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.

Educational Change - Parent Involvement

In reading the Ministry of Ontario policy resource entitled "Parents in Partnership~ A Parent Engagement Policy for Ontario Schools, 2010", I believe that a change in parent involvement and connection to the classroom is needed in order to make reading a more effective and meaningful life skill beyond the classroom.  This policy details the importance of having parents involved as a necessary component to building the learning culture that fosters successful readers beyond subjects.  

To begin with, our classroom barriers need to be removed as we extend our students awareness beyond themselves and issues within our classroom to those of our community and world as a whole.  The inclusion of our parents into classroom can be differentiated depending on the comfort level and flexibility of our parents. They don't all need to be physically in the classroom, but I believe it is necessary that all parents in a school community understand and take part in supporting reading. I believe that as teachers we can facilitate this with regular communication detailing events in our school/classrooms and connecting local/global community issues that, through discussions at home, can extend the learning to enrich student understanding. It is not necessarily having parents read text with children as the meaning making can happen from media analysis (pictures, news broadcasts, websites).  With parents connecting their own experiences from a different perspective comes the opportunity to connect to different organizations in our community that could continue the extension of the learning past what would normally be available in the classroom alone.  The teacher can also enrich this experience by connecting next steps for the individual students with the parents. 

Secondly, having parents as a partner in modeling and supporting students' continual application of reading skills in authentic experiences beyond the classroom will continue to develop the intrinsic motivation to read. It is so vital that students infuse effective reading strategies and meaning making into their DNA.  With support of the parents to apply the skills outside the bounds of the school walls, students will internalize all that is involved with reading as a mechanism for survival and this will be accompanied by intrinsic motivation. Reading needs to be integral to everything in the students' lives as they make meaning around themselves from so many different sources. Also, parents at home can support the student in connecting real authentic experiences that don't happen in the school environment. This will involve teachers again communicating with parents the strengths, needs, and next steps for their child. 

Lastly, through parent involvement in the school creates a partnership where the inclusion of everyone's voice will continue the evolution of the system to meet the constantly changing needs of the students. This part is actually two fold. The parents' voice as necessary to the functioning of the school makes everyone accountable as a team responsible to the greater community for the education of our students.  The second fold is that as society changes, the school will adapt organically as the members of the society (organizations, companies, support groups) are directly and continuously connected to the fabric that makes up the schools. This effective dichotomy is the sustainable new form education system that we are framing. 

The culminating influence of parent involvement will result in adaptive successful readers that are skilled, collaborative, and communicative partners in our community. 

Educational Change Processes

The process of educational change is diverse. It involves all components of the system to buy into the necessity for the change. The system is such that it incorporates so many different levels in the structure to accomplish the end task of teaching students in the class. Leadership making a system impactful change needs to coordinate and almost bring the whole vertical structure onto its side so that all levels are working collaboratively on a lateral plane to accomplish the change. This connects all the supervisory, administrative, support, and classroom teacher personal so that they directly establish their link to the front lines of change. This ownership of change brings individual accountability to the team for improvement and change. 

Reflection and evaluation of every component of the change needs to be monitored regular in order to make the changes early enough so that major landslides are avoided of people bailing from the mission. The coordinating team needs to be flexible enough to make changes on the fly that will steady the efforts to remain focused on the mission. So, this would required monitoring of the progress and efforts in order to determine effectiveness and better/best practices that would have more impact.

The process takes time in order to effectively have system wide changes that are institutionalized and grounded into regular practice. If the changes are given the necessary time to blossom into the desired outcome, the necessity will fizzle and motivation for change will sputter to a stall. The change needs to be supported by admin and support staff so that it becomes part of regular practice within the school/system. If it is done only on the individual level, there is such a limited chance that it will see its full bloom to impact change in the school and greater school system. This has more of a chance in changing the system as a whole through system support in planning and implementation with integration of community partners. 

Change involves almost everyone believing and motivated to take part in the vision guided by an advisory team. Even with everyone having a forum to input their ideas and build the vision, can the change still be effective and long lasting if there are still those that negate the efforts by holding on to what they perceive as the most effective environment/strategies/practices for teaching and learning?

Struggling Readers Past Grade 3

I am struggling with the statistics that students that struggle with reading after the 3rd Grade will continue to struggle throughout school. I personally struggled with reading right into high school. I struggled especially in decoding words and definitely in comprehension (well pretty much all aspects of reading)! I resisted even trying to get better wanting everything in the world to avoid tasks that involved reading (especially in school). This is no longer the case. The only struggle I have now is with speed. So, is it cognitive development, emotional, psychological, or simply motivation (as in the circumstances of struggling readers...socioeconomic status, parents reading difficulties, family learning expectations, cultural influences, hearing impairments, underdeveloped of spoken English ...etc.) Or, is it an intrinsic motivating factor (likely the lack of and a lack commitment to lifelong learning) that enables some to go against these statistics.

Bringing Assessment to Life

The 21st Century classroom continues to evolve not only as a physical space that isn’t bound by it’s physical walls with the infusion of ICT, but also with students that demand a challenging and customized learning experience. This individually tailored programming is driven by informed teaching based on a continual analysis of data gathered from many different forms of assessment. The main purpose of assessment is to illuminate each learner according to their unique strengths as well as clarify needs that require attention. This culminates in a differentiated learning environment where all students are successful in being active participants in their own learning.
Forms and Purpose of Assessment
Peering into this effective classroom, one would see a teacher committed to student learning.  To make this happen, the teacher is passionate in using diagnostic assessments in order to gather data that would be used to inform the course necessary for large group, small group, and individual instruction. Carmel Crevola, Peter Hill, and Michael Fullan reported their research in an article entitled “Assessment for Learning in Action” where they list the key principles of assessment for learning as:
1. be recognized as central to classroom practice,
2. be part of effective planning of teaching and learning,
3. focus on how students learn,
4. be regarded as a key professional skill of all teachers,
5. promote commitment to learning goals and shared understanding of the   criteria by which students are being assessed,
6. provide constructive guidance about how to improve,
7. develop learners’ capacity for self assessment so they can become reflective and self managing, and
8. recognize the full range of achievements of all learners.
Assessment for Learning can be in many different forms (group discussion, paper pencil test, student-teacher conference, anecdotal records, & KWL chart are just a few examples). The data will be used in the Teaching, Learning Cycle (TLC) where a Moderated Planning session involves teachers of similar classes (i.e., division) “engaging in discussion involved in assessing student work and collective sharing of effective strategies in planning next steps for instruction” (The Literacy and Numeracy Secretariat: Capacity Building Series). Teachers would agree on what the different skills will be the focus and what constitutes student performance level from this data. Read-aloud texts, shared reading texts, guided reading texts, and independent reading foci will address the needs of the large group right down to the individual. These strategies will support all student learning in meeting agreed upon curricular expectations. Diagnostic data will also be used by teachers individually to progress student learning as needs arise.  These formative assessments (i.e., checklists, running records, conversations, observations, etc.) provide feedback throughout a course of study indicating effectiveness of teaching/learning strategies. Some of these strategies may need to be tweaked in an ongoing effort to reduce learning stressors along the learning experience.  The cycle of learning weighs heavily on the effective use of diagnostic and formative assessment data guiding instruction.
This dynamic classroom would also incorporate intentional teaching strategies to include students in the setting of success criteria, rubrics, and checklists to make transparent what denotes success in their tasks. Throughout the learning, students will conference with teachers to set goals for individual improvement and underscore what students do well. This reflection process is Assessment As Learning where descriptive feedback empowers students to gradually assume responsibility for self assessment in setting their own goals and gagging their own learning. This process would take some time to evolve as continual formative assessment coaches the student along, but it does happen through the support and guidance of purposeful teacher instruction.
Assessment for learning and as learning serve the purpose of guiding effective instruction for efficient student learning. This will culminate at the end of a unit of study via authentic assessment tasks that include students in the making of success indicators and provide meaningful tasks for students to put their learning to use. This Assessment of Learning has a purpose in the classroom. Summative assessments are the completion of the learning cycle where the teacher uses the data to assess their effectiveness in designing learning paths for students that encompass individual strengths and needs. The data will help to identify what strategies and skills were transferred successfully and which students need further individual support to bridge the learning into the next course of study. The teacher will plan with this end in mind from the onset of the cycle and ensures that the learning was scafolded to prepare students for success. The less frequent summative data is combined with the more frequent formative data for reporting purposes to parents.  Standardized summative assessment of learning data (EQAO, DRA, PM Benchmarks) will provide system administrators data on the effectiveness of programs and initiative efforts.
In summation, pealing open the shell of this classroom reveals the living multi-cell organism of assessment with its many unique characteristics creating a productive flow of reflection, goal setting, and supporting learning paths dedicated to individual success.   Data perpetually drives instruction staging a commitment to the unique instructional needs of each student as well as providing early indicators for students that need direct intervention. The forms of diagnostic and formative assessment comprise Assessment For and As Learning culminating in much less frequent summative Assessment of Learning.  After in-depth analysis of the data, the cycle begins again with a new moderated effort focussing on the next curricular expectations along the progressing continuum of learning.