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?
Cognitively second and third graders are in transition from concrete to abstract thought. I think it is most effective to have them represent the problem with concrete objects (manipulatives). This also allows them to decontextualize – if the problem is about cats, or cookies, or coins, they have to first select a manipulative object to represent the items. If they are doing the problem orally, and using mental math, another decontextualizing strategy is to substitute a nonsense word (wugs, eg) for the word representing the real object. Moving to the conventional symbol level (ie numerals) then moves them to a different level of representation. Obviously, they can manipulate the concrete objects more easily than they can manipulate the numerals. However, manipulation of numerals can come by decomposing the number represented by the numeral and manipulating the component parts prior to composing an answer to the problem.
The other factor with these younger students is the power of narrative. Kieran Egan wrote a wonderful treatise, Teaching as Storytelling. Research will point to the fact that the human brain is wired for storytelling. The more compelling and realistic you make the problem, the more engaged and purposeful you will find the students.
Julie, I totally agree that the storytelling idea works well and from my experience it’s not just with younger students but middle, high school and even adults! Often I start a problem solving lesson by saying, “I’m going to tell you a story” and then I do a read-aloud but I stop short of asking the question at the end of the problem. So, instead I only read the “scenario.”
At that point I ask, “What did you hear?” It’s a non-threatening question. Each student (of whatever age) can tell me something because there is no right or wrong. It’s subjective.
Next I say, “I’m going to read the story again. As you listen think about if what you thought you heard the first time should be adjusted.” (With younger children I might choose different words, but that’s the main idea.)
After I read again, I ask if anyone wants to confirm what they first heard or if they want to change. (Depending on the problem, I might do this one more time!)
Then I say, “What are you wondering?” This prompt encourages students to come up with questions. More often than not they ask the question that I left off in the first place.
I believe that solving a problem in more than one way and being able to connect those various ways through discussion is a concrete example of reasoning that holds a child accountable to her learning and understanding of a topic. To clarify, when a student can solve a problem using an equation and then solve that same problem with pictures, she is showing a deeper understanding of the mathematical concept. Now have that child explain the relationship between the two or more methods, using words and definitions to synthesize the underlying concept, either coached (by a teacher or peer to explain further) or in a coaching role with another student (to help another student understand the concept), and that real-world connection to another person as an audience makes the understanding deeper.
At a conference I am attending, we have been watching videos of teachers that are using various methods (graphic organizers, learning routines, class discussion and meetings, etc.) and am excited to try them out in my own classroom, and also have felt somewhat validated for the times I have included this practice in previous classes but also hoping I make this a much more common part of my math instruction in the future.
When we are working on a problem, we make sure that all numbers are labeled – no naked numbers – and we go back to the original problem to make sure that the answer makes sense in the context. Someone in the class had a great idea that you could present the problem with no numbers at all and the students have to figure out how to solve it in a general sense. This also works well with older students because they need to come up with an algebraic formula without being distracted by the actual numbers. They can then go back and check the original problem using the formula.