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Developed by ERLC/ARPDC as a result of a grant from Alberta Education to support implementation [email protected] Twitter: @thescamdog Blog: www.thescamdog.wordpress.com This session is aimed at High School Math teachers and is intended to help support the implementation of the revised program of studies. Participants will see examples of how students can learn THROUGH problem solving. ERLC – Learning Through Problem Solving In the polling response section, please enter ◦ ◦ ◦ ◦ ◦ A if you are a High School Teacher B if you are a Junior High School Teacher C if you are an Elementary School Teacher D if you are a consultant or administrator E if you are some combination of the above ERLC – Learning Through Problem Solving In the polling response section, please enter ◦ ◦ ◦ ◦ A if you are from Edmonton B if you are from the greater Edmonton area C if you are from elsewhere in Alberta D if you are outside of Alberta ERLC – Learning Through Problem Solving There are seven mathematical processes in the front matter of the revised program of studies. These processes are to permeate teaching and learning. ◦ ◦ ◦ ◦ ◦ ◦ ◦ Communication (C) Connections (CN) Mental Math and Estimation (ME) Problem Solving (PS) Reasoning (R) Technology (T) Visualization (V) ERLC – Learning Through Problem Solving “The mathematical processes are the content, and the outcomes are the context.” Dr. Peter Liljedahl, Simon Fraser University – Secondary Math Education ERLC – Learning Through Problem Solving The description of the Problem Solving process contains the phrase: “Learning through problem solving should be the focus of mathematics at all grade levels.” ERLC – Learning Through Problem Solving We learned FOR problem solving. ERLC – Learning Through Problem Solving The top of Ken’s new desk is 3 1 m wide and 5 m long. Find the area of the top of his desk. ERLC – Learning Through Problem Solving We learned ABOUT problem solving. ERLC – Learning Through Problem Solving 1. Understand the Problem 2. Develop and 3. Carry Out a Plan 4. Look Back ERLC – Learning Through Problem Solving Learning through problem solving should be the focus of mathematics at all grade levels. - Alberta Program of Studies ERLC – Learning Through Problem Solving An example of learning THROUGH problem solving. http://www.amaranthpublishing.com/MozartDiceGame.htm ERLC – Learning Through Problem Solving Mozart's Dice Game Please roll your dice and enter the results. Mozart's Dice Game http://sunsite.univie.ac.at/Mozart/dice/collaborate.cgi?tables=yes ERLC – Learning Through Problem Solving Mozart's Dice Game If this is an activity used in class, the next thing I would tell the students is that I am incredibly confident that no one else has ever heard the minuet they just played. What questions might they have? Can we distill it down to one single question? ERLC – Learning Through Problem Solving Mozart's Dice Game Possible questions: Has the minuet we just composed ever been heard before? How many minuets could be created using the parameters of the problem? How long would it take you to listen to them all, assuming that each takes about 30 seconds? ERLC – Learning Through Problem Solving Mozart's Dice Game Debrief. Discuss. What mathematical concept is addressed? Could students arrive at this concept on their own? Where will they struggle? How will you scaffold for those who struggle? ERLC – Learning Through Problem Solving What makes for a good problem for learning through problem solving experience? ERLC – Learning Through Problem Solving The problem is given at the beginning of the learning rather than at the end. The problem is non-routine. The students can not solve it immediately. But, Every student has an entry. The problem is engaging and compelling so students will persevere. The problem invites multiple methods of solution. The problem fosters discussion and debate. The problem is at an appropriate level for the audience. The best ones allow the students to come up with their own problem to solve. A compilation from a variety of sources – Marilyn Burns, Marian Small, John Van de Walle, Dan Meyer, and me. “The definition of a good mathematical problem is the mathematics it generates rather than the problem itself.” “It's fine to work on any problem, so long as it generates interesting mathematics along the way - even if you don't solve it at the end of the day.” “I loved doing problems in school, I'd take them home and make up new ones of my own.” Another Example of Learning Through Problem Solving Laura has trained her pet rabbit, Jack, to hop up a flight of 7 steps. Jack can hop up one step at a time or two steps at a time. How many different ways can Jack hop up the flight of seven steps? ERLC – Learning Through Problem Solving Rabbit Problem Debrief Debrief. Discuss. What mathematical concept is addressed? Could students arrive at this concept on their own? Where will they struggle? How will you scaffold for those who struggle? ERLC – Learning Through Problem Solving What does your classroom look like while students are working on problems? How do you debrief the problems with students? Do you tell them the right answer? Where do you find these problems? When a problem is not particularly compelling, how do we engage students? ERLC – Learning Through Problem Solving An example of learning THROUGH problem solving. From Dan Meyer’s Blog (dy/dan) http://blog.mrmeyer.com/?p=5983 From worst to better… ERLC – Learning Through Problem Solving Worst •The problem claims to represent the real-world but its illustration is only clip-art or a line drawing. •The problem specifies the exact method of its own solution, usually in a series of substeps labeled "a, b, c, d.“ •The problem only gives information that the student will use in the solution. ERLC – Learning Through Problem Solving Bad The same as worst except: The real-world problem presents itself as it exists in the real world. ERLC – Learning Through Problem Solving Good The same as bad except: The problem reveals no information about itself — no measurements, especially — forcing the student to decide for herself what information is relevant to the solution. The problem doesn't hint at its own solution method with sub-steps. The student can develop that solution socially, in conversation with her teacher or her classmates. ERLC – Learning Through Problem Solving An example of learning THROUGH problem solving. http://vimeo.com/9552977 … to Best ERLC – Learning Through Problem Solving Best The same as good except: The problem hangs itself on a single, concise, intuitive question, one that any student can answer, regardless of mathematical ability. The teacher solicits guesses and records them publicly, investing the students in the outcome of the exercise, and refers back to them later, perhaps introducing the concept of percent error. The solution to the problem isn't read from an answer key. Instead, it's observed by the class together in a second multimedia artifact. The class compares the answer derived from their theoretical model to this practical answer. This is scary. The class will almost certainly be wrong but the conversation about sources of error should be embraced, not feared. ERLC – Learning Through Problem Solving http://www.ted.com/talks/dan_meyer_math_curriculum_makeover.html ERLC – Learning Through Problem Solving Math 10 Applied – Angle of Elevation and Depression 1. Students measured heights of their eyes. 2. Students found one object in the classroom that was the same height as their eyes. 3. Students moved around the room and listed objects they would have to look up to see (elevation) and down to see (depression). ERLC – Learning Through Problem Solving Math 10 Applied – Angle of Elevation and Depression 4. I took the class to the atrium and asked them, in teams of three, to estimate the height of the second floor railing above the main floor. 5. We returned to the classroom and recorded each team’s guess in a chart on the SMART ™ Board. 6. We made clinometers, and each team had a clinometer and a tape measure. ERLC – Learning Through Problem Solving Math 10 Applied – Angle of Elevation and Depression 7. They returned to the atrium to calculate the actual height of the atrium. 8. Each team recorded their calculated height next to their estimate. 9. We revealed the actual height, and celebrated the best estimate and the best calculation. ERLC – Learning Through Problem Solving A quick question that promotes thought and exploration is valuable. ERLC – Learning Through Problem Solving From Andrew Shores at http://andysunknownquantity.blogspot.com/ Looking at this illustration, what questions do you want to answer? Can the grasshopper start on any particular number (not counting shaded boxes) and be safe by taking hops of 4? What if it took hops of 5? 6? 7? What if the spider webs were on perfect cubes? What if they were on triangular numbers? Fibonacci numbers? etc. John Scammell can be reached at: E-mail [email protected] Twitter @thescamdog Blog www.thescamdog.wordpress.com ERLC – Learning Through Problem Solving