Schemas are neural networks, our brain’s way of organizing our thoughts and experiences. The way those networks get created and connected ends up defining your concept, or understanding of the topic. And when we talk about having “deep conceptual understanding in math,” we’re talking about building schemas of mathematical concepts that give students the ability to solve problems they haven’t seen before. The puzzles, activities, problem solving, and discussions in the ST Math Immersion program support students in revising, extending and building new schema around mathematics concepts. View this video to learn more about schema.
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All students have the potential to deeply understand and truly love mathematics. Brain research tells us students learn by doing. They need to be given the opportunity to engage in meaningful content, make predictions, test those predictions, receive immediate, informative feedback as a result of their testing, and analyze that information to refine their thinking around the content. This cycle of learning is known as the perception-action cycle. It is the foundation of our award-winning ST Math program. The perception-action cycle is this continuous flow of information and action between the brain and the world around it. On and on it goes: sense, predict, act, adjust. Sense, predict, act, adjust.
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Facilitation plays a pivotal role in creating a classroom rich with academic discourse. Effective facilitation promotes, deepens, and supports students' thinking as they grapple with concepts and build understanding. Students understand that they are accountable for their thinking, not just giving correct answers. As teachers continue to engage students in focusing on what and how they are thinking, students will build confidence, increase their communication skills, and deepen their understanding of concepts.
As facilitators, teachers help students understand that they are accountable for their thinking, not just giving correct answers. Teachers pose questions that encourage the students to think deeply and to ‘talk’ about their mathematical reasoning.
Facilitation engages students in focusing on what and how they are thinking. By asking questions to unpack student thinking, teachers are teaching students how to think and not what to think. This puts the onus on students to build confidence, increase their communication skills, and deepen their understanding of concepts.
ST Math Immersion uses a problem-solving process designed to support teachers as facilitators and students as authors of their own ideas and sense-makers of mathematics. It is aligned with the perception-action cycle (PAC) and helps students develop skills that can be used outside of ST Math.
This perception-action-cycle is embedded in our game-based learning design at MIND. ST Math games take students through this cycle over and over, giving them a safe place to fail, providing new information through immediate and informative feedback, and inviting them to try again until they find the solution.
As students engage in the perception part of the PAC, teachers can facilitate student thinking by inviting students to “Notice and Wonder.” They do this by asking questions aimed at helping students notice the information in the puzzle and discover or “wonder” what they may need to do to solve it.
Invite students to Notice and Wonder to focus students’ thinking about the problem.
What do you notice?
What do you wonder?
What question is the problem asking?
As students engage in the prediction part of the PAC teachers can facilitate student thinking by inviting students to “Predict and Justify.” They do this by asking questions focused on helping students identify a strategy and predict and justify what will happen when they try their strategy.
Invite students to Predict and Justify to uncover students’ thinking around how they plan to address the problem.
What is your hypothesis?
What strategy will you use to test it?
What do you think will happen when you test your hypothesis and why?
As students engage in the action part of the PAC teachers can facilitate student thinking by inviting students to “Test and Observe.” This important step gives students time to process what they observed when testing their hypothesis.
Invite students to Test and Observe by encouraging students to observe and process the results of testing their hypothesis.
Test your hypothesis.
Describe what happened.
Now that students have observed the outcome, teachers can facilitate student thinking by inviting them to “Analyze and Learn.” They do this by asking questions focused on students observing the outcome, analyzing what happened, and determining what they can learn from it. If they got it wrong students take what they learned, adjust their thinking, and start the cycle again. It is the process of building, revising, and extending schema.
Invite students to Analyze and Learn by facilitating students thinking in analyzing the feedback/results.
How does this compare to what you thought would happen?
What did you learn?
How will you use what you learned?
Once students ascertain the correct solution, it is important to help them connect it to their existing schemas to deepen their understanding of the concept. This can be done by asking questions that stretch students’ thinking; help them make connections, and apply their learning to novel situations.
This is the next step beyond the cycle that stretches students’ thinking
How does what you learned support your understanding of [the concept]?
What would happen if______?
How would you apply this concept to [this] situation?
By focusing on the problem-solving process, facilitation allows students and teachers to co-lead the learning. Students develop agency and accountability because they understand that their thinking is important. It is what will lead them to a deeper understanding. Facilitation is thinking-driven, not answer-driven. That distinction is what will ensure that the facilitation that occurs in the classroom is effective and impactful. Use the facilitation bookmark to support classroom facilitation.(MISSING LINK)
Facilitating in Action: Puzzle Talks
For facilitation to be effective we must purposefully plan the types of questions we are going to ask ahead of time. The types of questions you ask should make students accountable for their thinking. The questions should address thinking that results in correct solution strategies and thinking that results in incorrect thinking strategies. Students should be engaged in processing, communicating, evaluating, considering the thoughts of others, and refining their own thinking.
Problem Solving: Effective Strategy Discussions
The use of instructional stations in the classroom provides a great learning model to engage students in collaborative learning, build student agency and accountability, and provide opportunities for personalized instruction. ST Math Immersion engages students in 4 instructional stations as part of the curriculum.
To use instructional stations effectively, it is important for teachers to intentionally plan the purpose, goals/outcomes, and experiences they want students to have during the station time.
In ST Math Immersion, the content and experiences in the stations are designed to support students in practicing math concepts, engaging in critical and creative thinking, and communicating their learning.
Structuring the organization of stations and teaching students procedures are keys to implementing stations successfully. Here are some key things to pre-plan.
Click here for a planning sheet.
Organization
Establish procedures for movement, accessing materials, working together, getting started, cleaning up the station, completing the exit ticket, explaining the directions.
Set up opportunities for positive student interactions by creating norms.
Consider designating roles in the station (reader, materials manager, facilitator, recorder, encourager, time keeper, helper)
Content
Review the station content and anticipate student questions and areas of struggle during planning.
Provide clear directions. (e.g., table tent directions, visit each station to explain the task, make a recording of directions, etc.)
Identify strategies to support students during station time.
Communicate expected outcomes with students for each station time.
Monitor/Evaluate
Use the Exit Tickets and ST Math Journals to monitor student learning during station time.
Review the Exit Tickets and ST Math Journals daily to gauge student understanding and provide any needed support.
Remember to give students feedback and celebrate the work they do in the stations.
Best Practices
Communicate clear expectations to students.
Provide clear directions at the station and ensure that students have all the materials they need.
Make sure math tools are accessible to students.
Provide strategies for students who need help during station time.
Set a timer to help students pace themselves.
Give a 5 minute warning prior to the end of the station time to give students a chance to wrap up their work.
Problem Solving: Effective Strategy Discussions
Engaging students in discussions around the problem solving activities is a great way to explore connections, expand perspectives, and have students check and challenge each other. Have students share strategies, compare, and share their work. Click through the slide deck then watch the video to learn more about sharing student work.
The problem-solving activities provide a great opportunity to facilitate classroom discussions around student work.
Before students begin working on the problem solving activity think about what your students might do to solve the problem, and what mathematics you would want them to be able to see and describe.
What strategies would you want shared?
How would you order the work that is being shared to maximize the learning?
What questions would you ask?
Order the work you selected to share from least sophisticated to most sophisticated. Include some work that has misconceptions and/or errors in reasoning. Discuss both correct and incorrect reasoning.
How would you order these pieces of students' work?
A, B, C, D
C, D, A, B
B, C, A, D
Sharing the pictorial strategies is the easiest way to give all students access to the discussion. As you move from the more visual responses to the more abstract responses ask students to compare the strategies.
How are the strategies similar and how are they different?
Do you agree or disagree with your classmate’s strategy? Why or why not?
When students share work, ask questions like:
What is your level of understanding of your classmate’s strategy?
What questions can you ask to clarify your own understanding of the strategy?
How does your classmate’s strategy compare to your strategy?
Would you change any of your work? Why?
When students share work, ask questions like:
What is your level of understanding of your classmate’s strategy?
What questions can you ask to clarify your own understanding of the strategy?
How does your classmate’s strategy compare to your strategy?
Would you change any of your work? Why?
Math Notebooks: Thinking and Reflecting Time
My Thinking path, problem of the day, math journal and exit tickets are some of the resources that help students reflect, connect, and build understanding. They will keep all of these documents in a math notebook. At the end of every week provide time to engage students in gathering their thoughts from the week, reflecting on how they have grown in their thinking and confidence around math and discuss their learning as a class.
As you engage them in thinking about their thinking, some questions you might ask include:
Unpacking their thinking - how are they thinking about the math they learned this week?
Examining their mistakes - what were their aha’s this week? How did they learn from their mistakes?
Applying their hypothesis - how might they apply this learning to similar problems? What can they connect it to?
Click the image to see the Math Notebook
Final Week of ST Math Immersion
The ST Math Immersion program ends with a Learning Showcase and Celebration. The Learning Showcase and Celebration is a time for students to share what they have learned during the ST Math Immersion program. During the program, students will create two projects: a reflection poster and a math game. Students will present these projects at the Learning Showcase and Celebration. Click the image for the planning guide.
Reflection Poster
During the ST Math Immersion program, students keep a math notebook. During the final week of the program, they work with a group to create a reflection poster to show what they have learned. This poster is a culmination of the thinking, learning, and growth that students experienced during the program.
Math Game
Students will complete a design challenge in the ST Math Immersion program. They engage in the Engineering Design Process which is adapted from Engineering is Elementary ® from the Museum of Science in Boston.
Students work in a small group and complete tasks outlined in a design booklet. Teachers have content they teach once a week to support students in doing the work of designing a math game.
By the end of the ST Math Immersion program, each group will have a math game that they created that either teachers or engages students in practicing a concept.
*Parents, community members, school administrators, teachers, and board members should be invited to this showcase and celebration.
ST MATH IMMERSION: EMBEDDED PL 
