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Routine Octahedron to Develop Math Fluency in the Classroom (Ebook)

Do you want to develop your students' mathematical reasoning and their ability to solve problems creatively and efficiently? In this e-book you will find 8 routines that can be used from K to 8th grade, perfect to do before starting class, and an octahedral die ready to print and assemble, in case you want chance to decide today's routine.

What is math fluency?

Math fluency can be defined as the ability to work with numbers, operations and procedures with ease.

At the last annual conference of the NCTM (National Council of Teachers of Mathematics), held in Los Angeles at the end of September, math fluency was one of the main topics. In fact, some members of the teaching team were fortunate enough to attend a workshop by professors Jennifer Bay-Williams and John SanGiovanni (2022), two of the authors who have most extensively written about math fluency in the United States (2021). Attending their conference helped us to see that many of the activities that make up Innovamat’s proposal allow us to develop fluency. And it also helped us strengthen our theoretical framework of this idea.

In a nutshell, we understand that fluency has three levels of depth:

If we focus on procedural fluency, which includes the other levels, we can say that it is made up of three components and six related actions that allow us to better understand what we are talking about:

As you can see, although fluency refers mainly to arithmetic, which is found within numbers and calculation, we understand that we could extend these ideas further, and talk about fluency in other branches such as geometry or statistics.

How is math fluency developed in the classroom?

There are several ways to work in the classroom to promote math fluency, from productive or reproductive practice activities to the math conversation dynamics that we encourage in most sessions. A great way, is the routines, that we also saw at the NCTM. Routines, with their brief and repeated format, allow us to get into the habit of practicing strategies in an agile way. Because they are recurring, after only a few repetitions there is no need to explain to the students how each routine works, and this makes them an ideal way to start the class and get to work right away.

With Dr. Bay-Williams’ specific permission to reflect on this brief theoretical framework and translate it, we have been working on it and have come up with a list of routines we already do at Innovamat that could contribute to developing math fluency. From this list, we have created the octahedral die of eight routines that we present to you today. Well… in fact, we presented it at the XXIV Jornada ABEAM (Morera L. and Vilalta, A. 2022), which brings together math teachers of all levels from all over Barcelona. We talked about fluency and took the opportunity to do a few rolls of the dice and try out some routines.

Do you want to know what the routines are for, from a content and process point of view? Would you like to find out which eight we have chosen and take some examples back to the classroom with you? Do you want tips on how to create your own statements? Would you like to print out and assemble the octahedral die to choose the routines at random? Just download this document.


Bay-Williams, J. M., & SanGiovanni, J. J. (2022). Accessing Fluency through Routine and Opportunity. NCTM Annual Meeting 2022. Los Angeles.

Bay-Williams, J. M., & SanGiovanni, J. J. (2021). Figuring Out Fluency in Mathematics Teaching and Learning, Grades K-8: Moving Beyond Basic Facts and Memorization (Corwin Mathematics Series) (1st ed.). Corwin.

Morera, L. & Vilalta, A. (2022). We develop fluency through various routines. XXIV Conference of the Barcelona Association for the Teaching and Learning of Mathematics (ABEAM) 2022. Barcelona

  • Albert Vilalta

    Albert Vilalta has a degree in Telecommunications Engineering and is a PhD student of Teaching of Mathematics and Experimental Sciences in the Universidad Autónoma de Barcelona (UAB). With more than 7 years of teaching experience, he is currently a professor of mathematics in the Faculty of Education at UAB and a member of the Innovamat teaching team, where he carries out research, development, training and communication tasks.

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