As well as maintaining our website, which we hope reaches many educators around Ontario and beyond, we have also developed a 3part “Lunch and Learn” series to share a little more about the project with our colleagues at school.
In addition to our nifty, new Smart Boards, and all kinds of “days off”, our project includes an aspect of “sharing”, that is, we are asked to share our learning with colleagues, both immediate in the wider education community. As well as maintaining our website, which we hope reaches many educators around Ontario and beyond, we have also developed a 3part “Lunch and Learn” series to share a little more about the project with our colleagues at school.
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One highyield strategy that came to light a few years back is one that fosters metacognition. Students are given an assignment to complete, then shown samples of other students' work, discuss the merits and gaps in said work, and return to their own work to improve it before submitting it for assessment. Much as I dislike EQAO for a number of philosophical reasons, I do appreciate the bank of annotated student work exemplars provided from years' past tests. Today we looked at some examples of Grade 3 work from an open response question in geometry, which the students had completed for homework the night before. Click on the photo above to see a full screen image of what we did. Marian Small’s new book (this time with Amy Lin), “Eyes on Math: A Visual Approach to Teaching Math Concepts” offers teachers a bank of visuals that can be used as the basis for mathematical discussions with students. Organized – like her "Good Questions” book – by divisions (K2, 35, 68) and strands in math, this book provides a compendium of visuals, accompanied by a main “problem” question as well as rich extension questions with brief explanations of why they are important. The introduction discusses once again why building a “math talk” community is so critical to fostering a deep understanding of mathematics concepts in our students. And the questions provided with each visual help the teacher facilitate the “building on” process that research shows helps students deepen their appreciation of any given mathematical problem. I want to use bits of this resource with selected groups of students in my classroom. When and how can I build this into my alreadypacked program? Also, how can I develop my “rich questioning automaticity” so that I become more proficient in responding to comments students make, with a relevant, probing question that encourages them to think more or differently about a concept or idea? Finally, what do I do about the students who just won’t or can’t pay attention, even with visuals and rich questions? With the building of vocabulary “walls” within each lesson of a given unit, Dale and I are taking one step to address the ESL factor, and helping students to build their mathematical academic vocabulary. And my little checklist is helping me to ask some meaningful followup questions during the debrief phase of a lesson and as I circulate amongst pairs of students during problemsolving time. But I still have several students who – due to lack of sleep or proper diet, or because of medically unaddressed attentiondeficit concerns – continue to find it incredibly difficult to sit on the carpet and engage in a large group conversation about the mathematics we are supposed to be uncovering together as a class. It seems as though all the good books and monographs we’ve read about teaching math, while offering excellent insights into mathematics and how to make the teaching of it more meaningful for many students, fail to address the classroom management factors germane to most reallife classrooms in urban Ontario in 2013. Although I value the expertise offered by Marian Small and her contemporaries, it sure would be helpful to read something written by a practising teacher who successfully implements all or many of the things we’ve been reading about, in a “real” classroom! Barrie Bennett, in his book, “Graphic Intelligence”, suggests that many teachers may not have an intentional grasp of a taxonomy of thinking, and that this gap means that we do not always make thoughtful connections between curriculum expectations and the best instructional method by which to teach them. (page 164) It’s certainly true that even when I take the time to plan out a decent lesson, I am often at a loss for how to manage students “in progress” during a learning activity. This is especially true for me in math, as students may be working at different paces on a given problem. How do I scaffold for some of the learners as I wander the room with my clipboard, while stretching the thinking of others? Happily, a monograph exists, one which I skimmed and scanned several months ago, but have not really had the time to fully integrate. Asking Effective Questions is part of the Ministry’s Capacity Building Series, and was published in July 2011. The monograph points out that it is not enough to simply pose a “rich problem”, and hope that the students construct the learning. As my observations in my own classroom this year and in previous years confirms, many students will need to be continuously “prodded” and challenged through the teacher’s use of effective questioning to turn student experimentation with mathematics into observations, and to think about their observations and what they might mean (i.e. can conclusions be drawn and generalizations made? Why or why not?) I find myself particularly struggling when trying to support students who seem to be at a complete loss for what to do. Reihnhart (2000), quoted in the monograph, writes, “Every time I am tempted to tell students something, I try to ask a question instead.” Easier said than done, when you’re in the midst of it and – let’s be honest – don’t always understand the math as deeply as you should yourself! With this in mind, I used some of the suggested questions from the monograph to develop two documents which I hope will come in handy as I forge ahead with my teaching: The first is a list of possible questions to ask students as I circulate during the “action” part of the lesson. I plan to keep it on my clipboard for easy reference. The second document is one I will print on 11x17 paper and post for students to consult, especially when they are “done”, as they wait for the consolidate/debrief part of the lesson. When I am not immediately available to push them further, perhaps the questions for consideration posted on the wall will do so.
For those ready to be more flexible and customized in their teaching (ie, ready to move away from the scaffolds of the above documents!), the monograph suggests that if we listen carefully to students, and keep our lesson goals in mind while interacting with the learners, then we can quite effectively develop student thinking. (I would modify that statement somewhat to include space for “teachable moments”, which arise as students make discoveries we had perhaps not anticipated in a given math lesson.)
Thinking about thinking, considering Blooms’ or other taxonomies, and becoming familiar with specific stems from which to form open questions as students work through a lesson can help us to become more effective teachers. I want to do this, but am not ready, yet, to move away from the scaffolds of a prompt template like the one I created, above. This week, I will try to begin asking better questions during my math lessons, with the help of my new checklists! My last blog post promised to summarize two new Ministry Monographs related to math, but alas, I went off on a tangent… So here they are then, this time for real! 1. Trigonometry in Grade 3? This monograph, part of the LNS “Research into Practice” series, was published in August of this year. The authors suggest that we may be relying too heavily on Piaget’s stages of cognitive development, and thereby holding students back from what they are in fact ready and excited to learn about, were it presented in an engaging, ageappropriate fashion! This suggestion seems to support the argument put forth by Will Richardson (“A Web of Connections”) – Why are we holding students back when they are ready to move ahead?! In a video segment I watched earlier this week, Richardson speaks about a 14yearold boy in the US, who basically hosted his own TV show online during the 2008 election. What happens, muses Richardson, when this student returns to the classroom the next day and is given a textbook and paper to write on? Similarly, the Trig monograph suggests that by following a strictly developmental approach with our students, we may be prohibiting them from “achieving the fulfilment and enjoyment of their intellectual interest” in mathematics (and surely in other subject areas, too!) I am reminded of a lesson I recently did on growing patterns. The idea was for students to simply identify that when the numbers got bigger, that was called a growing pattern. But a few students noticed that there were predictable patterns in each of the tens and ones digits in the numbers, too, and they wanted to speculate about why that was. Several other students got excited about this once their peers pointed it out, and my lesson went out the window as I allowed them to explore this discovery. Perhaps we *are* holding students hostage by clinging too closely to the curriculum. Of course we need a roadmap of our intended mathematical journey throughout any given school year, but if we focus on mathematical processes, and encourage studentgenerated dialogue in math class—even when it diverges from our intended plan – then we can foster the sort of rich, authentic mathematical thinking this monograph supports. Even for students in Grade 3! 2. Paying Attention to Proportional Reasoning This 16page document is a support document to “Paying Attention to Mathematics Education”, and provides some specific comments and practical support for proportional reasoning, one critical factor in students’ understanding of mathematics. Don’t be alarmed by the size of the document; many of the pages are taken up by mathematical illustrations and/or samples of student work. It is a relatively easy and engaging read. The document begins with an overview of proportional reasoning and why it is important. The essence of proportional reasoning is the consideration of number in relative terms, rather than absolute terms. Students are using proportional reasoning when they decide that a group of 3 children growing to 9 children is a more significant change than a group of 100 children growing to 150, since the number tripled in the first case; but only grew by 50%, not even doubling, in the second case. (page 3) The resource then proceeds to consider some key concepts of proportional reasoning, as well as some examples and nonexamples. Starting points are offered for teachers, including practical illustrations across grades and strands. Following this, some specific examples from EQAO are disseminated at the Grades 3, 6 and 9 levels. It appears that developing a strong sense of proportional reasoning can help students with “reasoning” in general: The ability to think and reason proportionally is one essential factor in the development of an individual’s ability to understand and apply mathematics. (page 4) Interestingly, Dale and I have noticed that this tends to be a weakness of our students, this inability or unwillingness to reason. All too often I have noticed that students are far more eager to write – as an explanation for a solution – things like “I thought it in my head”, or “I remember my teacher from last year told me”, rather than really think through a solution, asking themselves, “hey, does this make sense? How do I know? How can I prove it?” On some level, we both believe that the students just struggle with paying attention in general, and not reasoning specifically. On the other hand, perhaps if we exposed them to more activities that developed their abilities in proportional reasoning, then our students might be better able to reason, period. Several “subconcepts” of proportional reasoning are explored in some detail, with illustrations for teachers who may be quite new to dissecting mathematics in this sort of detail. Scaling up or down, unitizing, and developing multiplicative reasoning, for example, are considered. For example, the monograph notes,
We have certainly noticed this “thin” understanding of place value in our own students, and indeed, have planned our crossclass differentiated math groups accordingly. Interestingly, the note about this concept not fully developing until a certain age seems to contradict the point made about developmental learning in the previously mentioned monograph. So, are students in Grade 3 not understanding place value deeply because it has (in some cases) not been well introduced in the early primary years, or are their brains simply not ready for it yet? One of the tips for getting started is to offer students problems that are both qualitative and quantitative in nature: Qualitative problems (e.g.,Which shape is more blue?) encourage students to engage in proportional reasoning without having to manipulate numbers. (page 8) Skimming and scanning this monograph reminds me of how very, very limited my own mathematical understanding is. Best practices for teachers include solving problems oneself before sharing them with students, in order to anticipate what sorts of errors and solutions might arise. I know that. And I even have the luxury of living with a mathematician (who scoffs at my pathetic abilities to reason and think!) Alas, when does the average teacher have the time to sit down and just do math?!
For more information about Ministry resources related to math, including links to all recent monographs and several videos, too, click here. Several new monographs came out from the Ministry of Ed recently… today’s blog posts will focus mainly on a two math resources, related to a document called “Paying Attention to Mathematics Education K12”, it seems. It is worth noting that regular visits to the Ministry of Education website is time well spent for Ontario educators. New material is constantly being produced and posted, and few administrators these days have time to properly peruse all of these and select salient resources for their teachers, let alone spend any length of time dissecting said resources with learning teams on staff in a way that speaks practically to teachers already overwhelmed with the minutiae of the daytoday grind in the classroom. Increasingly, teachers are on their own if they want to keep up with what’s current in terms of ministry publications in areas of interest to them. Happily, our school’s Early Literacy Resource Person, who is aware of our little project, stopped by to let me know about some of the resources currently on the Ministry website, and sent links to several. Our principal followed up by dropping a few hard copies of same in my mailbox. The first that caught my eye was, “Paying Attention to Proportional Reasoning”. It appears that this document, a concrete look at proportional reasoning with some ideas for classroom practice across grades and strands, was born out of the visionary work outlined in a document entitled, “Paying Attention to Mathematics Education K12”. The latter outlines seven principles to guide mathematics improvement across public education in Ontario. A support document, to guide discussion, was also produced. The Mathematics Working Group Hmmmm…. I wonder if the group included and real, live CLASSROOM teachers who are actually working with the material in question!
I read these visionary principles with interest, and was reaffirmed in my belief that elementary teachers – especially those of us who work in socioeconomically challenged school communities – have been burdened with an impossible task. Rather than simply complaining about “too many subjects to teach well”, or “most of my kids are still learning English” I have actually been doing a lot of thinking about how to improve the current situation, i.e. how to make teaching and learning more manageable and fun for students and teachers. I have come to the conclusion that the entire school day needs to be overhauled, and I have an idea for how to do it, which I will blog about on my own blog at www.verateschow.ca (LINK to relevant post coming soon… like, as soon as I get a chance to download my brain onto my laptop!) Ahhh, but I digress, hence the need for a Part Two to this blog post, which actually focuses on the two monographs I previously mentioned and claimed to be writing about in this post! :D Before I sign off though, let me encourage readers to visit the Ministry’s Website, in particular the “Leading Math Success” Section which includes links to a number of resources, as well as some thoughtprovoking videos segments from "A web of Connections: Why the Read/Write Changes Everything (Will Richardson)". Another TLLP day has flown by... We DID manage to edit, complete and/or post several lessons in the patterning and geometry strands, but the amount of work left seems overwhelming.
Creating the smart notebook files by far consumes the vast majority of our professional learning time: In addition to pairing appropriate warm up or "minds on" activities with the lesson proper and a followup or practice activity and homework assignment, we are attempting to really ensure we have met the specific curriculum expectations for the grade 3 level. Just because a problem is a good one, doesn't mean it is relevant for grade 3, and with so much to cover in such a short amount of time, we want to ensure we aren't doing all kinds of extraneous "stuff"; the textbook, while well laidout from a mathematical perspective, goes beyond the scope of the curriculum, and to be honest, one could ever fit it all in, especially in grade three, where nearly a month is lost due to early wrapup/EQAO! Once we've paired warmups and problems with one another, and matched them to the curriculum, we are trying, too, to lay them out in a sequence that makes sense and is developmentally appropriate. Have we succeeded? Next year, I think, would be the year to determine this. Having developed a comfort with the equipment and the process, it is then that we might have a chance to pause and reflect in a deeper, more meaningful way. For now, we are impressed with just how much work goes into such a project as this one, and are ever more aware of why some folks just don't teach this way. It is good, good stuff, but incredibly timeconsuming, especially in the preparation stages, and we cannot hold it against any colleague who does not have the benefit of multiple "days off" to plan and prepare such lessons. Hopefully, our bank of lessons posted on this site will help people who want to try teaching in such a way, but aren't sure where to begin. One thing about using a Smart Board to teach math is that I am becoming more comfortable now with technology, and building a more openminded attitude towards learning how to use new and unfamiliar technology. I see how motivating technology can be for some learners, and am eager to arm myself with the tools that can help these struggling students succeed and grow confident in themselves as learners.
I recently became the lucky recipient of an iPad, and I have been thinking about how to use this particular tool as a form of remedial support for some learners in my math classroom. As a first step, I started to research "math apps"  wow, is there ever a lot out there!!! Here is a review of three apps I decided to download. For anyone looking into iPad apps, may I reccommend Fred Ventura from Ventura Educational Systems. Lovely fellow, and very willing to share expertise. You can get a free download code (one per school) for one of their apps by visiting the Ventura website here. What a whole new world this project has opened up to me!!! One aspect of our project we have finally begun working on is the remediation of basic skills. Using a resource called "Leaps and Bounds", we gave each student in Grade 3 a diagnostic test to assess their understanding of number. Using this information plus our own observations over the preceeding weeks, we grouped our 45 students into six groups, some as large as ten students, others as small as two. Groups are based on their demonstrated ability to model numbers. Then we assigned a staff person to each group.
Staff included the two of us, our principal and three support staff (Spec Ed, ESL, etc.). For one double period per week, we meet with our small groups, and work specifically on the skills they demonstrated difficulty with. Today, for example, I worked with my group on modelling numbers up to 100, while another group worked with numbers up to 20, and a third group worked on a Puddle Math problem to extend mathematical thinking. After three weeks, we administer another diagnostic, and then regroup as needed for the following month. Our hope is that with this weekly intensive intervention, students will move through their mathematical areas of need more quickly and gain muchneeded basic understanding. Today marked our first TLLP day of the new school year. Trinder and I decided to work independently; we wanted to work together on making smart notebook slides out of our collection of math problems from last spring, however, we both felt that we needed to spend some time pouring over the stack of math resources that had been accumulating on each of our desks, thinking about said resources, and updating our blogs before we could collaborate with any degree of effectiveness! After completing a draft staff survey which we hope to administer to our schoolbased colleagues sometime before the end of the month, I set out to review some volumes of the GEIM, Nelson's Leaps an Bounds Resources, and a few websites, most notably the Edugains site. 
AuthorVera C. Teschow is a teacher, vegetarian, student pilot, drummer, and mother of monozygotic twin boys. Archives
June 2013
