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.
Connect Identifying Mathematical Connections Between Activities, by Sarah Dominick
Perhaps because students have not been given enough time or support to do so, they typically rely on the teacher to make the connections instead of themselves and each other.
To help students make their own quality connections, I implemented this routine:
- Have students fill out a Connect Table—an organizer for students to describe: 1) a given mathematical idea from the current lesson, 2) a given related idea from a previous lesson, and 3) some similarities and differences between the two.
- In a whole-class discussion, summarize the connections on an anchor chart.
With the class, use a rubric to evaluate the quality of the connections made by the class.
- Select activities where conceptual connections can be readily made; activities can be edited to support deeper connections.
- The discussion routine is essential: it offers quieter students opportunities to share their thinking and students often show deeper understanding in discussion than in their written work.
- It is difficult to measure the quality of connections students describe; I did so by including prompts for students to identify connections.
- Identify an activity from a previous lesson that has a mathematical connection to the activity students are currently working on (e.g., there are important connections between formal systems of linear equations and solving shapes puzzles).
- After students complete the current activity, ask them what connections they see between the activity and the previous activity. Allow at least 3 minutes for students to record their ideas in a Connect Table.
- Have students share their connections with a partner (1 minute share time, 30 second response time for each partner).
- In a whole-class discussion, summarize key connections from the class on an anchor chart.
Connect Eliciting Connections Through Informal Reasoning and Probing Questions, by Pamela Rawson
Students tend to see algebra as a self-contained set of procedures to be memorized and used to solve specific types of problems, rather than as an extension of arithmetic, numerical relationships. Making these conceptual connections is often discussed as something that is important for students to do, but it is not often explicitly pursued in traditional approaches to teaching algebra.
Students’ conceptual understanding would be better supported if they had opportunities to make connections between formal algebraic procedures and their prior understandings about arithmetic, numerical relationships. To provide these opportunities, I developed a routine that first allows students to solve problems using informal models and reasoning, and then pushes students to make connections to more formal procedures through probing questions.
- Selected tasks should provide rich opportunities for students to make connections between algebraic procedures and the underlying concepts involved.
- Simply asking students what connections they are making while solving a problem is ineffective for helping them develop an understanding of mathematical connections; rather, targeted questions should be used to elicit students’ thinking about connections.
- Introduce students to a new topic or concept, either through a pre-assessment or an exploration activity that can support connections to quantitative relationships and numerical properties. For example, card sorts (e.g., http://map.mathshell.org/lessons.php?unit=9210&collection=8) or scale models can be used when introducing equation solving; shape equations or mobile model activities can be used for introducing systems of equations.
- As students learn more formal algebraic procedures related to the topic, prepare a set of probing questions that push students to think about why a procedure works, drawing connections to their informal explorations in Step 1. For example, why does the substitution method work for solving linear systems? Can you explain it in terms of the mobile model or shape equations?
- Continue to pose these kinds of probing questions about connections as students explain their solutions to problems, either during small group or whole-class discussions, or as part of a written reflection or assessment.
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.
Justify Using Peer-Based Discussions to Support and Enhance Mathematical Reasoning, by Benadette Manning
Students often are reluctant to engage in math discussions that require them to think, question, and justify reasoning. This may be related to a lack of opportunity to engage in such discussions or a lack of confidence in their ability to use mathematical language.
The goal of this routine is therefore to support and facilitate mathematics discussions in which students justify, question, and critique their own and their peers’ thinking with the help of well-selected problems, a partner discussion protocol, and appropriate scaffolding by the teacher.
- Students will use a justification protocol, the Math Paired Conversation Protocol to guide their thinking, questioning, and critiquing during small-group problem solving and discussion.
- Students will be supported with scaffolds in the form of questions and response starters from the Math Constructive Conversation Skills Poster.
- The Math Paired Conversation Protocol has numerous steps; students might benefit from a simplified protocol with fewer steps. Consider selecting your own subset of focus prompts from the protocol each time the routine is used.
- Read the Math Paired Conversation Protocol with the whole class. Emphasize that each partner needs to contribute ideas for solving the problem, and that partners should talk about multiple ways to solve the problem.
- Have students read the Math Constructive Conversation Skills poster. Ask them to highlight a few prompts and responses they think would be most effective. Remind students to use the poster whenever they need help thinking of questions.
- Introduce the problem(s) students will be working on. It is important to select problems that have more than one way to solve them.
- While working in pairs, students solve the problem using the Math Paired Conversation Protocol, working through steps of: 1) clarifying the problem in partner discussion, 2) having each partner estimate an answer and justify the estimate to the other partner, 3) having each partner solve the problem and justify the solution to the other partner, 4) comparing solutions to estimates from step 2, and 5) comparing methods and justifying which method you would recommend.
- Support students’ discussions without answering their questions directly, but instead asking questions that will help them decide on their own pathways to solve the problem and justify their solutions. Encourage students to use informal language to express an idea; then you can re-voice their ideas with more formal mathematical terminology as a scaffold.
Justify Building Justifications Through Interactive Activities, by Lauren Hagadorn
Students often are unable to describe their reasoning when solving problems related to concepts and procedures they are familiar with. This may be because they have seldom been given opportunities to explain or justify their answers and are therefore not used to explaining their thinking in mathematical terms. For example, students’ explanations often include statements like, “because it is the next step,” rather than justifications based on mathematical relationships; other times students simply do not know how to begin a justification.
Students are provided with a scaffolded, interactive activity to help them understand how to develop and articulate justifications for algebraic procedures. Students engage in a card-sorting/matching activity in which they order the algebraic steps for solving equations, match justification statements to the algebraic steps, and eventually develop their own justifications for algebraic steps.
- Focus on one aspect of deep justification at a time (e.g., logical coherence, precise use of math language, definitions of terms and units, focus on mathematical relationships); this made it easier to understand how students are developing related to specific facets of justification.
- Examples of justification statements are beneficial for students who are new to the process of justification.
- Create a “Building Justification” activity by writing the solution steps to a familiar algebraic problem. For each step, write a justification statement. Scramble the steps to create a matching activity or make cards for each step to create a card-sort activity. As students gain confidence, consider adding to the challenge by creating superfluous justification statements/cards that do not match any of the steps.
- Have students place the algebraic steps into the correct order. Students should discuss their ordering with peers, the teacher, or in a whole-class discussion before proceeding.
- Next, students match the justification statements to the algebraic steps. As students gain confidence, they may be able to supply their own justifications for steps. To support this progression, replace some (and eventually all) of the justification statements/cards with blanks, on which students write missing justification statements themselves.
Justify Critiquing Worked Examples to Develop Deeper Conceptual Understanding, by Michael Refici
Students often do not engage deeply in the conceptual learning of algebra and therefore do not develop their understanding of algebraic principles—the “why” behind the “what.” Often, they are only able to solve problems by retracing procedural steps from previously solved problems. Students also have difficulty justifying their steps and providing evidence to support their work.
Students will develop deeper conceptual understanding and ability to justify their reasoning if they have opportunities to critique worked examples and justify their thinking in the process. In this routine, students are provided completed solutions to problems, either selected from their peers’ work or created by the teacher. Depending on the goal of the activity, solutions will either be mathematically valid or, in most cases, will include a common misconception or error. As students examine the worked example, they evaluate the reasoning contained in the sample work and justify their own conclusions about whether or not the work is mathematically valid.
- Use a well-defined protocol for the routine to ensure greater student engagement. Protocols should: provide students with individual think time before they work in groups; clearly specify what students will share in groups; and specify how much time will be allotted for each phase of the routine.
- Appropriate scaffolding tends to improve quality of students’ work over time. Scaffolds in this routine include showing worked examples to provide models for good written work (even in examples that include misconceptions); specific question prompts or sentence starters to create entry points to justification; and word banks with specific terms to include in justifications.
- Select a high-quality, non-routine problem related to a concept of interest. Create one or more worked examples for the problem (either teacher-created, or drawing from students’ previous work on the task), and include question prompts and/or sentence starters designed to help students evaluate the reasoning in the example, and justify their conclusions (e.g., Marlowe solved the problem this way… Is Marlowe correct? If so, justify why you think so. If not, Marlowe is incorrect because ____________.).
- Display a word bank with relevant, mathematical terms to be used in justifications related to the worked example(s).
- Distribute the worked-example activity and have students analyze the example individually for 3 minutes.
- Working in groups of 3 or 4, have students take turns sharing their thinking about the problem for 30 seconds each. Encourage students to justify their reasoning, trying to convince each other using mathematical concepts and terms.
- Have students work on the problem independently again to revise their reasoning based on the discussion.
- Repeat Steps 4 and 5 as needed to allow students to consider their peers’ arguments and revise their own thinking appropriately.
Justify Stimulating Quality Class Discussion, by Gina Sheehan
During whole-class discussions, typically not everyone contributes, and students’ conversations often do not demonstrate a depth of mathematical thinking. Improvement is needed in the quality of class discussions, such that students take ownership of their responses, spend time grappling with concepts that do not come to them immediately, and provide depth to their justifications.
I provided a structured routine for students to develop mathematical justifications, share them with others, and revise them based on discussion and feedback.
- Students were presented with a concise prompt specifically selected to support justification. Prompts focused on conceptually focused activities such as card sorts, matching equation rules to dot-pattern drawings, etc.
- After individual think time, students shared their justifications in small groups. They made edits to their own justification statement based on what they heard from others.
Then we engaged in a whole-class discussion. I hypothesized that giving students time to edit their responses both individually and in groups would help them to share their ideas more confidently and have more natural conversations about mathematics.
- Asking students to write down their ideas was helpful to increase the quality of what students were saying aloud.
- Individual think time allowed for more people to share their ideas with confidence and provided the teacher more time to circulate around the classroom.
- The routine allowed for more productive conversation because I had time to sequence who shared what, and when. I found that it made students feel special when you asked them to share their idea with the class.
- Provide students a relatively concise, engaging prompt or task that has a conceptual focus, includes multiple points of entry, and/or has the potential for important misconceptions to surface.
- Allow students 3 to 5 minutes to work on the task independently.
- Ask students to engage in small-group discussion about their solutions, emphasizing that they should support ideas with mathematical justifications. Circulate around the room and identify solutions/justifications you would like students to share in the whole-class discussion.
- Facilitate a student-led, whole-class discussion, asking questions about differences, similarities, and connections among the solutions/justifications that are shared.
Justify Implementing Tiered Checkpoints to Facilitate Understanding of Concepts and Support Justification, by Ben Winchell
“Checkpoints” are a means of preparing students for an upcoming assessment. The traditional way of organizing a practice assessment of this type is to give students time to work on it independently—without collaboration or opportunity for comparisons with one another. The teacher then provides correct solutions at the end of the class. Review activities might be more effective if they involved collaboration, comparison, and opportunities for students to justify their reasoning and critique the work of others.
I decided to change the structure of checkpoints to incorporate collaboration in partners and small groups, creating tiers of communication that culminates in a whole-class discussion guided by the teacher. Organizing review this way might enhance students’ understanding of concepts and support their ability to justify their reasoning more deeply among their peers.
- Collaboration during review can have a positive impact on assessment averages vs. a traditional approach to practice tests.
- Certain topics work better for comparisons/justifications than others.
- Students generally seemed to prefer collaborating and comparing their work over purely individual work.
- Students are given the checkpoint – a short practice activity including key soon-to-be-assessed concepts and procedures – and a certain amount of time to work individually (5-10 minutes).
- Students work in pairs to compare, justify, and revise their work as needed (5 minutes).
- Students join pairs into small groups of 4 to 6 students to further compare/justify their work (5 minutes).
- Students return to a whole-class discussion, where the teacher facilitates making connections among students’ ideas and formalizing concepts and solution strategies (5 minutes).
Justify A Graphic Organizer to Support Justification, by Jeannette Aames
When students have opportunities to verbally justify responses in group problem-solving activities, typically only some students are doing most of the cognitive heavy lifting during these discussions. Students’ justifications also often lack sufficient quality, depth, and mathematical meaning. In part, this may be because it is difficult for the teacher to assess students’ performance during verbal discourse, and therefore to provide formative feedback to students about their justifications.
Provide a routine that involves meaningful discourse among students, and which emphasizes strategy, evidence, and justification as part of the problem-solving process. The routine also involves students producing written justifications, clarifies for students what a quality justification is through use of an evaluation rubric, and provides students with feedback from the teacher on their written justifications.
- Problems should be open-ended and rich enough to support meaningful, worthwhile justification and argumentation.
- It is helpful if students do some planning for how to solve the problem before they participate in discourse; scaffolds can be useful to help students in their planning process.
- It can be useful for students to see models of high-quality justifications so that they develop a picture of what justification involves.
- Create the activity, starting with a rich, non-routine problem with multiple ways to solve it. Then add a three-part graphic organizer that helps students: 1) strategize (e.g., identify key information in the problem, understand the task, plan a solution path), 2) apply math knowledge to solve the problem, and 3) justify their reasoning.
- The first few times the routine is used, students receive samples of exemplar student work to discuss in groups and with the whole class.
- Launch the problem or activity with the whole class, ensuring students understand the problem context and what is being asked.
- Students work independently as they develop their plans for solving the problem(s) and complete the “strategize” part of the organizer.
- Students then work collaboratively in small groups to solve the problem and complete the “apply math knowledge” part of the rubric that shows their collaborative work and solution.
- For the “justify” part of the organizer, students write justifications for their solutions using the CER (claim, evidence, reasoning) format.
- The teacher then uses a task rubric to provide feedback to students. The rubric includes three parts to mirror each of the three sections in the task: strategizing, math knowledge, and justifying.
Justify Encouraging Justification over Answer-Getting, by Todd MacKenzie
School mathematics often emphasizes getting the right answer over mathematical reasoning. Students therefore have difficulty adjusting their thinking and justifying their reasoning on their own. This change idea involves use of open-ended investigations to help students develop their ability to justify reasoning about mathematical claims.
To help students learn how to justify their reasoning, they engage in an adapted version of the “Which One Doesn’t Belong?” routine. In this routine, students: 1) examine a set of four mathematical representations, 2) identify which one is different from the other three, and 3) explain why. Because several answers can be justified in these activities, students routinely engage readily in informal mathematical justification. Students then create their own “Which One Doesn’t Belong?” puzzle related to translations of parent functions, share their puzzles with peers, and discuss their reasoning.
- Students often approach algebra as an exercise of finding correct answers and thus find it challenging to engage in an activity in which there is no single right answer.
- Teachers should anticipate the wide range of ideas generated by students. In a single classroom, for example, some students may approach a task using informal reasoning, while others may apply formal principles.
Part 1: “Which One Doesn’t Belong?” Activity
- Select and display a set of four graphs or equations for a “Which One Doesn’t Belong” activity. See Which One Doesn’t Belong? for examples.
- Ask students to work individually to identify how various choices might be considered as different from the other three, thinking about their mathematical justifications for each choice. Allow enough time for students’ reflection and productive struggle, and to explore various reasons why one graph/equation may not belong with the others.
- On a half-sheet of paper, have students show which of the four graphs/equations they identify as different from the other three and write an explanation for why it is different from the other three. Ask them to justify their choice using precise vocabulary.
- Create a large printed version of each of the four choices and post one in each corner of the classroom. Ask students to move to the corner of the room where their choice is posted. Have students to share their justifications for their choice with the whole group and encourage peers to ask questions and discuss the choice in more depth.
- Summarize and formalize the most important mathematical ideas that were brought up in the discussion.
Part 2: Creating Your Own “Which One Doesn’t Belong?”
- Using the same four-box format, ask students to create a “Which One Doesn’t Belong?” activity with graphs of functions that are related in some way. Students should try to include graphs where any of the four can be selected as “not belonging.” To construct graphs for various function families, students could use the DESMOS applet.
- In small groups, have students share their puzzles with each other and try to identify which of the four is different, again providing justifications for their choices.
Justify Using Sentence Frames to Scaffold Justification, by Taylor Sabky
Students often find it difficult to explain their solution process, justify their reasoning, and critique the reasoning of others. Traditionally in mathematics classes, students are only asked to produce answers but are rarely asked to explain, defend, or critique their own or others’ reasoning. Often students know how to apply a procedure but do not understand why the procedure works.
To help students learn how to better explain, justify, and critique reasoning, I introduced a routine requiring students to more intentionally and systematically provide written and verbal explanations—not just of their answers but also for the mathematical processes they used to find their answers. Specifically, written problems and activities were supplemented with explicit prompts and sentence frames designed to scaffold justification. In addition, I consistently included prompts for explanations/justifications during class discussions.
- The quality of conversations and oral explanations were typically much higher than the quality of written explanations.
- It was extremely difficult to find a practical, reliable way to assess and track students’ oral explanations.
- Students provided better explanations when they understood the differences between medium- and high-quality explanations.
- Select a task that requires students to solve a problem. Problems that have more than one possible solution path may more readily support students’ explanation of their reasoning.
- Select a task that requires students to solve a problem. Problems that have more than one possible solution path may more readily support students’ explanation of their reasoning.
- Create a rubric that shows students the specific meaning of low-, medium-, and high-quality explanations. When first sharing the rubric with students, ask them to identify the differences between levels. In a whole-class discussion, decide how to turn a low or medium explanation into a high-quality explanation. Example explanations at each level can be helpful for this purpose.
- Allow students time to complete the task independently at first. Then have them work in pairs or in small groups to share their solutions, justify their reasoning to each other, compare their processes, and critique others’ reasoning.
- Have a few students share their reasoning with the whole class. During the discussion, include prompts to probe students’ explanations, asking for example, “Explain why you did that?” or “How did that help you move closer to a solution?” or “How is your reasoning different from the other method we saw?”
Justify Infusing Justification into Math Tasks, by Apolinario Barros
When probing students to explain their reasoning, their reply is often, “I understand it, but I cannot explain it.” When students do attempt to justify their reasoning, these attempts tend to be superficial or limited to one-word answers. Also, students seldom use mathematical terminology in their explanations.
If tasks included explicit opportunities for students to justify their work verbally and in writing, they could potentially develop deeper understanding of previously learned content. I developed a routine where students work in groups of three to solve problems collaboratively and provide justifications for their work. The premise is that the process of discussing their problem-solving processes and reasoning could lead to a deeper appreciation and understanding of the math content.
- Sentence starters can help students organize their thoughts as they prepare explanations of their work.
- Encourage students to use previous class notes and readily available related materials to promote referencing of relevant concepts in explanations.
- Select a problem or task that supports rich justifications of reasoning. Add question prompts and/or sentence starters that explicitly ask students to justify their reasoning related to their answers.
- Place students in groups of three.
- Have students read the problem individually first. Encourage them to annotate the task before initiating discussion.
- As students collaborate on their solutions, each student records the group’s work on their own worksheet.
- The teacher circulates, listening to students’ discussions and asking questions as needed to assess and advance students’ thinking.
Justify Strategies to Support Justification, by Donna Brink
Students often lean heavily on algorithms to solve problems and don’t connect their solutions to underlying concepts or relationships. Students need support to help them use mathematical reasoning in their solution strategies and justify their solutions using precise vocabulary. Specifically in group work, student would benefit from opportunities to propose a solution strategy (or a next step in a strategy) and justify why it is a valid or useful way to find the solution.
In this routine, students are explicitly asked to explain how they know their work is accurate – in the context of an opener routine –during problem-solving activities and comparison of worked examples.
- Using a “Which One Doesn’t Belong?” activity as an opener provides opportunities to explore what makes for a strong justification.
- Problems that allow for multiple solution paths are best suited to supporting students’ justification.
- As an opener, implement a “Which One Doesn’t Belong” activity (or similar activity) where there are multiple valid solutions that require justification. In this whole-class activity: 1) students identify which of four mathematical representations doesn’t belong, 2) students justify their reasoning about their choice using precise mathematical language, and 3) as many students as possible are encouraged to share diverse responses. Throughout the discussion, highlight specific aspects of students’ explanations that involve high-quality justification of reasoning.
- Introduce a problem or activity that has multiple ways of solving and allow students time to understand the problem and think about the solution individually (~ 5 minutes).
- In pairs or small groups, students compare their initial ideas and collaborate on a solution and document their thinking as a group (~ 8-10 minutes).
- Group work is supported by the teacher with clarifications and questions, but not in a way that steers students toward specific strategies or solutions.
- Use a rubric to assess – and to help students assess – one aspect of their work at a time. Use the rubric to help students reflect on how to improve their solution strategies and justifications.
Justify Increasing Student Participation to Support Their Justifications, by Joshua Parker
Students are often unwilling to share their thinking because they are not confident in their answer or because they are not used to justifying their thinking to others. Specific attention to supporting more participation in group discussions may help change this, especially if students are given multiple ways and/or opportunities to justify their thought processes. Discussing justifications in group work may also help students interpret whether or not their solution makes sense.
By introducing a group-based, problem-solving process, specific protocols can be enacted to increase participation by all group members. In the process, students’ collaborative thinking and group feedback may support stronger verbal and written justifications.
- It is essential that students get an appropriate amount of time to work on the task themselves.
- Group discussion helps students decide if solutions were reasonable, and how to revise and enhance their reasoning.
- Students need clear expectations of what comprises a high-quality justification.
- Provide an open-ended problem-solving task, read through the problem and ensure students’ understanding of the task, and provide students with a few minutes of individual time to begin thinking about a solution.
- Implement a small-group discussion protocol that ensures participation by all students in the group (e.g., using a timer, have each group member explain their process/solution to the rest of the group for 1 minute; for the first round, each student explains their reasoning without responses; for the second round, encourage students to respond to each other’s ideas with questions and responses).
- Give students the opportunity to put their ideas together to collaborate on a solution to the problem, building from the discussion in Step 2.
- Encourage students to use thought-provoking questions in their discussions and model these questions in your own interactions with small groups (e.g., “Can you explain why you did…?” or “Does that method always work?”).
- Have students review and strengthen the justifications they are writing. Provide written prompts to support this process. Examples include, Does your work follow a clear path? It is fluent? Can a reader tell what you did? Did you explain why?
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
In classes with a high proportion of special-education students, students can find it especially challenging to persevere in making sense of and solving challenging, non-routine math problems. Many students immediately ask for adult support, and the adults/resource teachers often provide too much problem-specific support (e.g., hints, strategies, answers) that reduce the cognitive demand of the activities.
In this routine, a collection of prompts was implemented for paraprofessionals and special educators to use in class to respond to student requests for help in a way that orients students towards their own thinking, their peers’ thinking, the task itself, or classroom resources.
- The special educator and paraprofessionals I work with were much more receptive to this change idea than I expected. Having specific tools to use when students get stuck was much more helpful than being told what not to do, as has previously been the case.
- Later in the year, after students had consistent opportunities 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 for adults in the room to refer to. These can include prompts like:
- What do you notice about this problem?
- What information seems useful for start to work on this task?
- Can you tell me (more) about your idea?
- Why do you think that?
- Is there another way you could organize or represent this information?
- Speak with administrator in charge of special-education services to ensure support of the routine.
- Meet briefly with all support adults about implementing a student-centered approach to learning with high-need students. Discuss ideas for supporting students in problem-solving activities, review the posted prompts together, and discuss some examples of how they can be used. If possible, ask supporting adults to read two short articles about the importance of letting students take ownership of their learning: Never Say Anything a Kid Can Say by Steven C. Reinhart and Telling You the Answer Isn’t the Answer by Rhett Allain.
- Remind adults of the prompts posted 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 allow 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.
- At the end of the 3 minutes, organize the students into pairs or small groups to discuss the task and solve the problem(s) collaboratively. During these discussions, adults are welcome to listen in but should aim for student groups to work as independently as possible. When students request help or if a group or pair gets stuck, adults should select a prompt to reorient students and try to help them persist in working on their own solution to 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.
Solve Giving Students the Opportunity to Solve Challenging Homework Problems, by Michelle Page
Students typically are not given the opportunity to solve challenging problems when completing their homework. Homework assignments usually consist of multiple rote problems and seldom provide opportunities to work on problem-solving. Students also have few opportunities to discuss homework problems after completing them.
Students engage in a homework routine that involves solving challenging, non-routine problems, with a routine for discussing and revising the homework in small-group discussion in class the following day. When completing the problems at home, students are provided with question prompts to support and encourage them in making sense of the problem, finding a solution path, and writing about their work.
- It is important to choose tasks that will be challenging but that also have multiple entry points. The goal for the students is to problem-solve, write about their thinking, and discuss their work with their peers.
- Provide students with a challenging, non-routine problem presented within a problem-solving template. The template should include the problem itself, as well as prompts to: 1) identify important information, 2) solve the problem while self-assessing “stuck-points” and strategies/questions for getting unstuck, 3) identify strategies being used, and 4) self-assess the level of challenge, time spent, and level of progress. The problem itself should relate to concepts from the lesson, but also extend those concepts into a novel problem or situation.
- Provide students with resources to help them solve non-routine problems. I created a poster with question prompts related to describing the problem, planning a solution path, monitoring progress, and making sense of and checking the solution.
- Share and discuss exemplars of student work from other non-routine homework problems, especially as students are getting familiar with the routine.
- Students work independently on the problem and write down any clarifying questions or alternative strategies they might consider if they get stuck. Students also reflect on the problem by checking off the problem-solving techniques they used, rating how challenging they found the problem and scoring their work using the problem-solving rubric.
- During the next class session, ask students to discuss their solutions in small groups. Ask students to edit their work using a different colored pen/pencil as they discuss. Students should not erase their original work even if they need to revise. If students solved the problem correctly, they should compare their work to their peers, and be able to make sense of each other’s strategies.
Solve Making Sense of Problems, by Casey Green
Students often have difficulty making sense of non-routine problems. As a result, students can become stuck and disengaged, and unable or unwilling to even attempt a solution. I often hear students say, “What is this even asking?” For students to be able to answer that question for themselves and- solve challenging, non-routine problems, they need targeted support to develop their sense-making ability.
By using built-in guiding questions as students work collaboratively on problem-solving tasks, students will be better supported in making sense of challenging math problems. Once they are able to make sense of what a problem is asking, they may be more likely to persist toward a meaningful solution.
- It is helpful to build the making-sense prompts into the task, as opposed to keeping the prompts on a separate form.
- Students benefit from ongoing feedback and gain confidence in making sense of problems.
- Students struggle to re-phrase the task in their own words in writing.
- Select a task with an appropriately high level of cognitive demand (i.e., includes problem-solving with non-routine tasks).
- Edit the task to include guiding questions at appropriate places to explicitly support students in making sense of the problem. Examples include, “What is the problem asking you to find?” or “What information in the problem will help you find a solution?”
- Give students 3 to 5 minutes of Private Reasoning Time (PRT) to preview the task and answer the guiding questions.
- Allow students to work collaboratively to decide what the task is asking and what important information is given. Note: Norms for group discussion should be established prior to having students work collaboratively.
- After students have made sense of the problem in group discussions, review their ideas in a whole-class discussion as necessary.
- Allow students to complete the tasks in pairs or groups.
Solve Small-Group Problem Solving, by Kristen Gervasio
Students are not often asked to solve non-routine problems that take more than a few minutes to solve. Solving challenging, non-routine problems is an import part of proficiency in mathematics.
In this routine, students solve rich, open-ended problems in small groups where there is explicit attention to and support for students making sense of the problem and collaborating on an appropriate strategy for solving the problem.
- The quality of the task itself is closely connected to students’ initial level of engagement.
- It is important to give students individual think time before they work in groups.
- Groups of 2 or 3 students generally work better than groups of 4 or more.
- During the planning process, choose a high-quality, non-routine problem for the students to solve. The problem should include multiple entry points and tasks/contexts that are interesting for students.
- Distribute a “task template” to each student, which is a simple advanced organizer for students to reflect and write about: 1) what the task asking is students to do, and 2) what information in the problem is important and why.
- Give students 5 minutes to read the problem independently and to answer the 2 questions on the template. Review students’ responses on the template as students finish writing, asking questions to probe and push their thinking.
- Ask if there are any clarifying questions and have students resolve any of the questions that arise in class discussion.
- Have students work in groups of 2 or 3. Provide each group with a single sheet of chart paper and different colored markers. Have each student record different ideas at the corners of the chart paper and discuss them aloud as they go. Once the group agrees on a solution path, it should be recorded in the center of the paper. Circulate as students are working to monitor progress and ask questions that help them to assess or advance their own thinking.
- After the appropriate amount of time has passed, ask a few groups to share and discuss their solution strategies with the whole class. Ask further questions and highlight key mathematical ideas as students discuss their solutions.