### Teaching Though Problem Solving

Research identifies 3 ways that problem solving might be incorporated into mathematics instruction. What are they?

1. Teaching for Problem Solving

2. Teaching About Problem Solving

3. Teaching Through Problem Solving

2. Teaching About Problem Solving

3. Teaching Through Problem Solving

How is a problem defined for learning mathematics?

-It must begin where the students are

-The problematic or engaging aspect of the problem must be due to the mathematics that the students are to learn (students are Doing the activity)

-It must require justifications and explanations for answers and methods (student is responsible for both)

-The problematic or engaging aspect of the problem must be due to the mathematics that the students are to learn (students are Doing the activity)

-It must require justifications and explanations for answers and methods (student is responsible for both)

What does it mean to teach mathematics through problem solving?

Problem-based tasks or activities are the vehicle by which the desired curriculum is developed. The learning is an outcome of the problem-solving process.

Why is the “teaching for problem-solving” approach not successful in supporting student learning and retention of math concepts?

it assumes that all students have the necessary prior knowledge to understand the explanations;presents only one way to do the problem, while communicating that there is only one way to solve the problem;puts students into a passive learner role;problem solving becomes a separate activity from skills and concepts; students accustomed to being told how to do math problems will not attempt math without explicit instructions on how to solve it

Examples of Problem-Based Tasks

1. Conceptual Mathematics

2. Algorithms and Processes

2. Algorithms and Processes

Conceptual Mathematics

* promoting mathematics as a conceptual tool, not limited to procedural thinking

* also promoting the educational reform necessary to help students see math as a creative way of thinking

* also promoting the educational reform necessary to help students see math as a creative way of thinking

Value of teaching through problem solving

1. S role is more demanding. They get to use various methods to solve problems in diverse ways.

2. Allows an entry point for a WIDE range of S, so all S can be successful. Teacher gives problem and S decides how to solve it.

3. S must be able to apply their concept in order to do it well and make sure their answer makes sense.

4. S must be able to make it personal and apply it to their lives. Give problems that are just right, not too easy and not too hard. Think ZPD development.

Material m/b interesting, better to plan a party than give a worksheet!

2. Allows an entry point for a WIDE range of S, so all S can be successful. Teacher gives problem and S decides how to solve it.

3. S must be able to apply their concept in order to do it well and make sure their answer makes sense.

4. S must be able to make it personal and apply it to their lives. Give problems that are just right, not too easy and not too hard. Think ZPD development.

Material m/b interesting, better to plan a party than give a worksheet!

Value of teaching through problem solving (2)

5. Builds students confidence..Math power.

6. Fun for students

7. Allows for extensions and elaborations

6. Fun for students

7. Allows for extensions and elaborations

Three phase Lesson Plan

Before phase

During phase

After phase

During phase

After phase

Before phase

1.Teacher activates Students prior knowledge. Think the estimate Raisins problem.

2. Make sure student understands the problem (KWL)

3. Establish clear expectations on how they will work (together, or alone) what the end product needs to be. Eg. Think, write, pair, share

2. Make sure student understands the problem (KWL)

3. Establish clear expectations on how they will work (together, or alone) what the end product needs to be. Eg. Think, write, pair, share

During phase

Toughest phase for the Teacher. Think Vogisty (ZPD).

1. Teacher has to let go and let the S do the work! Math write time!

2. Listen actively as this is the assessment piece…walk around

3. Provide appropriate hints to funnel towards right answer

4. Provide worthwhile extensions (requires T to plan ahead of time for the unexpected)

1. Teacher has to let go and let the S do the work! Math write time!

2. Listen actively as this is the assessment piece…walk around

3. Provide appropriate hints to funnel towards right answer

4. Provide worthwhile extensions (requires T to plan ahead of time for the unexpected)

After phase

Most overlooked phase due to time, and some teachers don’t see the value, but don’t miss this phase as it is the phase that brings things back around allowing S to get meaning out of the 1st two phases!

After phase (steps)

1. Teacher facilitates discussion only. Students do most of the talking. Talk time!

2. Include students at ALL levels.

3. Call on shy ones, after giving them a chance to prepare to foster a community of learners

4. Encourage students to ask questions

5. Is an assessment tool too.

6. Don’t allow misconceptions to go unaddressed

2. Include students at ALL levels.

3. Call on shy ones, after giving them a chance to prepare to foster a community of learners

4. Encourage students to ask questions

5. Is an assessment tool too.

6. Don’t allow misconceptions to go unaddressed

The importance of talk time

Students get to hear how others solved the problem, and hear diverse solutions to the problem. This also develops their social skills (socialization is important in learning)

Why is writing important?

1. Rehearsal time for the shy student

2. Prepares student to actually talk about his solution when the time comes

3. Used for assessment by teachers of the students work

4. Helps students to solidify their thoughts

2. Prepares student to actually talk about his solution when the time comes

3. Used for assessment by teachers of the students work

4. Helps students to solidify their thoughts

10 steps to planning a problem based lesson plan

1. Determine the mathematics and goals (state/county)

2. Consider your students’ needs

3. Select, design, or adapt a task (task has to accomplish the content goals (step 1)

4. Design lesson assessments. What you want students to know and how they will show you is important to know now.

5. Plan the Before phase- Activate prior knowledge

6. Plan the During phase-Monitor, assess, hint, keep on right track

7. Plan the After phase- Plan how to begin discussion

8. Check for alignment within the lesson

9. Anticipate student approaches. How to respond?

10. Identify Essential Questions-Quality of Teachers questioning is important.

2. Consider your students’ needs

3. Select, design, or adapt a task (task has to accomplish the content goals (step 1)

4. Design lesson assessments. What you want students to know and how they will show you is important to know now.

5. Plan the Before phase- Activate prior knowledge

6. Plan the During phase-Monitor, assess, hint, keep on right track

7. Plan the After phase- Plan how to begin discussion

8. Check for alignment within the lesson

9. Anticipate student approaches. How to respond?

10. Identify Essential Questions-Quality of Teachers questioning is important.

Planning a problem based lesson..eg.

Teacher chooses problem, based on ‘just right fit’. Teacher does problem herself, gives proper support to student, making sure problem is challenging.

drill

Repetitive, Non-problem based exercises designed to improve skills in an area (multiplication facts, +,-)

practice

Different problem based tasks or experiences spread over numerous class periods, each addressing the SAME ideas

4 step problem solving process

1. Understanding the problem

2. Devising a plan

3. Carrying out the plan

4. Looking back

We did an exercise in class where we had to look at the cost of making $5 in different coins and decide which to eliminate

2. Devising a plan

3. Carrying out the plan

4. Looking back

We did an exercise in class where we had to look at the cost of making $5 in different coins and decide which to eliminate

Problem-Solving Strategies

1. Draw a picture or create a model

2. Look for a pattern

3. Guess and check

4. Make a table of a chart

5. Try a simpler form of the problem

6. Make an organized list

7. Write an equation

8. Working backwards

2. Look for a pattern

3. Guess and check

4. Make a table of a chart

5. Try a simpler form of the problem

6. Make an organized list

7. Write an equation

8. Working backwards

Justify

Student gives rationale for the way they solved the problem. Using pictures, numbers or words. Draw, solve, write it down. Think of the students explanation of how many buses needed for the school trip 😉 This helps teacher to understand students thinking.

When selecting appropriate tasks consider..

Math needs to be conceptional, not procedural

Math understanding the student brings to the problem

Math understanding the student brings to the problem

Problems with multiple solutions

Students should be encouraged to solve in multiple ways, as this provides multiply entry points for students. This provides rich learning opportunities for math talk. Provides an easy way to tier assignments. Eg. A or B or C or D

Open ended problems

Allow for maximum student interpretation. Answers will vary. Eg. Planning a party and have X amount of money to spend.

Accommodation

Providing a different environment or circumstance made with a particular student in mind. EG. changing lighting in a room, allow to work alone

Modification

A change to the problem or task itself. Eg. simplifying the wording

Tiered Lesson

Teacher determines the learning goals for all students, but the level of difficulty of the task is adapted up or down to meet the range of the learners. Is not just about content, but can include the amount of assistance provided, structure of lesson, complexity of the task or process.

Flexible grouping

Think IK topped out of math in elementary school, and went to middle school for more intense training

Diversity in today’s classroom includes..

ELL, Special Needs, Gifted, Learning Disabilities, Cultural background. These are all things teachers must be aware of when presenting problems to students.

Barriers and solutions to learning for students with disabilities

1. STAR (Search problem for information,Translate into words or pictures, Answer the problem, Review your solution)

2. Reinforce key words/vocabulary. Create a word or symbol wall

3. Use friendly numbers. Round up from $6.13 to $6.00

4. Vary the task size. Ask student to solve fewer problems

5. Timeframe. Give reminders about time, and finishing tasks

6. Modify the level of support. Either through the teacher or helpers.

2. Reinforce key words/vocabulary. Create a word or symbol wall

3. Use friendly numbers. Round up from $6.13 to $6.00

4. Vary the task size. Ask student to solve fewer problems

5. Timeframe. Give reminders about time, and finishing tasks

6. Modify the level of support. Either through the teacher or helpers.

formative assessment

Assessment used throughout teaching of a lesson and/or unit to gauge students’ understanding and inform and guide teaching

Summative assessment

Cumulative evaluations that generate a single score. Eg. End of unit test, occurs at end of an instructional unit & document student learning

Multiple Entry Points

Students can approach from different angles, and can determine the simplest way to do the problem. Eg. Read a story and write a problem about the story.