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Teaching Secondary Mathematics

Teaching Secondary Mathematics. “The meaning of letters in algebra”. What do students think? Some students believe that: Algebraic letters are abbreviations for words or things Algebra is a sort of shorthand Algebraic letters stand for a secret code.

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Teaching Secondary Mathematics

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  1. Teaching Secondary Mathematics

  2. “The meaning of letters in algebra” What do students think? Some students believe that:Algebraic letters are abbreviations for words or things Algebra is a sort of shorthand Algebraic letters stand for a secret code.

  3. “The meaning of letters in algebra” • Students often have difficulty with algebra because of misconceptions in various areas.

  4. “The meaning of letters in algebra” Examples of Misconceptions • If I add 3 to x + 4 I get 7. Reason : Ignoring completely the presence of letters. l

  5. “The meaning of letters in algebra” Examples of Misconceptions • 8 m and 8m are the same. Reason : Not distinguishing between letters used as units of measure and as variables..

  6. “The meaning of letters in algebra” Examples of Misconceptions • Shirts cost s rupees each and pants cost p rupees a pair. If I buy 3 shirts and 2 pairs of pants, what does 3s + 2p represent? Reason : Treating letters as objects.

  7. “The meaning of letters in algebra” Examples of Misconceptions • c = 3 because c is the 3rd letter of the alphabet. y = 4 because in the previous questions y was 4. Reason : Believing there are rules used to determine which number a letter stands for..

  8. Teaching for understanding

  9. “The meaning of letters in algebra” Examples of Misconceptions • What can you say about p if p + q = 12 and p is a natural number greater than q? p = 7 Reason : Thinking that letters always have one specific value

  10. “The meaning of letters in algebra” Examples of Misconceptions • a + b cannot equal a + c. Reason : Thinking that different letters always represent different numbers.

  11. “The meaning of letters in algebra” Examples of Misconceptions • 6x = 13, then x = 2 Reason : Thinking that letters can only stand for natural numbers..

  12. Misconceptions aboutNotation • a + b = ab • 2x + 3 = 5x Reason : Combining letters and numbers incorrectly because they think that operation symbols answer cannot be part of an

  13. Misconceptions aboutNotation • The area of this rectangle is 3 x x + 4 or x + 4 x 3. Reason : Neglecting to use brackets when needed

  14. Misconceptions aboutNotation • –6x = 12 x=12/6 x= 2 Reason : Disregarding signs when manipulating expressions.

  15. “The meaning of letters in algebra” Students may have developed these misconceptions from: Copying other symbol systems letters are often used as: • Abbreviations for words in everyday life • Teaching – fruit salad algebra

  16. “The meaning of letters in algebra” Anything wrong with this reasoning?

  17. “The meaning of letters in algebra” Confusions about the meanings of letters: Chacolates ChcoC Students were asked the following question: Write an equation which describes the situation » “6 Chacolates cost 12 Rupees” Correct equation, written by nearly all students, » 6d = 12, but what does it mean? Instructions to students: After you have written the equation, say what quantity each of the numerals and pronumerals represents.

  18. Interpreting student work 6d=12 “6 Chacolates cost 12 Rupees”. * by far most common response

  19. Interpreting student work 6d=12 “6 Chacolates cost 12 Rupees” Only one correct! * by far most common response

  20. Interpreting student work 6d=12 “6 Chacolates cost 12 Rupees” Fran wrote this incorrect equation 2 d = 12 2 = cost of each doughnut d = number of Chacolates 12 = overall cost Unusual incorrect response, but Fran is one of the few students who thought carefully about what she really meant.

  21. 6S = P “The meaning of letters in algebra 4.25” Famous problem • At a university there are 6 students for every professor. Let S be the number of students, P be the number of professors,and write an equation. • Letter as object misconception again

  22. “The meaning of letters in algebra 4.25” Misconceptions about what a letter stands for in algebra affect formulating equations • Students need to understand that: • A letter stands for one quantity • The meaning is fixed through the problem

  23. “The meaning of letters in algebra” Illustration 2- Diagnostic item Write an equation that describes the following situation. Use b to stand for the number of blue pencils and r to stand for the number of red pencils.I bought some red pencils and some blue pencils and spent a total of 90 cents. The blue pencils cost 10 cents each and the red pencils cost 6 cents each.

  24. Explain the following student answers • 10b + 6r = 90 (correct) • b+r = 90 • 6b+5r = 90 • b=3, r = 10 I bought some red pencils and some blue pencils and spent a total of 90 cents. The blue pencils cost 10 cents each and the red pencils cost 6 cents each. Write an equation to describe this situation.

  25. “The meaning of letters in algebra” Diagnosing students’ thinking share Rs.47, but Surya gets Rs.5 more than Jothika. How much do they each get?

  26. “The meaning of letters in algebra” Main ideas: how many letters in my name • Letter stands for number – unknown to audience – possibly can be found by audience • Reinforces simple substituting, basic syntax, etc • Students may make harder equation than teacher expects – creativity, diversity • Some are equations and some identities: some equations can belong to one person, some to more than one person, and some to everyone • Equation solving by guess-check-improve.

  27. “The meaning of letters in algebra” How many letters in my Name? Let a = the number of letters in my first name. Let b = the number of letters in my family name. For Lini Marandri, a = 4, b = 8 Sample equation: a + b = 4 + 8 = 12 For Thy Voa = 3, b = 2 Sample equation: a + b = 3 + 2 = 5 .

  28. “The meaning of letters in algebra 4.25” Try it at your table: • Make up 3 equations for your name • Try to include the variety of equations which students might write (correct and incorrect) • Pool your equations and think about what different equations will reveal about students’ thinking.

  29. Sample Equations Probably a bracketing error b – (a+1) = 0 Place value confusion is common with beginners Some will be identities – true for everyone!

  30. Sample Equations No need to stay with linear equations These equations can be easily solved by guess-check, because there are only a few numbers to try.

  31. jUf;fKiw mikg;g[ • fzpjg; gpur;rpidiaj; jPu;f;Fk;nghJ xt;bthU goa[k; fhuzf;fhupaj;Jld; mikag;gl ntz;Lk;

  32. jUf;fKiw mikg;g[ • 200 igfspd; tpiy 20Qgha; vd;gij 200 igfs; = Q/20 vd vGJtJ jtW

  33. jUf;fKiw mikg;g[ • xU bgl;o ghjp jpwe;jpUg;gJk;. ghjp |oapUg;gJk; xnu bghUs;gLk;/ Mdhy; ,ij rkd;ghlhf vGjpdhy; jpwe;jpUg;gJ = |oapUg;gJ vd bghUs;gLk;

  34. jUf;fKiw mikg;g[ • 4 kP x 3 kP = 12 r/kP vd vGJtJ jtW (4 x 3 ) r/kP = 12 r/kP vdf; Fwpg;gp ntz;Lk;

  35. MATHEMATICS Technology Culture Mental Discipline Research(College Prep) QuantitativeLiteracy

  36. Putting the learner at the centre by; Assessment for Learning

  37. If we don't stand up for children, then we don't stand for much. --Marian Wright Edelman

  38. THANK YOU K.Bakthavachalam bakthabrte@gmail.com9944481639

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