1. Helping Students Learn
Lesson Design and Evaluation of Student Learning
Copyright ©2014 by Lynda Aguirre

2. Copyright ©2014 by Lynda Aguirre
Cardinality
Cardinality is a concept that students must master before they can learn any kind of mathematics.
They naturally develop this sense of what numbers are at around age 5.
You can test them to see if they have achieved Cardinality like this:
Test: Put some objects (let’s say 5) in front of them and ask them “how many objects are there?”
1) If they have achieved cardinality, they will say “5”, or they will count them and then say “5”
2) If they have not achieved cardinality, they will just count them.
In simple terms, achieving cardinality means that they understand that the “5” is the total number of objects as well as the 5th object.
If they don’t understand this concept, they can’t do addition like 2 + 3,
because they don’t understand what the symbols “2” and “3” mean.
Copyright ©2014 by Lynda Aguirre

3. Understanding
True understanding occurs when we see how something is related
or connected to other things we already know
Teaching in a way that promotes understanding involves more than just sharing facts
or methods. It must take the students attitudes and thought processes into consideration
Copyright ©2014 by Lynda Aguirre

4. Copyright ©2014 by Lynda Aguirre
Issues to consider
The way a student thinks about their relationship with mathematics will determine whether they understand and retain the material in your lesson
--Sex differences- Are Males assumed to be better at math than Females? Why?
--Is Math Anxiety an issue?
--What is the student’s societal attitude towards mathematics? Parents, peers and population
--Individual Interview helps to pinpoint the way a student thinks about mathematical learning
Copyright ©2014 by Lynda Aguirre

5. Copyright ©2014 by Lynda Aguirre
Issues to consider
The classroom environment, or social culture, can influence their ability to learn
--What tools are available in the classroom?
--Every student contributes their ideas without fear of ridicule
--Tasks are accessible to all students
--Ideas and methods are valued
--Correctness resides in mathematical argument
Copyright ©2014 by Lynda Aguirre

6. Copyright ©2014 by Lynda Aguirre
Interview Questions
When trying to determine a student’s learning strengths and weaknesses you can ask a series of Interview questions to narrow down your search
How did you get your answer (I may have missed something)
Why is your answer correct?
Could you explain it another way?
Can you make a drawing to show your answer is correct?
What kinds of errors do you usually make?
What do you do when you see an unfamiliar problem?
What kinds of problems are you worst at?
What can you do to get better at these?
Copyright ©2014 by Lynda Aguirre

7. How do they justify their answers?
1) Externally based-textbook or authority is cited
2) Empirically based- use of perception or concrete objects to show
their answer is correct
Analysis-students use counting strategies or cite
mathematical relations to justify their answers
Copyright ©2014 by Lynda Aguirre

8. 1)Nature of Learning Tasks
Use everyday objects or situations they can relate to
Use knowledge that they already have
Give them some valid reason to learn the material
Copyright ©2014 by Lynda Aguirre

9. Copyright ©2014 by Lynda Aguirre
2) Role of the Teacher
Select tasks that will achieve the learning goals
Share essential information
Establish classroom culture
Copyright ©2014 by Lynda Aguirre

10. 3) Social Culture of the Classroom
Ideas and input are encouraged and valued
Students feel safe sharing their methods
Mistakes are viewed as learning tools for everyone
Alternative solutions are examined for validity
Copyright ©2014 by Lynda Aguirre