Helping students generate an understanding and a method to remember the order of operations is important. When learning anything new it is important to remember the key concept that the learning has to be meaningful or of worth to the students. Just teaching a nmemonic such as P.E.M.D.A.S. or B.E.D.M.A.S or others does not always work for all students. Yes, that may work for the the very detailed and algorithm oriented students. The pyramid method only has three major ideas of Exponents, Multiplication or Division, and lastly Addition or Subtraction with always remembering to do what is in parenthesis first. That might be slightly easier but does it have that buy in hook? New technology games such as Order of Operations Millionaire Game. Games definitely might provide a hook. Also introducing interesting problems or investigations will help. I always find that success is the leading re-enforcer for learning anything. What that means is making sure that formative assessment makes clear the skills that students need to complete a problem successfully are present. Often the PEMDAS problems contain exponents, the order of m/d, multiplication or division themselves, or even a student difficulties with adding or subtracting...then add negative numbers, parenthesis and decimals which all might derail a student. If you want success prepare formative entrance tasks that focus on limited tasks. Have multi-leveled assignments that students must pass one section before moving on to a harder section, and build in class room norms that students are helping other students before yelling "Help" across the classroom (unfortunately a special educators Name). Teaching the order of operations should be a slam dunk and not a nightmare if you remember it is building on previously taught skills that may not have been mastered.
Three strategies suggested in the readings stood out for me. The first was Activity 23.1 (p. 585). While I am terrible at generating a story around any math problem more complicated than 2+2, this activity asks you to select a story that includes stacks of coins, bricks, or notebooks (notebooks?). With this context, students are asked to write a mathematical expression and provide an answer (tell how many). The discussion that follows should challenge students to think about their expression - can it be written differently yet still give the same answer?
Another way suggested by the book is to take a mathematical expression with exponents and multiplication and write it out as repeat addition. The example in the book was 4^2 + 3 + 2 x 5 would then become 4 + 4 + 4 + 4 + 3 + 5 + 5. With repeat addition, order doesn't matter. Could this repeat addition be written any other way if order didn't matter?
Activity 23.3 with its true-false equations is an interesting way for students to think about the order of operations and to decipher where a mistake was made. And this exercise 'sets up an excellent opportunity for students to debate, justify, and critique the justification of their peers.' I dream of classroom discussions that debate the validity of one's solutions.
Even Activity 23.2 is fun. I guess I like these last two activities because it turns learning into a game. Like Greg states, "Games definitely might provide a hook."
I agree that using games as a means to provide a meaningful context will lead to a more significant learning experience for students. I also think that using some of the games, like Dean mentioned, will allow students to explore the concept and develop their understanding of the material. Rather than simply memorizing a set order, students will have the opportunity to look closely at a problem and notice the impact of the structure.
PEMDAS has always been a great tool to help remember the order of operations. Students need to go beyond remembering the mnemonic and understand how the equation changes when you do not follow the order of operations. Write PEMDAS on the board and then show a problem such as 5 x 4^2 - 6 and ask students to explain why the exponents are computed first. Discuss how the results would be different if you multiplied 5 x 4 first instead of computing 4^2 first.
I also think that the 'Order Pyramid' is helpful for students to understand that addition/subtraction and multiplication/division are on the same level. Activity 23.3 is a good activity that gives students the opportunity to work with problems and prove that they are true or explain why they are false. During this activity they must think about the order of operations and apply that knowledge.
If students figure it out for themselves through guided games and activities, they will understand it better. Games and visuals can help remind them after they make the initial discovery.
Helping students generate an understanding and a method to remember the order of operations is important. When learning anything new it is important to remember the key concept that the learning has to be meaningful or of worth to the students. Just teaching a nmemonic such as P.E.M.D.A.S. or B.E.D.M.A.S or others does not always work for all students. Yes, that may work for the the very detailed and algorithm oriented students. The pyramid method only has three major ideas of Exponents, Multiplication or Division, and lastly Addition or Subtraction with always remembering to do what is in parenthesis first. That might be slightly easier but does it have that buy in hook? New technology games such as Order of Operations Millionaire Game. Games definitely might provide a hook. Also introducing interesting problems or investigations will help. I always find that success is the leading re-enforcer for learning anything. What that means is making sure that formative assessment makes clear the skills that students need to complete a problem successfully are present. Often the PEMDAS problems contain exponents, the order of m/d, multiplication or division themselves, or even a student difficulties with adding or subtracting...then add negative numbers, parenthesis and decimals which all might derail a student. If you want success prepare formative entrance tasks that focus on limited tasks. Have multi-leveled assignments that students must pass one section before moving on to a harder section, and build in class room norms that students are helping other students before yelling "Help" across the classroom (unfortunately a special educators Name). Teaching the order of operations should be a slam dunk and not a nightmare if you remember it is building on previously taught skills that may not have been mastered.
ReplyDeleteThree strategies suggested in the readings stood out for me. The first was Activity 23.1 (p. 585). While I am terrible at generating a story around any math problem more complicated than 2+2, this activity asks you to select a story that includes stacks of coins, bricks, or notebooks (notebooks?). With this context, students are asked to write a mathematical expression and provide an answer (tell how many). The discussion that follows should challenge students to think about their expression - can it be written differently yet still give the same answer?
ReplyDeleteAnother way suggested by the book is to take a mathematical expression with exponents and multiplication and write it out as repeat addition. The example in the book was 4^2 + 3 + 2 x 5 would then become 4 + 4 + 4 + 4 + 3 + 5 + 5. With repeat addition, order doesn't matter. Could this repeat addition be written any other way if order didn't matter?
Activity 23.3 with its true-false equations is an interesting way for students to think about the order of operations and to decipher where a mistake was made. And this exercise 'sets up an excellent opportunity for students to debate, justify, and critique the justification of their peers.' I dream of classroom discussions that debate the validity of one's solutions.
Even Activity 23.2 is fun. I guess I like these last two activities because it turns learning into a game. Like Greg states, "Games definitely might provide a hook."
I agree that using games as a means to provide a meaningful context will lead to a more significant learning experience for students. I also think that using some of the games, like Dean mentioned, will allow students to explore the concept and develop their understanding of the material. Rather than simply memorizing a set order, students will have the opportunity to look closely at a problem and notice the impact of the structure.
DeletePEMDAS has always been a great tool to help remember the order of operations. Students need to go beyond remembering the mnemonic and understand how the equation changes when you do not follow the order of operations. Write PEMDAS on the board and then show a problem such as 5 x 4^2 - 6 and ask students to explain why the exponents are computed first. Discuss how the results would be different if you multiplied 5 x 4 first instead of computing 4^2 first.
ReplyDeleteI also think that the 'Order Pyramid' is helpful for students to understand that addition/subtraction and multiplication/division are on the same level. Activity 23.3 is a good activity that gives students the opportunity to work with problems and prove that they are true or explain why they are false. During this activity they must think about the order of operations and apply that knowledge.
If students figure it out for themselves through guided games and activities, they will understand it better. Games and visuals can help remind them after they make the initial discovery.
ReplyDelete