The teaching ideas here are instructional routines teachers can implement in their classrooms to help students become more deeply and actively engaged in understanding algebra.
These ideas focus on how teachers can help students better engage, which we define as making deep mathematical connections, justifying and critiquing mathematical thinking, and solving challenging problems – or Connect, Justify, and Solve. We view the classroom as an under-utilized source for testing and refining instructional routines that are continuously informed by what teachers see day-by-day, class-by-class. These ideas have been tested and refined by our network members and may be promising strategies for your classroom, so take a look!
Making connections among mathematical algorithms, concepts, and application to real-world contexts, where appropriate.
Connect Using Big Ideas to Build Connections, by Michelle Allman
After students had completed activities that introduced concepts and were practicing the related skills, they made common errors that demonstrated disconnections between the skills and the underlying concepts.
I identified approximately four “Big Ideas” per unit and designed about six class activities per unit to highlight and build connections between those Big Ideas and the related skills.
- It is important to keep the Big Ideas focused and few.
- The task has to clearly demonstrate the Big Idea and the new algebra skill in relevant and meaningful ways.
- The Big Idea needs to connect to mathematics beyond the specific unit
- Students need constant reminders to connect to the Big Ideas when practicing and applying skills.
- Identify Big Ideas that highlight understandings of the unit’s concepts that, if internalized, would remediate common mistakes when applying related skills.
- Craft activities that connect the Big Idea(s) to the lesson objectives.
- At the beginning of the unit, list, read, and briefly discuss the Big Ideas for the unit.
- At the beginning of each lesson introducing new material, list, read, and review the Big Idea(s) related to the lesson.
- For each activity, introduce the Big Ideas, then include questions that raise how the Big Ideas connect to the activity.
Connect Making Connections through the Introduction of New Material, by Debbie L. Kowalczyk
Students were not independently making connections to prior knowledge, real life applications or patterns. I had been introducing topics myself and telling the students what the connections were.
I used guided discovery tasks to help students make their own connections to mathematical algorithms, concepts, and/or applications to real world contexts when introducing new material.
- It took time for the students to understand what was expected of them during these tasks.
- It is important to ask questions that are explicitly about the task and the connections involved, rather than asking more broadly, “what connections did you make?”
- Including the questions with the task rather than a separate piece of paper helped students answer more seriously, and more thoroughly.
Students benefited from mastery-oriented feedback regarding the depth of their connection as well as their engagement level. The deep connections didn’t always immediately surface through the task but over time, students are showing they’ve mastered the material and, at times, draw on connections they learned in a particular task to support their work on other problems.
- Choose a task related to the topic with multiple entry points for students to complete successfully (guess and check, table, graph, rule, pattern recognition etc.).
- The task itself should have leading questions to draw students towards pattern recognition, understanding the concept, and/or extending to a generalized rule. These questions are best delivered in the context of the problem and not separately on a card.
- Give time for productive struggle with the task.
- Provide scaffolding as needed.
- Take notes and/or audio record because the students do not always write down what they are saying or what they have learned.
Connect Exit Routines to Build Mathematical Connections, by Tara Sharkey
Students are not given regular opportunities to make explicit connections between what they are learning and previously learned concepts and algorithms.
I decided to create generic exit cards that asked students to make connections between that day’s big math idea and another math idea, concept, or algorithm.
- It is challenging to help students make deep connections if they do not know what a good connection looks or sounds like.
- Allowing students independent think time to develop their own understanding and connections is important.
- Students build better connections once they have the opportunity to talk about their initial thinking with a classmate.
- Often, a class discussion facilitated by the teacher will help the class as a whole get to deeper connections.
- The tasks students are asked to make connections about must be connected to previous tasks and learning experiences.
Phase 1: Setting up a Closure Routine
- Create a selection of generic exit cards, photocopied in bulk, that can be used as closure for any class. I created two exit cards asking students to make a connection and one open ended exit card that allowed for any question to be generated at the end of a class.
- At the end of each class, set aside about 5 minutes for students to complete an exit card.
- Utilize these exit cards for a series of classes to create a regular routine of completing exit cards.
Phase 2: Improving the Quality of Responses
- After students are used to the closure routine, begin to focus on the quality of students’ responses.
- Select from the following options to follow up on exit cards at the following class:
- Read all exit card responses aloud at the start of the next class.
- Select exemplar exit card responses, read or display these at the start of next class.
- Select a “favorite no” exit card response, have students discuss in pairs, then as a class what is incorrect about the response.
- Open class by working together as a class to create a quality exit card response for the previous class.
- As another option, create and use questions about making connections that are specific to a task. These may be separate from the exit cards.
- Have students think about responses independently, talk about them in pairs, discuss them as class, then revise or add on to their responses.
Phase 3: “Connect, Extend, Challenge” Template
- Students in Algebra I often struggle to make correct, quality connections, especially as the topics being learned change. Another way to improve student connections is to use the “Connect, Extend, Challenge” template.
- When first introducing this template, set aside about 15 minutes.
- Have students write about any connections, extensions, or challenges they have about the current big math idea, give them about 3-4 minutes for this.
- Next, have students engage in a structured math talk with a partner, give each partner 1 minute to share their connections, extensions, and/or challenges.
- Hand out post-its and have partners write their most specific connection, extension, or challenge on the post it, then put it on a chart paper in the classroom, with on chart paper for each of Connect, Extend, and Challenge.
- The next class follow up by:
- Having students walk around to the charts and read the post-its.
- Select a post-it they think is the most specific, then have students read them aloud.
- Read a few post-its aloud to the class that you select.
Phase 4: Keeping Making Connections
- Continue to regularly ask students to make connections between math ideas.
- Continue using exit cards or the Connect, Extend, Challenge template.
Communicating and justifying mathematical thinking as well as critiquing the reasoning of others.
Justify Using Small-Group Work to Improve Justifications, by Kerri Rogers
My students do not know how to vocalize/write down their mathematical thinking. Their justifications tend to be “surface level” with little or no mathematical evidence to support their claims.
I decided to try giving students open-ended problems, ask them to solve the problems in groups and then then ask them to justify their group work and problem-solving strategies verbally using a video forum called Flip Grid.
- It is important to give students time and a place to process problems and write down thoughts individually before moving into group work
- Student need to have access to vocabulary words in some sort of organized way and practice using them in meaningful ways to help them use the mathematical language effectively.
- Students need good examples of justifications to understand what strong justification look like and how they different from restating a series of steps Students need to have a good grasp of the concept and be able to justify their own work before being able to critique others.
- Research good tasks
- Decide which task to use which is a practice reinforcement for the unit you are teaching.
- Make a question template for students to discuss and think about their reasoning while solving the problem
- Give students about 5 minutes to look at the problem in front of them and make any notes to help when they start their group work, this allows for some individuals to process their thoughts before working with others.
- Have students get into groups of 3 to 4. Circulate around the room and listen to conversations and ask probing questions to help students through the solving process.
- Have students verbally explain their problem- solving process and justify their steps along the way and record them using Flip Grid
- Collect the student work and assess it along with the verbal responses.
- Watch videos, assess the depth of the justifications and give students feedback.
Justify Justifications Using a Partner Share Protocol, by Heather Vonada
Many students feel that by just showing their work they have justified their solution. Often there is no attempt to explain their reasoning, or it is limited and lacking logic or clarity. Another problem is that students don’t give appropriate feedback to each other on their justifications.
I decided to use sentence starters to help students write a conjecture and then use a partner share protocol that will elicit deep justifications using quality feedback.
- If the tasks weren’t broad enough, the conjectures, justifications and partner feedback were all weak.
- Students need support to write mathematically clear and correct justifications
- It took practice and guidance for students to provide useful feedback.
- Students are typically able to give higher quality spoken justifications than written ones.
- The engagement in writing a conjecture went up significantly when I added a sentence starter.
- Provide students with a task that requires them to state a conjecture, test it, and write a justification for why it was correct or incorrect.
- Provide a sentence starter for students to state a conjecture. For example: I think the graph of y = 3×2 + 4 will be a _________ because ________.
- Give 10 minutes of Private Reasoning Time to do the task (write a conjecture, test the conjecture, write a justification based on testing).
- Give 6 minutes for trading papers with a partner and giving feedback to each other (something they understand, are confused about and a question they have)
- Return papers to their owners and allow 10 minutes for students to revise their justification based on the feedback they received from their partner.
Making sense of and solving challenging math problems that extend beyond rote application of algorithm.
Solve Providing Problem-Solving Prompts for Support Adults to Use with Students in Math Class, by Julie St. Martin
I teach high-needs students who are supported by paraprofessionals and other adults during class time. I found that these adults tend to provide too much support (e.g. hints, strategies, answers) during problem solving, effectively removing the opportunity for students to persevere in the work of making sense of and solving the problem.
I decided to test a routine in which I post a collection of prompts for paraprofessionals and special educators to use in class to respond to student requests for help in a way that reorients students towards their own thinking, their peer’s thinking, the task itself, or classroom resources.
- The special educator and para-professionals I work with were much more receptive to this change idea than I expected. They said that being given specific tools to use when students were stuck, was much more helpful than being told what not to do as they had been in the past.
- This change is not enough to get students deeply involved in problem solving, but rather helps facilitate and encourage it.
- Later in the year, once students were more engaged and had learned more about how to reason with each other about mathematical ideas, the value of this change idea became much more apparent than when it was first implemented and tested.
Ahead of Implementation:
- Create a large print poster or two to post in the classroom with a list of prompts (e.g. What do you notice about this problem? Have you figured out anything that doesn’t work? What ideas do you have about what you could try next?) for adults in the room to easily refer to.
- Speak with administrator in charge of special education services, supervisor of adults who will be central to this change to this change idea, to ensure that they will support this effort. Also find out if time can be built into their professional development schedule to read two short articles (Never Say Anything a Kid Can Say by Steven C. Reinhart from Mathematics Teaching in the Middle School and Telling You the Answer Isn’t the Answer by Rhett Allain from wired.com) to help orient them to the purpose of this change idea.
- Ahead of our first class, arrange to meet for 15-20 minutes with all adults who will be providing student support in class.
- During this meeting orient the adults to the structure of the course and the philosophy & practical challenges of implementing a student-centered approach to learning, especially with high need students. Discuss how they can best support students in this learning experience. Introduce them to the prompts) and talk through some examples of how they can be used.
- As you begin a problem-solving task, remind students that after some independent think time they will be expected to work together to make sense of the task. Remind adults of the prompts in the room, and that it is expected that students will struggle as they work through the ideas raised by the given task.
- Distribute the task, read it out loud, and set a time for 3 minutes of private reasoning time where students will brainstorm any ideas and/or clarifying questions they have about the problem. Remind students they are not expected to solve the task during this time, but they all need to be prepared to share an idea, an attempt, or a question about the task. Adults should not interact with students during this time unless a student is disruptive to others.
- At the end of the 3 minutes, organize the students into pairs or small group (or pairs and then small groups) to discuss the task and begin to problem solve collaboratively. During these discussions, adults are welcome listen in but should aim for student groups to work as independently as possible. When students request help or if a group or pair has ceased being mathematically productive, adults should use a prompt or two to reorient students and try to help them productively engage in wrestling with the task.
Solve Daily Mindfulness Time in Math Class to Promote Problem Solving, by Julie St. Martin
My 2-year looping Algebra 1 class is designed for very high needs students. Students weren’t engaging with perseverance in the work of making sense and solving challenging math problems due to emotions, frustrations, anxieties, and stresses. Many students shut down in class or left the room. Students who aren’t physically or mentally present can’t engage in math problem solving, let alone engage deeply!
I wanted to try to implement a 3-to 5-minute mindfulness routine near the start of class using short mindfulness videos from YouTube.
- Mindfulness breaks made an incredible difference in the culture and productivity of this class.
- Students became much more likely to stay in class and engage in productive mathematical discourse. They also seemed more patient with each other.
- Students unanimously recognized that this time was helpful for our class.
- It is important to keep reminding students of the purpose and expectations for this time.
- 5-6 minutes was much more effective than 3-5 minutes.
- Providing various relaxing options during mindfulness time was important (video, audio, eyes open/closed, locations in room, doodling or coloring).
Ahead of implementation: discuss with special educators to plan for any accommodations that may need to be made. Select a 5- or 6-minute mindfulness video for class (be careful to listen for and avoid any religious references). You are welcome to explore the videos I’ve used successfully which are now included on my YouTube playlist: Daily Classroom Mindfulness Breaks. Print some coloring pages such as Mandalas etc.
In class: Implement mindfulness breaks after reviewing homework.
First time – explain to students the intention of this new part of our routine. Watch the brief intro video: Mindfulness: Youth Voices by Kelty Mental Health.
Then every time:
- Remind student of the purpose and to put away all technology.
- Let students know their options. Offer coloring sheets and colored pencils/markers to those who would like to color during this time.
- Turn off lights. Ask all to participate in 3-6 timed deep breaths as a full group following a visual clip such as Polygon Breathing by the School of Self or Breathr Mindful Moments: Three Breaths by BC Mental Health & Addiction Services’ Health Literacy group. Guide and model this from the front of the room.
- Play the selected mindfulness video and sit in the classroom among the students.
- When the video is over, turn on the lights and immediately dive into the warm up, problem solving task, discussion, practice, or other math thinking activity.