Conference Powerpoint v14 - fibonacci09-10
Using Problem Solving To Engage Students In Their Study of Linear Equations Options: D31 & F34 Presenter: Russell James 0011 0010 1010 1101 0001 0100 1011 While You’re Waiting ….. Peruse The Discussion Paper For This Option on pages 96-101 Of the Conference Handbook 1 2 4 Welcome 0011 0010 1010 1101 0001 0100 1011 Russell James Maths co-ordinator Fairhills High School 1 2 4 This year … 0011 0010 1010 1101 0001 0100 1011 I was given 20 days of professional leave by the education department to undertake a project that I have called. Using Problem Solving To Engage Students In Their Study of Linear Equations 1 2 4 Over the course of my project three units were developed & trialled 0011 0010 1010 1101 0001 0100 1011 Unit 1: Modelling linear data with formulae (and working with equations) Unit 2: Strategies for finding linear formulae from tables Unit 3: Strategies for finding linear formulae from graphs 1 2 4 My Objective Today is to … 0011 0010 1010 1101 0001 0100 1011 share some of the experiences, resources and pedagogical approaches that were trialled during my Professional Leave project . 1 2 4 I have given you … Booklets for each of the following: • the problem solving activities trialled • the ICT activities trialled • the complete activities of unit 1 • sample activities from units 2 & 3 • an Assessment Task from the SNMY project 0011 0010 1010 1101 0001 0100 1011 1 2 4 Plus a CD with electronic copies of the majority of the activities trialled in the project My Contact Details 0011 0010 1010 1101 0001 0100 1011 On the CD there is also the file “Presenter Contact Details” with my school contact details and: My Email: [email protected] Project Wikispace: 1 2 4 http://fibonacci09-10.wikispaces.com/ Fairhills High School 0011 0010 1010 1101 0001 0100 1011 We are situated in the City of Knox near the Dandenongs with a student population – – – – Around 850 Mostly born in Australia Mainly Anglo/European heritage Rated in the middle of the Education Department’s scale for learning potential. 1 2 4 Two of our major concerns in Mathematics across the school are.. 0011 0010 1010 1101 0001 0100 1011 1. Learning participation of 2. Students’ negativity students with a history towards new problems of disengagement in the or situations that Maths. require more thinking than just the repetition of a procedure. 1 2 4 0011 0010 1010 1101 0001 0100 1011 Hence My Project 1 2 4 I chose Problem Solving as a key element of the project because I 0011 0010 1010 1101 0001 0100 1011 believe well designed activities can: • Improve students’ thinking resilience and skills • Improve students’ engagement if the activity 1 2 4 – Gives them a realistic chance to explore a situation and feel a sense of satisfaction that they have thought a problem through – Provides a reference point and everyday relevance for the study of higher order mathematical concepts and procedures 0011 0010 1010 1101 0001 0100 1011 My Design Model For The Problem Solving Activities is described on pages 96 of your conference handbook 1 2 4 Some features of this model 0011 0010 1010 1101 0001 0100 1011 Problem solving activities should: – Have multiple pathways to a solution providing and entry point and challenge for students over a range of abilities – Provide a rationale for studying higher levels of mathematics – Have an investigative stage that is no more than 40minutes. 1 2 4 Also .. 0011 0010 1010 1101 0001 0100 1011 The activities should: – be presented with a problem solving toolbox to reinforce students problem solving skills – include a reflection session to enable students to develop their metacognitive skills and reinforce what they have learnt from their experience. 1 2 4 Let’s look at a problem This problem follows on from a Multiplicative Thinking assessment task that I will talk about later This is just as important as the task itself because a good lesson structure changes the task from a puzzle to a learning experience Table & Chairs Follow-Up Problem Purpose To give students: • A rationale for the study of linear equations • Develop students thinking resilience and problem solving skills (including increasing their familiarity with a problem solving toolbox) Lesson Structure 1. Introduce The Problem (5-10 min) 2. Solve The Problem (30-40 min) 3. Reflect On The Problem (5-10min) 4. Write a report (next lesson) Stage 1. Introducing the Problem After handing out the problem sheet (with the working page on the back): – I described Problem 1 to the class – 1 wrote the Universal problem solving method on the board • Understand the problem • Decide on a strategy • Solve the Problem • Check Your Answer – I invited possible strategies from the class and then referred them to our Problem Solving Toolbox A Common Strategy List Most references and websites I’ve looked up present between 8 and 10 problem solving strategies. Rob Vingerhoets (a very good consultant we have worked with) puts forward the following methodologies: – – – – – – – – – – Guess & Check Draw a diagram Make a table Make a model Work backwards Write a number sentence Look for a pattern Act it out Solve a simpler related problem List all possibilities Our Strategy List/Problem Solving Toolbox • Is an expanded version of Rob V’s • We broke the table approach into 2 • Included a strategy explicitly mentioning the use of a graph • Included 3 algebraic methodologies • Introduced 4 methods that utilise technology A hint from Rob Vingerhoets: Insist On A Starting Strategy From Each Student Before They Begin The Problem This sounds like it would take a long time but it is actually quite quick and it provides an impetus and a starting point for most students to get underway with the task. It also provides a way of identifying those students who need further help, either by revisiting the problem to make sure they understand it, or by helping them to choose a meaningful strategy. Stage2: Solving the problem As I moved around helping and encouraging students I constantly reinforced my mantra for all problem solving activities: – Record all your working (preferably under a heading) – Discuss the problem with your friends because this is a valuable way to learn – Feel free to change your strategies or combine them – I am asking you for a prediction with mathematical evidence. There is no right or wrong way to do the problem just follow a path that makes sense to you. Stage 3: The Reflection Phase This is perhaps the most crucial phase with respect to: a) Producing a long term improvement in students thinking skills b) Providing a meaningful rationale for the subsequent study of related maths skills. For this problem I asked selected students to describe how they solved the problem and recorded their approaches on the board. Solving with a diagram A small number of students solved the problem by drawing 30 tables (no one opted to build it with tokens and rectangles) ……… Using a table Lots of students liked this method because they appreciated the efficiency of getting an accurate answer with less effort than drawing 30 tables. Tables Chairs 1 6 2 10 3 14 4 18 5 22 .. .. .. .. 30 Writing number sentences A number of students tried writing number sentences to get an answer. Not all were right but there was some good discussion about the correct answer 1 + 30 x 4 + 1 = 122 1 + 4 + 4 + 4 + …… + 1 = 122 30 x 6 = 180 I am always very conscious about my language when presented with answers that do not give the “right answer”. E.g. “there is some good maths there, can you explain why you think this is right” Look For A Pattern A limited number of students recognised a pattern 2 + 4 x number of tables Or 1 + number of tables x 4 + 1 Building A Unit Around The Starting Problem From the primary and follow up problems I was able to give credibility to the statement “If we know how to find and work with rules (formulae) we might be able to make our life easier in the future” Formulae & Equations Skills Development & Assessment Tasks Problem Solving Activity 2 Word problems ICT Activities The Linear Equations Continuum Students who have progressed quickly through the skills and assessment tasks were directed to an exercise called “Year 8-9 Linear Equations Continuum”. Portfolio page for Unit 1 As well as including more problem solving into our teaching one of my other goals in my TPL project was to investigate alternative ways teaching and assessing linear equations so that students were offered some decisions about what and how they learned. Currently much of our curriculum could be classified as passive learning since it relies on the more traditional textbook approach that focuses on “mastery through practice” (rather than active learning). Intermission • Eat a frog • Have a read of the booklets I have supplied • Formulate some questions Evaluating our project This year Fairhills had seven classes at year 8 and they were all involved in the project. It is important to note that not all of these classes have the same course design. At year 7 & 8 students our classes are divided into two pathways An accelerated pathway, with two Maths classes aiming to attain a A second VELS level pathway of at leastwith 5.5 five by the end ofMainstream Year 8 andMaths henceclasses be able aiming move to attain a VELS levelthird of atyear leastat5.0 by into Year 10 in their the end of school. Year 8 and then move into secondary Year 9. Data we examined To evaluate our project we collected and analysed Student achievement data Student feedback sheets Teacher interview notes In general faster &/or motivated learners … appreciated the ability to move through the topics at their own pace performed at least as well as in these topics as with other topics. In many cases they were attempting higher order questions that they would not have been doing otherwise became more adept and resilient with problem solving situations as they undertook more such tasks become more active learners in other topics become more adept at making connections between their existing knowledge and the mathematics encountered in new topics. In general learners with a history of disengagement ….. found the problem solving tasks very difficult and tended to give up early or asked for guidance at every step. They did show some improvement in their problem solving resilience but were still heavily reliant on the teacher for instruction and motivation seemed to prefer the textbook approach because of its structure and the fact that they are lead into each question with a recipe for finding a solution. did not show any marked difference in interest or progress relative to other units undertaken. Teachers felt: a degree of satisfaction seeing their motivated students becoming more active learners and having the opportunity to progress to levels of understanding appropriate to the abilities. at times it was a lot of work managing the units, particularly when there was a spread of activities underway. Overall it was survivable if the units were kept to less than 3 weeks. They also felt … that the lack of success with hard to motivate students was largely due to frustration caused by their lack of thinking resilience and skills Compounded by the fact that these students were unaccustomed to this type of task because in the past they have been directed more to repetition, rather than cognition. Thus to achieve more success with the hard-to-engage students, we felt that we need to give them a lot more experience with cognitive and meta-cognitive exercises. Was the Project A Success? Overall, we would say Yes Because There were observable benefits for a significant proportion of students with no disadvantage to the remainder of the cohort It further highlighted the need for us to develop the thinking resilience and skills of the learners with a history of disengagement. So in 2010 we will ….. provide Year 8 students with some preliminary thinking tasks before we run slightly revised versions of the units trialled in 2009. incorporate more Problem Solving activities into the Year 7 curriculum so as to provide students with more cognitive and metacognitive experience for Year 8. SNMY Multiplicative Thinking Assessment Task Scaffolding Numeracy in the Middle Years (SNMY) refers to a research and evidencebased project under the direction of Professor Di Siemon, in conjunction with DEET, RMIT and the Tasmanian Dept Ed. The project aimed at identifying and refining a learning and assessment framework for the development of multiplicative thinking at this level using rich assessment tasks For more details Google SNMY project.