Model the Learning

When this resource was posted to Twitter a few weeks ago, it was shared quite a bit, so perhaps it needed a "home." Thank you to those of you who have shared stories and profiles of people who were not initially on this list - they have since been added. If students are to be successful in learning and exploring math, they need to believe in their intrinsic ability to think mathematically, an ability we ALL have. Part of this self-confidence develops (or is reinforced) when students can "see themselves" in others who have made great strides in maths and sciences, or who have made impactful contributions to their field. For many reasons, Indigenous learners are not often able to see themselves in that light in math and science. The following is the beginning of a list of Indigenous role models in mathematics and science, along with images and links to learn more about them and their work. The emphasis is on those residing/studying in Canada, however the list is ...

This post was featured in an episode of  This Week in Ontario Edublogs (Sept. 4, 2019), beginning at 35:54 . In Ontario, students need a minimum of three math credits to graduate high school, one of which must be at the grade 11 or 12 level. For students who are not pursuing a post-secondary path that requires mathematics, and/or who really struggle with math, the grade 11 college-pathway math course (MBF3C) is often the last mandatory math course they need to take to graduate. Teachers of this course face several challenges, such as the range of student abilities, range of student interest (particularly if a student is only in the course because they need it to graduate), and also the wide variety of topics in the course curriculum. Last year, we offered new support for MBF3C teachers in our board. At our first session with the teachers, we had round-table discussions on what they most wanted in the way of resources to help their stude nts succeed.  One of the...

Before we begin, take a moment and read the number sentence below out loud . Don't think about it too much, just read it as you normally would to a class or colleague. On Language This past year, I have been doing more work in looking at how aspects of literacy creep into (and affect) how we teach and learn math. One theme that reoccured in different contexts was language, and paying close attention to what we say when we are teaching math. Over time, many of us adopt "shortcuts" when it comes to talking about the symbolic representation of mathematics. We may know what we mean to say, but for students who struggle, the meanings of these shortcuts are not always apparent, and might even cause confusion. The following are three very small, deliberate changes in language I am trying to work into my practice moving forward. Though each is a simple change, I'm having a hard time undoing decades of bad habits! "Equals" Based on some of our work on...

One of my biggest learnings in my role as a board co-ordinator has been around mental math in elementary schools. To be honest, when I first heard the term, I assumed mental math had to do with memorization and just learning multiplication facts.  I now know it to be a procedure very rich in strategy, promoting flexibility of number, and the very important mathematical process  of representing  a problem. Being able to see the math not only contributes to the understanding of the question at hand, but also to assessing the reasonableness of the answer. Visualization and representation of mathematics does not come easily to me - I was the student who became good at math by memorizing procedures. I still have to put a lot of effort into picturing multiplication in an array or area model, or picturing factorization by splitting items into groups, or picturing what happens to fractions as they are operated upon. This past week, I had the opportunity to model the a...

In addition to working with grade K-5 teams on numeracy collaborative inquiries this year, I have been fortunate enough to also participate in literacy collaborative inquiries when they happen on the same days. I have very little literacy background, so I'm learning tonnes! I find it very exciting to see and hear how our students are learning, particularly in ways I've never stopped to consider. Earlier this year, I met with a group of grade K-3 teachers as they explored students' ability to predict what might happen in a story. Starting in Kindergarten, the teachers introduced a book to the students by showing them the cover, and then worked with students to access three factors in order to make predictions: they look at the picture on the cover, they consider the title, and they activate their schema . Aside...  Schema (SKEE-mah): relevant background knowledge, experience, or prior knowledge within a context.   I was blown away to learn that students as young as 4...

This morning, as I try to get back into a routine ahead of Monday's return to work from March Break, I started reading Kathy Richardson's How Children Learn Number Concepts - A Guide to the Critical Learning Phases . This was a book that was given to me earlier this year after my role was expanded from working with grade 7-12 mathematics teachers to a full K-12 mathematics co-ordinator role. With only a background in intermediate & senior math, I've learned so much from primary and junior math teachers this year, and I'm eager to learn more about how students acquire concepts of number, relation, and computation. In her introduction, Richards describes what she calls Critical Learning Phases - "crucial mathematical ideas that students must understand if they are to find meaning in the mathematics they are expected to learn." These crucial ideas are "insights, rather than facts or procedures," meant to be carried forward as students engage in ...

I've been speaking with many secondary school teachers recently about how we might track observations and conversations in the math classroom, for the purpose of assigning a level of understanding and formally contributing to a student's overall achievement in a course.  When it comes to keeping track of marks, many teachers feel uncomfortable with evaluating what they see and hear in the classroom, much more so than evaluating what they see on paper. We feel, perhaps because observations are not as tactile as a handed-in worksheet, that an evaluation of what we see in class is more subjective, and may be called into question more than an evaluation of a product. As teachers, we are often much more comfortable using products to evaluate student understanding.  We trust our professional judgment with products (without even questioning it) much more than with observations. But in reality, this process is not that different than when we create and evaluate paper prod...

This post was featured in an episode of  This Week in Ontario Edublogs (July 11, 2018), beginning at 34:42 . Over the past week, we have been piloting diagnostic questions with students to get an idea of their understanding of equivalence. We gave students a series of questions, all having to do with the understanding of what the equal sign represents. One question, though, really challenged my idea of what "higher order" strategies we were hoping to see in our students. Here's the question: If 4x + 8 = 52, what is 2x + 4? Take a moment to figure it out. What is your answer? How did you come to that answer? If you're like my colleagues and myself (and most of the students of whom we asked the question), your instinct might have been to solve for x using the first expression, ...and then substitute that value for x in the second expression: So in this case, 2x + 4 = 26. As students advance to higher order strategies when learning math, we ...

Last month, I was passing through the hallway one of our high schools, when I overheard a teacher delivering a lesson. He was discussing equations of some sort, where a fraction was multiplied by a variable.  In this particular case, he was asking about 2/3 x 6.  He checked the class for understanding: "Do we all know how to multiply a fraction by a whole number?"  Think about that for a second.  What is your go-to method for solving 2/3 x 6? Thinking about fractions Prior to this year, if someone asked how to multiply a fraction by a whole number, I would have given them the “algorithmic“ method of doing so: create a fraction out of the whole number by placing a numerator of six over a denominator of one.  Then multiply my 6/1 fraction by 2/3: multiplying the numerators together and then multiplying the denominators together. This would give me 12/3, which could then be reduced to 4.  Rethinking about fractions For the first tim...

In 2015-2016, I was fortunate to be involved with a TLLP team that looked into flipping the classroom: transferring the focus of our courses away from the teacher and on to the learners. Once we had successfully flipped, we became interested in how we could deepen our students' educational experiences, specifically through rich assessments. In 2016-2017, our same team took on a second TLLP project, which is just finishing up now, that had us digging deep into Rich Tasks. While the first project was very successful for us - everyone in the group was able to flip their courses in different ways and we were seeing success with the students - the second project was a much tougher go.  The learning curve was steeper, and it seemed the more we learned, the harder it was to implement GOOD rich tasks. We kept coming back to: what makes a rich task RICH? We came to an understanding that an ideal rich task should be broad , but have personal components; it should challenge the stud...

Last week, I was fortunate enough to attend three days of workshops with Cathy Fosnot ( @ctfosnot ) in North Bay. This was a great opportunity to see so many constructs in person that I had previously only learned about through her books, or the podcasts with #notabookstudy and #themathpod. It was amazing to see "math congress," "mini-lessons," and "gallery walks" come to life! Also last week, I wrote about how math teachers can change how they ask questions in order to probe for concept development, rather than probing for just the answer. Inspired by what I had seen in North Bay, I wanted to try out this different style of questioning first hand. While working with different learners this week, I tried going past just getting the answer, and encouraged them to show me how they got their answer, or to draw a picture, or to explain their thinking in words, or tell me how they know their answer could be right. The results were disastrous. Learners be...

Our final question in The Math Pod (#themathpod) twitter chat this past week ( see the archive here ), was built on Cathy Fosnot's challenge for us in the first podcast: What is the difference between questions meant to guide students to a specific answer vs. questioning to support development as a mathematician? So in the chat we asked: What kind of questions can we ask to support development as a mathematician? What would you actually say to your students? Here are some of the answers from the participants in the chat: What kind of questions can we ask to support development as a mathematician? What would you say to your students? Tell me about what you're doing. What do you wonder? Tell me more? How did you know that? What if i change this? Really??? (especially when they're right!) Can you explain what she just said? What made you think of that? How did you figure that out? What does this remind you of? Have you seen what (anoth...

One of my goals this year is to work with math teachers to better track observations and conversations in their classrooms, to better triangulate their assessments. I find this is a big jump for many who have been teaching for a while - we are very comfortable with marks on pencil-and-paper tasks, but we're unsure how to assess and track what we see and hear on an ongoing basis.  Just this past week, though, we've been playing with new ways of collecting data that I'm pretty excited about! (To try and wrap my head around this last year,  I asked kindergarten and primary teachers how they best track what they see , and got lots of great ideas!)  At one of our schools, we are implementing new strategies for improving numeracy skills. How can we track this?  We created a paper checklist that can be used by teachers or observers in the moment to track when and how students are demonstrating good numeracy skills: Our "good numeracy skills" checklist, base...