1. Make sense of problems and persevere in solving them
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
Representative IMP Lessons
Throughout the IMP curriculum, students are assigned Problems of the Week (POWs), which are open-ended problems that cannot be solved easily in a short period of time. POWs help students develop thoughtfulness and perseverance, and force them to focus on their own thinking processes. Students must explain and illustrate their strategies and solutions, and must justify their reasoning in clearly written reports.
Representative IMP Year 1 Lesson:
Corey Camel ( The Pit and the Pendulum), 150
Corey has 3000 bananas to bring to a market that is 1000 miles away. Corey can only carry 1000 bananas per trip and eats one banana for every mile traveled. Out of the 3000 bananas, what is the largest amount of bananas that Corey can bring to market? Initially, students may say “no bananas”. But they are asked to make sense of the problem and delve deeper into alternative solutions. To help students make headway to a solution, a mini-Corey Camel problem is presented to them. Students use simulations, pictures, tables of values, and alternative solutions to find an answer. Students work collaboratively, listen to each other’s solutions, and prove to each other that their solution is the correct one.
Representative IMP Year 2 Lesson:
Just Count the Pegs ( Do Bees Build It Best?), 304
Students recreate the problem of finding area on a geoboard similar to what confronted Georg Alexander Pick as he formulated his eponymous formula. Students look at different examples of polygons formed on a geoboard, gather data, and construct a formula. Two different approaches are considered and students are asked to support or refute the validity of the two approaches. Students are also encouraged to come up with their own approachas long as they can thoroughly support it.
Although an acceptable answer would be Pick’s Formula, that is not the point of this POW. Rather, the teacher looks at how students gather data to solve this problem and how they support and defend their own findings while they examine the work of their peers and try to prove them wrong by counter example.
Representative IMP Year 3 Lesson:
Let’s Make a Deal (Pennant Fever), 20
The classic Monty Hall dilemma from the game show, “Let’s Make a Deal” is presented to the students. You can win a great prize or a prize that you could do without. Monty Hall shows you what’s behind one of the three curtains and asks you if you would like to change your original curtain or switch. Students are asked which strategy, switch or stay, is better and why. Students are introduced to the problem through a simulation. Then they are asked to examine the probabilities and mathematics behind the problem. The solution flies in the face of intuition as the probabilities support the “always switch” strategy. Students are asked to explain the better strategy and use mathematics and probability theory to support their choice.