Selecting and Designing Tasks Math 423 Sept. 24, 2008

Differentiating Instruction “…differentiating instruction means … that students have multiple options for taking in information, making sense of ideas, and expressing what they learn. In other words, a differentiated classroom provides different avenues to acquiring content, to processing or making sense of ideas, and to developing products so that each student can learn effectively.” Tomlinson 2001

Open-ended Questions Open-ended questions have more than one acceptable answer and/ or can be approached by more than one way of thinking.

Open-ended Questions Well designed open-ended problems provide most students with an obtainable yet challenging task. Open-ended tasks allow for differentiation of product. Products vary in quantity and complexity depending on the student’s understanding.

Open-ended Questions An Open-Ended Question: should elicit a range of responses requires the student not just to give an answer, but to explain why the answer makes sense may allow students to communicate their understanding of connections across mathematical topics should be accessible to most students and offer students an opportunity to engage in the problem-solving process should draw students to think deeply about a concept and to select strategies or procedures that make sense to them can create an open invitation for interest-based student work

Open-ended Questions Working Backward Identify a topic. Think of a closed question and write down the answer. Make up an open question that includes (or addresses) the answer. Example: Quadratic Equations Sketch the graph of ½ y = (x – 3)2 The graph of a quadratic equation passes through the point (3, 0), what might the equation be? Sketch potential graphs.

Open-ended Questions Adjusting an Existing Question Identify a topic. Think of a typical question. Adjust it to make an open question. Example: Trigonometry Find the value of x in the triangle. A window is 12m off the ground. Explore different lengths of ladder that could be used to reach the window, if the angle of the ladder to the ground must fall within the range of 30º to 40º.

Open-ended Questions: Selecting a good problem A paper shredding company has an old shredder that can shred one truck load of paper in 4 hours. Because business has been good lately, they were able to purchase a new shredder that can shred one truckload of paper in 2 hours. If they run both shredders at the same time, how long would it take to shred one truckload of paper?

Common Task with Multiple Variations A common problem-solving task, and adjust it for different levels Students tend to select the numbers that are challenging enough for them while giving them the chance to be successful in finding a solution.

An Example of a Common Task with Multiple Variations Using Algebra Tiles show how to find the factors for one of the following:

Example Spaces: Fractions Two fractions whose product is less than ½. Think of another. Think of one that is really different than the first two.

Example Spaces: Algebra Think of a quadratic equation where one root is x = 5. Think of another. Think of one that is really different than the first two.

Think Multiple Representations Verbal: Explain it in Words Contextual: Write a Story Problem Concrete: Use Concrete Materials to Build It Symbolic: Write it in Mathematical Symbols Pictorial: Draw a Picture Model

Multiple Entry Points Based on Multiple Intelligences: Based on Five Representations: Logical-mathematical Bodily kinesthetic Linguistic Spatial Musical Naturalist Interpersonal Intrapersonal Concrete Real world (context) Pictures Oral and written Symbols Based on Learning Modalities: Verbal Auditory Kinesthetic

Task Selection Good problems Begin where they are Focus on important mathematics Requires justification and explanation Promotes doing mathematics and encourages understanding May be open-ended Open Process: many ways to arrive at the answer Open End Product: many possible solutions Open Question: can explore new problems related to the old problem Promotes the Big Five!

Levels of questions Level 1: Knowledge and Procedures Remembrance could be simple recall (defining a term, recognizing an example, stating a fact, stating a property) Questions within one representation (performing an algorithm, completing a picture) Reading information from a graph.

Levels of questions Level 2: Comprehension of Concepts and Procedures Makes connections between mathematical representations of single concepts (creating a story problem for an addition sentence, drawing a number line picture to show the solution to a story problem, stating a number sentence for a given display of base ten blocks) Makes inferences, generalizations, or summarizes ( makes inferences from a graphical display, finds and continues a pattern) Estimates and predicts Explanations

Levels of questions Level 3: Problem Solving and Application Multi-step, multi-concept, multi-task Non-routine problems Requires application of problem solving strategies New and novel applications Break Down Level 1 – 25% Level 2 – 50% Level 3 – 25%

Assessment Strategies Observations Observations of group work, problem solving, communication, etc. Conversations and Interviews Portfolios and journals Responding to open-ended questions, monitoring their own learning, reflecting on their learning, sharing and discussing with the teacher Projects and investigations Presentations Tests, quizzes and exams