When students arrive at school in kindergarten, “they have a lot of experience solving real-world problems, whether it is how to make a toy work, or how to fairly share limited snacks with friends. They come to school naturally curious about the world around them, asking questions and seeking answers.”
In grade 5 or 6, students develop Math anxiety because they feel they have to get the right answer. By the time these students have arrived at my Grade 9 Math Essentials Intermediate Classroom, they have been told by their elementary teachers “how to do things, and what the right way is to do them, that they start to doubt their own abilities and are expecting to be told how to do things, and what the right way is to do them. They have stopped asking their own questions, refrain from figuring out things on their own, and instead wait for me to tell them what questions to answer and how to figure things out.” They expect Mathematics to involve repetition and the memorization of disjointed facts.
Source: https://learn.etfo-aq.ca/d2l/le/content/49320/viewContent/842052/View
How can students be re-engaged in the mathematics in the intermediate math class?
Students have to get back to that exhilarating feeling of discovery when they were in kindergarten. The “Old” Math starts with the answer, never arrives at a real question and there is no real thinking going on. The focus is on memorizing by repeating the steps, covering the material and moving on.
The “New” Math starts with the question and not the answer. The open ended question is a real mystery so it feels authentic and compelling. The answer that is both beautiful and profoundly satisfying and the teacher provides sufficient time for debate and struggle.
Source: Thinking Math-ishly: Amy Lin at TEDxSixteenMileCreek
https://www.youtube.com/watch?v=0gW9g8Ofi8A
Traditional mathematics classes were typically structured into the following three parts:
The teacher taking up homework;
The teacher demonstrating examples on the blackboard and;
The teacher assigning questions to be done individually.
The “New” Math focuses on problem solving which the Ministry believes “is central to learning mathematics. It forms the basis of effective mathematics programs and should be the mainstay of mathematical instruction. Problem solving is considered an essential process through which students are able to achieve the expectations in mathematics, and it is an integral part of the mathematics curriculum in Ontario.” (The Ontario Curriculum, Mathematics, 2005).
Focusing mathematics teaching and learning on problem solving results in a radical transformation of the three parts of the mathematics lesson. The following is the wording form the Ministry’s TIPS4RM:
“Minds on… - Suggests how to get students mentally engaged, making every minute of the math class count for every student.
Action! - Suggests how to group students and what instructional strategy to use to support students in learning the specific concept or skill. This is the Doing phase of the lesson in which the teacher can facilitate and pose thought-provoking questions.
Consolidate / Debrief - Suggests ways to ‘pull out the math,’ check for conceptual understanding, and prepare students for the follow-up activity or next lesson.”
In the “New” classroom, teachers use the “5 Practices for Orchestrating Productive Math Discussions” to increase the amount of discovery and learning the students do on their own versus the amount of explicit play-by-play instruction they receive from the teacher is increased. The students are doing the thinking, not the teacher.” The five practices are as follows:
1. Anticipating
• Do the problem yourself
• What are students likely to produce?
• Which problems will most likely be the most useful in addressing the mathematics?
2. Monitoring
• Listen, observe, identify key strategies
• Keep track of approaches
• Ask questions of students to get them back on track or to think more deeply
3. Selecting
• CRUCIAL STEP – what do you want to highlight?
• Purposefully select those that will advance mathematical ideas
4. Sequencing
• In what order do you want to present the student work samples?
• Do you want the most common? Present misconceptions first?
• How will students share their work? Draw on board? Put under doc cam?
5. Connecting
• Craft questions to make the mathematics visible.
• Compare and contrast 2 or 3 students’ work – what are the mathematical relationships?
• What do parts of student’s work represent in the original problem? The solution? Work done in the past?
Source: 5 Practices for Orchestrating Productive Math Discussions Margaret S. Smith & Mary Kay Stein, NCTM & Corwin Press, 2011 www.nctm.org
http://www.mctm.org/mespa/5Practices.pdf
I wish to end this blog with a quote from Keith Devlin. He is co-founder and Executive Director of Stanford University’s Human-Sciences and Technologies Advanced Research Institute.
“The problems we need mathematics for today come in a messy, real-world context, and part of making progress is to figure out just what you need from that context.”