Textbooks and the Mathematics Curriculum

I have taught Intermediate math almost exclusively in my first 12 years of teaching and I did it strictly out of the textbooks that were provided for me. Eventually, I progressed to be very selective of the textbooks in influencing our school's purchasing so that the textbooks were a "best fit" to the current version of the Mathematics Curriculum. I didn't do any PD nor read any research. I taught math out of the textbook as that is what I thought had to be done. Differentiation as far as general lesson delivery wasn't much of a presence. Everyone received the same lesson independent of their current understanding. When the students were assigned independent work to do, that is when the grunt work began to get to as many students as possible in the limited time to help them see the main strategy that I was teaching. It wasn't about students developing conceptual understanding or learning it in their own way. It was about getting students to follow the set rules in math and just decoding the questions in order to connect the only correct procedure so that they can get the only right answer. Since, I switched to Primary (Grade 3) two years ago (this is my third year teaching Grade 3) and I immediately went to the textbook (Minds on Math) to run my math program. I quickly noticed that my set ways of teaching math (which I really wish I could go back and redo in a more inquiry based model) were my way of learning math and that this wasn't working for pretty much everyone. They needed more guidance as they worked through the task. Students needed more chunking into smaller parts in order to be able to handle the tasks. I was seeing past the procedures and seeing that I needed to get the students to buy into the concepts first in their own way. I tried for two years to run an inquiry model, but fell back onto the traditional textbook way quickly as I wasn't understanding how to run a program in that way. This year, I have been fully committed to an complete inquiry model with 3 Part lessons that focus on the individual student and I am struggling to build lessons and tasks for students that take all aspects into consideration. When I do this well, the students flow with the learning so smoothly, but when I make a mess of the partnered tasks, the students are left with so much confusion and lack of clarity in where to go, what strategies to use, and how to solve the problems. I have much more learning to do in so many aspects of my math program, but I am committed to creating the best math program possible for each of my learners and when it works well, it is so amazing to see the learning (again, I wish I could go back and redo my 12 years of traditional mathematics teaching as I am sure I'd be more effective then I was for those years).

As to how my use of textbook changes when implementing a newly released curriculum guideline, there wasn't a whole lot of change in the use of textbooks. The focus just shifted on the types of questions used for the independent tasks and often supplementing the base work with additional problem solving work or sometimes the use of manipulatives. But, the base of my program at that time built off of the textbook.

This year, I do not use a textbook at all (I actually gave them all to another teacher that doesn't run an inquiry program that was very much in need of the books). As of right now, I am very happy to not be using the textbooks. This is because of the work I did this summer in Mathematics Part 1 and now Part 2 where I have learned to more effectively analyze all aspects of the curriculum and construct very thought out long range plans for the whole year in Mathematics. This has given me a much deeper understanding of where the students should have gotten to in Grade 2 and where they will be going in Grade 4 to have a much clearer vision of what is necessary for Grade 3. I have never looked at the expectations and compared/connected across the strands and into other subjects to deeply. This has given me the base I need to stay focused as to what curriculum needs to be covered and when with connections to other subjects. From here, I use various authentic assessments to determine where each learner is in their understanding of math concepts and what misconceptions need to be clarified. At the beginning, this was a huge amount of work to build tasks with multiple entry points and parallel tasks giving students choice of meaningful rich tasks to explore their learning. I really like the inquiry approach and building unique authentic tasks for students as this engages them immediately in what they are doing and learning seems to be moving so fluidly.

I don't think the current textbooks we have in our school meat the teaching approaches in the curriculum guidelines as there isn't a whole lot of students building their learning using their own strategies and analyzing other strategies for effectiveness and efficiency. From what I have seen, the tasks are geared towards following the example strategy which can really limit creativity and stifle learner where students might discredit their own reasoning. Also, there isn't a whole lot of communication built into many of the lessons where students share their strategies and justify their reasoning. From personal experience, building this on my own from scratch helps to shape the lens of what I am looking for as indicators of student learning and this really helps me be more effective as I facilitate the learning throughout the 3 Part lessons.

Right now, I have the best understanding of the math curriculum then I've ever had and I owe it to having to shred the curriculum and make my own connections to build an overall plan for implementation. Having done this, my focus is very strong (may the math force stay with me!) when building rich authentic tasks for the needs of my students. I am not thinking of every going back to a textbook. But if one were available that I would consider, it would have to include samples of questions that would scaffold to meeting overall expectations. These samples would model the thinking and timeline of scaffolding to build the learning. I wouldn't need lessons on delivery, rather how the problems connect and relate to the culminating tasks that are coupled with direct learning intentions and support in building success criteria with my students. The problems don't even need to be fully written problems, rather prompts of things to consider in a diverse equitable classroom where I can plug in a context and vary the entry points for my learners. Samples of parallel tasks would also model how to differentiate for a group of learners. In addition, up to date assessment strategies (for, as, and of learning) and what to look for in the learning as indicators and flags to support descriptive feedback. This would be a great tool that would empower me to be more effective in building the appropriate program that keeps my learners' unique learning strengths at the forefront.

Assessment In Today's Mathematics Classroom

As teachers, we are entrusted to work with students of different ages, genders, cultures, experiences, learning styles, and learning experiences. For many years, these influencing variables have not played a role in what the mathematics learning experience looked like for each student. The lessons were taught in vacuums of segmentation where the only variable was the concept itself. For the majority of classrooms, mathematics learning was simplified to teaching concepts one way for everyone and having students fit into that conceptual framework. In a presentation excerpt entitled "Changing Education Paradigms", Sir Ken Robinson spoke of student creativity and how the pedagogical methods of years past are not effective anymore for the current and future education system. Sir Robinson is very clear that individuality in learning strengths and learning styles needs to be harnessed in order to fuel student creative thinking. As educators, we need to learn from students how they learn best and harness the schema and context of their understanding in order to facilitate students building connections, awareness, and new learnings of the world around them that is authentic, meaningful, and inveterate. This can only be accomplished through a student centred mathematics program that is open and receptive to using assessment for learning, as learning, and of learning to inform instructional strategies.

In discussing assessment, it is imperative to make distinctive the separation from evaluation. Assessment is the process and tools used to gather information about student learning for the purpose of illuminating thinking, reasoning, and/or misconceptions that will provide the information for descriptive feedback to support the student in meeting the learning intentions. Evaluation is the use of assessment data to make a judgement about student learning at the end of a course of study that will be reported as a level of performance based on ministry expectations.

Effectively individualizing the learning experience in mathematics uses diagnostic tools to gather information that will illuminate student learning on similar concepts from the previous year. Diagnostic assessment tools can also describe how a student learns, reasons, and communicates mathematically. Detailed information gathered can be used to better support differentiation of the learning experience through the use of appropriate accommodations (multiple entry points, parallel tasks, ELL support, contextual relevance problems) and scaffolding. As the learning progresses through the 3 Part Lesson, formative ongoing assessment continues to monitor student thinking and reasoning as they work in pairs or small groups to collaborate in solving problems. The tools chosen to monitor the learning will present information in order to provide descriptive feedback to broaden and deepen the learning experience. In checking in on student learning of new concepts, teachers can use instructional rubrics, checklists, and/or success criteria (a combination can work as well) that are clear to the students as to their performance in the learning categories. The students can also be part of writing these tools that will make the tool more meaningful when used for teacher assessment and student self assessment (reflection). Continuing with self assessment, reflection through well placed questions and prompts will model for and support students in thinking about their learning and how they are learning (metacognition). This is also part of the learning experience at the end of the 3 Part Lesson where students share the strategies and reasoning that encompass their solutions. This assessment as learning (some choices include Bansho, Math Congress, and Gallery Walk) is very powerful as it teaches students how to self assess and gauge their own learning building their responsibility and accountability. At the end of the lesson, the learning continues to be monitored using assessment of learning tools (performance task, independent problem solving, etc.) where the purpose of the assessment is to determine how effective the learning experience was in relation to the learning intentions (based on ministry expectations). This summative assessment will also be used to inform teaching as it will illustrate next steps that are needed in order to clarify any misconceptions or bridge gaps in learning leading into the next lesson.

Throughout the different forms of assessment (diagnostic, ongoing formative, and summative), the teacher needs to be selective in differentiating the tool used for assessment. We are aware how unique our learners are and the schema within which they function daily can present difficulties in performing via certain assessment tools. It is imperative that the teacher use their understanding of the learner to match the most effective assessment tool that will provide the necessary information to delineate the selection of teaching strategies that will guide the student in meeting the learning intentions (e.g., some students may not verbalize their thinking as well as writing or using illustrations/manipulatives/ICT while this may be the opposite for others).

Teacher's are excellent at capturing the learning opportunity as it surfaces, but assessment can be more effective when planned for from the beginning. Knowing the student through diagnostic assessment can better inform the selection of the assessment tool to check in on the learning. As well, knowing the learning intentions of the Unit and lessons also prompts building the summative assessment in order to keep students focused in meeting the learning intentions. This planning with the end in mind becomes an organic structure that is built for success by having checks built in that will help adapt the learning experience. When students reach the summative task, the scaffolded individualized learning experience will have prepared students for success. Assessment planed and embedded throughout the 3 Part Lesson will ensure a successful learning experience for students and teachers.

In wrapping up my thoughts on assessment in mathematics, I'd like to underscore the necessity to include parents into the classroom. It is imperative that parents and the community are aware of the mathematical learning environment. Students do not learn in isolation outside of the real world. The purpose of the learning experience is to support students making meaning of the world around them in a way that is authentic and individualized to their learning dynamic. We need to share the vision of mathematics with parents and foster a learning community beyond the classroom. Having the home aware of how their child as a learner will support parents continuing the learning in similar ways at home and supporting the problem solving model. We need to demonstrate how we use assessment to support developing skills in the 4 categories of the achievement chart and how they connect to the learning skills.

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.

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.