Intro to problem solving maps v. 8

Improving math

education with

Problem Solving Maps

Danilo Sirias, Ph.D.

What do students say?

The instructor moves too fast for students

The instructor knows his subject matter but cannot teach

I am not capable of doing math

I do not see the importance/relevance of math

The exams are too hard

I go blank on exams

The instructor is disorganized

Problems seems too difficult

www.problemsolvingmaps.com (c) Danilo Sirias 2012

Do these issues change by

Grade level?

Content?


Where are most efforts being focused

on? Most efforts are designed to improve the

teaching of individual content

Best way to teach fractions?

Best way to teach decimals

Best way to teach functions?

Best way to teach equations?

New curriculum, new books, new ways to

evaluate, new, …new….


Current model to improve math

educationConcept #1 Concept #2 Concept #3 Concept #4 Concept #5

Best

Practice

#1

Best

Practice

#2

Best

Practice

#3

Best

Practice

#4

Best

Practice

#5


Teachers keep trying to find

innovative ways to teach….


What is challenging about this

approach?

The methods used by students

to learn one topic may not

transfer to a different topic

Continuous changes in the

teaching approaches

It requires a lot of effort from

teachers and administrators

Math education is fragmented

(C) Danilo Sirias 2013

An ideal math learning process would

Be effective

Teach students tools that are transferable

from one topic to another

Be easily incorporated into the existing

curriculum

Allow teachers to teach a variety of topics

Not require enormous amount of training


Can we do something different?

We can learn from good students

They do well regardless of the topic

…the grade

…the teacher

WHY?

They have a generic process to solve

problems


Proposed model

Concept #1 Concept #2 Concept #3 Concept #4 Concept #5

Common set of

thinking skills


Hierarchy of math knowledge

Factual

Simple rules

Multi-rule problems

Application problems

Structured

Semi-structured


The relationship between the hierarchy

of math knowledge and their required

thinking process

Factual Memorization

Simple Rules Finding patterns (inductive

thinking)

Multi-rule problems Applying generic rules

to specific problems (deductive thinking)

Application problems Breaking down

problems into smaller sub-problems


Thinking process Problem solving map

Finding patterns Example-Conclusion

graph

Applying generic

rules to specific

problems

Multi-rule Branch

Breaking down

problems

Math Breaker

Thinking processes and their

corresponding problem solving map


Finding patterns –

Example- conclusion

graph

Example 2

Your own example

Example 1

Example 3

Example-conclusion graph

Conclusion


Example 2

Your own example

Example 1

Example 3

Example-conclusion graph

Explanation

Conclusion

1) Provide three

examples

2) So students can write

their conclusion

3) And then create

their own example


Example 2

Your own example

Example 1

Example 3

3X = 6

X = 2

5X = 15

X = 3

12X = 7

X = 7/12

Algebra rule

example

Conclusion


Applying generic rules

to specific steps---

The Multi-rule Branch

Multi-rule

Branch

Step 3

Step 2

Initial point

Step 1

Math Rule

Math Rule

Math Rule


Multi-rule branch explanation

Step 2

Step 1 Math Rule

2) And apply this rule

3) To move to this step

1) You start here


Multi-rule

Branch

X = 3

3X = 9

3X - 7 = 2Addition property of

equality

Division property of

equality


Learning to break

problems into smaller

sub-problems---

The Math breaker

Math breaker for teaching content

Description of step Description of step

Description of step


Math breaker

Description of step Description of step

Description of step

Space for students to

work on that step

Arrows denote

prerequisites,

what steps need

to be completed

before moving to

the next step


Each step has a brief

instruction of what to do

Find the cube root of the first term Find the cube root of the second term

Write the second termWrite the first term

Write first factor (FF) as cube root of the first term plus cube root of the second term

Use the FF to find the second factor (SF) by finding the square of the first term minus the product of the first and the second term plus the square of the second term

Final factorization is the product of FF and SF

Factoring sum of cubes (Standard form a3+ b3)


Structured problem: Graphing the equation of a line in standard

form

1. Identify the y-intercept (b)

3. Identify the slope (m)


2. Plot b on the y axis

4. Starting from the plotted point b, graph a 2nd

point using m, where m = rise/run and connect

the 2 points using a straight line

Semi-structured problems: Word

problems A group of 5 girls and 6 boys went to the fair.

Each girl has a medium soft drink at $1.00

each and a bag of fries at $1.80 each. The

boys decided to also buy soft drinks (large at

$1.10 each) and had ice cream at $1.10

each. How much more did the girls spend in

all than the boys?


How much more did the girls spend in all than the boys?

How much did the boys spend?How much did the girls spend?

How many girls are there?

How much did each boy spend?

How many boys are there?

How much did each girls spend?

How much is a medium drink?

How much is a bag of fries?

How much is a large soft drink?

How much is ice cream?


Starting from question

Putting all together

Example

2

Example

1

Example

3

Math

Rule

Math

Rule

Step 2

Initial point

Step 1

Description Description

Description


The goal is to use the problem

solving maps to teach as many

topics as possible so that

students internalize the thinking

skills to the extent that they use

them to solve future problems on

their own


Advantages

Problem solving maps can be used to teach a large variety of topics

For students, the skills are transferable from one topic to the next

They can be easily introduced within the current curriculum

It does not take a long time for teachers to learn them.


Additional advantages

Students have better notes which can lead to better performance.

The time it takes to cover the content is shorter.

Improvement efforts can be targeted to the right place.

Teacher can also use the diagrams as a tool for grading.


Summary


Questions, comments?

Danilo Sirias

[email protected]

Intro to problem solving maps v. 8

Education

Transcript of Intro to problem solving maps v. 8