Finding Meaning in Fluency

Contributed by RoseMary Hunt

Math Teacher, Turtle Lake Elementary, Shoreview

Math Instructional Coach, Mounds View Schools

MCTM Region 4 Director

“My mom showed me a trick,” came from a third grader in the middle of the room. The look of horror from four students at the table in front of me was priceless. They know “trick” is a banned word in my math class. They started whispering to me as the first student rambled on about zeros…

“…there are no tricks.”

“…there are short cuts, but they mean something.”

“…what was he doing yesterday!”

“…he wasn’t listening!”

We had been working on extended math facts (see photos: 5 x 40, 30 x 40, 4 x 200, 20 x 300). We took a few days and built models with base 10 blocks; created arrays using representations of the base 10 blocks; decomposed numbers and used commutative and associative properties; looked at patterns. In the example 30 x 40, students came to see that it was a 3 x 4 array which gave us 12 items in the array; the items each had a value of 100 (from 10 x 10), so the product must be 1200 (12 groups of 100).

Upon seeing the consistent patterning the students established the “short cut” to getting the product more efficiently. BUT, this short cut had been DISCOVERED in the exploration of the arrays and decomposition of numbers.

  

 

 

 

 

 

Where are the weaknesses?

During a cross-grade level meeting, a Kindergarten teacher had asked me about any particular weaknesses I noticed as students get older: Computation? Geometry? Problem solving? I pondered this for a while. Considering the history I have with math students in grades 3, 4 and 5, I can safely say that once students learn a procedure, they stop thinking. Students who come into my classes already knowing procedures (or short cuts) cannot explain why these procedures work—what is the mathematical structure? Getting them to explore is a challenge, sometimes, because, as they say, “I already know how to do it faster.” So, we work on it. In time, the “ah-has” come.

Let’s take this all the way back to the foundation…math facts (addition, subtraction, multiplication and division).

Jo Boaler, professor of math education and researcher at Stanford University, says that speed with numbers isn’t what makes a student good at math. It is the deeper understanding of numbers, the patterns, and the connections that foster fluency with numbers. She states that there is a “misunderstanding” of the word “fluency.” She argues while quick recall is certainly desired (as concepts become more complicated, students shouldn’t be spending time on recalculating simple facts), rote memorization should not be the focus in the classroom. Activities should foster use of numbers in different ways, repetition, and visuals. And forget those timed tests, she says—they create math anxiety for many students and can be counterproductive.

Finding a balance point is important.

When students learn to think about mathematics they tend to do better down the road with harder mathematics. They haven’t filled a mental file cabinet of procedures and rules (tricks if you will) that don’t have meaning. So that when they are confronted with something new, they are not trying to frantically (and sometimes randomly) search for “that” formula, “that” procedure, “that” trick which will work. They can reason out, “What do I know?”; “What do I need to find out?”; “What will help me here?”

Applying rhe Common Core Standards for Mathematical Practice can help.

Standard #7 — Look for and Make use of Mathematical Structure.

Encourage and coach students to find patterns and repeated reasoning in the numbers and facts they are working on. Encourage and coach students to decompose numbers and problems into smaller ones (compatible numbers for making tens, arrays within arrays, decomposing extended facts described earlier). Explicit modeling is important for the young ones just beginning to fill their math toolbox. And as for those timed tests? Consider alternative ways to assess a student’s fluency with facts: observation, student interview, number talks, problem solving, number puzzles, consistent accuracy. The goal is getting to those derived facts; and quick recall will come. The way in which we guide students and their journey to this end can make all the difference.

Read more about Jo Boaler’s ideas and research:

Should we Stop Making Kids Memorize Times Tables?  hechingerreport.org/should-we-stop-making-kids-memorize-times-tables

Fluency Without Fear  http://youcubed.stanford.edu/fluency-without-fear/

The Educators on BBC http://www.bbc.co.uk/programmes/b04gw6rh