How might we learn to show our work so that a reader understanding without having to ask questions? As we work with our young learners, we want them to grow as mathematicians and as communicators.
We ask students to show their work so that a reader understands without having to ask them questions. What details should we add so that our thinking is visible to others?
To show (and to assess) comprehension, we are looking for mathematical flexibility.
I taught 6th grade math today while Kristi and her team attended ASCD. She asked me to work with our students on showing their work. Here’s the plan:
- I can use ratio and rate reasoning to solve real-world and mathematical problems.
- I can show my work so that a reader can understanding without having to ask questions.
I can demonstrate mathematical flexibility with ratio and rate reasoning to show what I know more than one way using tables, equations, double number lines, etc..
I can use ratio and rate reasoning to solve real-world and mathematical problems.
I can make tables of equivalent ratios relating quantities with whole-number measurements, and I can use tables to compare ratios.
I can use guess and check to solve real-world and mathematical problems.
Sample student work:
All students benefit from using tools and learning how to use them for a variety of purposes. If we don’t make tools readily available and value their use, our students miss out on major learning opportunities. (Flynn, 106 pag.)
I’m taking the #MtHolyokeMath #MTBoS course, Effective Practices for Advancing the Teaching and Learning of Mathematics. Zachary Champagne facilitated the second session and used The Cycling Shop task from Mike Flynn‘s TMC article.
You can see the notes I started on paper.
Jim, Casey and I used a pre-made Google slide deck provided to us to collaborate since we were located in GA, MA, and CA. We challenged ourselves to consider n wheels after working with 8 wheels.
Here’s what our first table looked like.
Now, I was having trouble keeping up with the number of wheels and the number of cycles. So I did this:
This made it both better and worse for me (and for my group).
Here’s an interesting thing. I’ve been studying, practicing, and teaching the Standards for Mathematical Practices. Jennifer Wilson and I have written a learning progression to help learners learn to say I can use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem. (Sage, 6 pag.)
Clearly, I was not even at Level 1 during class. Not once – not once – during class did it occur to me how much a spreadsheet would help me, strategically.
The spreadsheet would calculate the number of wheels automatically for each row so that I could confirm correct combinations. (You can view this spreadsheet and make a copy to play with if you are interested.)
When making mathematical models, [mathematically proficient students] know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. (Sage, 6 pag.)
With a quick copy and paste, I could tackle any number of wheels using my spreadsheet. I can look for and make use of structure emerged quickly when using the spreadsheet strategically. (I want to also highlight color as a strategic tool.) Play with it; you’ll see.
[Mathematically proficient students] are able to use technological tools to explore and deepen their understanding of concepts. (Sage, 6 pag.)
There is no possible way I would have the stamina to seek all the combinations for 25 or 35 wheels by hand, right?
Students have access to a wide assortment of tools that they must learn to use for their mathematical work. The sheer volume of possibilities can seem overwhelming, but with time and experience, students can learn how to choose the right tool for the task at hand and how to use it strategically to reach their goal. (Flynn, 106 pag.)
Important to repeat, “with time and experience, students can learn how to choose the right tool for the task at hand and how to use it strategically to reach their goal.”
For this to happen, we need to have a solid understanding of the kinds of tools available, the purpose of each tool, and how students can learn to use them flexibly and strategically in any given situation. This also means that we have to make these tools readily available to students, encourage their use, and provide them with options so they can decide which tool to use and how to use it. If we make all the decisions for them, we remove that critical component of MP5 where students make decisions based on their knowledge and understanding of the tools and the task at hand. (Flynn, 106 pag.)
To be clear, a spreadsheet was available to me during class, but I didn’t see it. How might we make tools readily available and visible for learners to choose?
When we commit to empower students to deepen their understanding, we make tools available and encourage exploration and use, so that each learner makes decisions for themselves. In other words, how do we help learners to level up in both content and practice?
What if we make I can look for and make use of structure; I can use appropriate tools strategically; and I can make sense of tasks and persevere in solving them essential to learn for every learner?
How might we offer tools and time?
It’s about learning by doing, right?
Flynn, Michael. Beyond Answers: Exploring Mathematical Practices with Young Children. Portland, Maine.: Stenhouse, 2017. Print.
Flynn, Mike. “The Cycling Shop.” Nctm.org. Teaching Children Mathematics, Aug. 2016. Web. 03 Feb. 2017.
“Common Core State Standards.” The SAGE Encyclopedia of Contemporary Early Childhood Education (n.d.): n. pag. Web.
In his #CMCS15 session, Michael Serra challenges us to consider:
“Of all the Mathematical Practices, there is one that stands above the others: Make sense of problems and persevere in solving them.”
If our learners cannot make sense of tasks and persevere in solving them, will they even find opportunities to experience the other Standards for Mathematical Practices?
What actions do we take to develop and grow a collaborative culture of perseverance? How might we leverage gaming to foster perseverance, inspire struggle, and promote flexible thinking?
Can we demonstrate enough self-regulation to hold our solution long enough for our learners to persevere, productively struggle, and find a solution pathway for themselves?
How might we develop a community of learners that, when asked if they’d like a hint, say a loud, resounding NO! We can persevere; we can do it ourselves!
The trick is to choose a goal just beyond your present abilities; to target the struggle. Thrashing blindly doesn’t help. Reaching does. (Coyle, 19 pag.)
What if we teach how to reach? How might we offer targeted struggle for every learner in our care?
Investing time in teaching students how to learn is never wasted; in doing so, you deepen their understanding of the upcoming content and better equip them for future success. (Jackson, 19 pag.)
If we are to harness the power of feedback to increase student learning, then we need to ensure that feedback causes a cognitive rather than an emotional reaction—in other words, feedback should cause thinking. It should be focused; it should relate to the learning goals that have been shared with the students; and it should be more work for the recipient than the donor. (Wiliam, 130 pag.)
When people believe their basic qualities can be developed, failures may still hurt, but failures don’t define them. And if abilities can be expanded – if change and growth are possible – then there are still many paths to success.” (Dweck, 39 pag.)
What pathways to learning are illuminated in order to highlight learning = struggle + perseverance?
Coyle, Daniel (2009-04-16). The Talent Code: Greatness Isn’t Born. It’s Grown. Here’s How. Random House, Inc. Kindle Edition.
Dweck, Carol S. Mindset: the New Psychology of Success. New York: Random House, 2006. 39. Print.
Jackson, Robyn R. (2010-07-27). How to Support Struggling Students (Mastering the Principles of Great Teaching series) (Pages 18-19). Association for Supervision & Curriculum Development. Kindle Edition.
Wiliam, Dylan (2011-05-01). Embedded Formative Assessment (Kindle Locations 2679-2681). Ingram Distribution. Kindle Edition.