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As a teacher, one of our primary jobs is to support our students. This includes building in success by starting with simpler reasoning strategies that students can quickly grasp an understanding of how to use them. When a student finds themself succeeding, they are more likely to tackle a more challenging problem/approach. Discussing a student's strengths is another great way to boost their confidence. Furthermore, the idea of "Drill", which was discussed in a previous chapter, is not effective for students. Students need strategies that are explicitly taught to them using multiple means of representation. These supports are great for developing fact fluency among students, and lead to the development of automaticity.

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Assessment plays an important role in teaching, and observation is one form of informal assessment, even in mathematics. When observing basic fact fluency, teachers should be making sure that students are not only learning the facts they are focusing on during that time, but also that they are using strategies that are being taught. There are so many ways this can be done! For example, using equation observation tools such as tables that contain different fact sets. Additionally, having students write in a math journal about their thinking, or having them discuss with peers, or even one-on-one with the teacher! Keeping an eye on each and every student, and understanding their thinking, can go a long way.

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Growing up in school, I was given multiplication facts to memorize and we used to have "Fast Fact Fridays" where we would have to see how many multiplication facts we could answer in one minute. This could sometimes be stressful for students, and like the book mentions, memorizing facts does not allow students the opportunity to make sense of what they are learning and gain confidence in mathematics. Students can learn foundational fact sets, and then build off of that using a strategy-based approach where they can engage in discourse about how to solve certain facts. Multiplication and division facts are super important for students to learn, just as addition and subtraction are. Similarly, shying away from traditional approaches and moving toward learning strategies that promote procedural fluency is crucial to student development.

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Something that stood out to me in this chapter was that subtraction facts have been proven to be more difficult than addition facts when using reasoning strategies. This is most likely because in order to do subtraction successfully, students must first be able to do addition. For example, when a student is trying to solve the problem 12-7, they then have to think "What plus 7 equals 12?". Teachers should be moving away from the phrase "take away" when doing subtraction problems with their students, and instead saying "think addition!". This connects back to the idea of parts and whole, where students can use their comparing skills to see how far apart two numbers really are from each other.

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When working to develop fluency and automaticity in students, it is important to remember that there is a major difference between memorization and explicit strategy instruction. Research shows that having students simply just memorize facts is far less effective than the strategy based approach. There is a vast number of facts for children to memorize, which could lead to stress and a sense of feeling overwhelmed. With the idea of memorization, there is less room for creativity for students to use multiple strategies, create visuals, and use critical thinking skills. However, with explicit strategy instruction, students can explore and practice different strategies first hand. This supports student thinking and gives them the freedom to choose which strategy best works for them, and students love having choices!

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An important part of the planning process in math is understanding common misconceptions and challenges that come up for students when striving to solve problems. Taking the time to acknowledge the challenge, look at it from a student's perspective, and find ways to help students overcome the challenge, will lead to much more success in the classroom. For example, one of the common challenges mentioned in the textbook is that adding 0 is understood as making a number bigger and subtracting by 0 is understood as making a number smaller. Ex: 7+0=8 and 7-0=6. Teachers can help students work through this struggle by incorporating story problems that involve adding or subtracting 0 in meaningful context.

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Solving a math problem has many steps, one being actually thinking about the problem before you go in and try to solve it. This is something that teachers should be actively modeling for their students. Students should be encouraged to communicate with their peers and their teachers about the problem regarding ways they may want to solve it, the context of the problem, etc. A strategy from the book that I really liked was actually covering up the numbers in the problem so that students have to make since of it before they start punching numbers into a calculator to get to the answer.

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When thinking about teaching students multiplication, there are 4 different problem types that are each beneficial to students' learning in their own unique ways. The first of these types is equal-group problems, also known as repeated-addition problems. These types of problems focus on the number of groups, the size of each group, and the total of all the groups. Depending on which of those three things are known in the problem, that will determine whether it is multiplication or division. We discussed this idea of parts and whole in Dr. King's class. The second type is comparison problems, which involves two sets of groups. These types of problems have a focus on the product, the group size, and the number of groups. The third type is array and area problems, which involve using a representation of rows and columns, and making groups out of them to find the area. Finally, the fourth type is combination problems, which is about counting how many pairings can be created between two or more sets.

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Contextual problems are a key component in teaching addition and subtraction problems. For one, students are often focused on getting the correct answer as fast as possible in order to get the assignment over with. With contextual problems, these connect directly to recent experiences that students relate to. For example, these could be about a field trip the class recently went on, or related to a story they just heard during read aloud. Another great thing about contextual problems is that they support multi-language-learners because they connect to real life experiences. Including pictures/visuals next to the problem, along with incorporating key words that stand out to the reader, will ultimately be beneficial for these types of learners.

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The best way to teach students basic computational skills, such as adding and subtracting, is to build off of their instinctive conceptions. Adults often find it simple to solve an addition word problem by subtracting, and to solve a subtraction word problem by adding. However, research has shown that it is much more effective to use visuals, representations, manipulatives, drawings, etc... when teaching children to solve these types of problems. Students can use the information in the problem to make sense of the context, and reason through the problem accordingly, even if it takes some trial and error.