Hello ICTM members! My name is Angie Shindelar and I serve on the ICTM Board as the Vice-President for Elementary. My husband and I have two adult children who were both recently married. We keep telling them and their spouses how excited we are to join the grandparent world. No pressure. :)
I previously taught elementary and middle school math at Nodaway Valley CSD. I am currently a Math Consultant for Green Hills AEA. I enjoy supporting teachers in learning around math content and instructional strategies. This is an exciting time to be a math educator. Momentum has been building across the U.S. in recent years to improve math teaching and learning.
The teaching and learning of basic facts has been the theme of my two previous articles. You can read them here and here if you missed them. The most recent article examined the difference between memorization and automaticity. In this article I will discuss their cousin, fluency.
The Iowa Core Math Standards for K-3 use “fluently” to describe expectations for learning the basic facts. The specific standards are listed here:
•Fluently add and subtract within 5. (K.OA.5.)
•Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13). (1.OA.6.)
•Fluently add and subtract within 20 using mental strategies.2 By end of Grade 2, know from memory all sums of two one-digit numbers. (2.OA.2.)
•Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers. (3.OA.7.)
While the Iowa Core Math Standards provide clear expectations for learning the basic facts, questions are common about the authors’ intent of the word fluently. In thinking back to our own experiences learning basic facts, for most of us, the expectation was memorization. Therefore, we typically equate fluency with memorization. However, the use of the word fluently in the standards hints at a broader definition. For example, first and second graders are expected to demonstrate fluency for addition and subtraction facts by using various listed strategies and the inverse relationship of the operations. Third graders are expected to fluently multiply and divide by understanding the inverse relationship of the operations and by using properties of operations. So, if the intent of using fluency as a description in the standards was memorization, why would lengthy descriptions of strategies be given? You really don’t need strategies to memorize. You just memorize, right?
The National Research Council provided us with what has become the accepted research-based definition of fluency: “skill in carrying out procedures flexibly, accurately, efficiently, and appropriately” (National Research Council, 2001) This definition expands fluency beyond memorization. Below the four parts of the definition described in relation to the K-3 fact fluency standards. They show us that the authors likely had the National Research Council’s definition in mind.
Flexibly means the student can describe more than one way to solve the problem using deriving strategies. One strategy may be preferred, however s/he can describe another strategy when prompted. The standards describe a variety of strategies for the student to experience and be familiar with.
Accurately speaks for itself. No matter what strategy is used or if a fact has been committed to memory, the expectation is that it be accurate. This is inferred in the standards in the expectation to know the facts from memory, in that, when students have had enough experience practicing a strategy they commit the fact to memory.
Efficiently coincides with the developmental progression of how children solve problems described in Teaching Children Mathematics: Cognitively guided instruction (Carpenter T. P., et al, 2014). Children begin as direct modelers counting out objects for each addend and then recounting all of the objects to find the sum. With many experiences and discussion they move into counting on. In the past, students were expected to jump from counting on to knowing the facts. Research has helped us understand there is a critical phase between counting on and knowing the facts called deriving. Most of the strategies described in the fact fluency standards are deriving strategies. With many experiences and practice students will move away from counting on and use deriving. When looking at other standards in K-3 one can see expectations that support this progression. Many of the K-3 standards describe building understanding of number relationships which is foundational to understanding derived strategies. Solving a fact efficiently includes using a derived strategy with ease and knowing from memory.
Appropriately also coincides with the developmental progression described above. We expect that, with many experiences and practice, students will use derived strategies with ease. Note that counting on would not be considered appropriate. Counting on is at the beginning stages of the developmental progression. We do not want students to rely on counting as a fallback strategy for a forgotten fact. This would include skip counting for multiplication and division.
Thinking about these four parts of the fluency definition, consider the three student examples below to see what fluency might look like when achieved.
T: What is 7 + 8?
S1: Um…(brief think time)...15.
T: I noticed you thought about that one a bit. Tell me how you solved it.
S1: Well 7 and 8 are both close to 10 so had to decide which one
I wanted to make into a 10. I chose the 8 because it is closer.
So I took 2 from the 7 to make the 10. Then I added 10 + 5.
T: What is 7 + 8?
S2: Um…(brief think time)...15.
T: Tell me about your strategy. How did you solve 7 + 8?
S2: Well I was going to make a 10 with the 8, but then I saw that it would
be just one more than 7 + 7. And I know 7 + 7 = 14.
T: What is 7 + 8?
T: That was quick! You know that one automatically now!
S3: Yep, I have practiced that one a lot, so I know it.
T: What strategy did you use when you practiced?
S3: Doubles.7 + 8 is just one more than 7 + 7.
These students demonstrate fluency with flexible, accurate, efficient, and appropriate strategies. They either knew the fact or used a derived strategy with ease. The biggest benefit to deriving is you don’t lose the strategy over time. When presented with a fact that may have been previously committed to memory, you can recall the derived strategy typically used and solve the fact. Unfortunately, a common reaction to having forgotten a memorized fact is to count, or worse yet, guess.
Teaching derived strategies is not new. Many textbook lessons have included lessons on these strategies for many years. Why, then, do students still struggle with fluency? The answer to this seems to center around the time given to learn and the experiences provided.
Derived strategies take a considerable amount of time to develop. However, basic fact instruction is typically compacted into one chapter, early in the school year, expecting students to learn strategies quickly and with minimal practice. Spreading the instruction across the year focusing on one strategy at a time is an alternative way to think about achieving fluency for basic facts. More time per strategy can be given for instruction and practice. Time, however, will not solve everything.
It is also essential to consider the kinds of experiences students engage in to learn about each strategy. Students need many opportunities to think about how a strategy works and a lot of practice in using it. A variety of visual models and discussion, thoughtful ways to recording the thinking, word problems with number choices supporting the strategy, and games to practice using the strategy are all important to the learning and efficient use of each strategy. Over time the strategy becomes automatic and fluency is realized.
Recent research in fact fluency suggests that games are the best way to practice and reach fluency. Games that have been specifically designed to practice a strategy, such as using a double, provide multiple opportunities to practice a strategy. Games also provides practice without the time pressure often found in timed tests. If we want students to use derived strategies, we have to create settings that allow the student to practice the thinking and build fluency over time.
To summarize, fact fluency in the Iowa Core Mathematics standards has a much broader definition than memorization. To reach the desired fluency we so desperately want our students to achieve, we must rethink what it means to be fluent and give careful consideration to the time and experiences we provide. If you are interested in reading more about basic fact fluency check out Jennifer Bay Williams and Gina Kling’s work. They have written several articles for NCTM in recent years and have just published a book, Math Fact Fluency, 60+ Games and Assessment Tools to Support Learning and Retention (ASCD, 2019).
I hope you have found my thinking about fluency with basic facts helpful. I am always interested in others’ thoughts about this important topic. Contact me with thoughts or questions at firstname.lastname@example.org.