I have blogged a few times about problem posing using print materials and lately I’ve become really interested and excited about the potential for digital tools in this work! If you are new to problem posing, below are a few slides from Jinfa and my NCSM presentation for background – each image is linked to an associated research paper. What is problem posing? Many activities can easily be adapted to provide opportunities for problem posing by removing task questions (left) and replacing it with different prompt options (right). original curriculum activityproblem-posing prompt optionsHow can ..read more

One instructional strategy that I love for collaboration and public sharing of student ideas is a gallery walk. In a gallery walk, students create displays of their thinking on chart paper or white boards and then the small groups walk around the room and visit each other’s posters. And even though students create such beautiful displays of their ideas, it is always challenging for me to structure the walk in a way that actively engages them in one another’s ideas. Like any problem of practice, it takes trying out new ideas to see what works, when, and for whom. The Lesson Last week, it was th ..read more

In 3rd grade, students come to understand fractions as numbers. They count by them and locate them on a number line just like whole numbers. However, once they start operating with fractions in 4th and 5th grade, they tend to set aside everything they understand about whole number operations and treat fractions as numbers with their own set of ‘rules.’ I can think of many reasons why this happens, but my current wondering is how we can create more opportunities for students to make connections between their understanding of whole number and fraction operations. Why Connections? So many times I ..read more

Adapting math activities is one of my favorite parts of lesson planning. I love it so much because of the thoughtfulness, curiosity, and creativity involved in even the smallest of changes. In making any change, I have to think about what students know, the math of the activity, how the activity addresses the learning goal, ways students might engage in the activity, and questions to ask students along the way. Fraction Activity In this 4th grade activity, students were writing equivalent multiplication equations for a fraction multiplied by a whole number and then discussing the relationship ..read more

 “The widespread utility and effectiveness of mathematics come not just from mastering specific skills, topics, and techniques, but more importantly, from developing the ways of thinking—the habits of mind—used to create the results.“ Cuoco, Al & Goldenberg, Paul & Mark, June. (2010). Math curriculum lessons are often aligned to the Standards of Mathematical Practice. These practices can provide opportunities for students to develop the mathematical habits of mind described by Al Cuoco, Paul Goldenburg, and June Mark. Mathematical Habits of Mind Students Should Be Pattern Sniffe ..read more

I appreciate the coherent connectedness of good curriculum when I see students making connections as they move through a unit concept. At the same time we know that no one learns the same thing, in the same way, at the same moment in time, so the need for additional time with a concept and extension opportunities are always a necessity. In our school, we typically address the additional time element during small group time, but extensions seem to be more of that ‘back pocket’ in-the-moment teaching move that is challenging and fun at the same time. We have to think quickly about the purpose of ..read more

When I was teaching, I often felt overwhelmed by my own learning. The list of things I needed to know and be able to do felt never ending. And then, as I chipped away at my list, it seemed like the more I learned about teaching math, the more I didn’t know. I think David Cohen describes the root cause of my feeling perfectly: ‘To teach responsibly, teachers must cultivate a kind of mental double vision: distancing themselves from their own knowledge to understand students’ thinking, yet using their knowledge to guide their teaching. Another predicament is that although attention to students ..read more

This past week we planned for a 3rd grade lesson on arrays. The curriculum lesson goal was for students to build and describe arrays, in particular connecting the array structure to their understanding of multiplication as equal groups. The first activity in the lesson was written to encourage this connection, however having taught this lesson in previous years, we knew that the workbook examples could have come from students if we gave them the chance. Since we are always looking for ways to better amplify and leverage student thinking, we made some significant adaptations. The original lesso ..read more

In mathematics education, problem posing refers to several related types of activities that entail or support teachers and students formulating and expressing a problem based on a particular context, such as a mathematical expression, diagram, table, or real-world situation (Cai & Hwang, 2020). Because problem posing is so dynamic, multi-faceted, and varied between classrooms, I understand why it is hard to write into published curriculum materials. However, understanding and trying out the structure of problem posing makes it a really impactful teacher tool for adapting curriculum ..read more

Recently, I have been learning a lot about problem posing from my friend Jinfa Cai, in particular how to infuse these opportunities within the use of curriculum materials. Because, while there are rich problem solving experiences in a good curriculum, we do not often see explicit use of problem posing, especially in K–5. The Notice/Wonder routine is probably as close as it gets. Since I am in classrooms this year, I get to try some problem posing around curriculum activities and follow up with Jinfa. As with all learning, the more things I try, the longer my list of questions for him grows! Af ..read more