-Jean Piaget (Fosnot, 2016, p.5)
Dr. Jo Boaler from Stanford University and her colleagues at Youcubed focus on the need for a conceptual approach to teaching math. She maintains that we need to change mindsets about math, encouraging kids to develop capacity in number and conceptual understanding of number by giving them opportunities to create or construct mathematical understandings and concepts from their own engagement and experience in math tasks. The use of number talks allows students to construct their own understanding of number while teachers scaffold the language and thinking, guiding these student “discoveries” by giving them the language and communication tools that mathematicians use. In this way, students begin to think and communicate as mathematicians and are empowered to drive their own math learning. Last year we explored and used Number Talks (Parrish, 2010) to focus on developing flexible thinking about number. This year, we turned to the Concepts for Learning Units to build on these ideas across the math curriculum.
Catherine Fosnot and her colleagues have created units of mathematical inquiry that present real world challenges for students to engage in for a period of classes and/or weeks. The tasks evolve over time as students build knowledge and understanding. There is a combination of group work, documentation to communicate their mathematical thinking, math congresses, direct teaching, number talks, and conferring which allows the teacher to differentiate for individual students and the group as a whole. They are rich tasks with multiple entry points for students. I’ll consider how grade 5 used the unit on Field Trips and Fund Raisers last month to help us consider how we might use these units as a part of the curriculum and consider the questions that their use has provoked.
Field Trips and Fund Raisers is a 10 day unit that introduces the concept of fractions. It begins by having the students in groups of about 4 to consider the following problem:
A Fifth grade class traveled on a field trip in four separate cars. The school provided a lunch of submarine sandwiches for each group. When they stop for lunch the subs were cut as follows:
- The first group had 4 people and shared 3 subs equally.
- The second group had 5 people and shared 4 subs equally.
- The third group had 8 people and shared 7 subs equally.
- The last group had 5 people and shared 3 subs equally.
Groups then spent 1-2 class periods considering the problem and were required to communicate their thinking process on paper demonstrating in pictures, numbers, and/or words what they had done in order to justify their thinking. Students conducted a gallery work and made notes about their observations and questions of their classmates work. Then, a couple of key groups presented their thinking in the math congress. During the math congress, key strategies, models, and approaches were highlighted for kids to try in their future work. In the next days, students engaged in evolutions of the tasks in which they could apply and practice the concepts about fractions they are developing. It culminating in a Bike Race challenge in which they needed to plan a 60 kilometer bike race and place food, water, rest stops, and mile makers and certain evenly spaced locations around the track.
In our classes we found the 10 day timeline unrealistic and adjusted our implementation of this unit based on our experience using a similarly structured unit earlier in the year. A few key observations from our discussions as a team are below:
- The math congress is a key time to layer direct teaching on top of the students’ development of concepts as a group. Earlier in the year, when we experimented with the first unit like this, we struggled with where and how to intervene to advance thinking and understanding. The math congress is one important time. It is an opportunity to share the thinking of a student group that advances the understanding of the class through the use of a key model or the advancement of a key concept. The role of the teacher in this facilitation is to highlight strategies that are effective that mathematicians use so the students can try that in the next iteration of the task. This can be more efficient calculation strategies (moving towards efficiency is a goal in grade 5), moving towards models that better communicate the math completed or concepts developed, and/or mathematical terminology to teach specific vocabulary or approaches that mathematicians use.
- Conferring is a key opportunity to differentiate instruction and intervene with individual students or groups of students who might be struggling or be ready for more of a challenge. As groups of students are engaged in tasks, teachers can take the time to work with smaller groups of kids to problem solve on the spot. We have spent time as a staff reading and discussing Conferring with Young Mathematicians at Work which outline many key aspects that make conferring successful. These include identifying what students have done, paraphrasing their work to ensure you understand, acknowledging their success and progress, asking questions and proposing next steps to advance their thinking (Fosnot, 2016). It is not unlike the cycle of provocation and recasting found in Reggio-inspired approaches to teaching and learning. To do conferring well, teachers must have a very clear understanding of the landscape of learning for the concept being studied so s/he knows which questions to ask and which models or questions to introduce next. This is another opportunity for scaffolding concepts and direct teaching when students are stuck.
- The contextualized problems are great, but they are not enough. A significant part of the math curriculum, and one we are learning to do better, is the construction of mathematical concepts. However, it is still important for students to develop fluency and efficiency in these concepts and basic skills. These tasks do not provide space for the quantity of rehearsal that many students require to master concepts and become efficient. In this unit, based on our previous experiences, we knew it would be important to give students more experience and exposure to the underlying skills to develop efficiency. The Field Trips and Fundraisers experiences develop the concept of parts of a whole and models for representing this. There is not enough practice, though, for students to master and become efficient in this area. As such, we supplemented this unit work with many games from Name That Portion (Fractions, Percents, and Decimals) (Kliman, et al., 1998) and other resources. These gave the students lots of opportunities to practice and see the relationships among fractions, decimals, and percents and gave them repeated exposure and practice with the most common amounts. All of this helped them to add fluency and efficiency to the underlying concept development from the contextualized math unit. This was key in helping the students master the concept and for teachers to feel like they had fully prepared students in the concept area.
Even with these adjustments and greater success from the first unit, we found several challenges with which we are still wrestling:
- How might we better support students who do not have the background mathematical knowledge and understanding to adequately approach the task?
- How might we ensure all students in the group are actively engaged in the task?
- How might we better challenge students who can easily complete these tasks?
- How do we balance discovery with direct teaching for those who struggle? How do we get the balance right?
- How can we best intervene when the group dynamics impact the development of the math concepts we are studying?
- How do we document individual learning and progress within these group tasks?
- How do we assure mastery of concepts and the development of efficiency?
We don’t have answers to these, yet. We are still discussing them and experimenting. Some things we have tried include mixed ability grouping verses homogeneous groupings. They both have their advantages and disadvantages. For the culminating fund raiser experience in one class, we tried homogeneous grouping. We would not do this for every task, but in this instance, it allowed less confident math students who often allow stronger students to take control of the group, the opportunity to be in a group with comparable students and give them the chance to step up and be the leaders of the group. It was exciting to see this increased ownership and engagement in several students.
A key question for me as a learning support teacher in this context is how much to intervene when kids are stuck? How much do we let go and how much to we back up to develop strong foundations of underlying skills? There is no good answer to this. It is the age old support question. I reflect back to something David Ott said once. We want students to “own” their knowledge. This is the argument Jo Boaler and Catherine Fostnot make. Students need to discover math concepts so that they can “own” them. But when kids don’t do that because they are stuck, because they don’t understand or have the foundational skills, David Ott maintains that we need to help them “rent to own” . We must provide them with models and structures and skills to help them engage in and access the task so that they can put a down payment on owning it in the future. I don’t know if this is right or wrong, but it is the approach that I have been taking as I support students in these tasks, while we also practice and rehearse underlying, foundational skills that will solidify the skills they need to better develop these concepts to "own" them in the future.
If you are considering similar questions in your math classes, please let me know how you are approaching them. It is something we will be continuing to wrestle with as we try to embed these rich math tasks in our curriculum.
Resources:
Boaler, J. (2016). Mathematical Mindsets. San Francisco: Jossey-Bass
Fosnot, C. (2016). Conferring with Young Mathematicians at work – Making Moments Matter. New London, Ct: New Perspectives on Learning, LLC.
Fosnot, C.T. (2007). Field Trips and Fund Raisers – Introducing Fractions. Portsmouth, NH: firsthand, an imprint of Heinemann.
Kliman, M., et al. (1998). Name That Portion (Fractions, Percents, and decimals). White Plains, New York: Dale Seymour Publications.
Parrish, S. (2010). Number Talks: Whole Number Computations. California: Math Solutions.