Try a Teaching Strategy | Better Math Teaching Network
These teaching ideas are instructional routines teachers can implement in their classrooms to help students become more deeply and actively engaged in understanding algebra. The ideas focus on how teachers can help students better engage, defined as making deep mathematical connections, justifying and critiquing mathematical thinking, and solving challenging problems – or Connect, Justify, and Solve.
Connect - Making connections among mathematical algorithms, concepts, and application to real-world contexts, where appropriate
Justify - Communicating and justifying mathematical thinking as well as critiquing the reasoning of others
Solve - Making sense of and solving challenging math problems that extend beyond rote application of algorithm
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.
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.
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.
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.
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.
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.