-management problems prevent sustained engagement in high-level cognitive activities. 

5. Task is inappropriate for a given group of students (e.g., students do not engage in high-level cognitive activities because of lack of interest, motivation, or prior knowledge needed to perform; task expectations are not clear enough to put students in the right cognitive space).

 6. Students are not held accountable for high-level products or processes (e.g., although asked to explain their thinking, unclear or incorrect student explanations are accepted; students are given the impression that their work will not “count” toward a grade).

 

Teacher’s were able to relate to all of these a lot 

o   When used well, it should draw attention to what students are actually doing and thinking about during mathematics lessons. This focus on student thinking, in turn, helps the teacher adjust instruction to be more responsive to, and supportive of, students’ attempts to reason and make sense of mathematics.

Using framework as a tool to reflect on your practice

1.    Teacher’s should observe other teachers and work together,discuss, critique and make make suggestions for improvement. 

a.    Walk around the classroom during lessons to see how students are doing

b.   If the two of you agree that one or more tasks were set up at a high level of cognitive demand, go on to discuss whether those demands were maintained at a high level during the implementation phase or declined into less challenging work

 

2.    Teachers should observe themselves

a.    videotape