1. Math & Science Collaborative, © MSC 2010 1 Algebraic Reasoning Day 5 How Students Learn Mathematics

2. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry What is learning? What does it mean to have learned something? On your index card, write one sentence or statement that begins to answer this question Pass your index card to the person on your left. Each person is to add one more sentence/statement that builds upon or counters the statement on that card Keep passing the cards to your left and adding a statement until you get your index card back 2

3. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry 3 Fish is Fish by Leo Lionni

4. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry 4 At the edge of the woods there was a pond, and there a minnow and a tadpole swam among the weeds. They were inseparable friends.

5. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry 5 One morning the tadpole discovered that during the night he had grown two little legs.

6. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry 6 In the weeks that followed, the tadpole grew tiny front legs and his tail got smaller and smaller.

7. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry 7 And then one fine day, a real frog now, he climbed out of the water and onto the grassy bank.

8. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry 8 The minnow too had grown and had become a full-fledged fish.

9. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry 9 Then one day, with a happy splash that shook the weeds, the frog jumped into the pond.

10. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry 10 “Where have you been!” asked the fish excitedly.

11. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry 11 “I have been about the world – hopping here and there, “ said the frog. “and I have seen extraordinary things.”

12. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry 12 “Like what?” asked the fish. “Birds,” said the frog mysteriously. “Birds!”

13. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry 13 “Cows,” said the frog. “Cows!” “They have four legs, horns, eat grass and carry pink bags of milk.”

14. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry 14 “And people!” said the frog. “Men, women, children!”

15. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry 15 The frog had gone and the fish just lay there dreaming

16. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry 16 And so with a mighty whack of the tail he jumped clear out of the water onto the bank,

17. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry 17 He landed in the dry, warm grass

18. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry 18 Luckily the frog, who had been hunting butterflies nearby, saw and with all his strength pushed him back into the pond.

19. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry 19 He smiled at his friend the frog, who sat watching him from a lily leaf. “You were right,” he said. “Fish is fish.”

20. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry 20 How Students Learn Mathematics

21. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry 21 How People Learn Key Find ings Key Findings for Students Connection to Fish is Fish Story? 1 st Students come to the classroom with preconceptions about how the world works. 2 nd Students need a deep foundation of usable knowledge and understand facts in the context of a conceptual framework. 3rd Students must be taught explicitly to take control of their own learning by setting goals and monitoring progress toward them.

22. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry For HPL Key Finding # 1 preconceptions How does this key finding relate to teachers? What are some implications for you, for teachers? As a result, what are some implications for the mathematics curriculum and instruction? 22

23. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry How People Learn Key Findings: StudentsKey Findings: Teachers As a Result, Curriculum and Instruction Need To… Come to the classroom with preconceptions. Need to develop a deep foundation of usable knowledge and understand facts in the context of a conceptual framework Must be taught explicitly to take control of their own learning by setting goals and monitoring progress toward them 23

24. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry 24 How People Learn Key Findings Key Findings for StudentsKey Findings for Teachers First Students come to the classroom with preconceptions about how the world works. Recognize preconceptions and adjust instruction Second Students need a deep foundation of usable knowledge and understand facts in the context of a conceptual framework. Third Students must be taught explicitly to take control of their own learning by setting goals and monitoring progress toward them.

25. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry For HPL Key Finding # 2 conceptual coherence How does this key finding relate to teachers? What are some implications for you, for teachers? As a result, what are some implications for the mathematics curriculum and instruction? 25

26. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry 26 How People Learn Key Findings Key Findings for StudentsKey Findings for Teachers First Students come to the classroom with preconceptions about how the world works. Recognize preconceptions and adjust instruction Second Students need a deep foundation of usable knowledge and understand facts in the context of a conceptual framework. Understand the content and conceptual framework for a discipline Third Students must be taught explicitly to take control of their own learning by setting goals and monitoring progress toward them.

27. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry For HPL Key Finding # 3 metacognition How does this key finding relate to teachers? What are some implications for you, for teachers? As a result, what are some implications for the mathematics curriculum and instruction? 27

28. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry 28 How People Learn Key Findings Key Findings for StudentsKey Findings for Teachers First Students come to the classroom with preconceptions about how the world works. Recognize preconceptions and adjust instruction Second Students need a deep foundation of usable knowledge and understand facts in the context of a conceptual framework. Understand the content and conceptual framework for a discipline Third Students must be taught explicitly to take control of their own learning by setting goals and monitoring progress toward them. Teach metacognitive skills

29. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry How People Learn Key Findings: Students Standards for Mathematical Practice Math Task Framework Tasks Tools and Talk Framework Come to the classroom with preconceptions. Need to develop a deep foundation of usable knowledge and understand facts in the context of a conceptual framework Must be taught explicitly to take control of their own learning by setting goals and monitoring progress toward them 29

30. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry So far in the seminar, how have we… Explored effective strategies for uncovering and addressing student misconceptions and preconceptions in your classroom (finding 1) Explored effective strategies to help students retain and transfer new learning to novel situations (finding 2) Explored effective strategies for empowering students to take ownership of their learning (finding 3) 30

31. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry What is learning? What does it mean to have learned something? On your index card, write one sentence or statement that begins to answer this question Pass your index card to the person on your left. Each person is to add one more sentence/statement that builds upon or counters the statement on that card Keep passing the cards to your left and adding a statement until you get your index card back 31

32. Math & Science Collaborative, © MSC 2009 Science Teacher Leadership Academy, Part 1: Immersion into Inquiry Acknowledgement This material is based on work supported by the SW PA MSP 2010 funds administered through the USDOE under Grant No. Project #: RA-075-10-0603. Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the granting agency. Math & Science Collaborative at the Allegheny Intermediate Unit