The Hinge Question

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John Hayes, Eagle River, WI,

In my role as a CPM coach, I was planning a lesson with a teacher in Lesson 9.1.1 of Core Connections, Geometry (How can I build it?) a few weeks ago. I have written the conversation below. Note that this is not an exact transcript, and I took the liberty of numbering the dialogue lines for reference purposes.

  1. Me: What do you think your lesson goal is?
  2. T: I think I want them to be able to tell me the steps to move from a mat plan to one of the views, front, right, or top.
  3. Me: At what point in the lesson do you think the students will realize that goal?
  4. T: Probably not until they get into problem 9-3.
  5. Me: Okay, what question will you ask to determine if they have met your goal or is there a question already in the lesson that you can use?
  6. T: Well problems 9-3 and 9-5 both seem like questions that would indicate they know the goal, but I think I’m going to have to ask them directly “What are the steps you would use to draw the front view from the mat plan?” I also think that problem 9-5 would make a good question to lead into the closure.
  7. Me: When you ask your question about the steps, what answer would you consider “correct”?
  8. T: For the front view, I would hope that they tell me that they look at the first column, find the biggest number, and then draw in that column the correct number of boxes high. 
  9. Me: What errors in thinking do you anticipate?
  10. T: I think they are going to mix up the terms column and rows.

The conversation above illustrates a pretty typical planning conversation that I have with teachers. When I have worked with a teacher long enough, they often tell me their goal (line 2), their hinge question (line 6), and their success criteria (line 8) before I even ask. By identifying the learning goal, crafting hinge questions, and identifying the success criteria, teachers feel accomplished after the lesson and better in tune with what their students know. These three three things act as a formative assessment that help teachers decide what they are able to summatively assess students on in the future.

When I observed this teacher during the lesson, I noticed he asked several pocket questions and he patiently waited until students were doing problem 9-3 before he addressed the lesson goal. This teacher identified problem 9-3 (line 6) as the hingepoint in the lesson—that point where students begin to gain an understanding of the goal. At that point, he used his hinge point question (or hinge question) to assess their understanding: “What are the steps you would use to draw the _____ view from the mat plan?” He chose to circulate and ask each team the question.

Dylan Wiliam says that a hinge question should happen every 20-30 minutes in a lesson, and that it should happen as quickly as possible to minimize interruption during the learning process. I often view each team as a mini version of the whole class, and when a teacher asks the team a questions, often only one student replies. The hinge question is a question pertaining to the goal of the lesson and has all students share their understanding. In the example, the teacher asked each team to quickly jot down the steps they would take, and then had them share them within their team.

A hinge question is different than closure. In fact, the hinge question may influence the direction the closure takes. This teacher closed by having students record in their notebooks the steps for moving from the mat plan to all three views, and then had students peer edit their elbow partners’ steps. Then he had a short whole class conversation about how the three views and the mat plan were connected to finding volume and surface area. However, he confided in me later that in another section he used a Swapmeet to close his lesson because he was not satisfied with the answers he was getting from two of his teams when he asked the hinge question. In other words, he adjusted his closure Study Team and Teaching Strategy based on what he learned from the hinge question.

In a CPM lesson, we can often find an appropriate hinge question within the discussion questions, within the given pocket questions, or within the lesson problems. We do not always need to invent an original hinge question. The work supporting a hinge question is a little more sophisticated than asking a standard pocket question. Once the teacher has a hinge question, he or she should think about what a correct response would be and what errors in thinking students might encounter. Most importantly, the teacher should plan an action for the anticipated responses to the hinge question. I often ask the teacher, “What Study Team and Teaching Strategy are you planning to use if your students have errors in their thinking?” The teacher above used a peer edit in one section because he noticed a few students had not arrived at understanding yet. In another section, he recognized that whole teams had errors in their thinking, so he switched his STTS to a Swapmeet. Understanding the lesson goal, the hinge question and the success criteria not only create a powerful platform for formative assessment, they also energize both the teacher and the students to take ownership in the learning.

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Algebra Tiles Session

  • Used throughout CPM middle and high school courses
  • Concrete, geometric representation of algebraic concepts.
  • Two-hour virtual session,
  •  Learn how students build their conceptual understanding of simplifying algebraic expressions
  • Solving equations using these tools.  
  • Determining perimeter,
  • Combining like terms,
  • Comparing expressions,
  • Solving equations
  • Use an area model to multiply polynomials,
  • Factor quadratics and other polynomials, and
  • Complete the square.
  • Support the transition from a concrete (manipulative) representation to an abstract model of mathematics..

Foundations for Implementation

This professional learning is designed for teachers as they begin their implementation of CPM. This series contains multiple components and is grounded in multiple active experiences delivered over the first year. This learning experience will encourage teachers to adjust their instructional practices, expand their content knowledge, and challenge their beliefs about teaching and learning. Teachers and leaders will gain first-hand experience with CPM with emphasis on what they will be teaching. Throughout this series educators will experience the mathematics, consider instructional practices, and learn about the classroom environment necessary for a successful implementation of CPM curriculum resources.

Page 2 of the Professional Learning Progression (PDF) describes all of the components of this learning event and the additional support available. Teachers new to a course, but have previously attended Foundations for Implementation, can choose to engage in the course Content Modules in the Professional Learning Portal rather than attending the entire series of learning events again.

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Building on Instructional Practice Series

The Building on Instructional Practice Series consists of three different events – Building on Discourse, Building on Assessment, Building on Equity – that are designed for teachers with a minimum of one year of experience teaching with CPM instructional materials and who have completed the Foundations for Implementation Series.

Building on Equity

In Building on Equity, participants will learn how to include equitable practices in their classroom and support traditionally underserved students in becoming leaders of their own learning. Essential questions include: How do I shift dependent learners into independent learners? How does my own math identity and cultural background impact my classroom? The focus of day one is equitable classroom culture. Participants will reflect on how their math identity and mindsets impact student learning. They will begin working on a plan for Chapter 1 that creates an equitable classroom culture. The focus of day two and three is implementing equitable tasks. Participants will develop their use of the 5 Practices for Orchestrating Meaningful Mathematical Discussions and curate strategies for supporting all students in becoming leaders of their own learning. Participants will use an equity lens to reflect on and revise their Chapter 1 lesson plans.

Building on Assessment

In Building on Assessment, participants will apply assessment research and develop methods to provide feedback to students and inform equitable assessment decisions. On day one, participants will align assessment practices with learning progressions and the principle of mastery over time as well as write assessment items. During day two, participants will develop rubrics, explore alternate types of assessment, and plan for implementation that supports student ownership. On the third day, participants will develop strategies to monitor progress and provide evidence of proficiency with identified mathematics content and practices. Participants will develop assessment action plans that will encourage continued collaboration within their learning community.

Building on Discourse

In Building on Discourse, participants will improve their ability to facilitate meaningful mathematical discourse. This learning experience will encourage participants to adjust their instructional practices in the areas of sharing math authority, developing independent learners, and the creation of equitable classroom environments. Participants will plan for student learning by using teaching practices such as posing purposeful questioning, supporting productive struggle, and facilitating meaningful mathematical discourse. In doing so, participants learn to support students collaboratively engaged with rich tasks with all elements of the Effective Mathematics Teaching Practices incorporated through intentional and reflective planning.