Updated: Jul 21
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In episode 4, I talked about stopping timed tests because research has shown that they can increase math anxiety and do not fully help children demonstrate fluency. But what does it really mean, then, to be fluent? In Episode 5, we begin to unpack the four components that prove that fluency is more complex than just saying that a child simply “knows their facts”.
Here’s what a parent-teacher interaction without a true understanding of fluency and mastery might sound like.
Parent: How is Ben doing in math? Is he fluent?
Teacher: He’s doing well in math. He’s mastered his 2s, 3s, and 5s so far. Geometry is going to be our next focus area, but we will also continue to work on basic facts through 10.
Parent: Okay. I noticed on his homework that he didn’t know how to do some of the problems and started to get a little anxious. I told him to use his fingers to help. Is that okay?
Teacher: Yes, fingers for right now are okay. He gets stuck on the 7s a lot in class too, and isn’t able to recall those facts within 3 seconds yet. The goal is to be automatic by the end of the year. Try practicing those 7s more at home to help him become fluent.
Parent: Okay, thank you. We’ll work on that.
The parent said “okay,” at the end of the exchange- but are they really understanding? What plan of action are they leaving with for home practice? If that same parent had asked about reading fluency - how might the conversation have been different? I’m thinking that most likely there would have been a much more articulate and profound conversation-parents would have already been equipped with language to help begin the conversation, and maybe would have asked about comprehension- and the teacher would have been able to move the conversation along by breaking down Ben’s accuracy, expression, and his success with decoding words. And most likely there would have been specific strategies given to that parent to implement at home.
Over the past decade, and really probably longer than that, there has been a de-emphasis in schools about the language of mathematics. It is extremely common for schools to offer reading nights and to have multiple informational sheets go home to ensure that there is a solid understanding of a child’s path to reading fluency, as well as a child’s individual progress. There is an entire month devoted to reading in most schools, with many also having reading nights happening throughout the year. It is often a struggle to find this same level of interest and support for the path to math fluency. I have taught in 5 different buildings and now am a part of 9 as a math coach and over these 15 years, we have only had a handful of family math nights. They were never as well attended as the reading nights, but the joy in the rooms was obvious. Parents are yearning for more information about how to help their children with math. These surface level conversations, like the one heard earlier, are not productive for a child’s growth and it possibly means that the adults having these conversations are holding onto the unproductive belief that “mathematics learning should focus on practicing procedures and memorizing basic number combinations” (NCTM 2014 p.11). Teachers, let’s work together to develop a stronger and more positive conversation about math. This can start by educating ourselves on the difference between procedural fluency and mastery and then working with parents to create a common language and understanding around these concepts.
We’ll come back to this conversation about Ben at the end of the episode.
So why do we need a definition? Definitions are essential, otherwise we can’t be sure that we really know what anyone else is talking about, because people naturally bring their own experiences and backgrounds-their own identities- into definitions. Authors Jennifer Bay-Williams and Gina Kling state that “reaching the goal of basic fact fluency requires establishing a shared and complete understanding of the term” (Bay-Williams and Kling 2019 p.1) and thankfully, we have organizations like the National Council of Teachers of Mathematics to lead the way and provide us with this essential language.
In 2014, NCTM crafted a formal position statement where the beginning states that:
Procedural fluency is a critical component of mathematical proficiency. Procedural fluency is the ability to apply procedures accurately, efficiently, and flexibly; to transfer procedures to different problems and contexts; to build or modify procedures from other procedures; and to recognize when one strategy or procedure is more appropriate to apply than another (NCTM 2014).
You can read the full position statement by heading to the NCTM website.
This statement can be used as evidence that fluency is not a synonym for automaticity. A child can be fluent without yet being automatic and knowing facts within 3 seconds -which is a crucial understanding that is often missed. So I’ll say it again- A child can be fluent without yet having mastery. This is a powerful nugget of information to have when talking about the basic fact fluency progress of a child.
Parents and educators, instead of generalities, let’s develop specific talking points around fluency that break down the four components of accuracy, efficiency, appropriate strategy use, and flexibility.
Accuracy is the “ability to produce mathematically precise answers.” This component is pretty straight forward - either you have the correct answer to a problem or you don’t. (Bay-Williams and Kling 2019).
A simple question to get the conversation started with this component could be - “Is a child getting answers to these problems correct?”
The next component is efficiency. Part of what efficiency looks like in action is needing a reasonable amount of time to solve a problem correctly. It is important to point out here that efficiency is not the same as automaticity. Reasonable in this sense does not mean within 3 seconds.
A question you could pose to start to talk about a child’s efficiency could be - “Is a child able to move through a few problems in a work period or are they getting stuck on one?”
The answers to questions about accuracy and efficiency could be organized in some type of checklist system. This isn’t inherently wrong, and sometimes there is a need for this gathering of information. The misstep occurs, however, when this is the only type of information gathered about a child and that child is still deemed to be completely fluent in basic facts. Traditional timed tests are designed to only gather these two pieces of information.
Appropriate Strategy Use
The third component, appropriate strategy use, stems from efficiency. This is “the ability to select and apply a strategy that is appropriate for solving the given problem efficiently.” (Bay-Williams and Kling 2019).
The questions that you could ask here will give more than a simple yes or no answer and could sound something like, “How is a child thinking about the relationship between numbers in order to help choose a strategy?”
The final component of procedural fluency is flexibility. This is “the ability to think about a problem in more than one way and to adapt or adjust thinking if necessary.” Flexibility takes time and practice and adults have to give children the opportunity to build their number sense. High Yield Routines like Quick Looks, where a dot image is flashed and then covered, Number Talks, and Number Strings, are some ways to start. They all help a child to develop the skill of creating visual images and to see the same number in a variety of ways. (Bay-Williams and Kling 2019).
Reframing the Conversation
Let’s apply our new learning about the components of fluency to the parent-teacher exchange heard earlier. Instead of asking, is Ben fluent, let’s instead ask -
Is Ben accurate and efficient?
Yes, we could make the argument that he is for 2s, 3s, and 5s, but only if we make the assumption that the teacher’s use of the word mastery is interchangeable with fluency.
Another question to ask instead of is Ben fluent is -
What about Ben’s appropriate strategy use and flexibility?
The parent-teacher exchange isn’t in depth or specific enough so we don’t learn anything about either. If we only monitor two of the four components of fluency, then we can’t say with certainty if a child is fluent with basic facts. Only monitoring for half of the definition of fluency would also mean that opportunities to affirm a child and to show them that their thinking is valued would be missed.
We now have a working definition of procedural fluency, so let’s add another layer to our learning. I mentioned earlier that fluency is not the same as being automatic. But fluency does develop in three phases - Counting Strategies, Reasoning Strategies, and Mastery (Baroody 2006).
Phase 1 - Counting Strategies
No matter which operation we are talking about, children begin their path toward fluency and mastery with Phase 1 - Counting Strategies.
Children in this phase use strategies that involve object counting or verbal counting to determine an answer. These objects could be fingers, counters, blocks, or something else that is concrete. This is an important phase, as it helps children develop visual images.
Phase 2 - Reasoning Strategies
Once children no longer need to count out each individual item, they move into Phase 2 - Reasoning Strategies. This means that a child is using known information to logically determine the answer of an unknown combination.
For instance, using that fact that 8+2 equals 10, to help with figuring out the answer to 8+5.
A child could think about the relationship between the numbers 2 and 5, and knowing that they are 3 apart, and that 5 is a larger number than 2, conclude that the answer to 8+5 must be three more than 10, so the answer is 13.
Children need time and space for all of the sophisticated thinking happening in phase 2. As Baroody, one of the originators of these phases, states-this phase is where the discovery of patterns and relationships, the “flexibility, increased efficiency, and selection of appropriate strategies are all developed'' (Baroody 2006).
Children need meaningful practice over an extended amount of time in order to fully develop the majority of the components of fluency. They need teachers to engage in meaningful conversations with them to advance their thinking. Sufficient practice in this phase will create children who have a solid conceptual understanding and are fluent with basic math facts. Once children have cycled through the Phases for foundational facts, including the 2s, 5s, and 10s for multiplication for instance, they also need the opportunity to cycle through these phases for derived strategies, such as the 4s, 6s, and 9s for multiplication.
Phase 3 - Mastery
And once given more time, children will have made enough connections in order to master facts with recall within 3 seconds. Children only truly move to Phase 3 - Mastery, after they have had lots of meaningful practice with reasoning strategies where they are actively making sense of problems and developing their own strategies. When we talk about being automatic with facts, we are really talking about Phase 3 Mastery.
If children are rushed through this reasoning strategies phase, we run the risk of deflating a child’s confidence in themselves in later grades. That is to say, if facts are forgotten at some point for whatever reason, “students [won’t] have efficient, appropriate, and flexible strategies to fall back on which explains why some middle grade students are still counting.” They have slipped back into Phase 1 in order to solve basic facts (Bay-Williams and Kling, p.5).
This Phase framework provides adults with a much richer language and helps bring the NCTM teaching practice of building procedural fluency from conceptual understanding to life (NCTM 2014). It allows for a better understanding of what children currently know and what new specific opportunities to learn we can provide.
Imagine how we could help Ben, and really all children, if we as parents and educators frame our conversations around this definition of fact fluency and the phases in which fluency develops, instead of on speed.
The conversation might go something like this -
Parent: How’s Ben doing in math? What phase do you think he’s in right now for multiplication fact fluency?
Teacher: Right now he’s in Phase 1 - the counting phase - for foundational multiplication facts. Last week when I interviewed him, he was able to recall all of the 10s automatically, and needed a visual for 2s and 5s.
Parent: Okay. How do we help him move on to Phase 2?
Teacher: He has foam counters that he uses to help build visual models and explore new strategies during our work time in class. At home, have Ben do the same- you can use things like dry cereal or dry beans. Right now he needs that visual image for 2s and 5s. It’s important to let him do this in order to see the structure in numbers, to see relationships, and to develop strategies that will help him move into Phase 2... Here are some games to play at home and some sentence starters for you to use while playing with him to help Ben focus on building his understanding around multiplication as equal groups. This will also help to increase his flexibility with numbers.
Parent: Okay, thank you. I remember your last newsletter talking about creating visual images and flexibility being an important part of fluency. We’ll be sure to play those games at home too. Thank you for the suggestions on what to say to him while playing.
Wow. That conversation was so much more engaging and informative. This robust definition of procedural fluency gives adults a starting point for more informed parent to teacher and teacher to teacher conversations. It helps us move away from judgements and focus on the strengths of a child and the next steps in their instruction. While the teacher’s response did become much more focused and in-depth, the parent starting with a better understanding of fluency was the springboard for a more positive and informed conversation about Ben’s progress. Just having an overall awareness and knowing that there are 3 phases to fluency is extremely helpful as a parent.
Developing common definitions and understandings not only help us have more informed discussions, but it can also help to change our beliefs about the teaching and learning of mathematics. The adults in the second exchange do not sound as if they are focused on “practicing procedures and memorizing basic number combinations.” Instead, it sounds more reminiscent of the alternative belief that is productive - that “mathematics learning should focus on developing understanding of concepts and procedures through problem solving, reasoning, and discourse” (NCTM 2014, p.11).
Teachers - examine your current plan for basic fact fluency- Does it involve a robust definition of fluency? Are you accounting for the 3 phases-for both foundational and derived facts? Might you be holding onto any unproductive beliefs?
Regardless of whether we will have in person, hybrid, or remote learning in the fall, we need to have a solid plan in place for children to build fact fluency. If you don’t already have a plan from a previous year or if you want to start fresh, use the template on the Kids Math Talk website to help you get started.
Share your plan with parents at the beginning of the year. Give them the handout provided in the show notes for this episode, so that you can begin to partner together. Include quick math tips, definitions, and examples in your family newsletters to keep the conversation about math going and to build capacity. Talk with your colleagues and principal about incorporating a math night - maybe even a virtual math night - where parents and educators can get together to develop the language of mathematics together.
One resource to grab is the NCTM book Principles to Actions (2014) which has lots of user-friendly charts that describe productive and unproductive beliefs as well as suggestions on what teacher and student actions are for a variety of practices. I also suggest the book Math Fact Fluency (Bay-Williams and Kling 2019) to give even more background and to get suggestions on games that could be played to build conceptual understanding.
Parents, this Math Fact Fluency book is a great read for you as well.
Also, download the tips and questions sheet (see top) to help you reframe how you ask questions about your child’s basic fact fluency. Ask your school about plans for family math nights and about the school’s overall basic fact fluency plan.
The more specific and direct we can be, the better for our children. Let’s continue to partner together for active and positive Kids Math Talk.
Baroody, A.J. (2006). Why children have difficulties mastering the basic number combinations and how to help them. Teaching Children Mathematics, 13(1), 22-31.
Bay-Williams, J. M., & Kling, G. (2019). Math fact fluency: 60+ games and assessment tools to support learning and retention. Alexandria, VA: ASCD.
National Council of Teachers of Mathematics (NCTM). 2014. Principles to actions: Ensuring mathematical success for all. Reston, VA: NCTM.
-------------.2014.Procedural fluency in mathematics. Retrieved June 29, 2020, from https://www.nctm.org/Standards-and-Positions/Position-Statements/Procedural-Fluency-in-Mathematics/