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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Every now and then, you come across a really rich problem. Today was one of those days. When we first selected the following problem, Dale and I knew it would be a challenging one for the Grade 3s – some of them were still struggling with rounding, estimation and onedigit addition and subtraction – but we wanted to see what they would do with the challenge. I was amazed at the thinking that emerged with some students in my class this morning! Mr. Trinder has 23 model trucks and cars. He has 5 more trucks than cars. How many cars does Mr. Trinder have? At first, many students tried to add 23 and 5. Some added the resulting number to 23, and came up with a number in the 60s. Predictably, others subtracted 5 from 23. Although a few of them thought they had solved the problem with these strategies, many realised that something was not quite right. But they weren’t sure what it was. Today, I realised the importance of letting them wrestle with these incomplete solutions for a while, before stepping in to guide their thinking. And when I did step in, it was to encourage them to do more thinking, rather than to provide them with the “correct” path: After about 15 minutes of work time, several students had begun to realise that either their solutions didn’t make sense, or that they didn’t even really understand the problem itself, whereas others were well on their way to either solving the problem, or perhaps even already done solving it correctly. At this point, I asked students to be honest with themselves, and selfselect a partner who was clearly on a different learning path than they were, then meet to discuss their thinking to this point. I encouraged them to use “ask questions” and “ask for clarification” (from our “Rich Talk” chart), rather than “disagreeing constructively”. It was the first time I had explicitly referred to this chart during math class (we’ve used it extensively while dialoguing, in groups, about rich mentor texts in other subject areas). Students were encouraged to ask questions and refer specifically back to the problem. Many of them did this successfully, and the result was stupendous! “Ohhhh!”, remarked one student as the lightbulb went on, and, “I get it!” exclaimed another, as he realised his “answer” didn’t really make sense.
After this short, 5minute peer conversation, I sent students back to work on the problem, and after 3 minutes, I interrupted them again, as I could sense that – although many of them now realised they had done something wrong in attacking the problem, and why their original answers didn't make sense – they still really didn’t know where to go next. Now I presented the problem visually through hand and arm movements, "drawing" two groups of “vehicles” in the air, cars and trucks, and encompassing them with my hands to show that altogether, there are 23 vehicles, but that we do not know how many of each there are. “But what do we know?” I wondered out loud. Several students were able to correctly communicate that we knew that one number, the one representing the trucks, was 5 more than the other number. I confirmed this, and reiterated that both numbers, when added together, must total 23. The students went back to work. By the time the period was over, not only had many students solved the problem correctly (and using a variety of strategies), but one of them had even figured out that his strategy would work for all such problems, regardless of the numbers. He had discovered that if you deduct the “more than” amount from the total, than divide the resulting number in half, and add the number you originally subtracted to one of the halves, you would get the two numbers you needed to make the total correctly – it was very exciting; even I had not realised the general application of his solution strategy until we explored it together, after an “I wonder if...” on my part! The morals of the story? Offer a single, rich problem, provide lots of opportunity for independent thinking and then oral language to share thinking, and, most of all, ensure limited and very intentional teacher interference! A question that frequently arises is "how do you teach all that in a single math period?"
Many teachers do not have the luxury of a double math block in their timetable, and even when they do, it may not be long enough to do a "good" job on some concepts. Dale has long held the position and even blogged about taking more than one day to complete a single math lesson, and I am increasingly convinced that he is correct in this assumption. This year as we whip through a seemingly endless number of curriculum expectations, assemblies, performances, field trips, reading buddy meeting, etc., etc., etc., it has been tempting to leave off the "end" of the lesson, that is, the "practise/consolidate" question that we prepare for students to work on AFTER they have done the main lesson, and we have taken it up and recorded the "what we learned" comments. But this does not do students any favours. Those who are stronger in math do not get the opportunity to stretch their thinking by testing out new ways to approach a problem. And those who already struggle are robbed of an opportunity to try again. Today, I continued my morning math lesson after lunch, taking a full 40 minutes out of the hour I had with my students in the afternoon to record "what we learned" in the morning lesson in greater detail, and to give them time to apply what they had learned to the consolidation question. I had students hand in their responses, and was surprised at how much more detailed and accurate the responses were than their work on the lesson problem earlier had been. Does this mean we might run out of time before the end of the year, and that some units will be shorter or not taught at all? I will make a bold confession: It is very likely that neither Dale nor I will address every single expectation from the math curriculum this year. (Can I get fired for admitting that?!) But we know that if we teach more thoughtfully and slowly we will instill in our students a greater sense of precision and understanding of the expectations that we do uncover with them! Assessment doesn't always come at the end of a unit, and it doesn't always mean writing down a mark on a page... We all know the everpresent challenge of time (i.e. not enough of it), and the launch of our latest unit has been no exception!
Dale and I just began working on multidigit operations. We planned two lessons on addition (revisiting the concept multiple times throughout the unit, with an emphasis on mental addition and later, adding money). But we though an initial two lessons to construct, discuss, record and practise multidigit addition strategies would be sufficient before moving on to subtraction. Alas, after the second lesson, we both noted that many of our students are just beginning to grasp the concepts (making tens and compensating, rounding and estimating, add tens, then ones, then combine, etc.) we were hoping they would solidify in these lessons. So, we have decided to put together one more consolidation lesson. We hope that "Lesson 3AB..." will serve to summarize the dialogue from the previous two lessons, and solidify more students' understanding of various multidigit addition strategies. It is not a 3part or Bansho lesson in the same way we have been teaching other lessons, but we think that's okay! Hopefully this extra day spent on addition will help students become more confident with addition, so that when they return to it later in the unit, there will be a more solid foundation to build on. 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! 
AuthorVera C. Teschow is a teacher, vegetarian, student pilot, drummer, and mother of monozygotic twin boys. Archives
June 2013
