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

How are fourth and fifth 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.

Matt says,
My former 5th grade teammate let me hi-jack her Math class so I could try out a Math Forum PoW. The experience was very informative to me as a teacher and very engaging for the students.

The process of “noticing and wondering” seemed to perplex the kids. They weren’t quite sure how to respond. I gave them some examples to get things going but I can see that there is a need to do this on a regular process in order to establish problem solving routines and expectations. Before sending them on their way, we also talked about some different strategies for attacking a problem so they weren’t just left to “figure it out” without some ways in which to think about the problem.

Once students started working on the problem, it was very interesting to see who was comfortable with ambiguity and who really needed a concrete right/wrong answer. They would show me an answer and ask, “Am I right?” My responses were “Tell me your thinking. How did you arrive at your answer? Is there another way to solve the problem to see if you get the same answer?” Some really flourished and others floundered.

After a bit of time, some of the students were arriving at the right answer while others were still trying different strategies. At this point I stopped the students and explained that I didn’t want an answer to the problem but am more interested in how they solved/ tried to solve the problem. This is where I was pleasantly surprised. The students share a variety of ways they decided to approach the problem and all of them were viable/ plausible considering the problem.

After talking about the strategies I told the students that some of them had the right answers and that others still has some thinking to do. Instead of giving them the answer, I told them they had to show me their work and explain their thinking before I told them the answer was correct or not. This really got the students frustrated but excited at the same time. Many of them rushed up to me at the end of the lesson to explain their thought process. It added an edge of anticipation and actually got them excited about Math.

2. Reason abstractly and quantitatively.

How are fourth and fifth 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 fourth and fifth 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 fourth and fifth 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 fourth and fifth 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 fourth and fifth 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 fourth and fifth 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 fourth and fifth 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?