We want every learner in our care to be able to say
I can look for and make use of structure.
But…What if I think I can’t? What if I have no idea what “structure” means in the context of what we are learning?
How might we offer a pathway for success? What if we provide cues to guide learners and inspire interrogative self-talk?
I can integrate geometric and algebraic representations to confirm structure and patterning.
I can look for and make use of structure.
I can rewrite an expression into an equivalent form, draw an auxiliary line to support an argument, or identify a pattern to make what isn’t pictured visible.
I can compose and decompose numbers, expressions, and figures to make sense of the parts and of the whole.
Are observing, associating, questioning, and experimenting the foundations of this Standard for Mathematical Practice? It is about seeing things that aren’t readily visible. It is about remix, composing and decomposing what is visible to understand in different ways.
How might we uncover and investigate patterns and symmetries? What if we teach the art of observation coupled with the art of inquiry?
In The Innovator’s DNA: Mastering the Five Skills of Disruptive Innovators, Dryer, Gregersen, and Christensen describe what stops us from asking questions.
So what stops you from asking questions? The two great inhibitors to questions are: (1) not wanting to look stupid, and (2) not willing to be viewed as uncooperative or disagreeable. The first problem starts when we’re in elementary school; we don’t want to be seen as stupid by our friends or the teacher, and it is far safer to stay quiet. So we learn not to ask disruptive questions. Unfortunately, for most of us, this pattern follows us into adulthood.
What if we facilitate art of questioning sessions where all questions are considered? In his post, Fear of Bad Ideas, Seth Godin writes:
But many people are petrified of bad ideas. Ideas that make us look stupid or waste time or money or create some sort of backlash. The problem is that you can’t have good ideas unless you’re willing to generate a lot of bad ones. Painters, musicians, entrepreneurs, writers, chiropractors, accountants–we all fail far more than we succeed.
How might we create safe harbors for ideas, questions, and observations? What if we encourage generating “bad ideas” so that we might uncover good ones? How might we shift perspectives to observe patterns and structure? What if we embrace the tactics for asking disruptive questions found in The Innovator’s DNA?
Tactic #1: Ask “what is” questions
Tactic #2: Ask “what caused” questions
Tactic #3: Ask “why and why not” questions
Tactic #4: Ask “what if” questions
What are barriers to finding structure? How else will we help learners look for and make use of structure?
[Cross posted on Experiments in Learning by Doing]
Dyer, Jeff, Hal B. Gregersen, and Clayton M. Christensen. The Innovator’s DNA: Mastering the Five Skills of Disruptive Innovators. Boston, MA: Harvard Business, 2011. Print.
August 24, 2014 at 9:25 pm
“How might we uncover and investigate patterns and symmetries?”
This is what math is all about.
There are patterns and symmetries in the outside world, in architecture, in graphic art, in computer games, and also, in abundance, in nature. Even without knowing the jargon the students come with natural talents of observation, perhaps somewhat smothered by experience. They need stuff, real stuff, to look at, ask and answer questions about, such as “What do you see?”, “How many?”, and the disruptive questions as well. In elementary school it is essential not to break the links between different “subjects”. There is so much simple math in everything (almost). No 9 year old should ever be heard saying “Oh, no, that’s not math, that’s science, or graphics, or music”. Of course, there are patterns and symmetries in numbers, but these are fundamentally no different.
It’s a shame that the content of the CCSS math is so narrow and old fashioned, but then it is “only” a core!!!!!!!!!!!!
” What if we teach the art of observation coupled with the art of inquiry?”
We should be doing this all the time anyway.
About good and bad ideas:
The most important thing is to listen to and record all suggestions, ideas and opinions WITHOUT giving anything away. It took me a few years to get this right!
Besides, the classification into good and bad is definitely bad. It is better to use a more/less useful or a more/less productive or a more/less interesting classification.
August 26, 2014 at 5:32 pm
Hi Howard, I agree with you that math is all about uncovering and investigating patterns and symmetries. I also whole heartedly agrees about integrated studies and work against isolating subjects. I think our post is about clearing a path for more teachers to get to where you are. You say it took you several years to get this right. We want more teachers getting it right and we want to celebrate all who are already there.
I don’t think our post is about the content of CCSS math. I believe we are forwarding the standards of practice that we want to see illuminated, facilitated, and encouraged in every classroom – CCSS or not.
August 26, 2014 at 5:22 pm
Hey Jennifer… I didn’t want your readers to miss this great comment from Angél Kytle on my copy of our post.
I like the line of thinking here and am especially keen on working through the “what if I can’t” aspect of this post. Perhaps the art of questioning that you eloquently discuss actually provides the structure that learners can not envision at the beginning. I wonder if the process and act of questioning in and of itself can illuminate the structure for the learner.
I also am concerned about the value judgment we as humans place on ideas, and I wonder if we as teachers default to evaluating ideas as good or bad depending on what our objective is (especially in math?). We emphasize that we need to consider many bad ideas in order to unearth the good ones. How Might We suspend judgment of an idea (good or bad) in order to focus on the crucial generation of thoughts, wonderings, questions? Might this suspension of judgment actually encourage and facilitate such generation (kind of like having empathy for each idea to fully understand it in perspective)?
And my reply:
Thank you, Angél. I appreciate your support and your pushing and probing questions. I think Seth Godin’s point about “bad” ideas is actually just giving permission to take the risk to ship ideas. His post connects so strongly to the passage above from The Innovator’s DNA. It parallels the idea of Fail Up, Fail Fast, Fail Forward. When I was at the d.school, David Kelley said “Have the guts to hear that’s a terrible idea and then decide for yourself.” Tom Chi said “Fail Forward!? It’s really about “learning forward. Failing without learning is fruitless. We need to learn faster.”
I agree with you about suspending judgement. How might we teach how to suspend judgement about one’s own ideas to learn and share?