October 2009 Pro-D Day “Problem Solving Approach is Working”

1. October 2009 Pro-D Day“Problem Solving Approach is Working” Charles Schilling Highland Secondary School charles.schilling@sd71.bc.ca http://web.sd71.bc.ca/highland/?q=node/125

2. Who am I? • 20 years of experience • High school and middle school • Overseas experience as well • Math 7 to 9 recently

3. What are we going to do today? • How did I get started with PS? – survey • What am I doing? – Problem solving • What is working? – changed practice • What next?

4. Is this Problem Solving? • A wholesaler has 100 kg. of mixed nuts that sell for $4 per kg. To make the price more attractive, she plans to mix in some cheaper nuts worth $3.20 per kg. If the wholesaler wants to sell the mixture for $3.40 per kg, how many kgs of the cheaper nuts should be used? • Let x=kg of cheap nuts • $3.20x = total money expected by selling cheaper nuts • $4.00(100) = total money expected by selling $4 mixed nuts • 3.4(x+100) = total money expected to receive by selling mixture of $4 and $3.20. • Equation: 3.20x + 4.00(100) = 3.40(x+100)So …… x = 300 kilograms of the cheaper nuts needed.

5. I know think this is Problem Solving …

6. This is Problem Solving

7. PS Important? • The WNCP CCF K-9 says “Learning through problem solving should be the focus of mathematics at all grade levels.” • “Problem solving, reasoning and connections are vital to increasing mathematical fluency, and • must be integrated throughout …”

8. PS Important? • NCTM Principles and Standards says” Problem solving is an integral part of all mathematics learning. In everyday life and in the workplace, being able to solve problems can lead to great advantages. However, solving problems is not only a goal of learning mathematics but also a major means of doing so. Problem solving should not be an isolated part of the curriculum but should involve all Content Standards.

9. How did I get started with PS? • Returned to middle school after being overseas • Students told me they were struggling with Math • School results showed us that there were successes as well as struggles in Math • Math contests were running at the school and providing challenges for a small number of students but not overall • Learning support was well established

10. Wanted to Know More … • Put together a survey that asked students about their experiences in Math • It was done at the same time for whole school • Simple paper and pencil survey the first year that took a LONG time to collate into something that we could use – 400 surveys • Used an electronic survey the second year! • Asked about effort, attitude, textbooks, class examples, homework, etc.

11. Survey Results • Most wanted more explanation in class (Girls) • Everyone wanted less questions so there was more time on each question • Many said there were lots of topics that just went one after another – not enough connections • Some said Math class was boring or there was not enough hands-on • Boys said they wanted less homework ! • Boys were not doing homework

12. What did we learn from the Survey? • Needed more hands-on in Math • Needed to try more to build connections between topics (concept understanding as well as procedural) • Needed to look at what we were covering to make sure it was important and then spend appropriate time on it • Needed to look at homework and whether is was helping or not – no answer on that one • Textbook use was hit and miss at 8 and 9 with MOM but better at 7 where MMS was introduced

13. What did I do? • For me, problem solving seemed like a good way to get into this whole mess that we had brought on ourselves! • Designed a problem solving booklet that contained specific problems picked to look at specific, different strategies (besides guess-&-check) • We looked at problems that were best solved using specific strategies like draw a picture, work backwards or algebra (did PS right after Algebra in grade 9) • Then gave problems to students that required them to practice the strategies we had studied • Wanted to provide appropriate examples that students said they wanted more of in class!

14. Why PS? • I was introduced to the Network of Performance based Schools (NPBS) which gives $$$ support of school-based projects so they can be continued and shared • The Network of Performance Based Schools is a voluntary action research community designed to improve student learning • For more info, http://npbs.ca

15. Why Problem Solving? • Can be structured for all students in your class on some level • Can be “exact” to get an “answer” and you can have students struggle to get there (or not) • Can be open ended where it just sets the stage for future lessons • Can be just an investigation/exploration • Can be written questions or hands-on • Works well for communication between students • Works well for group work

16. What to worry about? • Can students read the problems and understand them? – some real struggles here and some problems were changed to make them more readable • CAREFULLY pick problems to use so that the entry point is right for your classes • Need to provide support for above with strategies for vocabulary and organization • Needed to work on “What is important?” in questions • Using tables or charts to organize questions really helped a lot of students

17. Where is the time? • Wanted to see if spending extra time on Problem Solving was going to make a difference! • Teaching strategies would have to come at the expense of other ILOs – mostly from Probability and Stats section • Did pre-assessment and post-assessment studies of my classes to see if students were actually building their “strategy toolbox” • Results were promising!

18. Results of PS Focus

19. How did I get my results? • Developed a rubric from the Gr. 8 Performance Standards that looked at 5 areas: • Persistence level for problem solving • Use of strategy to help solve problem • Conceptual understanding of problem • Accuracy level for problem solving • Communication level in problem solving • Gave some assessment-focus problems before instruction and same afterwards

20. What did I See Afterwards? • Students were much better at understanding the info in problems and were able to organize the question information better • Other strategies were used instead of G&C including tables, pictures and even Algebra! • Students persisted with the problems more and wanted to “get the answer” • Attitudes about PS were changing as well – there were fewer “I don’t get it”

21. Was I happy? Yes! • I had seen that the time spent on PS was worth it because it helped students to be more successful at this topic • Lots of NYM  MM which was great! • Some MM  FM which was my hope • Not much FM  EXC (hard to measure this difference with my rubric) • However, something was still missing …

22. Problem Solving Changed • When I started teaching Math, it was do some examples from the front of the class, assign questions and then have students do them • When I started with PS, it was give 1 or 2 problems then assign more for students to do • It was still giving students a recipe and then asking them to repeat that recipe • Noticed this with my honours class specifically because they were really good at transferring the solved examples to new problems of the same type

23. What else has changed for me? • Moving to a place where students need to know why just as much as how for concepts • Looked at and started to use assessment for learning techniques in my math classroom • More self and peer assessment • More use of exemplars and 1 to 4 rubrics • More use of “second chance” type strategies • Become more of a constructivist (Van de Walle)

24. What has PS changed into? • More hands-on problems than paper and pencil word problems now • More exploration type problems to get student brains working on a topic • Exploration first and then instruction and practice

25. Examples • Slice and Dice Cubes • What type of can holds the most? • Border Problem • Locker Problem • Match Stick Problem

26. Slice and Dice Cubes • Construct some cubes using Snap Cubes. These structures are called slice ‘n’ dice cubes, measuring 2x2x2 and 3x3x3 and 4x4x4. Once assembled, imagine you dip them in red paint and allow them to dry. Take them apart again and figure out the number of snap cubes with 0,1,2 and 3 painted faces • Great for SA, Volume, Data and Algebra • http://connectedmath.msu.edu/CD/Grade8/Painted/PaintedCubes.html

27. Locker Problem • There are 1,000 lockers in the long hall of Westfalls High. In preparation for the beginning of school, the janitor cleans the lockers and paints fresh numbers on the locker doors. The lockers are numbered from 1 to 1,000. When the 1,000 Westfalls High students return from summer vacation, they decide to celebrate the beginning of the school year by working off some energy. • Student 1 opens every locker. • Student 2 either opens or closes every other locker. • Student 3 opens or closes every third locker. And so on, ...http://connectedmath.msu.edu/CD/Grade6/Locker/ • Great problem to work with perfect square numbers!!! • 31 lockers are left open since there are 31 perfect squares < 1000

28. Where can you start? • Find and use good problems where you can • Teach specific strategies for getting started with a problem and for solving it • Let students struggle with problems • Don’t give them the answers sometimes • Have groups of students work on problems together and share their thinking