Math Matters

Within the math education community, a lot of confusion surrounds the word fluency. Yet state standards call for fluency at each grade level. Grades 2 and 3 standards specify that students know a standard “from memory,” phrasing which may further cloud our concept of fluency.

In the Ignite Talk “There IS a Difference,” K-5 math educator Graham Fletcher explains the subtle yet powerful difference between memorization and from memory.

Memorization
Many of today’s adults practiced facts using flash cards. They drilled and drilled and drilled. For some, the effort appeared successful; they could rattle off their facts within the two-minute time limit. For other learners, memorization did not happen. Instead, they arrived at the conclusion that they were simply bad at math.

Fletcher suggests we take a closer look at the definitions of “fluency” and “from memory.” He reflects on the use of “fluency” within language arts where some students “read” 120 wpm with no understanding while others read at 80 wpm and can fully explain what they just read. He notes that fast does not equal fluent.

Rather, true fluency means that students have developed efficient, accurate, and flexible ways of learning.

Efficiency - Efficiency implies that the student does not get bogged down in many steps or lose track of the logic in the strategy. An efficient strategy is one that the student can carry out easily, keeping track of sub-problems and making use of intermediate results to solve the problem.

Accuracy - Accuracy depends on several aspects of the problem-solving process, among them, careful recording, the knowledge of basic number combinations and other important number relationships, and concern for double-checking results.

Flexibility - Flexibility requires the knowledge of more than one approach to solving a particular kind of problem. Students need to be flexible and choose an appropriate strategy for solving the problem at hand. They can use one method to solve a problem and another method to double-check the results.

But if memorization doesn't promote fluency--efficient, accurate, and flexible ways of learning--what does?

From Memory
Fletcher suggests that it comes down to strategy. Memorization is void of strategy while learning facts “from memory” relies on strategy. When we practice strategies—when we use them repeatedly, practicing them over and over and over—they become automatic. Fletcher says, “Automaticity of a strategy can appear to be memorization…but it’s not.” Automaticity comes through “learning, repetition, and practice,” until it’s become a natural response. A student who can efficiently solve 6 x 7 by knowing that 6 x 6 = 36 and adding one more set of 6 has a mathematical depth of understanding that the mere memorization of facts will never equal. Fletcher reflects, “A student’s ability to memorize math facts reveals nothing about their mathematical competence.”

This reminds me of a former teaching partner. One day, in a teacher training, we were discussing math fact memorization. She broke down, saying that she’d never been able to memorize her facts. Yet her mathematical abilities far exceeded mine. As a child, I excelled in fact memorization. Highly competitive, I thrived on racing through the multiplication fact sticker chart. But, like the 120 wpm child in language arts, I had no clue what I was doing. I was simply memorizing something that had no meaning, no context. As an adult working with my partner, I had no idea she didn’t have her facts memorized. Her strategies were so accurate, efficient, and flexible—so FLUENT—that she could find answers to math facts as fast as I could. Yet, in the end, memorization was all I had to fall back on, while her bank of mathematical strategies was endless.

Accurate, efficient, and flexible strategies come from memory, not from mere memorization. Students will thank us for remembering the difference!

Watch the video here.

The teacher prepares problems and/or questions on individual cards.

Each team of students receives a set of all the question cards.

Students play in a teach of 4.

Student 1 holds the question cards in a fan facing him/her and says "Pick a card, any card."

Student 2 picks a card at random, reads the card aloud and allows 5 seconds of think time.

Student 3 answers the question.

Student 4 responds to the answer

• For right/wrong answers, student 4 checks and then either praises or tutors.
• For questions that have no right or wrong answer, student 4 does not check for correctness but praises and then paraphrases the thinking that went into the answer.

Students rotate roles one person clockwise for the new round.

Modification - students can play in pairs and alternate. Student 1 fans while student 2 picks and reads, then student 1 answers and student 2 checks, then tutors or praises.

Combine similar items and challenge students to choose which item in the group does not belong. Students are asked to justify their reason for selecting the number, symbol, representation, or mathematical term that does not fit with the others.

This probe requires students to analyze relationships between items in a group, to think about similarities and differences, and to use mathematical knowledge and language to justify their choice.

Click here for an example I made called Odd One Out Fractions then make one of your own for content coming up in your grade level and share it with me by email.

Teach students to ask themselves questions before, during and after they solve problems or work tasks. Their questions should be formulated so they focus on the problem at hand. Examples include: What technique did I use to solve a similar problem in the pas? What is the problem asking for? What information am I given?

Students should also ask themselves general questions to self-regular their performance such as: Is there anything I don't understand? Am I headed in the right direction? Is there any information given in the problem that is not immediately obvious? Have I made any careless mistakes?

In Math, the reasonableness of the answer obtained is not often considered. Students believe that make problems are contrived; therefore, the solutions do not really apply to the real world. This self regulating experience should be extended to this aspect of self-checking by considering not only the procedure used, but also the meaningfulness of the answer arrived at.

What do you predict will be the outcome? Will the answer be larger or smaller than what you started with? Why?

How can you describe what you were thinking about when you....?

How do you know when (something new) fits the rule or pattern?

Using Questions to Stimulate Math Thinking

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