Building Number Sense Using Collaborative Teams

“How do we review basic facts in middle school so that it actually does what we want: fill the gaps for those students who don’t know them and challenge the rest at their level?”

This is the question I’ve been asking for years and have yet to find a satisfactory answer. This year we are trying something new at my school and I will share our progress in a series of blog posts. Follow along, or better yet, try it along with us!

The difficulty is that students who don’t have any other strategies than finger counting and ‘lining them up’ (traditional paper and pencil algorithms) just continue to use these approaches and often don’t engage in learning strategies that are more efficient and help to develop their number sense because using the strategies involves breaking up numbers and putting them back together: a key foundation skill in mathematics.

I see so many middle school students, even grade 8’s, who can’t mentally calculate something like 38 + 47. They try to ‘line them up’ mentally and then get lost trying to ‘carry over’. A numerate student understands that they can add the 30+40 and the 7+8 and then add those sums.

This type of mental math is far easier on the working memory (which are not fully developed in our students) AND they help to build number sense because they are actually doing number arithmetic rather than digit arithmetic. This may seem like a small difference and to a person with strong number sense it is inconsequential but to students who are still developing number sense, it makes a big difference in how they develop their understanding of our number system. Our middle school students need a lot of work to develop a stronger understanding of our place value system as they often don’t really understand the connection (other than memorizing place value names).

In previous years, we first taught them strategies and then used Trevor Calkin’s Power of Ten sheets and Greg Tang's sheets and we tried to differentiate for students. These worked well for some students for sure but the problem we encountered was that some students needed to spend quite a bit of time on the basics as they seemed to not have much understanding of numbers other than counting, while others had memorized or had good conceptual understanding of these basics and flew through the sheets and were completed in a very short amount of time. How do we allow for the necessary understanding to be developed in those struggling learners while also challenging the rest of the class so they are learning too?

This year our approach is going to differ; we are planning to use collaborative math teams (see this blog post and/or this webinar if you want to learn more about how to use math teams in your classroom) to explore the basics.

Our plan is to have students in groups and look at each operation in multiple ways:

  1. In context
  2. Using various strategies for solving (2 or 3) by breaking apart the numbers and putting them back together,
  3. Visually using manipulatives and/or models and their connections to other operations.

Our hope is that those students that have memorized their facts will have the opportunity to learn strategies as a way of understanding how numbers can be manipulated (these are important pre-algebra thinking skills) and seeing them visually can also help deepen their conceptual understanding. Those students who are struggling will have lots of peer support and peer teaching to help ‘catch them up’ and the ones that have a good conceptual understanding can work on their curricular competencies such as:

  • Use reasoning and logic to explore, analyze, and apply mathematical ideas
  • Estimate reasonably
  • Demonstrate and apply mental math strategies
  • Use tools or technology to explore and create patterns and relationships, and test conjectures
  • Model mathematics in contextualized experiences
  • Apply multiple strategies to solve problems in both abstract and contextualized situations
  • Develop, demonstrate, and apply mathematical understanding through play, inquiry, and problem solving
  • Visualize to explore mathematical concepts
  • Use mathematical vocabulary and language to contribute to mathematical discussions
  • Explain and justify mathematical ideas and decisions
  • Communicate mathematical thinking in many ways
  • Represent mathematical ideas in concrete, pictorial, and symbolic forms
  • Reflect on mathematical thinking
  • Connect mathematical concepts to each other and to other areas and personal interests

For these students the challenge often isn’t in mathematical computations but in modelling, representing, etc. This approach is differentiated and because it is focused on more than just the basic facts, we are hoping that we can meet more of our learners needs. I will keep you posted on how we are doing and if you are interested, we are using the materials from our collaborative learning course to set up our math culture and get the teams up and running and we are using this as our first task: Click Here to Have the Attachment Emailed to You!

Educating Now was created due to teacher requests to have Nikki as their daily math coach. The site has lesson by lesson video tutorials for teachers to help them prep for their next math class and incorporate manipulatives, differentiated tasks, games and specific language into their class. Teachers who use the site can improve student engagement and understanding, in addition to saving prep time, by watching a 10 minute video tutorial and downloading a detailed lesson plan

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