Tag Archives: math

Assessment, Connecting Ideas, Information Age Learning, Learning, Questions

Deep Practice, Leveling, and Communication (TBT Remix)

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Does a student know that they are confused and can they express that to their teacher? We need formative assessment and self-assessment to go hand-in-hand.

I agree that formative self-assessment is the key. Often, I think students don’t take the time to assess if they understand or are confused. I think that it is routine and “easy” in class. The student is practicing just like they’ve been coached in real time. When they get home, do they “practice like they play” or do they just get through the assignment? I think that is where deep practice comes into play. If they practice without assessing (checking for success) will they promote their confusion?  I tell my students that it is like practicing shooting free throws with your feet perpendicular to each other. Terrible form does not promote success. Zero practice is better than incorrect practice.

With that being said, I think that teachers must have realistic expectations about time and quality of assignments. If we expect students to engage in deep practice (to embrace the struggle) then we have to shorten our assignments to accommodate the additional time it will take to engage in the struggle.  We now ask students to complete anywhere from 1/3 to 1/2 as many problems as in the past with the understanding that these problems will be attempted using the method of deep practice.

Our version of deep practice homework:
“We have significantly shortened this assignment from years past in order to allow you time to work these questions correctly. We want you do work with deep practice.

  • Please work each problem slowly and accurately.
  • Check the answer to the question immediately.
  • If correct, go to the next problem.
  • If not correct, mark through your work – don’t erase – leave evidence of your effort and thinking.
    • Try again.
    • If you make three attempts and can not get the correct answer, go on to the next problem. “

I also think that the formative assessments with “leveling” encourage the willingness to struggle. How many times has a student responded to you “I don’t get it”? Perhaps it is not a lack of effort. Perhaps it is a lack of connected vocabulary. It is not only that they don’t know how, is it that they don’t know what it is called either. It is hard to struggle through when you lack vocabulary, skill, and efficacy all at the same time. How might we help our learners attend to precision, to communicate in the language of our disciplines?

Now is the time to guide our young learners to develop voice, confidence (and trust), and a safe place to struggle.

Deep Practice, Leveling, and Communication was originally published on November 20, 2010

 

Assessment, Learning, Questions

Reflection, Attitude, and Efficacy (TBT Remix)

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How can we promote success-oriented behaviors to foster learning and self-efficacy?

I dare you to read the following journal entries but replace the word math with assessment or whatever you are struggling to learn right now.  Out of the mouth of babes…

 

I’ve been rereading journal entries from August to reflect on the growth of children I coach to learn algebra.  The point of this particular journal entry was to help assess disposition.

Can we effect their growth in algebra AND their growth as learners? Can changing our assessment practices and our approach to learning help them learn to embrace the struggle, to see that a “failure” is an opportunity to learn?  Does success breed success?  Does success change your confidence, efficacy, and disposition?

How can we help failure-avoidant students grow to become success-oriented learners?  Are most learners both success-oriented and failure-avoidant with a strong preference for one or the other?

Wait… I choose to revise my question.  How can we promote success-oriented behaviors to foster learning and self-efficacy?

What do you think?
Is QB success oriented, failure avoidant, or both?

 
”The reason why I chose a picture of a person repelling or climbing a mountain is because math is a mountain for me. A mountain is an object that you cannot go through or around. The only way to get to the top of the mountain is by climbing. Math for me is a mountain. I can only climb my way to the top. There will be slips and falls along the way but, that is the only way to get to the top of the mountain. Every step I take teaches me something about that mountain. When you climb to the top of the mountain you can look back and say all those little slips and falls taught me something about that mountain, but now I can see all those tiny steps added up.”

“Every step I take teaches me something about that mountain. When you climb to the top of the mountain you can look back and say all those little slips and falls taught me something about that mountain, but now I can see all those tiny steps added up.”

I love this child; he spends many hours with me learning and improving.  We have two classes together, and he chooses to work with me after school several days each week.  When I read his journal on the first day of class, I put him in the success-oriented category.  As I have worked with him this semester, I have seen him on a rollercoaster ride, struggling to not lapse into failure-avoidant behaviors.  I believe it is my job is to be his cheer-master, his coach, and his support.  I want to coach him to find his strenghts and successes.

The same day, CL wrote:

“I think this picture best describes my experiences in math for a lot of reasons. If you look at the girl’s face, it seems like she doesn’t know what she is doing. But if you look at her body, she seems to be doing the right thing. This is like me in math in a way. A lot of times I am doing the right steps, but I still think I am wrong. Like the girl in the photo, I don’t believe I am doing the right steps (or moves in her case). My feelings toward math are basic. I don’t love math, but I don’t hate it. Math also doesn’t come naturally to me. I have to work hard at something until I really understand it. I am more interested in math that we use every day than just random lessons. I also like to know the why in things. Like “Why do we use this trick?”. The why and how are keys words for me in learning math. I think my job in math is to learn new things, listed to the students and other students and the teachers, and to help others learn. I believe math is very helpful in everyday situations. I also believe math is hard, but if you work hard enough you will understand it. I want to learn from my mistakes in math. I also need different techniques to learn from if one doesn’t work. Lastly, my goal this year in math is to maintain a high grade by fully understanding the material.”

 

How often do we make curricular decisions based on what we think we see?  Are we looking at the face or the body?  How often do we assume that our students are learning?  Do we check for evidence of learning – not grade – really check for proof?   When we see the body doing the right things, do we ignore the face?  Do we check for confidence?  I fear that we may promote failure-avoidant behaviors if we are not careful.

CL wrote:

“If you look at the girl’s face, it seems like she doesn’t know what she is doing. But if you look at her body, she seems to be doing the right thing.”  

How do we give our learners enough feedback so that they know that they are doing the right work?  How do we build up their confidence so that they will either feel successful or know that it is safe (and encouraged) to ask questions to learn and grow?  How do we reward effort and willingness to struggle to learn without giving students a false impression of their achievement?

CL:

“I want to learn from my mistakes in math. I also need different techniques to learn from if one doesn’t work.”

Me too!  If we don’t assess learning and offer feedback in the midst of the experience, how will we know if we are promoting learning for all?  How will we know if some (or all) need a different approach? Again, we must be careful to promote success-oriented behaviors.

I also think that my team and I spend a fair amount of time in CL’s shoes.

“A lot of times I am doing the right steps, but I still think I am wrong. Like the girl in the photo, I don’t believe I am doing the right steps (or moves in her case).”

Am I doing the right things for my students?  My assessment plan is so different from what they will probably experience next year.  When I listen to others who are uncomfortable with this “radical” change, I question if I’m doing the right steps.  From what I read and study, I believe that I am doing the right things to help them learn and grow.

CL’s words where I have replaced math with assessment:

I don’t love assessment, but I don’t hate it. Assessment also doesn’t come naturally to me. I have to work hard at something until I really understand it.

My team experiments with me. Are we failure-avoidant teachers or success-oriented learners?  We collect data and ask questions; We refine our hypothesis and try again. We are learning by doing; we are making assessment and grading decisions based on what the data indicates.  Are we confident about our assessment work 100% of the time?  No…Does it cause us to ask questions, think deeply, risk, learn?  Yes…

It is certainly a work in progress.

Reflection, Attitude, and Efficacy was originally published on December 16, 2010.

Connecting Ideas, Learning, Questions, Reading

What we don’t remember about the foundation…

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I wonder if, when the house is finished, we forget the foundational infrastructure required for function.  How does water get into and out of my house? Who ran the wires so that our lamps illuminate our space? Who did the work, and what work was done, prior to the slab being poured?

When we recall a basic multiplication fact, it’s like flipping a light switch in our house. The electrical wiring allowing us to turn on the light is linked to sound, safe, and deeply connected infrastructure. (K. Nims, personal communication, August 30, 2015)

Just like the light switch is not part of the foundation, memorization of multiplication facts is also not foundational. It is efficient and functional.  Efficiency must not trump understanding.

We need people who are confident with mathematics, who can develop mathematical models and predictions, and who can justify, reason, communicate, and problem solve. (Boaler, n. pag.)

Screen Shot 2015-08-30 at 7.45.22 PMStudents who rely solely on the memorization of math facts often confuse similar facts. (O’Connell, 4 pag.)

Students must first understand the facts that they are being asked to memorize. (O’Connell, 3 pag.)

What if we have forgotten all the hard work that came prior to the task of memorizing our multiplication facts?

Do we remember learning about multiplication as repeated addition? Have we forgotten the connection between multiplication, arrays, and area?

Conceptual understanding of multiplication lays a foundation for deeper understanding of many mathematical topics.  Memorizing facts denies learners the opportunity to connect ideas, exercise flexibility, and interact with multiple strategies.

The goal is to have confident, competent, critical thinkers. Let’s remember that a strong foundation has many unseen components.  What if we slow down to develop deep understanding of the numeracy of multiplication?

Second, going slow helps the practitioner to develop something even more important: a working perception of the skill’s internal blueprint – the shape and rhythm of the interlocking skill circuits.”  (Coyle, 85 pag.)

How might we serve our learners by expecting them to show what they know more than one way?

Boaler, Jo. “The Stereotypes That Distort How Americans Teach and Learn Math.” The Atlantic. Atlantic Media Company, 12 Nov. 2013. Web. 30 Aug. 2015.

Coyle, Daniel (2009-04-16). The Talent Code: Greatness Isn’t Born. It’s Grown. Here’s How. Random House, Inc.. Kindle Edition.

O’Connell, Susan, and John SanGiovanni. Mastering the Basic Math Facts in Multiplication and Division: Strategies, Activities & Interventions to Move Students beyond Memorization. Portsmouth, NH: Heinemann, 2011. Print.

Assessment, Grading, Learning, Questions

Being Slow…Mindset…2nd Chances…Learning (TBT Remix)

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Rule Three from The Talent Code by Daniel Coyle is SLOW IT DOWN.

“Why does slowing down work so well? The myelin model offers two reasons.  First, going slow allows you to attend more closely to errors, creating a higher degree of precision with each firing – and when it comes to growing myelin, precision is everything.  As football coach Tom Martinez likes to say ‘It’s not how fast you can do it. It’s how slowly you can do it correctly.’ Second, going slow helps the practitioner to develop something even more important: a working perception of the skill’s internal blueprint – the shape and rhythm of the interlocking skill circuits.”  (p. 85)

We still take a lot of heat from our colleagues about 2nd chance tests.  It makes many people, teachers and parents, uncomfortable.

About our version of 2nd chance tests:

  • Our learners take the test; we mark (not grade) each problem as correct or incorrect, and return the paper to the child without a number-no grade yet.
  • Their job is to find, correct, and identify errors.  We ask them to categorize an error as either a “simple mistake” or “needs more study”.
  • We also ask them to complete a table of specifications and determine their proficiency on the assessed essential learnings.
  • After all problems are corrected, students write a reflection about their work.
  • Armed with the experiences of teamwork, feedback, and self-assessment, students are given a 2nd Chance test and are tested on only the problems missed during the first testing experience.
  • The final test grade combines the correct work from the first test with the work from the 2nd Chance test.
  • Yes, it is completely possible to bomb the first test and end up with a 100 in my grade book.

My assumption is that this discomfort comes from how non-traditional – radical – this concept comes across.  Just because it is different does not make it a bad idea, does it?  The discomfort comes from gut-reaction or theory rather than practice.  Shouldn’t you try it?  What do learners say?

Here’s what some of my learners say.

“If you give your best effort the first time around, you will have learned more in the process and the second time around will be less stressful therefore making the hard work the first time more rewarding. I think that the second chance test is a very valuable learning technique. Even after that unit is complete, it shows you where you need to improve before you start building on those concepts. So far this year, I have seen great improvement in my learning from my previous years in math. This year it has all started clicking, and I am excited about the new units to come.”
~CM

“Before we jump into a new chapter, our class usually takes a formative assessment to tell us where we are and what we know before we actually start learning from Mrs. Gough. I take these seriously because I think they really do help. If I can see where I am in the beginning and then where I am in the end, I can see how much I’ve learned and accomplished.”
~ MC

In Mindset, Dr. Carol Dweck writes

“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.” (p. 39)

More from my learners:

“Taking formative assessments and tests is something that I think is very important. I give my best effort, and work to learn from my mistakes. The second chance test is something that I think helps us actually learn from taking tests and making mistakes, rather than just getting tested on the material. Math has become one of my favorite subjects this year, and I have worked to learn from all my mistakes.
~VB

“I think that first chance tests and formative assessments are amazing because I can first understand my level and see what to work on and then really learn the material on the test to do better on the second chance. I do well in groups (except for the occasional random moments), and I love working in groups instead of taking notes the whole time. By helping others, it also helps me understand what I am doing wrong or just what I am supposed to do.”
~ HA

I feel the same as Daniel Coyle in the epilogue of The Talent Code when he writes

“Mostly though, I feel it in a changed attitude toward failure, which doesn’t feel like a setback or the writing on the wall anymore, but like a path forward.”

One more quote from our learners

“Overall, I feel as though I have done a pretty good job so far, but there is no one who can stop me from really stepping it up to an unbelievable level. The rest of the year I am going to fix any flaws I have, and show everyone what I can do when I REALLY put my mind to something.”
~ LM

In case this has been too broad for you, let’s go deep.  Here is one learner’s story from three perspectives.

From my perspective…

“GW came to me feeling that she is not very good at math and that she hasn’t been encouraged to like math.  She seeks an advocate and coach.  I strive to support GW as she becomes empowered to take control of her learning.  She is learning that it is great to struggle to learn; it is worth it to struggle to learn; and through the struggle she finds success.  Success leads to more confidence and more success.”   

From GW’s perspective…

“When I started out in math I had a really hard time and math was a definite challenge for me and my first test grade didn’t make it any easier. I was “in a hole” as my parents would tell me and I had to dig myself out. I started to go to extra help a lot more often and made solid B’s on my midterm and exam grades. What helped me through this process was the support. Support from not only my family but from Mrs. Gough and the faculty that really encouraged me to do my best.”

 From GW’s parents’ perspective…

“GW quietly got way behind in math first semester.  Partly due to an inner voice telling her she did not do well in math and partly a lack of commitment and time management. GW had given up.  Mrs. Gough communicated to us that GW needed to demonstrate the deep practice method on all homework. With our support and encouragement (not hands on help) GW began to do the deep practice on homework and began to “review and preview” every night. Our emphasis was ‘the process’ not the letter grade.

Her great success is directly attributed to the teacher/student relationship that Jill forged. Through encouragement (emails), support (office hours), an emphasis on deep practice and patience, Jill taught GW to try and try again, make the mistake, work through it, and get to the answer. Through perseverance, determination and resilience GW moved from failure and “not being good at math” to more than just passing. For us the 80 on her final exam was an A+ in effort, team work, student/teacher relationship, and determination.”

There are many take-a-ways for me…

  • If I can see where I am in the beginning and then where I am in the end, I can see how much I’ve learned and accomplished
  •  It’s not how fast you can do it. It’s how slowly you can do it correctly.
  • I have worked to learn from all my mistakes.
  • There are still many paths to success.
  • This year it has all started clicking.
  • I am excited about the new units to come.
  • There is no one who can stop me from really stepping it up to an unbelievable level.
  • Try and try again, make the mistake, work through it, and get to the answer. 

So here’s to being slow, making mistakes, and trying again.  It’s about learning content and skills.  It’s about learning perseverance and determination.  It’s about learning.  Period!

Time is a variable.

Learning is the constant.

Being Slow…Mindset…2nd Chances…Learning was originally posted on February 12, 2011.

Coyle, Daniel. The Talent Code: Greatness Isn’t Born : It’s Grown, Here’s How. New York: Bantam, 2009. 217.  Print.

Dweck, Carol S. Mindset: the New Psychology of Success. New York: Random House, 2006. 39. Print.

Algebra, Assessment, Connecting Ideas, Math-Science Connections

How do we use the December exam as formative assessment? (TBT Remix)

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“For assessment to function formatively, the results have to be used to adjust teaching and learning; thus a significant aspect of any program will be the ways in which teachers make these adjustments.”
Inside the Black Box: Raising Standards Through Classroom Assessment Black and Wiliam

How do students reflect on their work?  What opportunities are offered to help students carry the essential learnings from first semester through second semester and/or into the next level of learning?

I’m interested and curious about different strategies and methods used to help learners process and reflect on their exam experience and the accumulation of what they know.  Since each learner will have different bright spots and strengths, what strategies are used to differentiate for intervention and enrichment?

We aim to get “in the weeds” about reflection and intervention.  We want every child to reflect on what they could demonstrate well and where they need additional help.  We do not want them to move to the next year with any doubt or weakness if we can help now.  But, how do we know who needs help?  We collect data, and we let our learners gather data.  We need to be informed; they need to be informed.  We are a team working toward the goal of mastery or proficiency for all learners.

Our process:

  1. Return the exam to the learner on the first day back.
  2. Have each learner complete the exam analysis and reflection form (shown below) to identify strengths and areas of need.
        1. Circle the number of any missed problem.
        2. Begin, and possibly complete, correcting missed problems to review the material and determine if any error was a simple mistake or if more help is needed.
        3. Write a reflection about strengths, struggles, and goals.
        4. Report results on our team’s Google doc. (This is a copy; feel free to explore and “report” data to see how it feels. You can view the results here.)
  3. Meet in team to review all results and analyze for groups to design and provide necessary intervention and additional learning experiences.
  4. All assessments 2nd semester will have questions from first semester essential learnings to offer learners the opportunity to show growth and to help with retention.

We would love it if others would share methods and strategies for helping learners grow from an exam experience.  How do students reflect on their work?

What opportunities are offered to help students carry the essential learnings from first semester through second semester and/or into the next level of learning?

How do we use the December exam as formative assessment? was originally posted on January 4, 2012.

Assessment, Connecting Ideas, Learning, Professional Development Feedback, Questions

PD in Action: 4th Grade Math fluency and communication

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More results from PD that causes action…It makes me wonder about learning design.  Are we designing PD experiences for teacher-learners and lessons for student-learners that cause action and gain traction? Do we see products of our PD learning being translated into classrooms? Do we see products of our classroom learning being translated into action?

Last week I wrote about mathematical communication at an early age after co-teaching 4th grade math. In the comments, Kato helped me refine a learning progression for showing work so that it was more student-friendly for 4th graders. Kato commented:

I would love to experiment with these levels in Fourth Grade. I like the levels about showing your work, and that they never say “show your work.” I find that that phrase overwhelms Fourth Graders (of all abilities) because they don’t really know what it means. Level 3 and 4 are good. I wonder if they are too wordy or have too many action steps to follow.

I’ve revised the learning progression as follows.

Level 4
I can show more than one way to find a solution to the problem.
Level 3
I can describe or illustrate how I arrived at a solution in a way that the reader understands without talking to me.
Level 2
I can find a correct solution to the problem.
Level 1
I can ask questions to help me work toward a solution to the problem.

Arleen invited me back to 4th grade math this week. As I arrived, the children were working on a Math Message. On the page with today’s Math Message, Arleen included the learning progression that she designed with Kato during the #LL2LU Faculty Forum PD session last week.

I was thinking about Kato’s comment I find that [the] phrase [show your work] overwhelms Fourth Graders (of all abilities) because they don’t really know what it means. How do we communicate how to show your work when the phrase show your work is confusing or unclear?

Arleen’s outcome for the children was about computational fluency.  My target for the children was about mathematical communication.  As we worked – Arleen presented questions and I modeled math communication – we observed the written work and coached.

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Look at the children’s work.  Do we know that we are clearly communicating both learning targets? Can we see evidence of learning in the work? I know I said (over and over) how to organize work and show what you know? Did they receive the coaching? Did our work cause action and learning?

Are the targets clear? Do we do enough in-the-moment formative assessment and coaching? Do we offer feedback that causes action and learning? What if we collect evidence and analyze the products of our work? What if we use artifacts of learners’ work to formatively self-assess?

Algebra, Assessment, Conferences, Connecting Ideas, Information Age Learning, Learning, Presentations, Questions, Teachnology

Leading Learners to Level Up – #LevelUpMath #LearnFwd12 – the details

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Hello, I am Anne Conzemius, the host for your Learning Forward session.

Well, no pressure there, huh?  Actually, about 15 minutes prior to this quick introduction, I scanned the roster of participants and noticed Anne’s name on the list of our Learning Forward conference session..

My previous post, Leading Learners to Level Up – #LevelUpMath #LearnFwd12, was written prior to our presentation.  Here’s what we actually did after I got through the nervousness and shock of Anne’s presence.  (I used a quote from her book, The Power of SMART Goals: Using Goals to Improve Student Learning, in the slide deck for this conference session, in this blog post, to collaborate with Bo (@boadams1) on this rubric, and in many discussions with teachers.

To lead learners to level up – learners of any age – we want to find and highlight their bright spots.  We want learners working from a point of strength and climbing to the next level.  To introduce this idea, we used the YouTube video Dan Heath: How to Find Bright Spots, shown below.

I gave a 4-minute Ignite talk on the why we should lead learners to level up.

Jeff used the TI-Nspire Navigator for Networked Computers to assess our small audience so that we could adjust our plan to meet their needs.  We quickly learned that Algebra I could be our focus (whew!) and that teacher growth as well as student growth was important to our participants (yay!).

Jeff then shared the YouTube video Leah Alcala: My Favorite No, shown below, as a jumping off point for a discussion on turning mistakes into learning opportunities.  We then discussed how leveraging technology – we use TI-Nspire Navigator, but PollEverywhere, Google forms, and other tools could be used – to offer faster, more public feedback and discussion opportunities while redirecting the work to the learners.

Since Leah’s video was about multiplying polynomials, I shared our Algebra I leveled formative assessment to engage our group in a discussion about bright spot and strength finding.

How do we offer students voice to self-advocate for their learning?  The days of the negative self-talk “I don’t know nothing” must come to an end. Everyone needs to acknowledge what they know and what they want to know.  It is about empowerment – empowering the learner. It is about coaching.  How powerful for learner to approach the teacher and say: I can do XX; will you help me learn to YY?  I want to work in that environment, don’t you?

A question from our participants caused us to discuss our assessment plan. How did I handle summative assessments and what did my grade book look like?  I cannot post graded assessments here, because they might still be in play in Algebra I classrooms. I can, however, share How do we use the December exam as formative assessment? and the Google doc that we used to document progress on non-graded formative assessment work. (This is a copy; feel free to explore and “report” data to see how it feels. You can view the results here.)

Jeff asked amazing questions to facilitate the discussion.  Through his art of questioning, we talked about the philosophy of doing homework with deep practice, I can statements…, and leading by following.

My concluding remarks began with a quote from Anne Conzemius (and Jan O’Neill) which “outed” Anne as an assessment goddess to the rest of our participants.

“In order to engage in high-quality assessment, teachers need to first identify specific learning targets and then to know whether the targets are asking students to demonstrate their knowledge, reasoning skills, performance skills, or ability to create a quality product.

The teacher must also understand what it will take for students to become masters of the learning targets…

Equally as important, the teacher must share these learning targets and strategies with the students in language that they understand. It is not enough that the teacher knows where students are headed; the students must also know where they are headed, and both the teacher and the students must be moving in the same direction.” (Conzemius, O’Neill,  66 pag.)

To end the session, we quoted CL – an 8th grader in my care while beginning her journey to learn Algebra:

“I truly believe the formative assessments are helpful for using as study guides for tests. I use them as study guides and I learn from my mistakes through them.

I do like the fact that they are not graded because it takes the pressure off of taking them and makes me believe it is okay if you do not know the material at first. They are really helpful for going back and looking at what I missed, and then ask you for help on those questions.

Having the four levels really helps because I know what levels I need to work on so that I can keep moving up to a higher level.”

Notice her last sentence: … I can keep moving up to a higher level.

Lead learners to level up by empowering them to ask their own questions.

_________________________

Conzemius, Anne; O’Neill, Jan. The Power of SMART Goals: Using Goals to Improve Student Learning. Bloomington, IN: Solution Tree, 2006. Print.

Note:
2010-11 was the last year I taught Algebra I, but if you want to see the day-by-day plan for the entire 2010-11 year in Algebra I, it is still online as a resource.