Enhancing Mathematics Instruction Through Differentiation Techniques

1. Mathematics Support Differentiated Instruction

2. 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

3. Supporters of differentiation believe: • All students have areas of strength • All students have areas that can be strengthened • Students bring prior knowledge and experience to learning • Emotions, feelings and attitudes affect learning • All students can learn • Students learn in different ways at different times Gregory and Chapman 2006

4. Diversity in theClassroom • Using differentiated tasks is one way to attend to the diversity of learners in your classroom.

5. Differentiating Instruction • Some ways to differentiate instruction in mathematics class • Common Task with Multiple Variations • Open-ended Questions • Differentiation Using Multiple Entry Points

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

7. B12 calculate products and quotients in relevant contexts by using the most appropriate method C7 represent square and triangular numbers concretely, pictorially, and symbolically Choose three consecutive numbers, square them, and add the squares. Divide by 3 and record the whole number remainder. What happened? Why?

8. So why was the remainder 2? (n-1)2 + n2 + (n+1)2 = 3n2 +2 But it’s also true that n2 + (n+1)2 + (n+2)2 = 3n2 + 6n + 5 = 3n2 + 6n + 3 + 2

9. Or use a model

10. Plan Common Tasks with Multiple Variations • The approach is to plan an activity with multiple variations. • For many problems involving computations, you can insert multiple sets of numbers or have students select their own numbers. • You may also opt to give two or more choices of activities which relate to a common topic or outcome. • Common questions are carefully constructed so that students can contribute to the conversation no matter which variation they chose to explore.

11. A proportional example You used 240 g of rice. What was the total mass of the rice if… • Task A: It was 1/3 of the total mass of the rice. • Task B: It was 2/3 of the total mass of the rice. • Task C: It was 40% of the total mass of the rice. A4 demonstrate an understanding of equivalent ratios A5 demonstrate an understanding of the concept of percent as a ratio

12. Common Tasks with Multiple Variations • When using tasks of this nature all students benefit and feel as though they worked on the same task. • Class discussion can involve all students. • Questions should be phrased so that all students can offer comments and answers. • There is additional work prepared for any “early finishers”

13. What are some questions you could ask of students : • Choose a topic or an outcome(s) from your grade level curriculum and create a differentiated activity for your students.

14. Open-endedQuestions • Open-ended questions have more than one acceptable answer and can be approached by more than one way of thinking.

15. 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.

16. 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

17. Open-ended Questions Adjusting an Existing Question • Identify a topic. • Think of a typical question. • Adjust it to make an open question. Example: Ratios • The ratio of cats to dogs in a neighbourhood is exactly 2 to 1. There are 15 dogs. How many cats are there? • The ratio of cats to dogs in a neighbourhood is exactly 2 to 1. How many cats and how many dogs might there be in the neighbourhood? A3 write and interpret ratios, comparing part-to-part and part-to-whole

18. Sample question • Describe 25 as a percent of a number in as many ways as you can. Make sure some percents are big and some are little. A5 demonstrate an understanding of the concept of percent as a ratio

19. 100% of 25 • 50% of 50 • 1% of 2500 • 10% of 250 • 25% of 100 • 5% of 500

20. Other possibilities… • Name a fraction that is a bit less than 0.6. Explain how you know. Can you name another fraction that is between 0.6 and your suggestion? • The mean of some numbers is 14. What are the numbers? • The data can be shown using a coordinate graph with 4 quadrants. What might the data be? A9 relate fractional and decimal forms of numbers F8 demonstrate an understanding of the differences among mean, median, and mode F3 plot coordinates in four quadrants

21. Open-ended Questions • Use your curriculum document or other resource to find examples of open-ended questions. • Find two closed-questions from your curriculum document (or think of a typical question). • Change them to open-ended questions. • Be prepared to share both versions of your questions.

22. Differentiation Using Multiple Entry Points • Van de Walle (2006) recommends using multiple entry points, so that all students are able to gain access to a given concept. • diverse activities that tap students’ particular inclinations and favoured way of representing knowledge.

23. Multiple Entry Points Multiple Entry Points are diverse activities that tap into students’ particular inclinations and favoured way of representing knowledge.

24. Multiple Entry Points Based on Five Representations: Based on Multiple Intelligences: • Concrete • Real world (context) • Pictures • Oral and written • Symbols • Logical-mathematical • Bodily kinesthetic • Linguistic • Spatial

25. Sample - Ratio A3 Write and interpret ratios, comparing part-to-part and part-to-wholeA4 Demonstrate understanding of equivalent ratios

26. Sample – Rotational Symmetry 6E8 Students will be expected to make generalizations about the rotational symmetry property of all members of the quadrilateral “family” and of regular polygons * All activities demand some level of spatial sense.

27. Creating Tasks With Multiple Entry Points Using the outcomes for decimals, create tasks with multiple entry points. Take into consideration the five representations: real world (context), concrete, pictures, oral/written, and symbolic and multiple intelligences: logical/mathematical, bodily kinesthetic, linguistic, spatial.

28. Possible Uses for the Grid • Introduce some of the activities to students being careful to select a range of entry points. Ask students to choose a small number of activities. Other activities can be used for reinforcement or assessment tasks. • Arrange 9 activities on a student grid: 3 rows of 3 squares. Ask the students to select any 3 activities to complete, as long as they create a Tic-Tac-Toe pattern. • Other ideas?

29. Differentiating Instruction • When might you use each of these types of differentiation? Why would you select one rather than another type? • Common Task with Multiple Variations • Open-ended Questions • Differentiation Using Multiple Entry Points