My colleague Denise and I have been working in Grade 3 classrooms in two of our schools for the past few weeks, engaging students in problem solving using concrete materials. Our focus has been on concepts where our board has generally struggled , based on EQAO data over time. What's really interesting to us is that the data we used has led us to uncover a common thread. We realized that students are often asked to build and compare when problem solving. At the same time, we notice that many students don't build, which means they can't compare. Hmmm...I think we are on to something, here. We decided to use this common thread to help students to build a stronger understanding of equality. These photos document some of the learning we observed during lessons around equations and equal amounts of money.
What amazed us was the simplicity of the game that we used to help students to build their own equations. Each of us started with a stick of 30 cubes in two colours, with 5 cubes per colour in a repeating pattern (as described by the students). The students readily described our two lines as equal, and we were able to prove this by placing one line on top of the other. This creates a pretty cool visual. The cube lines became the equals sign. This led students to use the cube lines to solve for missing values in equations, by describing the actions they took to make their two cube lines equal lengths.
In the next lesson, we explored equal money amounts. At the beginning of this lesson, students made connections between the cubes lesson and equal amounts, which made it much easier to launch the idea of equal amounts of money. Pairs of students were given a paper bag with an amount of money between $5 and $7, which they needed to represent using different combinations of coins and bills. Using the mathies money app on a screen, we were able to model the different ways to represent equal amounts of money, in order to consolidate our understanding of the problem as a large group (see the image above for an example).
The power of connections across different models, strands and problem types really stood out to us as a result of these two lessons. The idea of equality resonates everywhere in mathematics, and yet it's an idea that seems to stump many students. That leaves me wondering why? How might we make more explicit connections between big ideas (like equivalence) in order to help all students see the idea more clearly?
Sitting here, listening to Cathy Fosnot talk about young mathematicians is creating quite the conversation in my kitchen. Conversation with whom, you ask? My 17-year old daughter. This year, all manner of opinions about education have surfaced at our supper table, and I've really enjoyed listening to the impact of learning and teaching on my own kids. As the last of my children prepares to graduate, I am happy to say that our kids have strong opinions on what makes the difference in education.
The conversation seems to revolve around the educator's learning stance. If the educator takes on the stance of learner, alongside the students, then my kids are much happier.
"Teachers don't know everything and they haven't run across all of the different types of learners. They never will." This is a powerful statement from a student.
Cathy Fosnot and Stephen Hurley start a great conversation, not just about ensuring that students have equitable access to math learning, but they also propose that it's not enough to just engage kids in having fun with math, but that we are compelled to help them to recognize themselves as mathematicians and powerful thinkers.
Cathy has many key ideas in this podcast, with a particular focus on what mathematicians do, and how mathematizing isn't just a process, but it's also content. She says that mathematicians are constructing meaning within the medium of mathematics. They are mathematizing the world around them, using mathematical models and mathematical lenses. This is how they construct mathematics and see relationships between ideas, proving conjectures. This is the process of building meaning.
Let's think about supporting the unfolding of mathematical ideas in the classroom, rather than transmitting our ideas to others. This stance really underlines the importance of educators engaging in the classroom as learners and observers, rather than just assuming they know how all of their students will see mathematical ideas.
If you haven't had the opportunity to listen in to Cathy Fosnot facilitating discussion about the first chapter of her book Young Mathematicians at Work, Constructing Multiplication and Division, click on the button below.
Not A Book Study has posted the first broadcast with Cathy Fosnot, originally aired on Tuesday April 11th. For more about this exciting learning opportunity, and to listen to the recording, please click on this link. Thanks so much to our friends at the Math Leadership Network for organizing this huge learning adventure for us here in Northeastern Ontario!!!
Idea #1 "Truly problematic contexts engage children in a way that keeps them grounded. They attempt to model the situation mathematically, as a way to make sense of it." (Pg 2)
This makes me think about those moments when we see a child really connect to the mathematics. Dr Lisa Lunney Borden talks about putting math into students' hands. We are noticing the difference that this makes. When I layer this idea onto Cathy Fosnot and Maarten Dolk's observation above, I am reminded that the story is what draws us in. It can't be just a story, it needs to be something we can connect to, in order to really engage us. The same can be said for a truly rich math problem. It's not just about building a context around a computation or set of numbers, it's about creating a logical, reasonable situation, which in turn leads students to be able to apply reasoning skills in an authentic way. Can your answer make sense? It can, if the situation is possible, or better yet, probable.
Idea #2 "Really doing mathematics involves working at the edge of your mathematical knowledge and enjoying the puzzlement." (Pg 4)
When we watch Terry encouraging her students to try a variety of strategies to solve their problem, we listen in to the questions and prompts that direct the traffic of learning, rather than just leading students to their destination.
Idea #3 "[Mathematicians] make meaning in their world by setting up quantifiable and spatial relationships, by noticing patterns and transformation, by proving them as generalizations, and by searching for elegant solutions." (Pg 5)
In the video clip shared during tonight's broadcast, Cathy Fosnot reminds us that all students are mathematicians and it's our role to ensure that they see themselves in this way. I can't underline enough the importance of this idea. At the end of the school year, I ask my students to write me a reflection called "My Math Learning Journey", using some guiding questions to lead them to think about their strengths as mathematicians. It's interesting to compare this definition to the reflections that I've read. As I think about this, the thing that really stands out is that the students are able to describe the relationships and patterns that have impacted their thinking. In conversation, they agree, disagree and build on each others' answers. It seems like they really did believe they were mathematicians. Now, that's worth a smile.
There's nothing like a little peer pressure to get me back to the keyboard. I haven't blogged in ages! This amazing online community learning opportunity was the spark I needed. If you haven't heard about Not A Book Study, I'll encourage you to check it out. It's free!!! Open access!!! Really, who wouldn't want to be able to listen to Cathy Fosnot on the "radio" for an hour at a time. For more information about this exploration of multiplication and division, see the flyer, or visit the Not a Book Study website, by clicking on the button below.
Jump in and ENJOY!
Last year, I decided to dedicate one of my two bulletin boards to our collective math thinking in my Grade 3 French Immersion classroom. Here's how we used the board. On the left, we posted our learning goals (attentes). I taught Grade 3 and 4 math, so we had some goals for each grade level. We highlighted a mental math strategy at the bottom left of the board. This was a strategy that we were trying out in our daily mental math workouts. In the middle, you will see a note that we co-created at the end of a lesson, to consolidate our thinking about a big idea. This was a whiteboard, so we could wipe it off and change it regularly. Students would put this note into their notebook, adding examples and other ideas of their own. We also wrote key words on the keys.
One of my goals with the class was to co-construct notes during consolidation, because it allowed students to hammer out the most important ideas we had uncovered. We knew we wanted to write fewer words, and have more images to help us to visualize the math. Students liked having the choice to use the examples we generated together, or to create examples of their own. Using the whiteboard allowed us to change the note as we wrote, which meant that students could build on each other's ideas and refine their collective thinking.
This was a really successful strategy throughout the year. We kept our gathering carpet in front of this bulletin board, so that the group could gather nearby, and everyone had a clear view of the board at all times, to use it as a reference tool.
This summer, I've been learning all about Twitter...as an educator. It's an amazing way to get lost in learning, and to listen in on the learning that's happening all around us. As part of the Math Leadership Network in North Eastern Ontario, I have been introduced to the use of social media to promote materials and resources, create a conversation, or spark an idea.
This has led me to think about how I can use social media, and my website, to support the administrators and teachers who I will be working with this year. Since I'm still thinking as a classroom teacher, these ideas would also work with students and their families.
1. Share a photo or video to spark conversation on Twitter. Followers can share their thinking.
2. Post a problem for followers to solve or discuss.
3. Post a video to model a strategy that students have been working with.
4. Share a link to a website, blog or other post that connects to an ongoing discussion.
5. Organize your website so that resources and links to practice are all available in one location.
6. Share a powerful quote, and a link that allows followers to read more. Encourage discussion.
7. Share ideas. Retweet. Quote.
8. Pose an open problem, "Marian Small - style", and ask for possible solutions. E.g. The answer is 12...what is the question?
9.Give a daily math workout suggestion. This will keep your followers in tip top mathematician form.
10. Connect your social media, so that people can access learning in different ways.
Just as we need to work out our bodies, we need to work out our math muscles! Imagine giving your brain a daily math workout. What would it look like? Is it a logic puzzle? A challenging problem? A series of mental math challenges? A pattern to continue? A shape to construct? Whatever your workout, allow your brain a solid 15 minutes a day to stretch and practice and you will see the results. Imagine turning this into a family affair!
Such an exciting time for math teaching and learning in Ontario's North East region. Thanks to an excellent LNS Regional team, we have embarked on a learning journey together, learning about math and thinking about how to shift the learning experience for all. We know that getting to the student desk involves a lot of steps, and that we need to support all learners (including educators) as they become more confident mathematicians. But the best part of all is that it's learning out in the open. Click on this button and check out the learning as it unfolds