Modifications to the Placement of Reflection on General Mathematical Principles

...by the time you go through all this stuff and you have these kids who are already at risk in terms of learning... By the time they get here, they forgot the point of the whole investigation. And it's like you keep trying to tie it all together and say well the reason why you did this, this, this to get here is because you wanted to... you know. And so sometimes I think Core-Plus can take too long to get to the point. And I think for some kids who are beginning to understand, that if you take that long, you frustrate them. And they want to throw their hands back up in despair. (Introductory Interview, 9/30/11)

In the Core-Plus investigations in this study, students were only asked to answer questions aimed at general mathematical principles toward the end of each investigation,

in a section called Summarize the Mathematics (STM). According to the developers, the

STM tasks are to be worked as a group then discussed as a class so that students can “construct a shared understanding of important concepts, methods, and approaches” (Hirsch, Fey, Hart, Schoen, & Watkins, 2011, p. 12). Ms. Spence used the STM sections at the end of investigation only once during the entire unit, despite her concern that students struggled to identify exactly what general mathematical principles they were meant to understand from working the Core-Plus sequences. In one investigation, she used the STM questions at the beginning of an investigation, because she felt that the questions fit better there as a review of the previous investigation. In the remainder of the investigations, she omitted the STM questions altogether, sometimes replacing them with supplemental summative activities.

When I asked why she tended to not use the STM questions as a summarizing activity at the end of each investigation, Ms. Spence indicated that she understood how the authors intended for the STM questions to be used but sometimes chose to position them elsewhere simply to attempt a better fit:

Summarize the Math is for them to go back and reflect back across the investigation and go for those big concepts. And then the Check Your

Understanding ends up being a practice problem for them to try. I don’t always use the Summarize the Math like the way they would have me use it at the end of the investigation. I think it depends on where it fits. And it may fit somewhere else. (Introductory Interview, 9/30/11)

In addition to critiquing where the tasks were placed within the lesson sequences, she also noted two other difficulties students had with the STM questions. In the following quotation, she explains that her students had trouble with the literacy demands presented

by the STM and she describes difficulties they had connecting these questions to activities that may have been done a few days before:

I think that they look at it and they just shut down. It’s too many words on the page. I hate to say that for kids that are in ninth and tenth grade but I think in how it’s all put together, they just still have trouble reading here, here, here, here [pointing to various tasks] and just connecting them. (11/02/11)

Instead of using the Summarize the Mathematics sections as intended, Ms. Spence frequently supplemented the written curriculum with tasks at the general level during the portions of the instructional sequences comprised of contextualized examples. The lesson map in Figure 12 illustrates the addition of activity at the reflection on general mathematical principles level. In the columns representing tasks posed to students on 10/27, Ms. Spence supplements their contextualized problem solving experiences with activity at the reflection on general mathematical principles level. This supplementation shows up in Figure 12 as both open and closed circles at the reflection on general

mathematical principles level in shaded regions (teacher supplements) surrounded by non-shaded regions (tasks from the written text) containing tasks at the contextualized example level.

Some of the supplements at the level of reflection on general mathematical principles were the result of a significant institutional factor: the school’s use of a school- wide lesson plan template. This template contained a number of elements, including essential questions, “activation strategies” to motivate lessons and activate prior knowledge, and an expansive library of graphic organizers. The template was meant to provide students with a consistent experience across classrooms and to make explicit for students the content that they were meant to learn. Ms. Spence supplemented the written

curriculum with additional tasks meant to address the required elements of the template; often, these tasks addressed general mathematical principles. As a result, the

supplementation of these elements increased the extent to which general mathematical principles were discussed throughout the lesson.

Additional activity at the reflection on general mathematical principles level also occurred due to the presence of “essential questions” as recommended by the

instructional template. The essential questions that Ms. Spence used to fulfill the

requirement in the instructional template were determined by the PLC. These questions were written to align with the Core-Plus curriculum but were not taken directly from the textbook. Ms. Spence introduced essential questions addressing general mathematical principles at the beginning of each lesson. Twice, out of six times she introduced new essential questions, she planned to go beyond simply stating the essential question for students. In these instances, she planned to have students discuss the answers to these questions. This action resulted in students engaging with activity at the general mathematics principles level earlier than would have been the case if she did not supplement the textbook in this way.

Additionally, Ms. Spence used graphic organizers provided by the school-wide instructional program to help students consolidate general mathematical principles. In the lesson-planning template, the graphic organizers were positioned at the beginning of instruction. Ms. Spence planned to use the organizers at various points in instructional sequences. In one investigation, she used the graphic organizer at the beginning of the instructional sequence, which resulted in the introduction of general mathematical principles before students had worked with the contextual examples. In another

investigation, she planned to introduce the organizer at the beginning of an investigation and to have students fill out only information they already were assumed to know; then, she planned have students complete the graphic organizer as a summative activity. In two other investigations, she planned to introduce the graphic organizers midway through the investigation, to consolidate general mathematical principles that students had engaged with during the earlier part of the task sequence. Taken together, the use of graphic organizers, as recommended by the lesson plan template, resulted in activity at the general mathematics level earlier and more frequently than would have been the case if she had not supplemented the written curriculum.

Ms. Spence also added activities targeting general mathematical principles that were not the result of following the lesson plan template. She tended to supplement the curriculum with tasks or questions targeting general mathematical principles after the book introduced a general mathematical principle. For instance, after students read a paragraph describing the concept of rate of change, Ms. Spence added a task asking students to write a brief description of what rate of change means and what it can be used to do. This task, and others like it, prompted students to actively engage with generalizing mathematical concepts earlier and more often than the text required.

A major factor influencing Ms. Spence’s insertion of supplemental general activity throughout the instructional sequence was her belief that the “discovery” aspect of the Core-Plus design was not working for some of her students, as demonstrated in the following quotation:

I see it more as Core-Plus wants the kids to go through the exercise and discover what's happening as they go through it on their own. I think that having taught this so long that it doesn't.. personally.. I don't see it working the way wanted.. the

authors intended for it to work... because I think that for the most part [to align with the authors’ intent] it should be more student-centered and I should be more a facilitator than a leader. And I think I end up leading more than I facilitate. But I think I also do that because I think that the kids sometimes have trouble

interpreting what is meant by what they're asking and also I think sometimes there's so much that they want the kids to figure out on their own I think the point of the investigations sometimes gets lost if I don't find a way to lead them where they need to go. (Introductory Interview, 9/30/11)

Ms. Spence felt that the mathematical ideas were not made explicit enough and took too long to develop in the text. Because of this, she felt her students were distracted from the mathematics they were meant to learn. In other interviews, Ms. Spence distinguished between how the curriculum worked for different students in her classes, particularly in regard to the “discovery learning”. She felt that students who were well-prepared were able to make the necessary connections. But for a majority of the students who had been less academically successful in the past, she felt that she needed to take an active role in mediating students’ experiences with the written curriculum. To mitigate what she felt were mismatches between the curriculum as written and her particular students, Ms. Spence intentionally took on a more directive role that she understood to be in conflict with the authors’ intentions.

In summary, Ms. Spence saw the value in a discovery approach, but she also believed that her students were not prepared to make these discoveries without additional support. As a result, she supplemented the textbook with activity focused on general mathematical principles throughout the task sequences, rather than at the end, in order to compensate for a curriculum that did not adequately match the level of preparedness possessed by her students.