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A strategy that I can use to build a student’s mathematical intuition is the context clues/prior experiences approach. The context clues/prior experiences approach uses a student’s previous math knowledge. That ties into building a student’s mathematical intuition because intuition is a trace left by our experiences. All students have had past experiences with things that contain quantities. Students could know that a can of soda has 12 ounces, the average doorway is 6 foot 8, a large pizza typically has eight slices, and that there are twelve eggs in a carton. That would allow a student to know that 5 soda cans are 60 ounces, estimate that 4.5 large pizzas are 54 slices so you could give 27 friends two slices, guess that the teacher is about 5 foot tall if they fill up three-quarters of a doorway, and know that 2 egg cartons contain up 24 eggs. All of that entails using a student's mathematical intuition! By using context clues/ prior experiences, I can help a student learn because it brings in past experiences personal to them and it uses real-world aspects of their lives. Tying into a student’s experiences can help them guess, estimate, and even know a correct answer through an informal method.

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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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In chapter 9 the author went over some strategies that we as future educators can use when teaching math. The one that resonated with me the most was probably listening to your intuition when solving a problem. Sometimes when I am solving a math problem I initially think the answer is one thing but then see another answer that could be right. Usually I end up changing my answer to the second choice, and its almost always wrong. With the correct answer being the original answer I put. I need to learn to trust my intuition, because usually I am right. I know that I am not the only student who struggles with this. So teaching this as a strategy in my classroom I think could only benefit the students who struggle with this. I think this would also help in the long run boost confidence when it comes to problem solving, collaborating with peers, and confidently defending their work. As a future teacher, understanding your students' intuitions and tendencies in math class, will help us use this strategy to the best of our abilities.

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A strategy that stood out to me from this chapter was "Mathematician Reuben Hersh argued that our experience manipulating objects, symbols, representations, and mental images is how we develop mathematical intuition". I see this a lot in my math observations, teacher use a lot of symbols in there teaching. For example having a fun song for students to learn the different shapes and different patterns. Also another example is a song to learn the value of coins.

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A strategy I can use to activate a student's mathematical intuition is to help students listen to intuition during problem solving and to ask live rapport questions. During problem solving, students get caught up in the problem-solving process to find the correct answer that they end up losing their understanding. In the Chapter, the author emphasizes the concept of helping students make sense when it comes to estimation and problem solving. A way to do this is through asking live rapport questions. Live rapport questions are meant to help students internalize the dialogue when solving problems. Such questions can look like, "What's going on here?" when initially solving a problem or asking questions to help students redirect their thinking back to the problem like "Does that seem possible?" With practice and modeling students can begin to make sense of estimation and ask themselves questions throughout the problem to follow their intuition.

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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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The quote, "What’s extra lovely about this
relationship is that each side strengthens and supports the other." It really stood out to me. I have never thought about how these two things could support one another. A strategy that I can use to build a student's math intuition is manipulating objects, symbols, representations, and mental images. The more students get the hands-on activities the more their intuition will build. There are so many ways to apply this to your classroom. You can do it by bringing objects for students to manipulate or by creating a catchy song that will help students remember the content. All of these are quick and easy ways to help build students' intuition.

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A strategy from Chapter ( that I can use to activate a student's mathematical intuition is the context clues and prior experiences approach. I also liked the idea of building personal referents for measurement benchmarks. Encouraging students to connect to familiar objects or experiences is great. It is also beneficial to relate math to real-world situations so students can see that math is all around them in their everyday world. relating new concepts to students prior knowledge can help build understanding, estimations, and ideas.

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The author gave many wonderful examples of how to instill this math intuition within your students. I think a way I plan on implementing this is pushing students to feel valid through their thinking and processes. Instead of just saying the way they did a problem is incorrect, commend them of their process or let them know that not knowing how to solve is completely okay. I believe this allows students to not only trust themselves but also you as their teacher. I also struggle with my intuition and find myself second guessing my answers on assignments and tests. I wish I would have stopped this happen long ago so it did not carry with me to college. I think it definitely hurts my math confidence as well as other subjects and I would hate for that to happen to my students as well.

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I think I would use the context clues/prior experience to activate my students' mathematical intuition. I think they are more likely to make connections to things they have experiences with. I also think this would keep them more engaged overall. By working with what they know, I think they would feel more comfortable using their intuition as they can use their common sense.

I really liked this quote, too.

"The point of rigor is not to destroy all intuition; instead, it should be used to destroy
bad intuition while clarifying and elevating good intuition."

This made me reflect a lot on how I understand intuition in mathematics. I think it's easy to say "go with your gut" when problems are easy/comfortable, but that feeling likely goes away when problems become more difficult. Allowing students to connect things to their own experiences might help them feel more confident in using intuition while also strengthening their good intuition at the same time.

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The author gave many ideas on how to activate students' thinking. The main idea he followed along with is by teaching the students to follow their intuition or, as some would say, "follow your gut." I would follow this strategy all the time. It helps students realize that they typically know the answer and are sometimes afraid to say it because they don't want to be wrong or are just simply thinking the answer is too simple. A way to allow this in the classroom is by providing group work, discussions, and even casual one-on-one discussions with the teacher or even a peer in the classroom.

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The author had a lot of ideas and stratgeys that allowed students to activate thier learning. All of the strageties that the author had given in the chapter will help us as future educators, my favorite was trusting your first answer and not changing it when you see something differnt or second guessing yourself. I think this is very important to teach, becuase i had a teacher in highschool who taught me to never chnage my answers or you will most likely second guess yourself and get it wrong. normally, after we were done with tests my teacher would call us to her desk and grade our tests with us. almost every answer i second guessed i got wrong. this chapter was important in that every student has sa differnt way of thinking, most of the time we second guess not because of the answer but the way that we got the answer. Teaching this to students at a young age will make them more confident in thier work, but allow them to do work in thier own way and not second guess the way they do it. It will also allow educators to see how exactly they got it wrong instead of them not knowing how to do it and just copying others work.

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I would like to use Andrew Stadel's Estimation 180 website. He has visual estimation problems to last all year. The problems include estimation, capacity, distance, time, weight, height and more. He also incorporates different standards into his problems. One of the problems he has on the website right now is "how many pieces of candy corn are in the cup?" I think this would be a fun activity for students as Halloween is this Thursday and they will also be practicing estimation. You could show students the video Andrew made or you could do it in person and the students could eat the candy corn after!

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a strategy that I would use to activate my students mathematical intuition would be the context clues/prior experience. I feel like students are more interested and inclined to learn and participate when they are able to make personal connections to things that they have experienced in their lives. When students are able to feel connected to a subject in school they are more able to make problem solving connections in the real world. I also want my students to be able to explain their own thinking process while they are solving math problems. I think that it's important for students to know how they are actually solving the problems instead of just memorizing how to solve problems.

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I would use Andrew Stadel's Estimation 180 strategy in my classroom. Estimation 180 includes fun and engaging ways for students to practice estimating. Estimation 180 provides estimation for height, distance, and length. For each challenge, students are instructed to give two estimations, one that they know will be too low and another that they know will be too high. These seemingly silly estimations actually introduce them to and make them comfortable with creating estimation ranges, which is part of our everyday lives!

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The strategy that stood out to me was "Sampling/Disaggregating/Layering/Chunking/Grouping/Scaling/ Decomposition-Recomposition". In this strategy the author talks about equipping your students with practical information that works beyond the classroom. This strategy also helps students think about groups as part of a whole and getting comfortable with thinking conceptually. I really appreciate how all we are learning draws back to using real world experiences. This makes students so much more engaged and gives them motivation that their learning matters outside of the classroom.

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A strategy mentioned in this chapter that I would use in my classroom is Sampling/Disaggregating/Layering/ChunkingGrouping/Scaling/Decomposition-Recomposition. This strategy stood out because it is what most adults use when estimating, which means its practical for students to learn and use in the real world. It allows students to think creatively by taking small groups of an object and multiplying it to get their estimate.

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A strategy I thought was important that I plan on using in my classroom is the idea of angle estimation. Angles are hard to understand as an adult, let alone as a child. Using elbows and knees to represent angles and explaining them to children in an easy to understand way is so much more beneficial than just using numbers and terms they don't know. Once they learn the concept of angles, it is so much easier to estimate based on the look of the angle. If they can look at an angle that's a little less than a right angle and understand that it's close to 90 degrees, it makes it so much easier to check their work later and grasp larger concepts that are related. Learning how to estimate is effective in general because it allows you to get a general idea of where the answer should be, which helps you to make sure you're on the right track and check your answer.

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A strategy that I liked is Estimation 180. A quote about it that I really liked says "These students are using what they know to estimate something they don’t know. That’s a powerful problem-solving approach, across the board." It is important that they understand that estimating is not just guessing, but reasoning. It can also be a great way to help students develop number sense. Estimation is a way for students to make math a part of their lives. It gives them an opportunity to use their lived experiences and common sense in the math classroom.

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In chapter 9, one strategy that stood out to me to help students build their mathematical intuition is letting students do more discovery based activities before the worksheet. One quote that stood out to me was, "when we build true, deep understanding of mathematics and why it works, our intuition gains wisdom." This quote stood out to me in the beginning of the chapter and I feel like it really connects with this strategy. Allowing students to discover and explore a new concept allows them to think outside the box about a new concept. I always felt like worksheets restricted students being able to explore and try to figure out a new concept. When I imagine a math work sheet I think of a paper where each question has a right answer and that is it. I want students in my classroom to feel like they understand the concept before being given a worksheet and allowing them to do discovery based activities first helps them do so.

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After reading chapter 9 and being able to see all the strategies, the one that stuck to me the most and the one that I would like to use in my classroom would be Andrew Stadel's Estimation 180. I think this is a great strategy to get students really seeing where we can get different measurements while also making a variety of estimations. I have always believed that it is important to incorporate numbers into math problems were the students can really relate to. Andrew Stadel seems to focus of estimating height, capacity, time, distance amount, weight, proportions, values, and area. All of these can be numbers that are customized by that student individually or as a class because they are all estimates. Students tend to be motivated and eager to learn. I would love to be able to see what different types of problems and or estimations my students would have while also watching them stay engaged in the activity we are doing.

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In this chapter the author emphasizes the importance if mathematical intuition for students in the classroom. a quote that stood out to me a lot was "Intuitive experiences must be acquired by the student through his/her own activities—
they cannot be learned through verbal instruction." I noticed this because of the straightforwardness of the fact that intuition cannot be taught. I think it is important to teach students to follow this because it teaches the students to trust their own minds instead of constantly second-guessing themselves. One thing I often find myself doing on exam is using my intuition, then quickly moving on so I can't change the answer. When I was younger I always saw when I second-guessed my work, it would greatly impact my grade. In conclusion, I love teaching students to trust their brains instead of constantly second-guessing.

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A way that we could increase mathematical intuition is by encouraging students to follow their own intuition. Although many people tend to think that they are not a math person, with a natural intuition for math, this oftentimes comes from experiences and practice. Creating a classroom culture that encourages students to always persevere and accept failure as a natural part of the learning process is essential. We need to allow students to do the thinking and reasoning themselves, rather than just tell them what to do. That way, it becomes a part of them, a thinking process and connections unique to them, that later becomes a part of their 'intuiton'.

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A strategy mentioned in this chapter that we can use as educators to build students' mathematical intuition is context clues/prior experience. Students are more likely to make connections based on their own personal experiences. This will make them more interested in learning the subject when they can make a personal connection to it. This will also help students see math in their everyday lives and practice doing math outside of the classroom.

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While reading the chapter, I remember reading about the different questions that are asked in the classroom. I believe that through asking these questions is vital to a child's learning because it causes them to go beyond the original question. They are able to understand further what they did and allow other students to understand what they did and didn't do on their own.