Building Mental Math & Computational Strategies

Rebeca Itzkowich, M.A., Senior Instructor, Erickson Institute, Early Math Collaborative

Students are memorizing mathematical rules and procedures to compute mathematical problems, but they do not understand the numerical relationships and why these rules work. They don’t learn the fundamentals of math and don’t understand the main ideas. So students think that math is just a series of rules and procedures to memorize. They are taught that being good in math means doing computations using these rules accurately and as fast as possible. For most students, this means they have no understanding of what the numbers of an equation stand for. Therefore, even if they memorize the formulas, they don’t understand why they work. As mathematics content becomes more complex, many students who were not taught mathematics as a sense-making discipline, cannot persevere in problem-solving. They eventually lose any confidence in themselves where math is concerned and often choose not to pursue more complex math classes.

We need to teach students how to make sense of numbers, see their relationships, think flexibly to develop efficient computational strategies, explain how they approached and worked through the computation to prove their solution.

Fluency in math includes four components.

1. Accuracy – solving problems accurately

2. Efficiency – solving them in a reasonable amount of time

3. Flexibility – using a variety of strategies to solve problems

4. Appropriately – choosing best strategy to solve each problem

According to Piaget, there are three kinds of knowledge

1. Physical knowledge: These are physical attributes of things such as water is wet, snow is cold, summer is hot, the cat is furry, etc. Physical knowledge is knowing physical attributes about something by exploring it and noticing its qualities.
2. Social knowledge: These are names and conventions, made up by people. The names of the week, how you behave in certain social situations, the laws of Kashrut, mathematical rules and procedures. Social knowledge is arbitrary and knowable only by being told or demonstrated by other people.
3. Logico-mathematical knowledge: This is the creation of relationships. The brain builds neural connections which connect pieces of knowledge to one another to form new knowledge. Logico-mathematical knowledge is constructed by each individual, inside his or her own head. It doesn’t come from the outside. It can’t be seen, heard, felt or told. *(http://http://blogs.psychcentral.com/always-learning/2010/01/three-kinds-of-knowledge/)

Most people learn math by heart, with rules and procedures. In other words, they learn it as social knowledge. They don’t know why it works. This learning is often fragile because it depends on memory and does not support understanding.

Number sense is being able to:

• Think and reason flexible with numbers

• Use numbers to solve problems

• Spot unreasonable numbers

• Understand how numbers can be taken apart and put together in different ways

• See connections among operations

• Figure mentally

• Make reasonable estimates

Why is there a need to develop Number sense? Memorizing facts and rules does not work well. People can forget, and then they are at a loss. Or memorizing is not their strength in learning.

Four relationships that build number sense are:

1. Spatial Relationships – recognizing how many without counting, by seeing the visual pattern. You can use subitizing cards, dot cards, and  5 and 10-frames.

2. One and Two more, One and Two Less – knowing which numbers are one or two more or one or two less that any given number.

3. Benchmarks of 5 and 10 – knowing how numbers relate to 5 and 10. 5 and 10 frames are very useful here.

4. Part – Part – Whole – ability to conceptualize a number as being made up of two or more parts is the most important relationship to develop.

Number sense needs to develop gradually. It can’t be taught. Students need multiple varied experiences to develop relationships amongst numbers.

How do we help students develop number sense? Number Talks are five to fifteen minutes of purposely chosen mental discussions about the number relationships and strategies used for computations. Number Talks follow Standards for Mathematics. They do not use pencil and paper. This encourages students to think through approaches using number relationships rather than procedures and rules.  There can be many appropriate strategies to solve a problem. Number Talks allows for differentiation by encouraging students to find the strategies to solve the problem that bests suits their way of thinking and learning.

Start with a problem all children can be successful with. 

• The problems should be designed to elicit specific strategies that focus on number relationships. 

• Students are given problems in either whole or small group settings.

• Students are expected to mentally solve them accurately, efficiently, and flexibly.

• By sharing and defending their solutions and strategies in a safe environment, students have the opportunity to collectively reason about numbers while building connections to key conceptual ideas in mathematics. 

Number Talks occur in a strong risk-free classroom community, where the teacher:

• Uses intentionally selected problems that will lead students to discover a mathematical property of an operation (say the communitive property) or build a numerical relationship.
• Records student thinking.
• Supports growth of logico-mathematical knowledge

Number Talks focuses on the mathematical process rather than getting the answer.

• Students are asked to defend or justify their answers to prove their thinking.

• Students have a sense of shared authority in determining whether an answer is accurate.

• The teacher is not the ultimate authority, but rather a facilitator.

• Wrong answers are used as opportunity to unearth misconceptions.

• Students investigate their thinking and learn from mistakes.

• Mistakes play an important role in learning and provide opportunities to question and analyze thinking, bring misconceptions to the forefront, and solidify understanding.

Students have the opportunity to:

• Clarify their own thinking.

• Consider and test other strategies to see if they are mathematically logical.

• Investigate and apply mathematical relationships.

• Build a repertoire of efficient strategies.

• Make decisions about choosing efficient strategies for specific problems.

How do teachers execute Number Talks? By:

• Creating an environment that is safe and risk-free, for making mistakes and promoting student participation in discussion through mutual respect.

• Providing purposeful computation problems.

• Promoting mental math.

• Facilitating classroom discussion

• Asking honest questions

• Listening to students’ thinking

• Learning with the students

• Modeling student thinking to make the mathematics explicit.

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