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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Best Practices: Building Community in a Virtual Classroom
Building community is an important aspect of teaching in a virtual environment! While things might feel a little different, many routines and tools are effective regardless of the environment. ST Math Immersion’s virtual program begins each class with an opening meeting. Use this time to build community and set the tone for the day’s lesson.
Welcome your students to your virtual class each day with an opening question or activity. There are several strategies that can be used for welcoming students.
ST Math Immersion Virtual curriculum provides a slide deck for teacher use during the opening meeting.
Pose an opening question that everyone can answer and have them type their answers into the chat or let them unmute and share. Focus on learning one another’s names along with interesting information about one another!
Collect the data. You might collect data through a Google Form, Google Slide, Survey Monkey, Jamboard, Virtual Whiteboard. Display the data and create different ways to represent the data (tallys, graphs, tables, etc.) Use the data to tell the “math story” of the class.
Explore numbers and their relationships to support students in developing a strong sense of numbers and operations. Show students 1-2 numbers and have them tell you what they know about the numbers. You can do each number separate or you could have students do both numbers and make comparisons.
One way to engage students in exploring numbers is to use the same set of numbers for the week. Connect them to number lines, ten frames and see how many equations students can come up with to equal one of the numbers.
A great way to generate conversation is Christopher Danielson’s Which One Doesn’t Belong? In this routine, students choose one of the images that they believe does not fit with the others in the group. Any answer can be correct - as long as it is justified with mathematical reasoning. Iit is a great way to generate critical and creative thinking.
Doing activities like these helps build community. It is important that in addition to these activities norms and routines are established in the virtual classroom. This sets clear expectations for participation and interactions among students.
Goal setting should also be part of a class meeting. Students develop confidence and recognize their progress and growth by setting goals.
The data trackers are good resources to use to support goal setting.
Goal setting:
Increases opportunities for student agency.
Strengthens students’ ability to communicate what they have accomplished.
Helps students identify areas for improvement.
Teaches time management.
Encourages focus on strategies to overcome challenges.
One way to empower students and promote a positive mathematical identity is to celebrate their brilliance. Take time to call out and celebrate the brilliance of students from the week. Invite other students to share brilliance that they have felt themselves or seen in others. This is a great opportunity to call out those “soft skills” that students are developing in this program.
Facilitation plays a pivotal role in creating a classroom rich with academic discourse. It can be tricky to do in a virtual classroom, but it is important to engage students. 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. See strategies for engaging students in discourse in the virtual setting.
Throughout ST Math Immersion teachers will facilitate student thinking as they engage in Puzzle Talks, complete My Thinking Path, engage in pre-work, 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 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.
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?
he 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 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 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 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.
Facilitation in Action: Puzzle Talks
Puzzle Talks are a way to get students thinking about and solving ST Math puzzles as a group. They're a wonderful way to bring in the math practice standards as students discuss, problem solve, justify their thinking, connect the math to what they already know, and provide you with formative data. Let's see what they look like with Meagan, a first grade teacher from Ohio.
In this video, Meagan discusses using ST Math remotely with her students and how it's a great way to focus their attention and start talking about their thinking.
Having something on the screen that students can focus on is a great engagement tool. Using it with small groups allows everyone to talk and engage in conversation. Having some tools for students to share their answers (other than shouting them out) is a great idea. Here, Meagan asks students to use white boards and thumbs up.
In this video, Meagan explains and then shows how she uses ST Math puzzles as pre-work. This technique allows her to build on their individual experiences playing the game for a discussion about the relationship between addition and subtraction.
Puzzle Talks don't stop with the puzzle. Here, Meagan explains how she ends the Puzzle Talk with a question that uses what they've learned to solve a problem that they might encounter on a written test.
Teaching is hard enough without being remote. In this video, Meagan reflects on her students not "getting" the point of the lesson, using go-to in person strategies remotely, and the importance of giving yourself some grace.
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?
ST MATH IMMERSION: VIRTUAL EMBEDDED PL 
