DEVELOPING MATHEMATICAL THINKING THROUGH LESSON STUDY: INITIAL EFFORTS AND RESULTS Soledad A. Ulep National Institute for Science and Mathematics Education Development University of the Philippines

ANALYZING EXISTING CONDITIONS Identifying Usual Practices: parts of a lesson, use of AGONA Determining the Need for a Research Lesson: develop a lesson on solving problems involving subtraction of whole numbers Examining the Usual Practices: use of AGONA and with teacher’s guidance, use of clue words, use of the “take away” meaning of subtraction

DEVELOPING A LESSON THAT ELICITS MATHEMATICAL THINKING Using a Framework on Mathematical Thinking - Teachers must engage in mathematical thinking. - Teach mathematics through problem solving to integrate mathematical thinking in the learning of content and enhance pupils reasoning and communicating skills.

- Through questioning, encourage pupils to think. - Ask pupils to discuss their ideas and formulate their own problems. - Use non-routine tasks such as open-ended problems to develop mathematical thinking in routine tasks. - Enable pupils to use their previous knowledge and skills to unfamiliar contexts. - Connect concepts and procedures. - Anticipate various pupil responses. - Develop persistence in solving problems and verifying results. Using a Framework on Mathematical Thinking

- Teachers were asked: In which word problem would pupils have more opportunity to think: (1) finding one of the following, given the other two: difference, subtrahend, minuend, or (2) finding the missing digits in the minuend, subtrahend, and difference which are all given Engaging Teachers in Mathematical Thinking

Example: Mr. Jose saves money for his house repair. The repair costs P_246. He has already saved P238_. So he still needs to save P3_ _7. How much does the house repair cost? How much as Mr. Jose saved already? How much more does he need to save? - A teacher’s work: Engaging Teachers in Mathematical Thinking

- A teacher realized that the problem that he formulated is open-ended. Engaging Teachers in Mathematical Thinking

Problem 1: A fruit distributor has 76_4 mangoes. There are 388_ green mangoes and 3_86 yellow mangoes. How many mangoes does the distributor have? How many mangoes are green? How many mangoes are yellow? Preparing the Task

76_4 76_4 388_ 3_ _ - 3_86 + 3_ _ 3_86 388_ 76_4 76_4 Possible number sentences

Problem 2: Mr. Jose keeps a record of the income of his store. The income two years ago was P18,_9_. The income last year was P6_,5_4. The income last year was P_3,7_8 more than the income two years ago. - What was the income two years ago? - What was the income last year? - How much more was the income last year than the income two years ago? Preparing the Task

6_5_4 6_5_4 _ 37_8 18_9_ - 18_9_ - _ 37_8 + 18_9_ +_ 37_8 _37_8 18_9_ 6_5_4 6_5_4 Possible Number Sentences

If one systematically substituted the digits 0 to 9 in the minuend, then the digits in the difference would be the same. Why? The higher the digit that was substituted in the minuend, the higher would be the digit in the difference. Why? The higher the digit that was substituted in the difference, the higher would be the digit in the minuend. Why? Would these results still be true if the given digit was no longer 9? Why? Possible Questions to Pursue

Why was it that it was only in the tens place of the minuend and difference that the digits differed? Why were the digits the same for all the other place values? What would be the answers if instead of the digits missing on the minuend and the difference they were missing on the minuend and the subtrahend or the subtrahend and the difference? What if instead of two digits on the same column, two consecutive digits on the same row were missing?

ELICITING MATHEMATICAL THINKING IN PUPILS Relating lesson to pupils’ daily life Making the pupils read numbers with missing digits Focusing the pupils’ attention on having to find the missing digits to get answers to the problem Leaving the pupils on their own to understand and analyze the problem

Asking pupils to form a number sentence representing the situation Making pupils discuss with a partner how to solve the problem Making the pupils explain their work by giving reasons for how they arrived at their answers Requiring the pupils to verify their work Relating relationships learned before to the new lesson ELICITING MATHEMATICAL THINKING IN PUPILS

Enabling the pupils to realize their own mistake and giving them the opportunity to correct them Making the pupils realize that a problem may have the same correct answer but different correct solutions or different correct answers and different correct solutions Comparing problems and their corresponding solutions and answers Eliciting Mathematical Thinking in Pupils

REFLECTING ON NEEDED IMPROVEMENTS Processing of pupils’ responses Helping pupil make connections Making the most out of pupils’ work

CONCLUSION When the teachers become more at home with their changed practices and engage more in mathematical thinking, possibly pupils could engage better in mathematical thinking.