Math, Teaching and Instruction

Inquiry in Mathematics

In Saskatchewan our provincial curricula are built around one central core…inquiry.  Inquiry is a philosophical approach to teaching and learning that empowers students to explore and discover.  Through inquiry students are active participants in the creation of understanding and knowledge.   They are asked to be curious, to wonder, to question, to reflect, to share and to think

In Mathematics inquiry is often seen through problem solving.  By collaboratively solving and creating mathematical problems students construct and deepen their understandings of concepts, strengthen their strategic competence, and develop their logical reasoning skills.  As students engage in problem solving they need to attend to 4 important stages in the inquiry/problem solving process:

Stage 1: Understand the Problem–  This very important stage is often overlooked in the classroom.  It is often assume that students understand the problem in front of them but this assumption can by costly.  Without a solid understanding of what they are being asked to do, students are often stumped before they start.  Time spent carefully dissecting a problem and ensuring understanding can be critical to success.

Stage 2: Make a Plan– Just like the construction of a building, the construction of a mathematical solution requires forethought and a plan.  Having a carefully considered plan of attack can help students construct their solution successfully.  Problem Solving plans or strategies can include:

  • Acting it out
  • Using a model
  • Drawing a picture
  • Guessing and testing
  • Looking for a pattern
  • Making a chart or a graph
  • Working backwards
  • Making an organized list
  • Using logical reasoning   

Stage 3: Carry out the PlanIn the words of  Nike – Just Do It!

Stage 4: Look Back –  This stage involves careful reflection, checking to see if your answer makes sense, and considering the solutions of others.  Considering the solutions of others and comparing them to your own provides an opportunity for understanding and learning to be collaboratively constructed in the mathematics classroom.

For more information on Problem Solving in Mathematics check out these great resources:

Making Math Meaningful to Canadian Students by Marian Small

Making Math Meaningful

Introduction to Problem Solving by Susan O`Connell

 

Math

Comprehending Math through Problem Solving

This spring I am championing a book club for the book Comprehending Math by Arthur Hyde. I was initially drawn to this book because it attempts to link reading comprehension strategies, which have been a focus of my professional learning for many years, with ways students can make sense of and understand mathematics. Hyde refers to this connection as BRAIDING- the weaving together of reading and mathematics comprehension strategies. I firmly believe if we can make these connections as teachers, we can help our students make them and in turn, enhance and strengthen their learning experiences in our classrooms. This opening comment by Ellin Keene says it best “teaching will be clearer, bolder, more connected. And for the ultimate beneficiaries, the help(the learners), they will have a chance to understand just how integrally our world is connected.” p. xii.

In this book Hyde emphasizes problem solving by demonstrating math-based variations of common reading comprehension strategies such as:

  • K–W–L
  • visualizing
  • asking questions
  • inferring
  • predicting
  • making connections
  • determining importance
  • synthesizing

One of my favorite take aways for this book is the KWC chart which is a modified version of the reading KWL chart. In this math version students take a math problem or situation and break it down into 3 separate area.

What do I Know?

– what information is given to us within the problem or situation

This could be a specific list of items such as:

  • 14 girls
  • 12 boys
  • 28 spots on the bus

What do you Want to figure out?

Basically, you have the students restate the question, problem or situation in their own words. This can be a brief sentence or a longer statement depending on the situation

Special Conditions– are there any special conditions we have to watch out for

This section can be a little tricky to fill out because sometimes the conditions are also things that we know. When doing the KWC chart with your students, feel free to go back and forth between the K and the C when you first start. Eventually, you can move onto the discussion of is that a K or is it a C? A condition would be something that must be remembered in order to complete the problem. For example, a condition might be

  • You cannot create the same shape more than once or
  • it is a 3 digit number

Sometimes with students I talk about the conditions as things we have in our schema that need to be brought forward to aid in the solving of the problem. For example if the problem is about money it might not specifically state that there are 100 pennies in a dollar, but this information must be drawn forward in order to complete the task.

When using this tool in the classroom I would recommend the “gradual release of responsibility model” as a guide. Begin first by using it with the whole class with the teacher being the guide, then gradually move into students using it in small groups, partners and finally own their own. Once students understand the mathematical problems they encounter they are more able to move into solving them with enhanced confidence and understanding.

Happy Problem Solving!!!