What’s the Problem?Loria Allen K-2 AMSTI Math Specialist

Laura Clemons 3-5 AMSTI Math SpecialistCarrie Warden K-2 AMSTI Math Specialist

Sheila Holt 3-5 AMSTI Math Specialist

“It is the story that matters not just the ending.” ― Paul Lockhart, A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form

Young children begin learning mathematics

before they enter school.

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence (NCTM 2006) continues to emphasize the importance of developing conceptual and procedural understanding of addition and subtraction.

Word problems are often broken into:

1-step word problems multi-step word problemsCharles and Lester (1982) call this type of problem a translation problem.

Two teaching strategies for problem solving used by many teachers are:

1. The key word approach 2. The problem-solving

steps approach

Altogether there are 24 children on the playground. 14 of them are boys. How many are girls?

The key words approach prepares students to solve only a very small portion of

problems on assessments as well as in the real world.

in some instances the use of key words may ultimately prove detrimental to success in solving word or story problems (Clement & Bernhard, 2005). As with mathematical exercises, success in solving word or story problems typically involves a great deal of automaticity. That is to say, for success in the mathematical operation, very little cognition actually occurs rather than simply recalling a formula or a fact and executing it with the provided numbers or data.

Mathematical Problems That Optimize Learning for Academically Advanced Students in Grades K–6Chamberlin, S.(2010)

Key Word Strategy…

Students understanding of a problem continues as planning

and solving are underway. Problem solving is not an

algorithm. There is not a series of steps that produce success.

What does problem solving look like in your

classroom?

Problem Solving for K - 5

1. 37+ 8 = 452. 24 - 11 = 133. 26 x 18 = 4684. 252 ÷ 14 =18

Write 1 word problem for each equation

Use Table 1 Appendix A Addition and Subtraction Situation Types

1. Identify the problem situation type and unknown for each word problem.

2. What situation types and unknowns were not shown?

3. Be prepared to share your findings.

Switch problems with another group

Appendix AMultiplication and Division Situations

Identify problem situation and unknown type

Which problem situation and unknown were not used?

Be prepared to show your findings.

Switch problems with another group

Yesterday I visited the gym during a physical education class, and I saw 25 of the students playing in a soccer game, 13 of them playing in a basketball game and 16 playing in a baseball game.

Retell the story.

Retelling and Making Sense

Yesterday I visited the gym during the physical education class, and I saw, 25 of the students playing in a soccer game, 13 of them playing in a basketball game and 16 playing in a baseball game.

1. What questions can be answered by the information in the problem?

2. Solve some of the problems generated.

Predicting

Yesterday I visited the gym during the physical education class, and I saw 25 of the students playing in a soccer game, 13 of them playing in a basketball game and 16 playing in a baseball game.

What proportion of the students played soccer?

Solve the REAL Problem

What are the possible benefits are there to using this word problem strategy with students?

Discussion

Developing number sense takes time; algorithms taught too early work against the

development of good number sense. Children who learn to think, rather than to

apply the same procedures by rote regardless of the numbers, will be empowered.

Fosnot (2001)

Counting all Counting on

Kindergarten

Counting all Counting on Doubles Near Doubles Making tens Making landmark or friendly numbers Breaking each number into place value Adding up in chunks

1st Grade - Addition

Adding Up Removal or counting Back

1st Grade Subtraction

Counting all Counting on Doubles Near Doubles Making tens Making landmark or friendly numbers Breaking each number into place value –

Partial Sums and Regrouping

2nd Grade- Addition

Adding Up Removal or counting Back Place Value and Negative Numbers – Partial

Differences and Regrouping

2nd Grade - Subtraction

Open Number Line Bar Model Breaking each number into place value Adding up in chunks Compensation Doubles/Near Doubles Making Tens

3rd- 5th grade- Addition

Adding up Number line Removal or Counting Back Place Value and Negative Numbers (Partial

Differences) Compensation

3rd- 5th Grade -Subtraction

Modeling counting by ones Counting by sub groups Repeated Addition

Equal Groups Equal Groups in an Array

Arrays Open Area Model Partial Products Distributive Property Doubling and Halving Powers of 10 Traditional Algorithm

3-5 Grade Multiplication

Sharing out by ones Sharing out in equal groups Repeated subtraction or adding up Skips count Array Model Trial and error Inefficient partial products Partial products Distributive Property Inverse relationship Treats the remainder appropriately Traditional algorithm

3-5 Grade Division

Teachers often tell children that division is nothing more than repeated subtraction. This idea not only is insufficient but also hinders children’s ability to construct and understanding of the part/whole relationships in multiplication and division.

Ex: Socks are on sale at 3 pairs for $12; how much is this per pair? Where is the repeated subtraction?

Ultimately, how you teach the curriculum has a greater influence on student learning than

the curriculum itself (Stein & Kaufman, 2010).

“Pedagogy trumps curriculum. Or more precisely, pedagogy is the curriculum,

because what matters is how things are taught, rather than what is taught.” Wiliam (2011)

Caldwell, Janet H. Developing Essential Understanding of Addition & Subtraction PreK-Grade 2, NCTM, 2011

Charles, Randall (1988) Solving Word Problems Developing Students’ Quantitative Reasoning Abilities, Research Into Practice.Pearson