1. Make sense of problems and persevere in solving them.

How are second and third grade teachers helping their students make sense of problems? During a recent online course focused on problem solving in elementary classrooms, these thoughts were shared in our discussions. I’ve received permission to quote them and provide them here as starting points we all might use to have a conversation.

Barbara says,
I think that if we teach our students that there are multiple ways to solve a problem (our goal—flexible thinkers) then they will become more independent in their problem solving. If they have a bag full of tricks…lots of different ways to solve a problem they will be able to explain why one way works over another way for them (or for that problem). I also think that if they become comfortable with lots of different strategies they can dialouge with other students about the stratey each one chooses to use. While I think the Brainstorming Representations is a bit wordy for 2nd graders to use, I like the idea of having a paper to help students think about HOW they are going to solve the problem before they solve the problem. I also think that the idea of a group working together would encourage constructing viable arugments….after all to decide HOW to solve it the student would be talking about all of these strategies that they have learned….Maybe they will draw a picture, act it out, build a model, use blocks or a number line. We have introduced many of these strategies but we continue to focus SO much on getting the problem figured out and a lot less time on how we might solve the problem. I think that is where I struggle….giving students the time to think about HOW to solve problems.

2. Reason abstractly and quantitatively.

How are second and third grade teachers helping their students reason abstractly and quantitatively?

How can students be helped to:

• make sense of quantities and their relationships in problem situations?
• decontextualize — to abstract a given situation?
• represent a problem symbolically?
• manipulate the representing symbols as if they have a life of their own?

The CCSS states:
Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

What are you doing to help students develop this practice? What makes it hard? What challenges are you encountering?

3. Construct viable arguments and critique the reasoning of others.

How are second and third grade teachers helping their students construct viable arguments and critique the reasoning of others?

How can students be helped to:

• understand and use stated assumptions, definitions, and previously established results in constructing arguments
• make conjectures and build a logical progression of statements to explore the truth of their conjectures
• analyze situations by breaking them into cases
• recognize and use counterexamples
• justify their conclusions, communicate them to others, and respond to the arguments of others
• reason inductively about data
• make plausible arguments that take into account the context from which the data arose
• compare the effectiveness of two plausible arguments
• distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is

The CCSS states:
Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

What are you doing to help students develop this practice? What makes it hard? What challenges are you encountering?

4. Model with mathematics.

How are second and third grade teachers helping their students model with mathematics?

How can students be helped to:

• apply the mathematics they know to solve problems arising in everyday life, society, and the workplace
• write an addition equation to describe a situation
• identify important quantities in a practical situation
• map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas
• analyze those relationships mathematically to draw conclusions
• interpret their mathematical results in the context of the situation
• reflect on whether the results make sense, possibly improving the model

What are you doing to help students develop this practice? What makes it hard? What challenges are you encountering?

5. Use appropriate tools strategically.

How are second and third grade teachers helping their students use appropriate tools strategically?

How can students be helped to:

• consider the available tools when solving a mathematical problem
• use pencil and paper, concrete models, a ruler
• identify relevant external mathematical resources
• use technological tools to explore and deepen their understanding of concepts

What are you doing to help students develop this practice? What makes it hard? What challenges are you encountering?

6. Attend to precision.

How are second and third grade teachers helping their students attend to precision?

How can students be helped to:

• communicate precisely to others
• use clear definitions in discussion with others and in their own reasoning
• state the meaning of the symbols they choose
• careful about specifying units of measure
• calculate accurately and efficiently
• give carefully formulated explanations to each other

What are you doing to help students develop this practice? What makes it hard? What challenges are you encountering?

7. Look for and make use of structure.

How are second and third grade teachers helping their students look for and make use of structure?

How can students be helped to:

• look closely to discern a pattern or structure.
• notice that three and seven more is the same amount as seven and three more
• sort a collection of shapes according to how many sides the shapes have
• step back for an overview and shift perspective

What are you doing to help students develop this practice? What makes it hard? What challenges are you encountering?

8. Look for and express regularity in repeated reasoning.

How are second and third grade teachers helping their students look for and express regularity in repeated reasoning?

How can students be helped to:

• notice if calculations are repeated
• look both for general methods and for shortcuts
• maintain oversight of the process, while attending to the details
• evaluate the reasonableness of their intermediate results

What are you doing to help students develop this practice? What makes it hard? What challenges are you encountering?