Four-Phase Instructional Model For Mathematical Investigations
(adapted from Manouchehri, 2001)

 


Phase 1:  Problem Posing 


     In this phase, the teacher sets the stage for the mathematical investigation. The investigation may be based in a real-world context, anchored on prior concepts and knowledge, or posed as an interesting investigation that will lead students into constructing important mathematical understandings. The mathematical problem(s) or task(s) are either posed by the teacher or generated from students' questions.

Ex: The teacher asks students to generate a list describing what they know about quadratic functions of the form f(x)=Ax^2+Bx+C and how they are similar to or different from linear functions of the form g(x)=Mx+B.  Based on what students know about the effects of M and B on the graph of a linear function, students are asked to predict how they expect A, B, and C to affect the graph of the quadratic function. The teacher poses the task for students to use a technological tool (e.g., Explore Math interactive applet) to investigate how the parameters A, B, and C affect the graph of the quadratic function.
 

Phase 2: Small-Group Investigation


     In this phase, students work in small groups on the task posed while the teacher monitors their progress and poses questions to guide students' thinking. The teacher can assume several different levels of facilitation during this phase. For example, students may have a guided worksheet that has pre-planned questions posed that they should investigate. In this case, the teacher may monitor group work, join small groups and only interject questions when needed. The teacher may also choose to leave the task open-ended and pose guiding questions to the individual groups or the whole class as needed. The students should be recording their work in some form. This may be on paper, using a technology tool (e.g., saving work in a file), or on a white board, overhead transparency, or poster paper.  Students will need their records for use in phase 3.

Ex: Students use the Explore Math applet to change the values of A, B, and C, and observe the effects on the graph. As the teacher circulates, he or she notices that one group has the value of B set to 0 as they changed A and have concluded that A makes the graph "skinnier or wider". The teacher asks the students if this conjecture is true for A for all values for B and C.  Another group uses the "show vertex trail" option in the applet and notices that when A changes, the graph not only gets "wider and skinnier," but that the vertex seems to follow the path of a straight line. Seeing what the students have done, the teacher asks the group to think about how the path of that line is related to the values of A, B, and C.  The students in each small group record their conjecture for the effect of A, B, and C on an overhead transparency.
 

Phase 3: Whole-Class Discussion of Investigation


     In this phase, the teacher leads the class in a discussion of the findings during small group work. The role of the teacher in this phase is NOT to summarize what students should have learned in the small-group work. Rather, the teacher's role is to facilitate classroom discourse to provide students with an opportunity to explain their findings, raise questions about what they may not understand and share interesting "surprises" that may have raised additional questions or lead to further investigation. Students should use their recorded work as a means of reference during this discussion or to display to the whole group. The teacher should pose questions to students to have them justify their reasoning and provide mathematical warrants for their claims. The teacher may also pose questions to have students anchor their findings in previous learned concepts.

Ex: The teacher asks for groups to share how C affects the graph of the quadratic function. A student shares that his group found that changing C only moved the graph "up and down.". The teacher asks if that conjecture was true for all values of A and B. Other groups confirm that it seemed to be true for all values of A and B. The teacher asks "what about if A and B are both 0?" All the groups admit that they did not test that condition. The teacher gives the groups several minutes to discuss this with each other, only reasoning from the symbolic form of the function, f, with rule f(x)=0x^2+0x+C.  After groups share their reasoning, the teacher uses the Explore Math applet on a class display, models changing C, and asks students to explain what it means in terms of X and Y values for the graph to shift vertically when C is changed. The class discussion continues in this vain with students explaining their conjecture, justifying their reasoning, and the teacher posing critical questions to focus students on connections and important mathematical ideas.
 

Phase 4: Summarizing and Extending


     In this phase, the teacher leads the class in a summary of the key mathematical ideas and poses follow-up tasks that stem from this investigation. These tasks may be homework problems, or they may be topics of additional investigations to be done the next day in class.

Ex: The teacher uses the students' findings from the investigation to summarize how each of the parameters A, B, and C affect the graph. The students record these generalizations in their notebook. During the whole class discussion, the issue of the path of the vertex was raised by one group. They were not sure how to describe these paths.  For homework, the teacher asks the students to focus on the path of the vertex when they change C, when A and B are non-negative. They are to use their graphing calculator and graph several functions that have the same values of A and B but have different C values. Using the table or trace features they should look at the coordinates of the vertex and try to describe the equation for the vertical path of the vertex in terms of the parameters A, B, and C. The teacher also plans to have students explore the paths of the vertex when A and B change in a class investigation the next day.