Chapter 4 A Problem-Solving Platform
I was very interested in reading this chapter, as I am always trying to improve my use of problem solving.
According to the authors, in a differentiated classroom, the problems presented to the students will allow students to differentiate for readiness for themselves.
The authors define a “problem-solving platform” as “a math curriculum that consistently draws students into mathematical inquiry through stories, situations, or scenarios that challenge students with intriguing problems.” These types of problems allow students to “grapple” with the math at their own readiness level while also collaborating and listening to other perspectives.
“Good problems” are needed for this type of thinking to occur; they must be open problems, can be a variety of forms, and they invite persistence (which, in my opinion, needs to be greatly encouraged – I have noticed a general decline in persistence over the years...anyone else?).
An example of a good problem that one of the authors likes to use at the beginning of the year: holding up a chessboard, ask students how many squares are on the chessboard. Many students will begin by counting the 1 x 1 squares, but others will help to expand their vision by counting the very largest square, or the 2 x 2 squares, and so on.
A second example is called “Petals Around the Rose,” from an article by Marie Appleby. Five dice are rolled, and each time, an answer to the five dice is given by the teacher. The teacher continues to roll and give answers to each set of 5 dice. Students must find the connection between the numbers on the 5 dice and the answer. The authors suggest doing this activity periodically over several days, giving students time to think, brainstorm, and discuss throughout this time.
To incorporate this type of problem solving into your program, authors suggest that each unit could begin with a problem that has the upcoming content “embedded” in it. Good problems could be situations, puzzles, games, or questions.
A “good problem”:
*leads to significant math
* is open-ended; can be approached with various strategies
* is accessible to students with different strengths, needs and backgrounds
* has various solutions
* leads to higher-level thinking and discussion
* has constraints that provide direction but don’t limit thinking
* strengthens conceptual development
* allows teachers to assess how and what students are learning and to identify needs
The authors discuss the idea that the amount of time used for such problems will vary according to the needs of the classroom. The problem could be a warm-up or an introduction to a specific topic. It could be in the beginning or middle of a unit…it doesn’t necessarily need to take a great deal of time.
The authors reference the work of Sullivan and Lilburn’s Good Questions for Math Teaching (2002) to explain the steps for creating a good question for any math topic:
1. identify the topic
2. think of a closed question
3. open the closed question by including the answer and working backward to situations that might give that answer.
Or, rather than follow those steps to create your own question, adapt a standard question from something like – What is the volume of a 2 in by 3 in by 4 in box? To something like – The capacity of a box needs to be 24 cubic inches. What are the possible dimensions?
Another given example – instead of asking students to solve a fraction multiplication question that results in the answer of 2 ¾ - give the answer of 2 ¾ and ask students to find what two numbers would result in that product.
An example I came up with – instead of asking students the perimeter of a rectangle that is 9 feet long and 6 feet wide, pose this question: “Ben is getting a puppy and wants to build a rectangular pen in the yard for the puppy to play. Ben’s parents told him that there is 54 feet of fencing in the garage. If he uses all of the fencing, how long and how wide can the pen be?”
To me, this question is accessible to all students – if they know how perimeter works, they can use that knowledge to determine possible lengths and widths; if they do not understand perimeter yet, they can draw the rectangles, guess the side lengths and then add the sides to check. For those students who quickly find a possible answer, they can extend to find all of the possible answers and then determine which one would give the puppy the greatest area in which to play. This allows all students to solve, but allows for differentiation based on student strengths and background.
I have done quite a lot of problem solving in my classroom, giving situations like the puppy one above, but many of my actual questions were closed; a few were open. I need to take some time to go back to these and see how I can revise them to make more of them open.
Do you have any favorite "open" problem solving scenarios, puzzle, games?