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There are so many ways to create an environment that encourages intuitive thinking in math. Some examples include taking your students outside and creating a challenge for them to find objects that match the measurements you give them. Another fun activity is calculating the amount of pencils and crayons the class has a whole and how many they need if running low on supplies. Using real-world problems with your students also connects ways for them to use math in day to day life outside of the classroom. This helps them to develop a deeper understanding of math and encourages intuitive thinking in math.

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Teachers can create an environment that encourages intuitive thinking in math by allowing students to work with the math problems regularly and freely. Students can learn a lot by testing and simply playing with the math they are given. I feel like always having the teacher watching you or expecting something specific from you can make students feel overwhelmed. If students feel overwhelmed then they will not really learn the material, but only memorize it to simply pass. Another way could be to practice regularly with students using different methods for the topic. Worksheets can be helpful in the classroom, but it is also important to allow students to get up and explore the real world so they can make connections to the math they are working with. If students make connections to their lives they will easily remember the concept and learn more about it because they value it.

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One of the most important parts of being a teacher is creating an environment that encourages intuitive thinking. A few ways to do this are by holding off on worksheets, rather letting students do hands-on, individual play with something such as magnetic tiles. This increases their creativity. interest, and allows them to develop a few ideas about the topic before discussing it as a class. Having student led discussions can also help encourage intuitive thinking. Having students use tools in the classroom and prompting them to do deeper thinking is also very helpful. One way this is mentioned is to have students see if they can find the angles of shapes based on information they were given. This encourages forward thinking and creativity in the classroom. Another great way to encourage intuitive thinking is to use real world problems that students may come across in their daily lives. By doing this, students become more likely to notice math in everyday life and begin to think about it unprompted. They will then transfer this knowledge to other areas of their lives, inside and outside of the classroom. Teachers can also use activities such as would you rather questions (i.e. would you rather have a stack of quarters as tall as you or $225). This makes lessons fun and engaging for students, and can lead to great discussions between students. The teacher can then take these discussions further, asking students how they may figure out how much money they would have if they had a pile of quarters as tall as them. This creates room for intuitive thinking and creativity!

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The chapter includes a quote that I see as a great segway to this topic, "Intuitive experiences must be acquired by the student through his/her own activities—
they cannot be learned through verbal instruction."
—Erich Wittmann, “The Complementary Roles of Intuitive and
Reflective Thinking in Mathematics Teaching” This quote highlights that intuition is not taught but acquired through students doing their own thinking and reasoning. Teachers can help activate intuitive thinking by giving meaningful activities and worksheets. For example, based on the topic being learned teachers could involve a challenge or incorporate manipulatives into an assignment that will make students build mathematical intuition. Teachers can also encourage students to really be thinking and building rapport when problem solving as opposed to only trying to get the answer. Ask questions like "what's going on here? what do you notice? what do you wonder?" etc. I think it's very helpful for teachers to know that for students to build intuition, they must be able to prove something themself, not just be shown how.

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I think one way teachers can encourage intuitive math thinking is by letting students be able to do math freely. I think something teachers can do is give students math activities were they can chose what they would like to do, and let them do math problems freely and let them experience different math problems. I think that it lets them problem solve on there own and build that math intuitive thinking.

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I have always loved the idea of taking real world problems and applying them to mathematics. I think it is one of the most natural ways to encourage intuitive thinking in math in a way that is fun and interactive. I love the idea of using objects like we read about in the Pencils and Crayons article. When students are given a problem at hand that has an actual impact on their life, it allows them to work with intention and a goal in mind. When students are excited about a goal, I believe their work would yield better results. I even love the idea of having students incorporate objects from home, like counting collections so that they feel apart of something that is directly connected to their personal life.

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A strategy that I would like to use in my future classroom to help build students mathematical intuition is the example given by Jennifer Clerkin Muhammad from chapter 9. Jennifer made a point that once she brings a worksheet out students are only focused on the worksheet, even when they might have really good questions that would help build their intuition. In her classroom, she gives her students time to explore new materials and math topics on their own. The students were able to share their discoveries and connect what they were noticing to concepts that were previously learned. I think that I would like to include this in my classroom because students are able to build mathematical intuition on their own and then share with their classmates/teacher.

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I believe hands on experiences are so important when it comes to intuitive thinking, this seems to be a lot more long lasting rather than trying to learn from memorization and procedural learning. Manipulating objects, creating patterns, and engaging with visual and physical representations helps students form mental models. Estimation exercises is a good way to encourage students to make predictions and then verify, this strengthens their intuition and develop reasonableness in their answers. When the teacher asks the students questions like "what do you notice" it makes them think step by step how they got their answer and understand what they really did.

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It is so important for educators to create a classroom environment to build and encourage intuitive thinking in math. As stated in this chapter, intuitive thinking is not taught but it is acquired by the students. Some ways as teachers that we can help aid this is by giving students hands-on activities and staying away from giving your class worksheet after worksheet. Although worksheets are not inherently bad, they do not give students a way to be creative with their solutions. Having math lessons that students can relate to and touch can inspire intuitive thinking in the classroom.

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In this chapter of the book it talks about how students cannot learn intuition through verbal instruction, they have to learn it through their own act of doing things. I really like the idea of Intuitive thinking in the classroom because I feel like all of the lessons and activities that I remember the most from elementary school were the ones where I got to be creative and hands on. It allows students to be able to complete things in the way they are thinking about them in their own heads. Teachers can create an environment for this by ensuring that their students have room to solve problems freely and in the way that works best for them rather than forcing the traditional way of solving things. Teachers can also create this environment by making sure that their students are being able to do hands on activities that allow them to be creative and keeps them more engaged on what they are learning. They could do activities like the design challenges we are learning about in our STEM class. It allows for the students to use their problem solving skills and creativity.

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Personally, I think the best way for teachers to create an encouraging environment is to make the classroom a safe space. It's important as teachers to embrace student ideas regardless of if they are right or wrong/good or bad. Every student idea should be listened to and respected, because even if it is a good idea, if the student can't embrace it, it does not actually matter to them. A great way to promote mathematical intuition is to let students see teacher's personal challenges when thinking. Having students see that their own teacher has struggles lets them be okay with having difficulty in class. If anything, welcome difficulty since without it, students aren't necessarily learning.

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I think that math can sometimes be a very scary subject growing up. I think the way teachers can create an environment that encourages intuitive thinking in math can be by allowing students to solve math problems by using the materials that better help them understand or visualize problems. There may be some students that are great with numbers but there may be others that need more hands on materials or even hand on activities. Students have a lot of energy and having them sit in a chair for long periods of time usually results to them not fully paying attention to what they are supposed to be doing. Sometimes teachers have to get out of their comfort zone to try new things to see if that will help their students be successful.

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To create an environment that encourages intuitive thinking in math, teachers should focus on promoting curiosity, collaboration, and a growth mindset. This involves helping students see challenges as opportunities to learn and understand that mistakes are part of the process. By using open-ended questions, teachers can encourage multiple solutions, and collaborative discussions can encourage peer learning. Connecting math concepts to real-life situations makes them more relatable, while hands-on activities help students actively engage with the material.

Incorporating various hands-on activities can enhance students' mathematical intuition. Keeping math journals can also encourage students to reflect on their learning and thought processes. By using some of these strategies, teachers can effectively support intuitive thinking in math.

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Teachers are able to promote intuitive thinking in math by allowing the students to work on their problems using their own strategies. By allowing them to implement their own ways into their problem-solving, they are taking what they already know and making it their own. Teachers can have different ways for the students to learn that best supports their learning. For example, they can use different tools for hands-on learning, videos explaining what they are learning about, and written instructions with tips and tricks to help. I would implement worksheets into the classroom to use a resource to see where the students are at and to help show them the different types of problems they may see in the real world. In my classroom, I would like to incorporate lots of activities that keep the students interested and entertained. From personal experience, when I was interested in the activity, I was more focused on what we were learning.

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Teachers can make math fun by letting students try ideas without worry about making mistakes. Using real-life things, like snacks or toys, shows how math fits in everyday life. Asking questions like “What do you notice?" or "What do you wonder?” helps students think for themselves. Simple activities like guessing, finding patterns, or making predictions help students feel good about their ideas. Working in pairs lets students talk and learn from each other. Focusing on exploring, not just getting the right answer, helps students build their own math thinking.

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I think teachers can provide hands on and open-ended activities that can keep students engaged and interested. Teachers can create activties lead by students or activities that could be worked at in small groups. I think not only is it important to have engaging activites I also think it is important for the teacher to have a positive attitude towards to class and lesson.

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Interactive activites are great ways to get students more engage students in a math lesson. Something I noticed during my observation is when students are doing something hands on or interactive, they are a lot more engaged versus when they are sitting on the carpet. I also noticed students get the solution correct more often in interactive activities. Another great way to get students more engaged is real world problems. If students can relate to the problem being asked, they are more likely to understand the problem. It is so easy to make a simple math problem like 5+4 more exciting for the student.

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I believe that that teachers can create an environment that encourages intuitive thinking in math by incorporating real world problems that relate to the students lives. I believe that this encourages an inclusive environment and engages the students. Also creating problems around things they enjoy. This can be creative to the students and allows for more fun in the classroom.