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3 ideas that reaffirm my prior knowledge:
- STEM teaching is important. I see it all over the place in every school. I did not grow up with STEM classes or a focus on those topics so seeing it as such a big deal in today's classrooms has always been interesting to me. I even thought that maybe it was being overdone and that was all they were focused on these days. I knew that being good at math could get you any job really but never realized just how much our world is changing and making the need for those jobs even bigger. Our world is becoming reliant on technology. It is everywhere you look. We have to teach our students to navigate this new world and to build up the future generations to thrive in it. STEM teaching is the new way to prepare students for their careers and lives as adults.
- The chapter discusses the need for mathematics instruction to not be rushed and studied deeply. They state this was fixed in 2006 with the NCTM "Curriculum Focal Points", but I feel we are regressing somewhat in this especially in the immediate grades because of assessment pressures. I would really be interested in seeing and studying how this is still being used today.
- The 5 process standards remind me of the social studies standards. Both have a focus on inquiry. Students can listen and repeat, but do they understand what they are doing? Are they making real connections and building their skills? Meaningful learning requires inquiry to build your knowledge.
2 concepts that make me question what I thought:
- "Families' and teachers' attitudes toward mathematics may enhance or detract from students' ability to do math." (Van de Walle, Karp, & Bay-Williams, n.d.) This statement was in the opening paragraph to the chapter. It really resonated with me because I had never thought about how the attitudes the adults in my life may have impacted my own math experience. I am one of those people who struggles with math and always jokes that math is like a foreign language to me. I have to change my mindset on math as a teacher. Talking negatively about my own abilities or struggles does not contribute to the positive growth mindset that I wish to instill in my students. My students need me to build their confidence so I must change my attitudes about math so they can succeed.
- The chapter states that it is our job as the teacher to educate our students and their families about the research behind the standards. I have always thought that educating the families was not necessary and they would have whatever opinion they had about it. I really like the idea of opening that conversation with my student's families. They deserve to know why I am teaching the way I am and how they can support their student. This is something I will strongly consider adding to my open house and parent conferences.
1 ah-ha moment:
-" You don't want to work on the brink of your knowledge base." (Van de Walle, Karp, & Bay-Williams, n.d.) This statement really spoke to me. I have always struggled with math and my biggest fear for becoming a teacher was my concern of how I was going to be able to learn it on my own to teach it. It is important that I take the time to learn everything I can, not just the bare minimum. Students will come with a variety of understanding issues, that as the teacher, you need to be well-versed in the subject to help them. It is much more than knowing my times tables. I need to understand the why. I need to understand the processes. I need to understand the variety of strategies. It is my job to fully educate myself to be the best teacher that I can be.

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3 ideas that reaffirm my prior knowledge:
- Chapter 2 discussed the importance of the verbs you use in making goals for teaching your students math. In previous classes, I have learned the need for higher-order thinking. That should always be your goal for your students. You want them to apply the skills they learn so they will carry on throughout their education and lives. Some verbs to promote this in math are: analyze, apply, formulate, model, predict, etc..
- Chapter 3 defines inquiry as, "a disposition of openness, curiosity, and wonder." (Clifford & Marinucci, 2008). The goal of teaching is to instill inquiry. This means that when we plan our lessons, we need to make sure students get the opportunity to explore different strategies for solving math problems and that we make it interesting and allow for predictions as they learn.
- Chapter 3 discusses how math curriculum goes from practicing to solving story problems. When this occurs, students do not think about the story scenario and just lift the numbers out of the problem. I never realized this was a problem because that is what I have been taught to do and seen other teachers tell their students to only focus on the numbers. This does not make sense now that I think about it because what is the point of even making a story problem if you do not even read the words or make the connections. At that point, you might as well not even give students word problems if you think they should only focus on the numbers. Word problems show students the significance and use of math in real life. We need to stop telling students to only focus on the numbers. They need to understand the why behind the process.
2 concepts that make me question what I thought:
- Chapter 2 was all about how learning math is more than just procedural fluency. To truly understand and benefit from mathematics instruction is to gain conceptual understanding from it. This understanding will carry on to later life skills through problem solving. I had never thought about the importance of understanding math on that level. Through my own education, it always felt like math was just learning a new procedure and rules for math every week. We never understood why we were doing it and would forget about it the following week. The skills we teach students need to last beyond the weekly assessments or the standardized tests. This is done through teaching conceptual understanding.
- Chapter 2 states that calculators should be a readily available tool for students to use. In my own education, I was often told by my teachers that we had to learn how to do math without calculators because we would not have them in real life. 15 years later and now we do have access with our phones, so not true anymore. This is our new reality. Sure, it is important students learn the process behind math problems, but they should be given access to calculators after that has been met and understood.
1 ah-ha moment:
- "In the real world of problem solving and doing mathematics, there are no answer books." (Van de Walle, Karp, & Bay-Williams, n.d.) This statement really stood out to me. Students focus too much on getting their teacher's approval or looking in the back of the textbook for the right answer. This will not help them in the long run out of school. As an adult on your own, you have to figure things out on your own. You have to keep trying until you get it right.

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3 ideas that reaffirm my prior knowledge:
-"You must be sure that students understand the problem before setting them to work." (Van de Walle, Karp, & Bay-Williams, n.d.) This is crucial in mathematics teaching. We set our students up for failure if we do not give them the support. Sure, your students can complete the routine, but they may not understand what is required of them for their independent work. Taking the time to help them understand what they will be doing makes their chances of success much higher. The chapter also stated the importance of doing this for EL's and struggling readers. This method accommodates all students.
- Chapter 4 brought up how some statements can be damaging when trying to help students. Saying "It's easy" or "Let me help you" tells the student that you think they are not very smart or that they need you. This hurts their self esteem and their confidence in their academic abilities. It is important to think before you speak when helping a struggling student.
- Chapter 6 discussed equity in mathematics instruction. One important statement they made was that equity is achieved by reaching equal outcomes, not by equal treatment." Even though you may make accommodations/modifications, your students should still be meeting those learning objectives. Your goal is to make sure every student is succeeding and meeting those objectives.
2 concepts that make me question what I thought:
- Avoid ability grouping. This is discussed in chapter 4. I have regularly seen this method of grouping done in classrooms. I can easily understand why it is not a good idea for self-esteem in the students, but had not thought about how it also keeps them behind.
-Thinking on my field placement classroom, she has math instruction and then math centers. I wonder if the math centers are put in place as a Tier 2 approach so she can support those students more closely. That makes sense, I just had never considered it and just thought they did math centers because they had to. I will have to ask her next time that I am there.
1 ah-ha moment:
- Math is not a universal language. This shocked me. In chapter 6, they state that while conceptual knowledge is universal, procedures and symbols are culturally determined.

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3 ideas that reaffirm my prior knowledge:
- Chapter 7 discusses the importance of children "subitzing" and recognizing patterns. I had never thought about its importance before but understood that I have always just easily recognized certain patterns. It makes a lot of sense why it would be so significant in teaching children counting. I wonder if it is an active part of instruction in the primary grades. If it is not, then it definitely should be.
- Chapter 7 stated that initially children do not see a numeric pattern in the numbers between 10-20 and see them as simply ten more words in the number sequence. I have observed this many times working with children. They seem to struggle with making sense of this. I found it so interesting that they stated that in other languages it is read as "10 and 1" instead of eleven.
- Across the chapters, they reinforce the idea of adding to student's understanding over just solving the problem. For example, in Chapter 13, it continually says to ask the question, "Why is it true?" across different algebraic thinking situations. It is important to expand on student's thinking with higher order thinking such as this. Students need to understand the how and why behind their math.
2 concepts that make me question what I thought:
- In chapter 7, it discusses the importance of learning mathematics early and how it is a predictor of later success and reading skills. It mentions that in first grade, students begin to count up to 120 and then they will work on place value and then addition/subtraction. I find this interesting because in my first grade placement class, they have been focusing on addition/subtraction the whole school year. Most of them cannot count that high and I have never seen them do place value work. I wonder if the rush to meet standards is negatively affecting students. I have felt that over the past 20 years, education has changed a lot and forcing kids to do higher level content at earlier ages. Everyone knows it is not developmentally appropriate, so why do we continue to let it happen?
- Decomposing numbers is earliest form of algebraic thinking in primary grades. I never thought of algebra being present in kindergarten. Looking now at how we decompose numbers and present those problems, they are using algebra. This adds on to my next statement below.
1 ah-ha moment:
- Algebra is generalized arithmetic. I have never made that connection and had always thought of it as two very separate things. Arithmetic is just the foundation for algebra and algebra makes big computations much simpler.

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3 ideas that reaffirm my prior knowledge:
- Number talks are a part of the daily routine in my field experience class at Atkinson so I have had some knowledge of what it looks like. In the chapter it states that it should only last 5-15 minutes and is meant to develop procedural fluency. One reflection I had reading that was the chapter discusses the importance of understanding the why behind the math more than the procedure, but yet they do still need to know the procedure to be successful. This is why number talks are important. Procedural fluency is meant to be reviewed and done in short bursts to work on their skills, not to be the whole lesson. Number talks should be more widely used to combat this so that math lessons can focus more on the importance of making connections and meaningful learning.
- Think-Pair-Share is a teaching strategy that I am really familiar with and have seen so often especially with math lessons. Discussion is an important part of any lesson but especially useful during math as students share varying strategies and solutions together.
- The JCPS handout reaffirmed some of what I know about what math time looks like and consists of in their classrooms. They are required to do a number talk, a math lesson, and math learning centers. The handout specified how much time should be spent on these and what their expectations are for what teachers should be doing with math everyday.
2 concepts that make me question what I thought:
- 3 act math tasks is a new concept that I have never heard of before. Students come up with the problems and collaborate to solve them from a visual context. I did not realize it before but this is the same concept that Dr. Marin was trying to introduce us to on the first day of class. Students look at pictures or videos and are asked to observe and discuss/write what they notice or wonder. They then ask questions and make estimates to solve their questions. They then collaborate to figure how to solve their question and show their thinking. When I first read about it in the chapter, I did not understand. I went to https://gfletchy.com/3-act-lessons/ and found many examples. This is a great method to teach students and to use real-life examples to engage them into their learning teaching real problem solving skills.
- I had never thought on the significance of using worked examples. I can recall doing problems like these throughout my own school experience but never really thought about why my teachers had us do them. In the chapter, they state that worked examples bring light to alternative solution strategies and common misconceptions. This would be a great way to highlight misconceptions to help children who struggle. At first, they may assume the solution was correct due to this. Allowing students to problem solve and take a deep look at the possible strategies will let them better understand the process. If they are still struggling with the worked examples, it still leads to a meaningful discussion about those misconceptions and different strategies used.
1 ah-ha moment:
- All of the readings struck me as important because they are clear routines that I can and will be implementing in my own classroom. There will come times when some or all of the routines described will be a part of my lesson plans. It is important to understand them and the reason for using them.

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3 ideas that reaffirm my prior knowledge:
- Flash cards and timed tests are not the best way to develop fluency. I have heard a lot about this in my teaching. I am trying to move away from this being my go-to, but it is hard being this is what I had done in my school years and all I had ever known.
- I understand the three phases of learning facts but never saw it listed out and broken into parts. Phase 1 begins in early childhood and kindergarten and is the first foundational step to the fact fluency.
-Number talks develop phase 2 and help students by discussing strategies and others use of them. Through this, students may find strategies they had not thought of to use.
2 concepts that make me question what I thought:
- The chapter discusses the timeline for the different fact fluency mastery by grade levels. They say that subtraction should be mastered by grade 2 and then multiplication and division by the end of grade 3. I do not think that should be expected by grade 3. How can you spend so much time on addition and subtraction, but then expect mastery of multiplication/division in only one grade level? I think this is a big problem with standards and such because all students develop their math abilities at different levels and need more or less time. You cannot set a timeline for mastery. It should continue to be developed over time.
- I wonder if it is acceptable to use timed tests and other memorizations strategies after they have mastered strategy use. I feel there needs to be some practice using timed practice for fact fluency so students can quickly answer when taking tests and working through more complex math problems.
1 ah-ha moment:
- "More drill is not the answer." ((Van de Walle, Karp, & Bay-Williams, n.d.) This is so important in developing fact fluency. So often teachers just continue to pass out worksheets or other activities in hopes that students will eventually successfully answer each problem. They do not go back to the strategies to help get the students to fluency. Students who struggle need extra support on developing appropriate strategies.

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3 ideas that reaffirm my prior knowledge:
- Chapter 10 discusses how kindergarten students view 18 and 18 ones rather than 1 ten and 8 ones. They are not able to separate the quantity into place value groups yet. This explains why the students in my first grade field placement are struggling to understand how to easily find 10 more or 10 less than a number without sitting there and counting back 10. I would love to find out more information about how to help them understand this because many children were not even after my attempts at getting them to understand.
-Chapter 10 discussed how many students in early grades may write multiple digit numbers like 21 as "201". I observed this in my class today as my students I interviewed wrote three digit numbers like 1003 for 103. It was very interesting to see and I had wondered why they were doing that but to be honest, it definitely makes sense why students may think that is correct.
- Ch 5 states, "To be formative, assessment must include a recipe for future action." I have always been told this especially from Dr. Finch. Your assessments must be meaningful. From your assessments, you should be able to gather data to figure out next steps for instruction for students.
2 concepts that make me question what I thought:
- "Assessment should enhance students' learning"(Ch 5) This is a concept I had never truly thought of before. Assessment is not just for the teachers and data. Assessments should challenge students to showcase their learning and add to their thinking as a result of.
- Ch 5 says "by challenging students who are secure with their answers or thinking helps them focus on their reasoning." This is something I had never thought of but shows the importance of always pushing higher order thinking on students. Challenge is healthy with a good growth mindset in your classroom environment.
1 ah-ha moment:
- Be precise in your math language. Refer to its place value location. This helps keep a universal math language in your classroom and helps not confuse students. This is also very useful when you have ELLs in your class who may need that distinction from their own languages use of two digit plus numbers.

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3 ideas that reaffirm my prior knowledge:
- In Chapter 8, it brings up how when a child records their ideas on paper, it should explain what they did well enough to anyone who reads it. This brings up the importance of teaching students strategies more than just giving them multiple problems to complete. Seeing a students thinking and understanding of the use of strategies is more important and shows they have learned the concepts than if they can get all 20 problems solved correctly.
- Chapter 11 discusses using number talks. They explain it as an opportunity to engage in solving and discussing alternative strategies to solve computational problems. This is so important for children because it allows them to hear a variety of strategies and points of view of a problem. They may have used one strategy but then as they listen to their peer explain another, they may try that out another time or see the problem in a different way.
- Students who count by ones in addition have often not developed base-10 grouping concepts. This is something that I continuously observed while working in my field placement. Most students would sit there and count on one by one. This would take a long time and they would get visibly upset. It is important to put a focus on teaching place value for this reason. It is much more efficient and will allow children to be more successful in solving two digit and up math equations.
2 concepts that make me question what I thought:
-"Children's initial conceptions differ from adults." (Ch 8). This is something I had not thought of or fully understood until seeing the example word problem. Adults may read a problem and know how to solve it within seconds and with the most efficient strategy. Children may not make the addition/subtraction connection as easily and will try counting on or doing some other strategy that may work, but is not efficient. It is important to teach strategies but push them towards the most efficient ways. Also, let children reason through a word problem, but model how to read it for understanding and the best strategy to use.
- Ch 11 discusses how if students only understand computation as a digit-by-digit exercise and not the value of the numbers involved, they will make more errors and are often unable to judge the reasonableness of their answers. This highlights the importance of place value teaching and its connection to operations.
1 ah-ha moment:
-"Addition and subtraction are taught at the same time to reinforce their inverse relationship." I put this as an ah-ha moment though it could have gone in either of the other categories for me also. I knew that both are taught during same times but never thought about how that was specifically for that reason. It makes so much sense though. It is more than just being the basic operations. Their connection is significant and should be taught and understood. Thinking of them as together versus separate is helpful to students.

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3 ideas that reaffirm my prior knowledge:
- Early strategies are often based on repeated addition. This is not efficient. I have not yet witnessed students learning multiplication, but I can recall from my own education having started off essentially using strategies with that in mind. Teachers will even reinforce how it is repeated addition, which it is, but sitting there and adding groups one by one is not efficient and will take a lot of time when students get to two digit multiplication. I would discuss the correlation, but make sure you encourage students to look for other strategies to solve.
- Ch 12 says that when solving division problems and probably this applies to any operations problems, when you first begin asking students to show more than one strategy for a problem, they will often use an inefficient method for their second approach. I can recall doing this as a student myself. I would be frustrated with having to show my work not once, but twice. I also did not have many different strategies to think of. This is also why it is so important to provide students with a variety of strategies to work with.
-Ch 8 discusses the importance of teaching what a remainder means. This is something I struggled a lot with in school and we would only write the answer as "R4" or whatever digit remainder. We never talked much on what a remainder signifies and the context of a problem. This would have been much more helpful rather than leaving me confused and worried if I got the right answer.
2 concepts that make me question what I thought:
-Ch 12 says that you should introduce students to a variety of different representations of multiplication strategies until they have a collection of useful ideas. This made me question my thoughts of teaching one strategy for extended period of time. This makes sense though because giving students many different strategies allows them to find the one they are most comfortable with or that helps them understand the concept better. Some strategies may not be helpful for certain students. I do wonder how you go about introducing strategies though. One a day? One a week? What is best?
- I have never heard of compensation strategies for multiplication. I have seen it used in addition/subtraction, but never thought about it being used in multiplication. I like the strategy, but I worry students would forget the step where they subtract or add in what they took away from a digit and get the wrong answer.
1 ah-ha moment:
- Chapter 12 says that, "at least half of the grade 3 standards in the common core state standards involve understanding multiplication." This was surprising to me that so much of that grade year focuses on multiplication specifically. It is more difficult concept to grasp so it makes sense why so much of that year has to slow down and focus exclusively on multiplication.

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3 ideas that reaffirm my prior knowledge:
- Ch 20 explains how "statistical literacy is critical to understanding the world around us, essential for effective citizenship, and vital for developing the ability to question information presented in the media." This is so important especially in this day and age with media presence so involved in our lives with technology. I remember taking statistics my senior year of high school and my teacher showing us how important it is to look at graphs and data with a critical eye. He would show us how graphs can be skewed to look certain ways in the favor of those trying to persuade us by making the values unproportionable or spread apart, when in reality they results are very different or very close in range. It is important to teach kids how to read graphs accurately and look at them with a critical eye.
- Ch 20 explains that students should be involved in the full process of collecting and modeling data to gain full understanding of statistics. I think this would also really engage students in their learning when they are able to create something of their own. Students can be shown graphs and such but making their own and working through the steps to see for themselves how data is gathered and modeled makes it more concrete and understandable.
- Ch 14 discusses the importance of fraction understanding and how it translates into algebra, percent, and decimal understanding. This explains why I had so many issues with those in school because I did not have good fraction understanding past the basics. It is important to build that foundation for students so they can thrive later on down the road. I think that often teachers do not think about the importance of the things they teach and how it may impact their students later on, but they really should. Every concept and day of learning is important and needs to be taken seriously.
2 concepts that make me question what I thought:
- Statistics and mathematics are different fields. I never would have thought this. In my mind, anything that involves numbers is math. When I think about it , it makes sense. I just never knew they were technically in two different categories.
- Ch 20 brought up a great way of teaching students how to gather data effectively. A group of students wanted to know how many of their class were 6 years old. They went around asking everyone and when they went to present their data findings, they all had different numbers. This allowed the teacher to explain the importance of having a carefully planned method of collecting data.
1 ah-ha moment:
- This was not in the chapter necessarily, but when ch 20 started discussing formulating questions and how students want to get to know eachother naturally, I thought of the idea of a weekly graph warmup. It would be fun and get students learning without them realizing too much. Maybe once a week, the teacher or students formulate a question to answer and students have to quickly collect the data and form a graph of some type. This would essentially be a math review warmup and get students involved in statistical thinking.

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3 ideas that reaffirm my prior knowledge:
- Division is connected to fractions. Teaching the connections of shared parts also allows for meaningful connections to real-life usage of fractions. This is often not emphasized though it should be and may even help students understand the two concepts better individually.
-" It is important to move among fractions greater than, equal to, and less than one." This helps students understand fractions as being values that come between whole numbers. This will then help them as they get to decimals and percentages as well. It may be difficult at first to get students to understand the value of fractions due to this, so emphasizing this is important.
- Manipulatives and visuals should be more than just area models. There are a variety of options to use to model fractions. Anything can be made into a manipulative almost. Students learn best through visuals when learning fraction concepts.
2 concepts that make me question what I thought:
- Fraction instruction should begin with use of the words rather than symbols. I never would have thought about this before. It allows students to first focus on making sense of fractions as a part without the complication of symbols. The chapter says that you should explain notation after story problems and not before.
-Length models. I have very little experience with using these as a student. Especially cuisneaire rods. I struggled a lot with understanding these in my mathematics for education class a few years back due to no experience with them as a student. The chapter says that length models such as that are very important in developing students understanding of fractions. I would like to explore them more and how to use them in my own classroom if I teach fractions.
1 ah-ha moment:
- Do not use the term "reducing" when simplifying fractions. It makes it sound like the fraction is being made smaller, when that is not the case. Use the term simplify only.

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3 ideas that reaffirm my prior knowledge:
- Students make mistakes when regrouping mixed numbers. It is best to change the mixed numbers to fractions greater than 1 instead. This may help some students better understand the value of the fraction versus just looking at the whole number.
-"Percents are simply hundredths." This needs to be understood by students as they learn about percentages and their values compared to decimals and fractions. This just shows again just how important place value understanding is and its effect on later concepts.
- Students need to understand the role of the decimal point and how it marks the location of the ones place. Adding zeroes to the left of a decimal point are unnecessary and do not change the value of a decimal. It is not shown on calculators so it is not necessary when writing out.
2 concepts that make me question what I thought:
- Use of estimation. This concept was also brought up in the last chapter as well as ch 15 when working with fractions. I never thought about estimation as being very important when learning math. When using operations in fractions, estimation keeps the focus on on the meanings of the numbers and operations, encourages reflective thinking, and helps build number sense with fractions. Estimation leads to the development of strategies.
- It is important to let students explore multiplication of fractions conceptually before introducing the rules. They already have the ideas of addition/subtraction and whole-number multiplication internalized so introducing these rules too early will just confuse them. I really struggled with this when I was in school because I would get all the different rules mixed up. I never did fully understand this.
1 ah-ha moment:
-Common denominators are not required. I always thought this was the one and only strategy for solving with unlike denominators. There are many other strategies students can use including number lines.

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3 ideas that reaffirm my prior knowledge:
- Understanding decimals is important because our monetary system uses decimals. This is a great way to teach decimals in a contextual way for students to better understand and carry on their learning.
- Helping students understand that decimals and fractions are the same can be difficult. They represent the same quantity. One way to support this is by saying for example " five and two-tenths" versus "five point two". I am guilty of saying the latter, but we must be concious of how we use vocabulary to support student learning.
- Using a number line to teach fractions, percents, and decimals can be so effective to show their value as a whole "1". This is a great visual to teach students value.
2 concepts that make me question what I thought:
- I never thought about how to use manipulatives to teach decimals, but I still am not sure how to do so with the examples in the text. I know how important visuals are though for learning. I just feel their examples would confuse the students too much in thinking about fractions versus decimals. I would like to explore online options as well to demonstrate decimals better.
-Counting how many decimal points in problem to figure out where decimal point is in answer is not effective when blindly told to do. Students need to be able to explain their reasoning. Therefore you must do more than tell students the rules, they need the strategies and visuals to compliment and explain their thinking.
1 ah-ha moment:
- When reading about the misconceptions many students have with percentages, I found that many were some I still have. This inspired me to look into my misconceptions and why I have them and what I can do to change those thoughts.