Failure, Learning, and Progress

I'm a month in to the start of another school year.  This is my third year coaching and I am constantly amazed at how the same "ideas" come back to the forefront over and over again.  It is so easy to get stuck in the daily grind and to forget what amazing researchers and educators have told us about students and learning.

First, I want to attribute this image to the incredible blogger, author, artist, mathematician Jessica Hagey and her website thisisindexed.com.  She publishes a new "index card" weekday mornings that are incredible commentaries on the world around us.  I find so many of them are useful for instruction, analysis, sense making, and general life lessons.  Check her out, follow her, and see what you find!

Grade Level: ANY
SMP: MP1, MP2, MP3, MP4, MP5, MP6, MP7, MP8

So, you may ask, is this a lesson?  No.  It's not meant for kids, although I know my students could engage in a thoughtful discussion around the ideas.  This is meant for ME, for my teachers, for those I work with and coach, and for you, if you are anyone who has experienced failure and growth.

Why this?  Why now?
Just this week one of the incredible teachers I work with brought up the idea of feedback over grades.  She had just read an article that talked about never giving a grade, but constantly providing feedback. She was intrigued and wondered how it might work in her own class.

"Mindset" isn't just a buzzword in education today.  It's a buzzword in parenting, in coaching, in sports, and in business.  Carol Dweck, author of Mindset, is one of the major names in the field today, but is far from the only person pontificating on the power of positive thinking.  (If you haven't read the book, it's incredibly accessible, quick to read, and useful!)

Let's get down to it.  Here is an incredible resource to use if you are providing PD around feedback and mindset to teachers.  The MARS/Shell Centre has some really nice resources including one around feedback for students.  The research is clear:  providing students with feedback (instead of a grade or score) will increase the opportunity for students to reflect, revise, and improve.  A score is too permanent for students and decreases the chances that students will see their learning and performance as fluid and open to growth.

As I told my teacher, I can't imagine a classroom where EVERYTHING is open to revision and growth and I never award a final performance score (proficient, partially proficient, etc.).  However, I love the idea and the meaning behind it.  At least 4 times over the last two years I have provided written feedback to students regarding their thinking on assessments (in lieu of a score) and DID see an increased effort to respond to my feedback and revise, expand, or elaborate on their thinking.  This is an incredible tool for educators to use and I know I need to do quite a bit more of this in the future.

Not only that, but if we take a minute to reflect on the Standards of Mathematical Practice, think about how essential this idea of FEEDBACK is to helping your students become proficient in using the SMPs.  Of course, SMP 1 is obvious.  "Make sense of problems and persevere in solving them" is closely linked to providing feedback and opportunities for students to explore and improve their work.  However, what about the other SMPs?  SMP 2?  Reason abstractly and quantitatively?  Isn't this your chance to encourage your students to contextualize or decontextualize (as needed) in problem solving situations?  This is your chance to ask students to explore further, to apply numbers and symbols to their solutions, or to back off of specifics and begin to answer for generic cases.  SMP 3?  Construct viable arguments and critique the reasoning of others?  Your feedback could center around asking students to make a stronger argument for their answer, or to say, "I saw several students say the answer should be _____, what do you think?"  Your feedback and questions can push students to really work on their argumentation skills when it comes to mathematics. I could continue, but I think it is clear, FEEDBACK is an obvious solution to "How do I teach the SMPs?  How do I engage students in this kind of thinking and reasoning?"  

A "failing" grade doesn't lead to learning.  Progress occurs when failure and learning overlap, and I believe that the progress can only come from timely, specific, relevant feedback with a chance for students to try again.

If you are looking for more ideas around growth Mindset, let me know!  I've been gathering a lot of resources and have been using them with my 8th graders this year, and I think it is beginning to pay off!

Please feel free to use any of these ideas and modify them to meet your needs.  However, please acknowledge the original source of the items and my own lesson outlines.  ©NatalieRSprigg 2015

Exponents...Rule?

Personal Reflection:

Oh my goodness, I love it.  I'm pretty sure I found this on http://math-fail.com/ (which is a pretty fun site if you have time to search for the good stuff).

I love that this shows someone, who looks like a teacher, making the same conceptual errors our students do!  What a perfect way to get kids engaged in discussing not just the "rules" but the "whys" and "hows" of exponential notation.

Grade Level: 6-9

Course: 6th Grade, Pre-Algebra, Algebra

Standards:  6.EE.1, 6.EE.2, 8.EE.1, N-RN.1, N-RN.2, A-SSE.3, 
Skills: Algebra, Exponents, Exponent Rules, Powers, Bases


How to use this as a mad minute:
You have 60 seconds.  Explain why this teacher's simplification is incorrect.

How to use this as a warm up:
You could ask the students to consider one of the following:
1. What is the meaning of an exponent?
2. What is the difference between 3-squared and 3x2?
3.  Where in real life do we use exponents?  Why?
4.  What is the difference between the original expression and what the teacher wrote?  (Note:  Only for students who more experience with exponents!)

How to use this as a mini-lesson:
Some might wonder why I listed this as a 6th through HS level standard or lesson.  Truly it is because of the depth of thinking and analysis you could ask each level to bring to the table.  Ideally, the skill of simplifying this expression is an eighth grade standard.  However, exponents and the use of them is introduced in sixth grade and is, of course, expanded through high school.

If I were teaching middle school, I'd begin by revisiting the meaning of an exponent and might even ask students to write examples and expanded forms.  I'd continue by asking them to replace g-squared with another substitute or variable.  If they realize that the replacement should expand to x*x*x*x*x*x*x and if they also can say that g-squared should expand to g*g, they can quickly arrive at the idea that this is really g*g*g*g*g*g*g*g*g*g*g*g*g*g.  What a great review!

I'd return by asking kids to create their own "mistake" problem and prove the right answer.


How to use this as a full lesson?
I wouldn't use this as a full lesson unless you were knee-deep in your exploration of exponents and their properties.  If that is the case, you are probably teaching an eighth grade math class!  And if that is the case, you probably have a district-mandated curriculum.

This is a great supplement to that!  If you have used your primary curriculum to build understanding of exponents and their properties, you could use this as an exit slip for your lesson and simply ask students to explain the mistake in the teacher's thinking.

If you'd like, use this to launch the lesson.  Your students should already understand the meaning of exponents, but have probably not experienced "nested" exponents.  You can simply ask students to make sense of the original problem, make sense of what was written, and compare their answers.  Kids would have to dig deeply, with scaffolded questions, to get there, but I'm confident they could, as long as they have a solid understanding of exponents and their meaning.  (See the mini lesson above for some scaffolded questions.)


 How to use this as an assessment?
If your students are ready for an assessment, I would definitely put this photo on an exit slip, quiz, or test with a simple, "Explain the error in thinking shown here."

*Remember to think about what a proficient answer would entail, and what might a student to go beyond your expectations!

Please feel free to use any of these ideas and modify them to meet your needs.  However, please acknowledge the original source of the items and my own lesson outlines.  ©NatalieRSprigg 2015