1. TEACHING FOR UNDERSTANDING
Maths Support Team

2. (rote learning and rules) Teaching by telling
Concrete materials and instructional variety
Problem solving and investigations
Active learning and more open ended questions
Routine worksheets
Template exercises
Exams
One method /one answer
Memorization
(rote learning and rules)
Teaching by telling

3. WHY EDUCATE FOR UNDERSTANDING?
Knowledge and skill in themselves do not guarantee understanding. People can acquire knowledge and routine skills without understanding their basis or when to use them. And, by and large, knowledge and skills that are not understood do students little good! What use can students make of the mathematics they have learned unless they have understood it?

4. Do you feel confident in explaining and transferring your understanding to others?

6. What do we understand by understanding?
New learning
Existing concepts
Procedural
Asks HOW
Isolated skills, concepts, rules and symbols
Ideas memorised and may be at risk of being forgotten
Learning Maths for the test
A child knows a rule or procedure and has the ability to use it.
Conceptual
Asks WHY before HOW
Connecting existing concepts to the new idea
Concept is learnt and likely to be used for constructing new ideas
Owning your Maths
A child knows what to do and can explain why. vs

7. Classroom practices promoting relational understanding
Teaching Maths for Understanding
On-going assessment
Effective Questioning Techniques
Using visuals and manipulatives
Multiple Approaches
Mistakes and Misconceptions
Opportunities for Problem solving, discussion and dialogue.
I hear I forget
I see I remember
I do I understand
Ask children to justify own answers and reasoning.
Children are more likely to challenge each other’s ideas and reasoning, rather than the teacher’s.
Focusing on criteria, feedback and reflection
Purposeful questions to check for understanding and scaffold children’s learning.
Children’s individualities are respected.
Children discuss the effectiveness of each approach.
Mistakes are opportunities for growth and learning.

8. Ans ques 2-4 for both of the lessons
TASK 1
Which lesson leads to instrumental understanding and which lesson leads to relational understanding?
Ans ques 2-4 for both of the lessons
2. Who determines the procedure to use in each lesson?
3. The learning intention ‘understand two-digit addition’ is common for both lessons. What is captured by ‘understand’ in each setting?
4. How accessible is the lesson in meeting the needs of different students?

9. Children in both classrooms will eventually learn how to work out addition sums but what they learn about addition and mathematics is quite different.
Children in the first class are more likely to:
Develop richer mathematical thinking
Become more flexible thinkers
Become better problem solvers
Remain more engaged in learning
Develop more positive attitudes towards learning mathematics

10. Effective Questioning How many times do you ask the question:
DO YOU UNDERSTAND?
The fallacy with this thinking is that sometimes the students do not understand that they do not understand, so there is no way that they can ask a question about it.

12. Effective Teaching requires two vital qualities:
the teacher is a patient listener
knows how to ask questions effectively.
A good teacher doesn't tell his students what to think. A good teacher teaches his students how to think. Questions achieve that goal.

13. Quoted by Wiggins and McTighe, Dr. Bloom explains:
" some teachers believe their students should 'really understand,' others desire their students to 'internalize knowledge,' still others want their students to 'grasp the core or essence.' Do they all mean the same thing? Specifically, what does a student who 'really understands‘ do which he does not do when he does not understand? Through reference to the Taxonomy teachers should be able to define such nebulous terms."

16. Remembering : Recite numbers when counting in 2s, 5s and 10s.
Understand: : Identify whether numbers are skip counting forward or backward by 2s, 5s, or 10s.
Apply: Show how to skip count by 2s, 5s and 10s orally when starting at any interval. Also show how to count backward by 10.
Analyse: Distinguish between patterns on a 100s chart when skip counting by 2s, 5s and 10s.
Evaluate: Determine whether to skip count by 2s, 5s or 10s in a real life situation.
Create: Create a hopscotch board to represent skip counting by 2s, 5s or 10s all the way to 100.

17. Remembering : what are fractions
Remembering : what are fractions? What do the words half, quarter and third mean?
Understand: What fraction is shaded? Shade a quarter of the square.
Apply: Name fraction smaller than ½?
Analyse: How can you work out one quarter of something? Can you work it out another way?
Evaluate: Mum splits a sandwich in 2 and gives Tara 1 piece. Mum splits a sandwich in 4 and gives Mark 2 pieces. Tara complains “It’s not fair. He has 2 and I have 1” Is she right?
Create: Design two shapes; one which can be split in halves and one which can be split in halves and quarters?

18. Changing the way a question is phrased can
make a significant difference.
Good Questions:
Are self differentiating
Promote dialogue
Lead to more good questions

20. TASK 2
Sort these questions into two categories; Less effective and more effective questions; and pair them up with each other.

21. Mistakes and Misconceptions using Concept Cartoons
Research shows that many of the misconceptions held by children are retained during adulthood if they are left unchallenged.
Concept cartoons are often used as a stimulus for discussion, to identify area of uncertainty and questions to be answered.

22. Which shapes show how the girls can share the cake equally between themselves without cutting it further?

24. 8 concept cartoons are presented.
TASK 3
8 concept cartoons are presented.
Discuss as a group and add another entry in the blank balloon targeting a different misconception.