How are **kindergarten** and **first **grade teachers helping their students make sense of problems? During a recent online course focused on problem solving in primary level 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.

one Maryland teacher says,

“Strange as it may seem, I like the idea of introducing a problem that is too difficult for my students in an effort to get them to analyze the process of problem solving, rather than focusing on finding a solution to the problem. The strategy was presented in the Problem Solving and Communication Activity Series article. The strategy forces students to move away from resolving the problem (since it is too difficult for them to know where to start) and to think about the process. It also confirms that the teacher truly does not want an answer; that we want them to think about strategies not solutions. Many of my kindergarten students have not spent significant time considering how they solve problems. They are not familiar with guessing and checking, working backwards, estimating, and asking themself if their answers make sense. They focus on product, not process.”

Elaine says,

“Does this make sense?” has only recently become meaningful to students in my class. Quite frequently, I get a unified “yes” or “no” from the class, and the response is given so quickly (and usually incorrectly) it is obvious that they are not really thinking about the reasonableness of their answer. So, we’ve been discussing a lot about how to think reasonably before predicting or searching for an answer. Now, when I prompt students “Does this make sense?”, I immediately follow with a quick “Stop…think…now answer” and I’m finding that their responses are more aligned with where I think they should be when they have truly thought about their answer.”

one Maryland teacher says,

I agree that it’s a good habit for students to learn to continue improving their work. They learn to revisit the question and allow themselves to see it differently the second, third or fourth time. It slows them down, too. Many of my students just want to be finished with their assignments. They figure that as soon as something (anything) has been submitted, their work is done. With the PoW mentoring scenario they also learn 1) to take more time with the first submission, 2) to be prepared to revise and resubmit their work, and 3) that there is value not only in correct answers but also in effort.

How are **kindergarten** and **first **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?

]]>How are **kindergarten** and **first **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?

]]>How are **kindergarten** and **first **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?

]]>How are **kindergarten** and **first ** 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

How are **kindergarten** and **first **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

How are **kindergarten** and **first **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

How are **kindergarten** and **first **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