Recently a great conversation regarding computational fluency flew across the internet highways. I thought it was worth sharing with all of you here on this blog I’d like to start by sharing a quote from the National Math Panel Report that encapsulates the essence of the growing body of research calling forth discussion around this topic.
“The mathematics curriculum must simultaneously develop conceptual understanding, computational fluency, and problem-solving skills. The development of these concepts and skills is intertwined, each supporting the other and reinforcing learning.
Teachers can help by providing students with sufficient practice distributed over time and including a conceptually rich and varied mix of problems to support their learning. In addition, teachers should encourage and support students in their efforts to master difficult mathematics content. Students who believe that effort, not just inherent talent, counts in learning mathematics can improve their performance.”
For more discussion on this topic, I’d like to point you to another great resource. The very first newsletter on the Teacher to Teacher website (October 2008) was called Developing Computational Fluency. I’d encourage you all to check it out and also read through the question and comments below.
Do you have any other questions or comments that haven’t been explored here? I’d love to hear your thoughts on this topic. Please post your comments or questions below.
Now, here is the question as it was originally posed:
From: “Hall, Kathryn B (TDO Eng.)” <email@example.com
I am working with the elementary teachers in Lebanon throughout this year to help them transfer their instruction to the new standards, work on concept retention, and make sure that the focus of their curriculum are the new standards and focal points. They have a question for OMEC that I’ll bet occurs in many other districts.
What does “fluency” mean? How is it measured? For example, there are targets for reading speed (40 wpm, 100 wpm, etc. that grow as the grade grows). Writing has targeted length of essays, by grade. Math just says “fluent” in multiplication, or addition or whatever, depending on the grade. It is clear that different teachers have wildly different ideas of what fluent means.
What did the authors of the standard have in mind with this phrase?
The rest of this blog is devoted to sharing all the comments that came in regarding this topic from OMEC members and other math professionals from around the state.
I do not have a precise answer. I don’t like a quantitative measure like 50 facts in 2 min. as that assumes fluency for the fast thinker/writer who in fact may be counting in his head or on his fingers, and assumes non-fluency for the slow methodical kid who can’t do anything quickly, but may be fluent in his thinking and strategies. To me fluency implies a non-effort comfort. I can find the answer using my strategies and it isn’t a problem. Non-fluency would imply a struggle, that finding the answer is work. Perhaps our progression is in the skills–addition facts in first (or at least most of them), subtraction in 2nd, multiplication (most) in 3rd, all mult. and div. in 4th, expand this knowledge to decimals in fifth. Just thinking as I write. I personally hate the idea of time as it penalizes the slow thinker, also the ADD or ADHD if the task is more than about a minute as they can’t hold concentrated attention under stress for very long. I’m sure you hear my special ed. bias.
Kathy Reed, Retired Teacher and Math Consultant
I totally agree with your thoughts Kathy. For kids, taking out the writing piece is very important. As math TOSA in my district, I have been helping with first grade fluency interviews. If they KNOW it, quickly (verbally) or use a smart, quick strategy like our district adopted material Bridges teaches—I think that is fluency. Just had another thought. When I taught 3rd grade, we used a rule of 3 seconds per problem. Somewhere I heard someone important say “they should be able to do the problem in the time it takes to say the problem—like five plus seven is —-12. Hope that helps. I will ponder this some more and get back to you if any more thoughts occur to me.
Jane Osborne, Hood River School District Math TOSA
I have been working with teachers and ODE folks to help write items to the 3rd grade standards for 2007. Fluency has been a tricky one to write to because it does mean exactly what you said, that students no longer need calculators to help them to quickly recall their facts, be it addition, subtraction, multiplication or division.
If you look at the Number and Operations, Algebra and Data Analysis standard, 3.2, in Oregon, it uses verbs such as: represent, apply models or apply increasingly sophisticated strategies based on the number properties to solve multiplication problems involving basic facts.
In the Test Specifications for grade 3 (page 4) it says:” Central to this Standard is the development of number sense – the ability to decompose numbers naturally, use particular numbers like 100 or 0.5 as referents, use the relationships among arithmetic operations to solve problems, understand the base-ten system, estimate, make sense of numbers, and recognize the relative and absolute magnitude of numbers.
In these Standards, understanding number and operations, developing number sense, and gaining fluency in arithmetic computation form the core of mathematics education for the elementary grades…”
So this is looked at as a holistic approach to arithmetic and mathematics. Check out the Curriculum Focal Points from N.C.T.M. and you will see the same approach, using the terms “developing understandings of multiplication and division and strategies for basic multiplication facts and related division facts.”
Hope this helps!
Ann McMahon, Retired Educator and Math Consultant
From: Nicole Rigelman, PSU Math Education Faculty
Thank you for sharing your thoughts on this and pointing out the importance of solid number sense as a means to support fluency… not to mention, sense making.
A colleague of ours, Tanya Ghattas, former math specialist in Salem-Keizer now administrator in Portland Public gave me a number that is helpful – 3 seconds per problem on average. What this allows for is both facts one knows with automaticity and facts that one may use a known fact to get to (i.e., 6 + 7 can be solved by knowing the fact 6 + 6 = 12 and adding 1 or by knowing the fact 7 + 7 = 14 and subtracting 1… along with others like knowing 6 + 4 = 10 and needing to add 3 more since 4 + 3 = 7, additionally knowing 7 + 3 = 10 and add 3 more since 3 + 3 = 6.). I’m sure you can imagine the corollary with subtraction or multiplication/division. You should note that role of strong number sense in accomplishing this work with known facts not to mention the connections to important number properties (associative and commutative).
What I like about this number too is if a teacher gives an occasional 100 problem practice sheet, easily generated from resources on the internet (AKA Mad Minute), he/she would only need to dedicate 300 seconds (or 5 minutes) to this practice and then know which kids have met this level of “fluency” and which kids need further support. If a teacher is also supporting kids by coaching them to go through and do all that they know automatically first and then work on those that are more challenging… then he/she has gathered more formative assessment data and can share with parents what they might like to work on at home with their child… or share with school support individuals number facts to target, etc.
I think that we also need to keep our eye on the standards advocated by the National Governors’ Association and the CCSSO, the Common Core Standards.
Thanks for asking this question. I was not there when the standards was drafted, so I can’t speak to the conversations that occurred around the concept of fluency when the document was drafted. But I’d be happy to share my understanding of the concept of fluency within math education research, since it sounds like an open discussion on the topic.
The concept of fluency is more than just a rate, just as fluency in Spanish is more than how fast one can say words. It is a larger idea that has more to do with the concept of automaticity, that is how automatic, or effortless, it is to access the needed facts to carry out a procedure. The National Research Council in Adding it Up, defined fluency as:
“Procedural fluency – skill in carrying out procedures flexibly, accurately, efficiently, and appropriately” (National Research Council, Adding it Up, 2001, p. 5)
Thus four aspects of fluency include:
Flexibility – The ability to carry out a procedure using more than one method, or strategy. I like Bruner’s model of concrete, pictorial, and symbolic representations, and like to think that students should able to calculate answers with each method. This may include such methods as base-10 blocks and 10 frames (concrete), sets and base-10 grids (pictorial), in addition to symbolic methods including both alternative (e.g. partial sums, partial products) and traditional algorithms.
Accuracy – There is no fixed number here, but I’d image the goal would be to have students as close to 100% accurate as possible. Thus the accuracy of a “mad minute” is part of fluency as much as the speed.
Efficiency – This is a reference to how well a student can go straight to the correct answer. Researchers tend to stay away from defining a specific rate (such as problems per minute), since the rate can certainly vary. It is fine to calculate and track how many problems were correctly done in a fixed time, recognizing there is a range of rates that would be acceptable and should change with age and experience. With the goal of automaticity in mind, a better question would be “how laborious is finding an answer for the student?”, rather than simply how fast they can calculate. I would encourage more conceptual, perhaps less efficient, approaches at first, with the goal to refine methods to more efficient ones over time.
Appropriately – The final aspect of fluency refers to a student’s ability to select the appropriate method for the task. That is, they can pick an appropriate math “tool(s)” to find a solution. BTW, using a calculator is another algorithm (a very efficient one at that!), and teachers cannot simply ignore or “outlaw” the method. I think teachers within a district need to decide how, and when, will using a calculator would be appropriate. I image that students should be able to use calculators more as they get older and demonstrate fluency within arithmetic. It is also a practical method when needing to carry out repetitive tasks such as calculating the mean of large data sets.
This is as close to a working definition of “fluency” that I’ve been able to come up with, and is what I used when I taught my methods courses. I’d be happy to know if it is helpful, or if the definition needs clarification.
Mark R. Freed
Mathematics Education Specialist
Oregon Department of Education
255 Capitol St. NE, Salem, OR 97310
Office 503.947.5610 | Fax 503.378.5156