## Effective Math Programming

## Big Ideas and Problem Solving in Junior Math Instruction

“Let the main ideas which are introduced be few and important, and let them be throwninto every combination possible.”(Whitehead, 1929, pp. 1-2)

The idea of focusing instruction around ‘Big Ideas’ is not a new one. According to Wiggins & McTighe (2005) big ideas allow for what has been learned in one lesson to be applied to other situations and problems (p. 40). Likewise, as students come to an understanding of a big idea in mathematics, they make connections that let them use math more effectively (Ontario Ministry of Education, N.D., p. 19). Concentrating on big ideas helps students to gain a deeper understanding of mathematical concepts. (Ontario Ministry of Education, 2008, p. 15).

The problem-solving approach to mathematics instruction, as its name indicates, involves using problems to teach mathematics content. Throughout the mathematics unit, teachers pose problems to students to engage them in developing and applying concepts in math (Ontario Ministry of Education, N.D., p. 7). While teachers model problem-solving strategies through think-alouds, they also encourage students to use their own strategies as well (Ontario Ministry of Education, N.D., p. 26).

The problem-solving approach is well-suited to developing understanding of big ideas in mathematics. Students need to be confronted with authentic problems to call upon them to develop understanding (Wiggins & McTighe, 2005, p. 48). Solving meaningful problems with multiple entry points requires students to integrate knowledge from many different strands of the curriculum (Ontario Ministry of Education, N.D., p. 18; The Expert Panel, 2004, p. 9). While the use of predetermined steps and algorithms is efficient, it can result in a lack of understanding of important concepts. Using the problem-solving approach and allowing students to determine their own problem-solving strategies allows for deeper learning of concepts related to big ideas in mathematics (The Expert Panel, 2004, pp. 11-12).

The classroom structures and procedures that are required to support problem solving include those related to the learning environment, scheduling, and physical resources. The problem-solving approach to mathematics instruction accepts mistakes as a part of the process of learning, and teachers need to provide a risk-free learning environment in which students feel confident taking risks and sharing their ideas (Ontario Ministry of Education, N.D., pp. 15-16). Students also need enough time to experiment with different strategies and chart their own courses towards the solutions. According to the Ontario Ministry of Education (N.D.), students should be given at least one hour per day to study mathematics at the Junior level (p. 14). Finally, a classroom rich in resources and manipulatives supports student experimentation and visualizing of problems (Ontario Ministry of Education, N.D., p. 19). Some commonly used manipulatives include colored counters and unit blocks.

A final point to consider when examining the use of big ideas and the problem-solving approach in mathematics instruction is the importance of communication. When students communicate their ideas to their classmates and try to follow other students’ reasoning they are more likely to understand math concepts on a deeper level (The Expert Panel, 2004, p. 13). This communication allows students to use mathematics terms and to see how various big ideas in mathematics are linked to one another (The Expert Panel, 2004, p. 15). Furthermore, the strategies used by students to solve problems in mathematics follow a loose progression that is unlikely to appear if instruction does not stem from student ideas (Ontario Ministry of Education, N.D., p. 20). As such, students should collaborate with the teacher as well as their classmates (Ontario Ministry of Education, N.D., p. 18). That being said, in order for the teacher to accurately assess each student’s learning, students should each record their own work (Ontario Ministry of Education, N.D., p. 19).

To begin approaching mathematics instruction using big ideas and the problem-solving approach, teachers can implement a number of strategies. Teachers should:

The problem-solving approach to mathematics instruction, as its name indicates, involves using problems to teach mathematics content. Throughout the mathematics unit, teachers pose problems to students to engage them in developing and applying concepts in math (Ontario Ministry of Education, N.D., p. 7). While teachers model problem-solving strategies through think-alouds, they also encourage students to use their own strategies as well (Ontario Ministry of Education, N.D., p. 26).

The problem-solving approach is well-suited to developing understanding of big ideas in mathematics. Students need to be confronted with authentic problems to call upon them to develop understanding (Wiggins & McTighe, 2005, p. 48). Solving meaningful problems with multiple entry points requires students to integrate knowledge from many different strands of the curriculum (Ontario Ministry of Education, N.D., p. 18; The Expert Panel, 2004, p. 9). While the use of predetermined steps and algorithms is efficient, it can result in a lack of understanding of important concepts. Using the problem-solving approach and allowing students to determine their own problem-solving strategies allows for deeper learning of concepts related to big ideas in mathematics (The Expert Panel, 2004, pp. 11-12).

The classroom structures and procedures that are required to support problem solving include those related to the learning environment, scheduling, and physical resources. The problem-solving approach to mathematics instruction accepts mistakes as a part of the process of learning, and teachers need to provide a risk-free learning environment in which students feel confident taking risks and sharing their ideas (Ontario Ministry of Education, N.D., pp. 15-16). Students also need enough time to experiment with different strategies and chart their own courses towards the solutions. According to the Ontario Ministry of Education (N.D.), students should be given at least one hour per day to study mathematics at the Junior level (p. 14). Finally, a classroom rich in resources and manipulatives supports student experimentation and visualizing of problems (Ontario Ministry of Education, N.D., p. 19). Some commonly used manipulatives include colored counters and unit blocks.

A final point to consider when examining the use of big ideas and the problem-solving approach in mathematics instruction is the importance of communication. When students communicate their ideas to their classmates and try to follow other students’ reasoning they are more likely to understand math concepts on a deeper level (The Expert Panel, 2004, p. 13). This communication allows students to use mathematics terms and to see how various big ideas in mathematics are linked to one another (The Expert Panel, 2004, p. 15). Furthermore, the strategies used by students to solve problems in mathematics follow a loose progression that is unlikely to appear if instruction does not stem from student ideas (Ontario Ministry of Education, N.D., p. 20). As such, students should collaborate with the teacher as well as their classmates (Ontario Ministry of Education, N.D., p. 18). That being said, in order for the teacher to accurately assess each student’s learning, students should each record their own work (Ontario Ministry of Education, N.D., p. 19).

To begin approaching mathematics instruction using big ideas and the problem-solving approach, teachers can implement a number of strategies. Teachers should:

- provide students with appropriate and challenging problems with multiple entry points by determining students’ current level of knowledge and connecting it to the learning goals (Ontario Ministry of Education, N.D., p. 26; The Expert Panel, 2006, p. 11).
- avoid explaining the process of the solution in small steps (The Expert Panel, 2006, p. 9).
- provide sufficient time for students to solve problems, at least one hour per day (The Expert Panel, 2006, p. 14).
- create a classroom community in which students collaborate with one another and the teacher, and feel safe making mistakes and sharing their ideas (Ontario Ministry of Education, N.D., pp. 15-18).
- base instruction on student ideas and encourage students to solve problems using their own strategies (Ontario Ministry of Education, N.D., pp. 20, 26)

References:

Ontario Ministry of Education (2008). Measurement, grades 4 to 6: A guide to effective instruction in

mathematics, kindergarten to grade 6. Toronto: Queen’s Printer for Ontario. Retrieved from:

http://eworkshop.on.ca/edu/resources/guides/Guide_Measurement_456.pdf

Ontario Ministry of Education (N.D.). A Guide to Effective Instruction in Mathematics: Kindergarten to

Grade 6. Retrieved from: http://eworkshop.on.ca/edu/resources/guides/Guide_Math_K_6_Volume_2.pdf

The Expert Panel of Mathematics in Grades 4 to 6 in Ontario (2004). Teaching and learning mathematics:

The report of the expert panel of mathematics in grades 4 to 6 in Ontario. Toronto: Queen’s Printer for Ontario. Retrieved from: http://eworkshop.on.ca/edu/resources/guides/ExpPanel_456_Numeracy.pdf

Whitehead, A.N. (1929). The aims of education and other essays. New York: Free Press.

Wiggins, G. & McTighe, J. (2005). Understanding by design, expanded 2nd edition. Alexandria, VA: ASCD.

Ontario Ministry of Education (2008). Measurement, grades 4 to 6: A guide to effective instruction in

mathematics, kindergarten to grade 6. Toronto: Queen’s Printer for Ontario. Retrieved from:

http://eworkshop.on.ca/edu/resources/guides/Guide_Measurement_456.pdf

Ontario Ministry of Education (N.D.). A Guide to Effective Instruction in Mathematics: Kindergarten to

Grade 6. Retrieved from: http://eworkshop.on.ca/edu/resources/guides/Guide_Math_K_6_Volume_2.pdf

The Expert Panel of Mathematics in Grades 4 to 6 in Ontario (2004). Teaching and learning mathematics:

The report of the expert panel of mathematics in grades 4 to 6 in Ontario. Toronto: Queen’s Printer for Ontario. Retrieved from: http://eworkshop.on.ca/edu/resources/guides/ExpPanel_456_Numeracy.pdf

Whitehead, A.N. (1929). The aims of education and other essays. New York: Free Press.

Wiggins, G. & McTighe, J. (2005). Understanding by design, expanded 2nd edition. Alexandria, VA: ASCD.