1. Laura Brake, M.Sc. Mathematics Achievement Specialist

2. Tactics versus algorithms Focus on algorithms What research indicates Moving algorithms from Conceptual Understanding through to Procedural Fluency Session Target

3. Tactics: general rules that govern an overall approach Ex. To read data from a chart: –Determine what the rows report –Determine what the columns report –Determine the relationship between the two When we read a chart, we use that general pattern but it is not necessary to follow it in a rigid or set order. Tactics

4. Algorithms are skills that have specific steps for specific outcomes –Ex. Traditional long division The steps in a tactic do not have to be performed in a set order; the steps in an algorithm do. Algorithms

5. At your table, discuss your answers to the following questions: 1) Are tactics and algorithms explicitly differentiated between? 2) How are tactics and algorithms taught to students? With what strategies? 3) What questions do I have about using tactics and algorithms? Existing Practice

6. 1) The discovery approach is difficult to use effectively with skills (tactics and algorithms) 2) When teachers use discovery learning, it is effective to organize examples into categories that represent the different approaches to the skill 3) Skills are most useful when learned to the level of automaticity Research — 3 main points

7. It is a common misconception that allowing students to discover how to perform a skill or process is always better than directly teaching the skill or process Both “discovery” and “drill-and-practice” techniques have their place in the classroom Some concepts are better taught through discovery and some are better taught through direct instruction 1) The discovery approach is difficult to use effectively with skills (tactics and algorithms)

8. 2) When teachers use discovery learning, it is effective to organize examples into categories that represent the different approaches to the skill

9. 3) Skills are most useful when learned to the level of automaticity Research relative to skill learning is that skills must be learned at a level to which they require little or no conscious thought. Automaticity To do this, students must engage in practice. At first, with practice sessions that are close together; and, then distributed over time. Conceptual Understanding comes before Procedural Fluency.

10. Without an initial model, learning a skill or process can be chaotic and time consuming because it’s essentially trial-and- error. A model, or step of steps can be created for almost any skill Some students prefer a written set of steps In math, algorithms are models Modelling and think aloud

11. + and – of Integers Using a $ representation First: identify the sign on each number. Does each value represent money or debt?

12. Negatives are money you owe (debt). Combine your negative values. Positives are money you have. Combine your positive values. Debt: Money:

13. Algorithm: “ Subtract and keep the sign of the biggest.” …because…why? Subtract: find the difference Did you have more money or debt? You had more debt, so your final situation must be debt. -$91. Ninety one dollars of debt. Now, you have $237 debt and $146.

14. Integers Algorithm “ subtract and keep the sign of the biggest” Debt: Money: Solution:

15. Apply it to rational numbers and the connection becomes stronger Find the difference between the Absolute Values More negatives (debt) than positives (money), so the answer must be Negative (debt). Answer

16. Once you begin to use a skill or process, it will be altered from the original. This process is called shaping. Shortcuts and tricks are a shaping of the process, once fluency is established. The importance of shaping a new skill or process cannot be exaggerated. Inattention to this aspect of learning skills and processes is a primary reason for students’ failure to effectively use them. At first, students need to show their work, identifying the sign on the number; subtotalling the positive values, and the negative values….etc. As they become proficient, shaping is an important step—after they are proficient. Shaping

17. Automaticity is learning to the point where a skill no longer requires much conscious thought. This frees the thinking process for application and problem solving, at a higher level of learning. Conceptual Understanding leading to Procedural Fluency. Automaticity

18. At your table, define “Division” without referencing any form of the word “Divide.” Division…what is it?

19. Division Algorithm “to ÷ fractions, x by the reciprocal” Steps of the algorithm: ▪To divide fractions, X by the reciprocal ▪ Cancel diagonally ▪ X numerators; X denominators ▪ Change improper fraction to a mixed numeral Because… why…?

20. 24 split into 8 equal groups = 3 in each group “dividing” into 8 groups = taking “of” the total.

21. Expressed symbolically:

22. 24 split into 3 equal groups = 8 in each group “dividing” into 3 groups = taking “of” the total.

23. Expressed symbolically:

24. When students have a solid understanding of meaning the same thing as “of”, then it makes a lot of sense for why Conceptual Understanding

25. split into 5 equal groups =

26. Dividing by a number means the same thing as multiplying by the reciprocal of that number.

27. means the same as split into equal groups

28. Note: dividing by a number less than 1 makes the number bigger, because it is then multiplying by a number bigger than 1.

29. How do you teach algorithms? What do you do to shape the skills, once students are proficient? How do you move students to automaticity? In your groups—on flip chart paper Present back to full group

30. Formative assessment to direct differentiated instruction How to then differentiate? Guided Practice Full-group Instruction Independent Practice

31. What algorithms will I teach this year? What will I do to help students build models for skills? What will I do to help students shape the skill or process? How will I monitor how well students are using the skill or process? What will I do to help students who are struggling with the skill? Planning for algorithms

32. Formative assessment to direct differentiated instruction Return to differentiation Guided Practice Full-group Instruction Independent Practice Guided Practice Full-group Instruction Independent Practice Guided Practice Full-group Instruction Independent Practice

33. Marzano, R. J., Pickering, D. J., & Pollock, J. E. (2001). Classroom instruction that works: Research-based strategies for increasing student achievement. Alexandria, VA: Association for Supervision and Curriculum Development. Text is taken directly from the resource. Works Cited