# Misconceptions about Mathematical Literacy &

Instruction that Inhibits Change

Questioning is the beginning of understanding!

Franklin

## Discussion for Mathematical Pedogogy Misconceptions &

the Eleven Question survey

### Question 1

Algorithms (traditional procedures for adding, subtracting, multiplying, and dividing) are (necessary / not necessary) to be a mathematician.

#### Answer

Yes and no!

A mathematician doesn't value traditional algorithms to be used to calculate solutions to mathematical operations. A mathematician values traditional algorithms for the beauty of their theoretical elegance that represents a solution created by mathematicians to solve similar problems no matter how complicated or difficult the calculations.

< Question 2 >

### Question 2.

The sooner a person learns the basic facts the better. (True or False)

#### Answer

Yes and no!

Knowing the basic facts provides a person with reliable, accurate, and efficient solutions to basic operational facts. However, the research suggests that once people know the basic facts they discontinue their quest for understanding relationships of numbers and how they combine to create solutions. Those who struggle to learn, or are not rushed, develop different strategies to combine basic facts, which they add to their repertoire of strategies for understanding both number and operational sense. The benefit is better mathematical understanding and skills in the long run.

< Question 3 >

### Question 3.

The (earlier / later) children start to learn the traditional algorithms the better.

#### Answer

The research seems stacked against this.

Not teaching students a traditional algorithm seems to increase student achievement. Students that invent algorithms have better mathematical abilities and problem solving skills and some national curriculums that don't introduce traditional algorithms until third grade or later out perform countries that introduce algorithms earlier.

< Question 4 >

### Question 4

Countries that score (lower / higher) in mathematics than the United States spend more time on algorithm instruction and start earlier.

#### Answer

The United States students spend more time practicing algorithms than most other students in the world.

< Question 5 >

### Question 5

Children and some animals (are / are not) born with the ability to recognize the value of 1, 2, and 3.

#### Answer

Yes they are!

Studies suggest that infants less than one distinguish these number values. Studies with birds and mammals, other than humans, can also.

< Question 6 >

### Question 6

Learners; inaccuracies with addition, subtraction, multiplication, and division usually (are / are not) due to procedural errors.

#### Answer

Most computational errors are related to lack of number or operation sense; less often to inaccuracies or not knowing procedural steps.

< Question 7 >

### Question 7.

Students who use calculators have (higher / lower) math achievement.

#### Answer

Not only do learners who use calculators have higher achievement results, they also are better at higher order thinking skills, do better on mental computation, have more positive attitudes towards math, and are less likely to use a calculator when solving a problem.

< Question 8 >

### Question 8.

Rote learning (does / doesn't) interfere(s) with number and operation sense making.

#### Answer

Learners that have memorized basic facts and traditional algorithm procedures become very efficient in solving problems that are presented to them to calculate. They are less willing to look for alternative solutions, even if such solutions are significantly quicker, and less willing to question why or how a particular solution works or has failed.

< Question 9 >

### Question 9.

Practice without conceptual understanding (will / will not) enhance learning.

#### Answer

Learners invented algorithms are often more successful, particularly when they build on their thinking about operations.

< Question 10 >

### Question 10.

Practice without prior experience with other methods (will / will not)

enhance understanding.

#### Answer

Absolutely not.

Practice without understanding is like repeating a phone number until you dial it. As soon as the practice stops, remembering stops. Practice without understanding isn't connected to any other mathematical understanding. Without this connection memories fail and no ability to know when to use the information is created. The student struggles to memorizie a procedure, but isn't sure when or how to use it. Hence, the value of the information and the learners' self-esteem is lowered. Not a way to develop self-efficacy in mathematics.

< Question 11 >

### Question 11.

Practice seems to legitimize a single procedure and (enhances / limits) students computational fluency.

#### Answer

Absolutely.

The proof is in the masses of American Citizens that get outraged by the fact that students may not be able to use a traditional algorithm.

However, those same masses seem not to be outraged by larger numbers of American Citizens that are unable to solve story problems, or explain the basic operations of addition, multiplication, subtraction and division, or being able to use fractions for everyday measuring.

< Question 1 >

## Research

What every person should know about how our children and adolescents are being miseducated. ... Further readings!