Teaching For and Via Problem Solving: The Case of Addition and Subtraction David McKillop, Consultant Pearson Education Canada

What is meant by Teaching For Problem Solving

Teaching For Problem Solving Mathematics concepts and procedures in school mathematics are used in the world. These applications should be front and centre in the study of any concept or procedure, NOT treated as an afterthought. For example, all units on addition and subtraction should centre on the different structures of story problems and their solutions.

As teachers, we cannot over estimate the role word problems play for most students to see the relevance of a math concept, and the very essence of the concept.

What is a strip diagram? This strip diagram could be the drawn to represent two addition situations: = = 60 and two subtraction situations: 60 – 18 = – 42 = 18

Mr. Smith drove 112 km from Halifax to Bridgewater on Friday morning. Friday afternoon he drove another 279 km. How far did Mr. Smith drive on Friday? ?

The farmer had 50 eggs. Sixteen of them hatched. How many eggs didn’t hatch?

4 327 people moved into the city last year. The population was then What was the population before last year?

9 What is the same and what is different about these three story problems? 1. Tristen had 24 comic books in his collection. He bought 12 more at The Comic Bookstore last Saturday. How many comic books did Tristen then have in his collection? 2. Tristen had 27 comic books in his collection. After he bought some more at The Comic Bookstore last Saturday, he had 40 comic books in his collection. How many comic books did Tristen buy last Saturday? 3. Tristen bought 10 comic books last Saturday at The Comic Bookstore. This brought his collection to a total of 35 comic books. How many comic books were in Tristen’s collection before last Saturday?

10 Same and Different All three questions involve Tristen having a starting quantity of comic books, changing his collection by the action of buying more on a Saturday, and resulting in his having an increased collection Action is a GAIN The difference lies in which is unknown: the start, the change, or the result.

11 Why are all three story problems called Join problems? They all involve an action of gain. This action causes a change in a starting amount to produce a result. Examples of action words in Join story problems: buy, get, deposit, win, find, join, given, etc.

Tristen had 24 comic books in his collection. He bought 12 more at The Comic Bookstore last Saturday. How many comic books did Tristen then have in his collection?

Tristen had 27 comic books in his collection. After he bought some more at The Comic Bookstore last Saturday, he had 40 comic books in his collection. How many comic books did Tristen buy last Saturday?

Tristen bought 10 comic books last Saturday at The Comic Bookstore. This brought his collection to a total of 35 comic books. How many comic books were in Tristen’s collection before last Saturday?.

CHANGE: Join – Action of Gain At 7 pm, there were 357 people seated in the auditorium. Between 7 and 8 pm, 239 more people were seated. How many people were seated by 8 pm? At 7 pm, there were 357 people seated in the auditorium. Between 7 and 8 pm, more people were seated until there were 596 seats filled. How many people were seated between 7 and 8 pm? At 7 pm, there were many people seated in the auditorium. Between 7 and 8 pm, 239 more people were seated. By 8 pm, there were 596 seats filled. How many people were seated before 7 pm?

CHANGE: Join – Action of Gain At 7 pm, there were 357 people seated in the auditorium. Between 7 and 8 pm, 239 more people were seated. How many people were seated by 8 pm? = ? At 7 pm, there were 357 people seated in the auditorium. Between 7 and 8 pm, more people were seated until there were 596 seats filled. How many people were seated between 7 and 8 pm? ? = 596

CHANGE: Join – Action of Gain At 7 pm, there were many people seated in the auditorium. Between 7 and 8 pm, 239 more people were seated. By 8 pm, there were 596 seats filled. How many people were seated before 7 pm? ? = 596

CHANGE: Join – Action of Gain Mary had $56 in her bank account on Thursday. On Friday, she deposited another $35. How much did Mary then have in her bank account? Create a Join – change unknown problem and a Join – start unknown problem involving this same situation.

19 What is the same and what is different about these three story problems? 1. Tristen had 36 comic books in his collection last weekend. This week he gave away 12 of them to his friends. How many comic books did Tristen then have in his collection? 2. Tristen had 30 comic books in his collection last weekend. After he gave away some of them to his friends this week, he had 22 comic books left in his collection. How many comic books did Tristen give to his friends? 3. Tristen gave 11 comic books to his friends this week. Then he had 26 comic books left in his collection. How many comic books were in Tristen’s collection last weekend?

20 Same and Different All three questions involve Tristen having a starting quantity of comic books, changing his collection by the action of giving some away, and resulting in his having an decreased collection. Action is a LOSS The difference lies in which part is unknown: the start, the change, or the result.

21 Why are all three story problems called Separate problems? They all involve an action of LOSS. This action causes a change in a starting amount to produce a result. Examples of action words in Separate story problems: sell, give away, withdraw, stolen, lose, burn, eat, etc.

22 1. Tristen had 36 comic books in his collection last weekend. This week he gave away 12 of them to his friends. How many comic books did Tristen then have in his collection?

23 2. Tristen had 30 comic books in his collection last weekend. After he gave away some of them to his friends this week, he had 22 comic books left in his collection. How many comic books did Tristen give to his friends?

24 3. Tristen gave 11 comic books to his friends this week. Then he had 26 comic books left in his collection. How many comic books were in Tristen’s collection last weekend?

25 Activity Ellen had 12 marbles before recess. She lost 5 marbles during recess. How many marbles did Ellen have after recess? Create two more story problem about Ellen and losing marbles at recess: one with the change unknown and one with the start unknown.

Structures of +/- Situations CHANGE Join (result, change or start unknown) Separate (result, change or start unknown)

27 What is the same and what is different about these story problems compared to CHANGE (Join and Separate) story problems? 1. Tristen has 12 new comic books and 19 used comic books in his collection. How many comic books does Tristen have altogether? 2. Tristen has 38 comic books in his collection. If 10 of them are new, how many of his comic books are used?

28 Same and Different The stories are about Tristen and his comic book collection You add or subtract to get an answer His collection does not change There is no action involved Statements about two parts that are combined to make up a whole COMBINE Whole missing OR One part missing

Structures of +/- Situations CHANGE Join (result, change or start unknown) Separate (result, change or start unknown) COMBINE Part Unknown Whole Unknown

30 1. Tristen has 12 new comic books and 19 used comic books in his collection. How many comic books does Tristen have altogether?

31 2. Tristen has 38 comic books in his collection. If 10 of them are new, how many of his comic books are used?

32 Why do you think some children might find these COMBINE Story Problems difficult? Five girls and four boys at a table. How many children? Six cats and three dogs. How many pets? Three novels and two poetry books. How many books? Three daisies and two delphinium. How many flowers? Eight forks and six knives. How many pieces of cutlery?

33 What is the same and what is different about these story problems compared to CHANGE and COMBINE problems? 1. Tristen has 36 comic books in his collection. Mary has 32 comic books in her collection. How many more comic books does Tristen have than Mary? 2. Tristen has 38 comic books in his collection. Mary has 6 more comic books in her collection. How many comic books are in Mary’s collection? 3. Tristen has 42 comic books in his collection. He has 12 more in his collection than Mary has in hers. How many comic books does Mary have?

34 1. Tristen has 36 comic books in his collection. Mary has 32 comic books in her collection. How many more comic books does Tristen have than Mary?

35 2. Tristen has 38 comic books in his collection. Mary has 6 more comic books in her collection. How many comic books are in Mary’s collection?

36 3. Tristen has 42 comic books in his collection. He has 12 more in his collection than Mary has in hers. How many comic books does Mary have?

37 COMPARE Story Problems Two sets compared, NOT changed or combined You do add or subtract Sometimes both sets are given and you are to find the difference Sometimes the larger set and the difference are given and you are to find the smaller set Sometimes the smaller set and the difference are given and you are to find the larger set

38 Three Questions – One Solution Tristen has 36 comic books in his collection. Mary has 32 comic books in her collection. How many more comic books does Tristen have than Mary? Tristen has 36 comic books in his collection. Mary has 32 comic books in her collection. How many fewer comic books does Mary have than Tristen? Tristen has 36 comic books in his collection. Mary has 32 comic books in her collection. What is the difference between their collections?

39 Create COMPARE Story Problems Use the context of a math quiz marked out of 10 and two students’ marks on it Create a comparison story problem that will ask for the difference. Create a story problem that will ask for the student’s mark who made less Create a story problem that will ask for the student’s mark who made more

40 Create a story problem that will ask for the student’s mark who made less Pat made 7 out of 10 on the math quiz. Her friend Beth made 3 less than Pat. What did Beth make on the math quiz? Pat made 7 out of 10 on the math quiz. That mark is 3 more than her friend Beth made on the same quiz. What did Beth make?

41 Create a story problem that will ask for the student’s mark who made more Bill made 7 out of 10 on the math quiz. His friend Andy made 2 more than Bill. What did Andy make on the math quiz? Bill made 7 out of 10 on the math quiz. That mark is 2 less than his friend Andy made on the same quiz. What did Andy make?

EQUALIZE Problems Gary had 25 marbles and his friend Marty had 32 marbles. How many marbles does Gary need to get to have the same amount as Marty?

Structures of +/- Situations CHANGE Join (result, change or start unknown) Separate (result, change or start unknown) COMBINE Part Unknown Whole Unknown COMPARE/ EQUALIZE Difference Unknown Smaller Unknown Larger Unknown

Structures of +/- Situations CHANGE Join (result, change or start unknown) Separate (result, change or start unknown) COMBINE Part Unknown Whole Unknown COMPARE/ EQUALIZE Difference Unknown Smaller Unknown Larger Unknown

What is meant by Teaching Via Problem Solving

NCTM Principles and Standards, 2000 Solving problems is not only a goal of learning mathematics but also a major means of doing so…

Teaching Via Problem Solving Van de Walle’s Three-Part Lesson Planning: Teachers: Before, During, and After Students: Explore (Show and Share) Connect Practice (Reflect)

Teaching Via Problem Solving Teach in ways that students tell you what you used to tell them!

Teaching Via Problem Solving Each learner constructs meaning in his/her own unique way. A learner, therefore, must be actively interacting with the idea. That is, he/she must be thinking (inducing, deducing, reasoning, sequencing, classifying, finding patterns and relationships). It is the quality of this interaction that is crucial to the development of understanding.

Teaching Via Problem Solving What does it look like in units on addition and subtraction at all grade levels?

Moving From Situation to Operation Situation A.Represent with a model using role play, materials, diagrams, or numbers B.Solve by counting Represent the model and solutions with a number sentence Situation Represent with numbers and symbols (operate) Solve using a computation (compute)