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
education
Concept #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 = 2
Addition 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
DaniloSirias@problemsolvingmaps.com

Intro to problem solving maps v. 8

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