There is a lot to learn in this area as I work to deepen my math instruction approach with greater problem solving and use of mathematical vocabulary. I've started to chart this course below, and will be adding notes in the days to come. I welcome helpful links and tools in this regard.
Make Sense of Problems and Persevere in Solving Them:
- Choose relevant, meaningful problems with "just right" levels of challenge.
- Tell students to expect "cognitive discomfort" as that's the path to learning.
- Team students with similar-ability grouping so that they have to think about, talk together, and persevere to solve the problem.
- Be available to coach, trouble shoot, and examine.
- When the problem is complete have students reflect on the following questions:
- Was this problem too easy, just right or too difficult and why?
- What strategy(s) did you use to solve the problem?
- What worked well with regard to your team's collaboration?
- What did you learn?
Reason abstractly and quantitatively.
This area led me on a search related to abstract reasoning. I came up with a few helpful links:
- Abstract problems
- How to Solve Logic Puzzles
- Fourth Grade Logic Puzzles
- Area and Perimeter Problems that Require Reasoning
- Scholastic: Building Abstract Thinking Through Math
- Quantitative Reasoning Exercises and Information
- Note that Poptropica has some good logic problems online with animation.
I want to search for more apt problems and exercises in this area, but I'll start with these.
Construct Viable Arguments and Critique the Reasoning of Others
- I'll share the video on this page which shows fifth graders critiquing each others' work, and we'll talk about what they noticed with regard to the discussion.
- I will look for a couple of really good, real-life math problems to incite this type of discussion. I want to find problems that relate to my current math concept goals, and problems that offer a range of discussion topics from very basic to enriching so we can have a great classroom discussion.
- Prove It: Viable Arguments Project
- Nice example of a proof.
Use Appropriate Tools Strategically
- I'll focus on this learning with the units we're now covering including fractions, decimals and geometry. We'll study and use fraction bars, pattern blocks, protractors and clock/circle models in this regard.
- It's important to neatly and strategically arrange the tools in the classroom (I still have to work on this).
- At the start of each unit, give students time to play and explore the tools and what the tools can do. Ask questions such as:
- Have you seen this tool before? If so, where, when and what was it used for?
- What do you notice about the tool's shape, material, structure? Does it remind you of other tools?
- What can you do with this tool? Why is it useful?
- Why did mathematicians or others create this tool?
- Bonus: What is the history of the tool?
Look and Make Use of Structure
- Google table is one of my favorite structure tools
- Other structure tools that are helpful are lists and venn diagrams.
- Use the questions: How can we give structure to this problem or question? How did you organize your work?
Look for and Express Regularity in Repeated Reasoning.
- What patterns do you see here?
- How does your work here remind you of other problems you've solved in the past?
- What are symbols, numbers, relationships that help you make meaning?
The talk and work of math is the talk and work of any job well done. It is work that involves observation, stamina, multiple perspectives, trial and error, connections, response, analysis and presentation/communication.
To foster the standards of mathematical practice is to foster the essential mindset of apt thinking and learning, and that's worth the effort.