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 Summer 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.
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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.
Throughout ST Math Summer Immersion, teachers will facilitate student thinking as they engage in Puzzle Talks, complete My Thinking Path, solve and discuss word problems, and work through ST Math puzzles.
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 concept.
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.
ST Math Puzzles take students through the perception-action cycle over and over, giving them a safe place to fail, providing new information through immediate and formative 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.
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.
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.
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 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.
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 invite them to "Connect and Extend" 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.
Stretch students’ thinking by asking questions that help students connect and extend schemas to deepen learning.
How does what you learned support your understanding of [the concept]?
What would happen if______?
How would you apply this concept to [this] situation?
The problem-solving process allows students and teachers to co-lead the learning.
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.
Students develop agency and accountability because they understand that their thinking is important.
Asset-Based Approach: Making Math Accessible
ST Math Summer Immersion uses an asset-based approach to instruction through diversity, equity, and inclusion. An asset-based approach focuses on the student’s strengths and talents instead of their deficits which is crucial to bringing equity in education. It supports students in seeing how they think about and engage in math. It is important that every student, teacher, administrator, family, and community see themselves in math. Math is from everywhere, in everything, and for everyone.
ST Math Summer Immersion uses an asset-based approach to instruction by focusing on the students' strengths and talents instead of their deficits.
One of the goals of ST Math Summer Immersion is to help students build a positive math identity and instill confidence, joy, and wonder in their mathematics ability. It was designed to equip students with the agency to be a knower, doer, and sense maker of math.
The lessons provide opportunities for student voices to be heard. Through the sharing of strategies, thoughts, and perspectives, students are positioned to see each other as mathematical resources and build on each other’s ideas.
The activities and resources build from students' potential and use what they know to help them work toward what they don't know yet.
The teacher’s role is to facilitate student thinking:
teachers center student thinking and instead of teaching the puzzle by asking facilitation questions
use the Problem Solving Process to promote student thinking
Facilitation should:
promote, deepen, and support students' thinking as they grapple with concepts and build understanding.
make students thinking visible.
hold students accountable for their thinking, not just giving correct answers.
The teacher’s role is to position students as authors of their own learning.
Students learn that they can construct their own knowledge.
As students engage in discourse they share ideas, ask each other questions and see each other as mathematical resources.
Discourse should engage students in
checking and challenging each other.
asking clarifying questions and building off each other's ideas.
center students as owners of the learning.
support students in seeing each other as mathematical resources.
The teacher’s role is to: support students in developing the “soft” skills.
Students learn that mistakes are not only okay but they are expected and are part of the learning process.
Perseverance and stuckness are seen as a challenge to conquer instead of a judgment on ability.
Focus on fostering assets like:
Perseverance
Resilience
Growth mindset
Agency
Self-motivation
Goal setting
These are strengths that students can carry into the school year.
Puzzle Talks: Facilitation in Action
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.
Instructional Stations: Collaborative Learning
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 Summer Immersion engages students in four 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 Summer Immersion, the content and experiences in the instructional stations are designed to support students in practicing math concepts, engaging in critical and creative thinking, and communicating their learning.
Structuring the organization of instructional stations and teaching students procedures are keys to implementing instructional 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, and explaining the directions.
Set up opportunities for positive student interactions by creating station norms.
Make materials and directions easily accessible to students.
Consider designating roles in the station (reader, materials manager, facilitator, recorder, encourager, time keeper, helper).
Content
Review the station content and create a plan by anticipating student questions and identifying ways to address areas with which students may struggle.
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 My Thinking Path, Exit Tickets, and ST Math Puzzle Reflections to monitor student learning during station time.
Review the Exit Tickets and ST Math Puzzle Reflections daily to gauge student understanding and provide any needed support.
Provide students with feedback and celebrate the work they do in the instructional stations.
Best Practices
Communicate clear expectations to students.
Provide clear directions at the station and ensure that students have all the materials they need.
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?
Assessments: Monitoring Learning Growth
ST Math Summer Immersion offers different types of assessments that are optional. During the first week and last week of the program, an optional pre/post assessment is available. There are also optional weekly pre/post quizzes that could be administered on Day 1 and the last day of each week. The purpose of these assessments is to monitor the progress of your students and make instructional decisions based on their results. Your team should decide whether or not you will administer these assessments and how you will use the results.
Student Portfolio: Building a Mathematical Schema
The Problem Solving Journal serves as a student portfolio. It is a compilation of the academic work students complete during the program which includes: My Thinking Path, Problem of the Day, Exit Ticket, and ST Math Puzzle Reflection. It serves students as an artifact to track, evaluate, synthesize, and communicate their thinking around the mathematics they are learning. ST Math Immersion helps students build strong schema around mathematics, develop perseverance, confidence, and agency as they engage in the lessons and activities.
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?
Final Week of ST Math Immersion: Presenting and Sharing Learning
ST Math Immersion concludes its program with the Learning Showcase and Celebration. The purpose is to provide students an opportunity to show what they know. It will be a great way to take all the information students have learned during ST Math Immersion. We recommend teachers extend an invitation to families, community members, district leaders, teachers, peers and school board members to celebrate and hear from students about what they learned. The Immersion Debriefing time allows for students to communicate their understanding to an authentic audience. We have designed a Learning Showcase and Celebration Editable Invitation you can use.
Reflection Poster
The reflection poster is a culmination of the thinking, learning, and growth that students experienced during the program. Throughout the ST Math Immersion, students write their reflections and responses in the Problem Solving Journal. During the final week of the program, they work with a group to create a reflection poster to show what they have learned.
Math Game
In 4-Day ST Math Immersion, there is an optional mini-game design that teachers can do with students. This activity involves having students re-design a game they know (e.g., Twister, Concentration, etc.) and add mathematical components to the game. Note: This is optional and can also be used in 5-Day Immersion in lieu of the design challenge for time purposes as suggested.
In the 5-Day ST Math Immersion, students will be designing a more elaborate math game using the design process. They will engage in the Engineering Design Process which is adapted from Engineering is Elementary® from the Museum of Science in Boston. Students will work in a small group and complete tasks outlined in a Design Challenge Station Booklet at the design challenge station. 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 teaches or engages students in practicing a concept.
ST MATH SUMMER IMMERSION:EMBEDDED PL 
