Some Common Preconceptions About Mathematics

1. Some Common Preconceptions About Mathematics

2. Some Math Preconceptions Discussed in “How Students Learn: History, Mathematics, and Science in the Classroom”

3. Why are associations with mathematics so negative for so many people? Food for Thought

4. Mathematics instruction often overrides students’ reasoning processes. Too often we replace our students’ reasoning with sets of rules and procedures that disconnect problem solving from meaning.

5. A True Story from John Holt

6. A good student was working on this problem: “If you have 6 jugs and you want to put 2/3 of a pint of lemonade into each jug, how much lemonade will you need?” The student answered, “18 pints”.

7. I said, “How much in each jug?” The student: “2/3 of a pint.” I said, “Is that more or less than a pint?” The student said, “Less”. I said, “How many jugs are there?” The student said, “Six”. I said, “Well that doesn’t make any sense”. (6 x something less than 1 = something less than 6)

8. The student shrugged his shoulders and said, “Well, that’s the way the system worked out.” Holt argues: “This student has quit expecting school to make sense. They tell you the rules, and your job is to put them down on paper the way they tell you. Never mind whether they mean anything or not.”

9. Preconception #1:Mathematics is about learning to compute.

10. For example, answer the following question.. What, approximately, is the sum of 8/9 plus 12/13? Many people immediately try to find the lowest common denominator for the two fractions and then add them, because that is the procedure they learned in school.

11. What, approximately, is the sum of 8/9 plus 12/13? Some people take a conceptual rather than a procedural approach and realize that 8/9 is almost 1, and so is 12/13, so the approximate answer is a little less than 2.

12. The point here is that all math problems do not necessarily have to be addressed by using computation. If you need an exact answer to this problem, then computation is the way to go. In this case conceptual understanding provides a way to estimate the correctness of computation.

13. Preconception #2: Mathematics is about following rules to guarantee correct answers.

14. Mathematics is a constantly evolving field that is far from cut and dried. It involves pattern finding and continuing invention.

15. “I want you to solve this problem by using this procedure.”or“I want you to do this problem like this.” How often did we hear this in school?

16. Example: When I do Praxis math tutorials, I allow students to problem solve using their own methods. Then I let them share their problem solving strategy with the entire group. One of the “beauties” of mathematics is that often there is more than one way to solve a problem.

17. Consider this problem.

18. A rectangle has an area of 36 sq.units and a perimeter of 30 units. What is its length and width?

19. The typical algebraic solution: xy = 36 2x + 2y = 30 A rectangle has an area of 36 sq.units and a perimeter of 30 units. What is its length and width?

20. However, Why can’t I do this problem by using “guess and check”? A rectangle has an area of 36 sq.units and a perimeter of 30 units. What is its length and width?

21. A rectangle has an area of 36 sq.units and a perimeter of 30 units. What is its length and width? 9 12 6 6 x 6 = 36 6+6+6+6=24 4 9 x 4 = 36 6 6 9+4+9+4=26 4 3 12 x 3 = 36 3 12+3+12+3=30 9 6 12 That’s it!!!

22. Preconception #3:Some people have the ability to “do math” and some don’t.

23. This preconception is widespread in the United States. How often have we heard this? • Boys learn math better than girls. • Children and youth from low socioeconomic situations and underrepresented groups can’t do well in math. • American children and youth have less mathematical ability than Asian youth. (These differences seem to be more related to teaching and expectations rather than ability.)

24. Teachers in some countries believe it is desirable for students to struggle for a while with problems. Whereas teachers in the United States try to simplify things so that students need not struggle at all. In the U.S., we tend to believe that ability is more important than effort.

25. It is imperative that we, as teachers, engage these preconceptions, as well as the preconceptions of our students.