Differentiation in Mathematics

In the webcast "Differentiating Mathematics Instruction", Dr. Marian Small begins by identifying elements of successful mathematics classrooms. She recounts situations where teachers created types of tasks that had a relative sense of importance to the learner. As teachers interact with students, they have demonstrated care and knowing their learning by choosing words that would prompt learning and deeper understanding. Dr. Small recognize that effective teacher facilitation of student learning responds differently to each learning being sensitive to what they need creating a safe and inclusive environment for all learners. Throughout her observations, Dr. Small also notes that effective mathematics classrooms are lead by teachers that focus on what the learner is saying, thinking, and how they are responding to one another. From there, she examines the knowing and learning that happens and how time to answer, perseverance, appropriately built inquiry at the right challenging level for the learners, allowing for diversity in student thinking, and creating meaningful tasks all contribute to an inclusive differentiated successful mathematics classroom.

With these understandings of successful classrooms, Dr. Small builds the conceptual framework of what a differentiated mathematics classroom will comprise.

Big Ideas with Open Ended Approach -- Dr. Small feels that students need to deal with Big Ideas at different levels of sophistication to allow them to perform successfully and safely where they are in their learning. In order to do this, the teacher needs to know where students are at in their learning in the intended content area through diagnostic assessments (these can be in different forms...conferencing, performance task, paper/pencil test).  This will give a window into how well prepared the students are for the intended learning (it may turn out they are well ahead and can tackle more diverse inquiries as well).  Diffferentiation also includes focusing on inclusivity where the inquiries are open ended enough for al students to engage in meaningful ways. The entry points may look like giving students options to select their own objects, build their own designs, select their own number set, selecting their own data group for graphing, etc. where they are empowered to make as much of the inquiry safe and familiar to themselves while not detracting from the learning intention connected to the big idea.

In addition, Parallel Tasks can be incorporated to give students the responsibility of choice to select the most appropriate entry point for their engagement. The two tasks can be more structured then the open ended tasks, but are varied in levels of complexity but still meet the learning intentions of the lesson.

Differentiating Mathematical Process is something that can also contribute to deeper understanding by allowing students the power to use their understanding of concepts to decompose procedures and communicate solutions.

Reflecting on My Practice

I would begin by saying that I don't come near some of these recommendations especially with regards to parallel tasks. I am intrigued by this diversity in inquiry design and feel that this will definitely provide even more opportunity for students to engage in meaningful learning experiences. With regards to open ended inquiries, I have always felt that this kind of flexibility is always a positive accommodation for the learner as they are empowered to customize some of the elements to make it their own. This has always brought amazing results as students expand from this starting block to naturally connect other ideas and understandings they have to the task. I believe that Dr. Small's chronicle is more structured and effective as I have always just kept everything open to anyone's ideas. I will change my practice of open ended questions to purposefully build threads of openness in order to stay the focus on the intended learning.

Beyond task design, I also feel very passionate about Dr. Small's notions on treating our learners differently as to how they would respond best (public praise might not be appreciated by all learners) and on listening to our learners. I have always believed in the power of students making their own meaning and that each person might have different solutions and/or different explanations that deserve our attention. I think having students share their ideas fully can also provide an opportunity for us to facilitate their learning even further.

Supporting Diverse Learners

In the article"Teacher Collaboration in an Inclusive Classroom", we read about how an inclusive class can be a difficult goal to accomplish because of how students with special needs have not performed well in an inquiry model mathematics program. The reading then moves along to identify 4 very important variables that need to be attended to differently and with varied intensities support these learners (Time, Structure, Support, Complexity).  We know that students are encouraged in the inquiry model to construct mathematical understanding through problem solving, cooperative learning, and mathematical discourse (this includes debriefing/reflection). So, how then can this challenging task be accomplished?

One of the first components that will need to be attended to is planning with differentiation in mind. Differentiated planning will build a successful learning experience for special ed. students from the very beginning of the inquiry. In addition, this planning construct will extend beyond the identified students and also be a supportive and successful experience for all learners. One element of differentiation is the use of flexible groupings. This is something I thought I did well until I analyzed my practice through the lens of this article. Flexible groupings allow for the teacher to facilitate the learning differently for the members in the group. Some members of the group may need very little support as they understand much of the inquiry and are ready to branch off into smaller threads where they can continue the inquiry. Other members may require additional support through further breaking down of the inquiry to construct their understanding of the inquiry components. This then provides those students with the additional teacher facilitation needed to deconstruct the problem developing mathematical comprehension skills in the process with teacher support. While in this group, student unique learning needs/styles will be taken into consideration in using specific strategies (models, manipulative, verbal/visual explanation of the problem, further breakdown of the inquiry into smaller more manageable pieces, or continued teacher facilitation).  Effective use of flexible groupings provides inclusivity as students are grouped differently with each inquiry through teacher facilitation. This will avoid having students in the class stuck in their own low group. Through inclusivity, every learner is equal and a contributing member of the whole group. I have misunderstood the flexible groupings strategy and plan on reshaping my classroom practice to make my time with students more effective.

Another component of supporting diverse learners is creating a mathematics classroom that has opportunities for choice embedded into it. As I move through the literature on this topic, I am constantly thinking of my literacy work stations where students make choices on different independent tasks that provide them the opportunity to take responsibility for their learning in making choices on what they need to work on. The tasks are specifically built by the teacher taking into consideration the mathematical needs of the group. This gives students time to practice their learning in consolidating their understanding and reinforcing emerging skills. Continuing with embedded opportunities for choice, the teacher can also provide students with the decision on how to work (independently, partner, teacher facilitated group). As students work through their choice, the teacher can facilitate students reflecting on the appropriateness of their choice. While students work on their tasks, the teacher can also assess student learning by conferencing with individuals, facilitating the learning of a small group, extending the learning for other students/groups. This as well is something that I haven't done before in my class. I believe very much in the use of literacy stations in my class as an important part of the literacy program and I had wondered in the past how this could be used in mathematics. Now I know and intend very much on implementing Math Stations.

In summation, through the differentiation provided via flexible groupings and the choice provided by math workstations, an inquiry mathematics model is built for success for all learners. This student centred approach to inquiry based math has inclusivity entrenched into it as all learners are monitored, assessed, and accommodated for on an ongoing basis in order to meet their needs supporting engagement in successful authentic learning experiences.

The Inclusive Mathematics Classroom

In reading A Guide to Effective Instruction in Mathematics, Volume One: Foundations of Mathematics Instruction Pgs.23-34, it is clear that student learning in mathematics needs to go beyond procedural instruction (actually it needs to come first and be the learning intention of lesson). The research base of this reading identifies that students lose procedural understandings overtime, but the conceptual understanding...the part that makes sense to students, lasts. This is only logical as it becomes an ongoing evolution of how students see their world around them. This "making sense" of their world is life long because they live it everyday. So, as teachers we need to have the greatest impact on student learning in the area of conceptual understanding. Inclusion is connected to focusing on conceptual understanding because we NEED to see each and everyone of our learners as unique people that make meaning of their world in their own way using the power of what they are good at and sometimes inhibited by what they struggle with. This understanding needs to be active in our minds as we plan inquiries that have many connecting points for the stages our learners are at and what schema they bring to the table. We need to maximize the connections for each of our students so that we can put them in a position to be successful in making meaning for themselves.

In addition, we need to use reflection through conferencing and debriefing to support student metacognition. This solidifies/reshapes their understanding and helps to bridge any gaps that they may have had in their understanding of concepts.