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Chapter 4: Seven Significant Strategies for Developing Fluency

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Chapter 4: Seven Significant Strategies for Developing Fluency

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Please post 1 or 2 ideas that you took away from this chapter or that you would like to explore further.

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Chapter 3: Good (and Necessary) Beginnings For Fluency

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Chapter 3: Good (and Necessary) Beginnings For Fluency

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Please post 1 or 2 ideas that you took away from this chapter or that you would like to explore further.

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Chapter 2: Fluency Fallacies & Related Truths

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Chapter 2: Fluency Fallacies & Related Truths

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Please post 1 or 2 ideas that you took away from this chapter or that you would like to explore further.

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Chapter 1: What Does Fluency Really Mean, and Why Does it Matter? 

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Chapter 1: What Does Fluency Really Mean, and Why Does it Matter? 

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Please post 1 or 2 ideas that you took away from this chapter or that you would like to explore further.

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Fluency is much more than speed and spitting out facts, there has been a shift where the focus is now on strategic thinking. Students become problem solvers and ultimately have a deeper understanding of math concepts. It is about selecting the right strategy for the problem and being flexible where they can try other strategies.

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This chapter opened my eyes to a more detailed definition of fluency. I will sadly admit that I used the words fluency and automaticity interchangeably in regards to a student's ability to recall facts quickly. Fluency clearly has more layers. What stood out to me was having knowledge of strategies and an understanding that selecting the appropriate, time efficient strategy is key to a students ability to have math fluency.

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1/2nd grade Option A Team - Barbara, Jen, Delia and Ciara

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1/2nd grade Option A Team - Barbara, Jen, Delia and Ciara

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Our 1st/2nd grade teams are considering a student and a teacher checklist using the procedural fluency chart to help us in our Option A project. We are looking at slide 12 and also incorporating the check for reasonableness chart that is included in the slideshow.

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What have you come across in your reading that you've implemented or has changed the way you approach fluency in your classroom?

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What have you come across in your reading that you've implemented or has changed the way you approach fluency in your classroom?

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This chapter and the discussion we had inspired me to think about implementing games and routines that focus on real fluency--especially in terms of guiding students to be more thoughtful and deliberate in selecting appropriate/efficient strategies. More often than not, my students are used to simply using whatever strategy is being taught that day because the problems presented in the lesson and workbook lend themselves to that particular strategy, and it is likely assumed that we should be practicing that particular on that particular day. There is not enough practice with considering the problem and selecting an appropriate, efficient strategy. The "Just Right" and "Strategories" games presented in chapter 1 can be adapted for second graders. Strategories seems like a valuable "exit ticket" activity after a strategy is taught; can students come up with problems that are examples and non-examples for a given strategy? I also love the "strategy analysis" idea that came up during our discussion. Students are used to "error analysis" questions; developing strategy analysis questions would provide more opportunities to practice taking the time to consider what strategies are appropriate for a given addition or subtraction problem.

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It is important to help students make the distinction between a representation (a tool) and a strategy (how one is thinking about numbers). While I understood this, I really appreciate the book's suggestions around language to use with students to help clarify this distinction. "Did you use a tool to help you use your strategy?" is one example. I also plan to set up the suggested interactive "Math Reasoning" bulletin board for addition and subtraction with "Tools" on one side and "Strategies" on the other.

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I found the example of including a Tools vs. Strategies bulletin board/chart to help students distinguish between the tool they used (number lines, base ten blocks, drawings, bar models, etc.) and the strategies they implemented to solve efficiently will be very helpful for students. We are starting the unit on addition strategies for 2 digit numbers and will be exploring 2 to 3 digit addition/subtraction using the same strategies. This might be a great visual for students to have available to them so they can identify what tool and strategy they feel most comfortable with.

Another issue that we are encountering this year is, being an ICT class, we have students that struggle with multiple strategies taught and while we are exploring these strategies and exposing them to this, it is evident they are becoming confused and tend to gravitate to the one strategy they are most comfortable with.

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I found it interesting that the US standard is not internationally standard. Meaning, procedures vary from country to country. This is something to keep in mind when thinking about a culturally responsive classroom.

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In second grade we have begun to explore addition strategies for 2-digit numbers. As we add more strategies to our repertoire, I want to emphasize considering the problem to select an appropriate, efficient strategy. The book's suggestion to play "Just Right" will be very helpful.

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I find it interesting to think about that standard algorithms in the US are not standard everywhere. "While mathematical concepts are universal, procedures and the related notations can vary from country to country" (p. 31)

It is so important to stand by the fact that procedural fluency doesn't equate to fluency of understanding - metacognitively).

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So relatable to hear that the definition of fluency oftentimes is very different when it relates to the parents. So many of our students are enrolled in Kumon and Russian math. They also have workbooks that their parents purchase to practice math. These are all tools where the goal is a correct answer, not a deep understanding of math.

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I connected to the Fluency Fallacy #8 on pages 36-37. I feel that some of my students use procedures without really understanding the reason behind the procedure and miss out on making connections. Students feel successful when they memorize facts but fail to understand how to build on that skill to connect it to generalize methods and explain their math thinking.

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I appreciate the table on page 37 that speaks to student engagement and procedures that create engaging math opportunities for kids. High-level engagement procedures allow for inquiry and critical thinking in a way that can elevate learning in the classroom.

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The goal used to be to calculate accurately but now the math goal is how to think mathematically. Teaching kids to think, not just to calculate- to problem solve and be flexible, not only about getting the correct answer. This leads to having better number sense.

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There were many important parts in this chapter, but what made me ponder was written on page 26 TRUTH: " Mastery and automaticity mean getting skills ( and facts) down. They are outcomes of practice. Fluency includes mastery of skills and automaticities, but also includes DECISION-MAKING. Fluency practice attends to efficiency, flexibility, accuracy and reasonableness in mathematical terms."

This means there must be a 180 degree shift in the way we teach our students, the way we model and utilize strategies. In addition to this, when it involves special educators, there must be a shift in the way they write goals in the IEPs.

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I was happy to see estimation distinguished as a big idea, along with the various strategies and procedures under its umbrella. Looking at the numbers first or the problem's context will dictate the appropriate estimation tool.

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I have a student who is not yet comfortable with "distance from a 10." Not having 10 as a benchmark really impacts her fluency--including her ability to generalize this benchmark to problems like 397 + 435. I plan to use the Distance to 10 Walk (maybe on a small scale).

Decomposing numbers flexibly is also an essential to second graders as they work to add and subtract efficiently. I'm glad this is built in to many of our Number Routines. I will continue to emphasize alternatives to decomposing by place value. Many are already doing this flexibly.

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One idea that resonated with me is how important the number 10 is and how understanding relationships like 9 being one less than 10 can help students build strong number sense. Teaching facts of 10 isn’t just about memorization, it helps students see patterns and understand why the facts matter. Another idea I took from this class is the importance of flexibility in math, and how giving students choices with strategies, open tasks, and tools like strategy cards and estimation activities helps them become independent problem solvers rather than relying on rote memorization or timed tests.

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In second grade we focus a lot on developing a comfortability with different addition/subtraction strategies all with the focus on making friendlier equations. We just looked at adjusting numbers for subtraction and it is quite interesting seeing the lightbulb go off for the students when they recognize how powerful these strategies are. There number sense is strengthened especially when they are able to fluctuate between strategies depending on the numbers they are adding or subtracting.

When they discussed students who have a strategy knowing their facts better than students who do not approach math work with a strategy in mind that was powerful. It is very interesting how in reflecting on the use of fluency, it is more important to focus on fluency with strategies than fluency with facts, which should be referred as a child's automaticity with facts.

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The chapter emphasizes that true mathematical fluency starts with students understanding what numbers and operations mean. When students build strong conceptual foundations, efficiency and accuracy develop naturally. A major takeaway is that number sense—flexibility with numbers, awareness of relationships, and comfort decomposing and recomposing quantities—is essential. The chapter highlights strategies that nurture this sense-making.The authors reinforce that fluency is not about speed or getting answers immediately; it’s about thoughtfully choosing strategies. Allowing students to think, try, adjust, and reason strengthens their long-term fluency. I found it interesting how important it is to build multiple approaches to solving problems rather than relying on a single memorized procedure. Having options empowers them to choose efficient strategies in different situations. We see this in the Reveal Math program here in second grade as we continue to teach students how to add and subtract with multiple strategies.

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I thought the section on estimation was very interesting. Exploring different ways to estimate is so important and it was a real reminder that this skill develops over so much time. I enjoyed reading about the language such as "it's about that" or it's close enough". Estimation is a life skill and is used in so many real life situations. We do a bunch of work with our students on estimation in our class with the little ones and we try to teach them that it is much more valuable than a precise number.

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I found the part on estimation to be very interesting. It was a reminder that estimation is to be explored in many different ways and that it is a skill that develops over time. It is a life skill and estimation is more valuable than a precise number. We do a lot with estimation in our classroom with our little ones and I really liked the use of language such as "it's about that" and "it's close enough".

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On page 77, Developing Strategic Competence, the author lists a few things that work against strategic competence. I found it interesting that one the things to avoid was, #3 Do not require students to show all their work.

I totally understand that some of the computation can be done mentally but I like to see my students show all their work, especially when they are first learning a new strategy. I find that when students show their work they can find a computation or operation error quicker.

On the other hand I can understand that requiring students to show all their work at times can become laborious.

I also found #5 Do not ask students to solve a problem two or more different ways, would be included on the list. While I don't regularly ask students to show 2 ways, I do ask them to use another way when they get the wrong answer so they can start to recognize where a mistake was made.

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It is reassuring that the seven significant strategies are so prominent in Reveal Math. At the same time, I realize that there could be more built into the program to help students think about which strategy to use and why certain strategies work better for certain problems. I was particularly interested in the idea of explicitly teaching the pre-solving pause to look closely at the numbers and decide on an efficient approach. This feels like something that would fit well into second grade number routines. After reading, I’m motivated to create number routines where students compare strategies and decide which one makes the most sense. I'd also like to make anchor charts with the class that support students in choosing strategies more independently.

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I am interested in exploring further how helping students distinguish between breaking apart numbers and compensating by adjusting numbers can support deeper mathematical understanding. When students move beyond viewing strategies as isolated procedures, they begin to reason about number structure and equivalence, which is essential for developing true fluency.

I found Chart 4.13 particularly helpful, not only for teachers but also for students. This chart could serve as an effective scaffold for students who struggle to choose or apply different strategies, as it helps them organize their thinking. For example, when solving a multiplication problem, students can decide whether it makes sense to break apart numbers, use doubling and halving, or compensate.

Additionally, I appreciated the reflective questions posed in the text, such as “When might you use a counting-up strategy for subtraction?” These types of questions would be valuable discussion prompts in the classroom, as they help students connect specific strategies to the operations and problems they are solving. This kind of discussion reinforces strategic reasoning and supports the development of flexible fluency.

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Improving visuals and supports- Vertical Number Lines

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Improving visuals and supports- Vertical Number Lines

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I thought the idea of using vertical number lines on page 79 was great! It makes me think more carefully about the visual supports I use and make them better aligned to the problems my students need to tackle.

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We are teaching our students to show their work due to the State tests, but it is interesting to see students who have evolved to do the work mentally based on the numerous strategies we have shown throughout the year.

I find it very reassuring that Reveal is helping us help students to learn all of these new strategies.

The seven strategies are:

1. Count On/Count Back-using incremental counting to solve addition and subtraction problems.

2. Make Tens- structuring numbers around 10 to simplify addition.

3. Use Partials- breaking numbers into parts to support addition, subtraction, multiplication, and division.

4. Break Apart to Multiply -decomposing numbers to make multiplication easier.

5. Halve and Double -adjusting factors to make multiplication simpler.

6. Compensation- adjusting a number to create an easier calculation and then correcting the result.

The strategies should be taught explicitly and conceptually, including when to use each one, so students build strategic competence rather than simply memorize procedures. Teaching these strategies helps students become more efficient, accurate, and flexible in their mathematical thinking.

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Chapter 6: Fluency Practice is Not a Worksheet

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Chapter 6: Fluency Practice is Not a Worksheet

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Please post 1 or 2 ideas that you took away from this chapter or that you would like to explore further.

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While there is alignment between Reveal and the idea that fluency is not simply about speed but about deepening strategy use and number sense, I keep coming back to the need to be more intentional about discussions and even dedicated lessons around efficiency and practicing when to use certain strategies. It was encouraging to recognize some of the suggested strategy structures in Reveal, such as open number lines and number strings. I am thinking about using more number strings with a student who is still not generalizing combinations of 10 with larger numbers. I also see how incorporating quick images with ten frames could further support students in making tens when adding  2-digit numbers. I would love to have dedicated time to put these structures in place so that fluency practice is purposeful and more frequent.

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I agree with Dara, it is nice to see that we already are using number routines in our classrooms that build these fluency strategies. I appreciated reading about facilitating routines (partners vs triads and constant partners) it's helpful to reflect on my current practices. I am excited to try some of the games and the Make 100 learning center presented in this chapter, it's a great way to build skills during morning meeting and during CCT time.

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Chapter 6 reinforces something that is reassuring to see in our own classrooms, that many of the number routines we already use truly support meaningful fluency development. It’s affirming to recognize that effective practice is not a worksheet filled with multiple problems solved in the exact same way.

Fluency practice needs to be substantial and intentional so students can build both strategies and automaticity. The goal is not just speed or repetition, but helping students develop flexibility and confidence. One thing I would like to try more intentionally in my 4th grade classroom is incorporating short daily “strategy share” routines. After solving a fluency problem, students could explain the strategy they chose and why it was efficient. We could compare different approaches and discuss when one strategy might be more effective than another. I also think it would be powerful to occasionally present the same numbers in a slightly different format to push students to adapt their thinking rather than rely on a single method. This would help reinforce flexibility and deepen their understanding while still building automaticity.