All posts by Jill Gough

Learner, Love Questions, Problem-finding, Math w/technology. Interests: Collaborating, PLC, Formative Assessment
Assessment, Learning, Questions

Reflection, Attitude, and Efficacy

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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?  And, is it really that black and white?  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 than 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. 

_________________________

I dare you to go back and read both journal articles on this page replacing the word math with assessment or whatever you are struggling to learn right now.  Out of the mouth of babes…

Assessment, Learning, Questions

2nd Chance Tests, Effort, and Assessment

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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?

Listen to what my learners are telling me. 

 I chose this picture because this is me after a few hours of trying to figure out a math problem. I get frustrated when I don’t understand the concept or material. This picture shows you that I hate math because it is obviously not my best subject. Math obviously doesn’t come easy to me, but what this picture shows is only after a little while of focusing. What this picture doesn’t show is that sometimes certain material comes easy to me and I do well on that certain section. This picture is mostly for the harder concepts that I do not understand and wish I didn’t have to do. My goal : My goal is to do the best I can, meaning studying more than usual and not racing through homework, but really taking my time so I make sure to understand everything. I expect to accomplish this goal by pushing myself to study and take longer on homework and not turning on the computer, TV or phone.  

KW wrote

“I get frustrated when I don’t understand the concept or material. This picture shows you that I hate math because it is obviously not my best subject.” 

I want to say that there is nothing in this writing that gives evidence that math is not her best subject, however, I infer that she hates math because she has not experienced success from struggle.  Could it be that she has never been given the opportunity?  Would a different testing philosophy change her attitude and efficacy?

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 specification 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. 

To answer the most frequently asked question… Yes, it is completely possible to bomb the first test and end up with a 100 in my grade book.  

 CM wrote:

 
Math can be a challenge for me because sometimes I feel that I give the effort and it just doesn’t reach the results that I wanted. Like this picture it seems as though I climb towards a better way to understand it and achieve my goals but never reach there. Sometimes I feel that others make it to the top before I do. That can be discouraging, when I am ready and prepared to climb the mountain I feel better about the outcome, whatever it may be. I think that I am good at math just sometimes I need a little bit of encouraging to reach my full potential. I often need help from peers and my teacher to show me easier ways to reach my goals.My job in this class is to participate in class and offer my ideas to help me and my peers to better understand a math problem. My obligation in class is to learn math in new ways to that I can better understand math in a whole and be able to apply it to life. I believe that every day is a learning experience and every day even if I don’t succeed that I am trying towards my goals and get one step closer every time I make a mistake. I think that math is a real challenge for me but if I work hard I can do well this year. I feel that this class is going to beneficial to my learning this year. For my year to be even more successful I need in a class, projects, communication, opportunities to make up grades so that I can learn from my mistakes. What I want is a way to make math more fun and enjoyable.

Goals for Math
My goals in math are to really understand the problems and not only memorize them but to learn from my mistakes and apply them at the next chance. I hope to get a good grade in this class and learn new things I didn’t know about math.  

 CM wrote

“My goals in math are to really understand the problems and not only memorize them but to learn from my mistakes and apply them at the next chance.” 

How often do we provide the opportunity for students to learn from their mistakes and apply them?  When do they get a next chance?  We, the Algebra I team, take a lot of flak from our colleagues and parents about our 2nd Chance Test commitment.  This flak comes in many forms, some subtle, some not so subtle. And, most arguments fall apart when we point out that you can take the SAT multiple times and bank your scores.  Hmm….

The bottom line is that we have seen positive learning results on our semester exam that we contribute to allowing children to learn from their mulligans.

CM also said

“Math can be a challenge for me because sometimes I feel that I give the effort and it just doesn’t reach the results that I wanted.”

Our grade books are sparse; we have very few grades.  We assess quite often; we grade little.  We are rewarding effort, but not because you are sweet, try really hard, and/or are compliant.  We expect proof that the effort yields results.

In January, we will ask our students for feedback our course and assessment strategies to get a feel for their perception of their learning.

Learning, Questions

Deep Practice in practice

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Let’s think more about the idea of deep practice for homework and what it might look like.  The children that I am coaching to learn Algebra I are encouraged to complete the homework the idea of deep practice. 

We are trying to move learners from “If at first you don’t succeed, hide all evidence that you tried!” an attitude of “If at first you don’t succeed, try, try again.”

When homework is assigned, part of the homework is to complete the assignment in deep practice.  All complete solutions are posted my webpage and the answers only are posted on the Algebra I team’s MOODLE course (password protected). 

Students are to work a problem, check the answer, and try again if necessary.  To keep them from getting stuck, if they are not successful after three attempts they should move on to another problem, question, or task.  So that we can diagnose their error in class, the student should simply x out incorrect work and try again. 

When students come to class, they should know that their answers are correct or know where they have questions.  As our class is getting settled, students should be asking and answer each other’s questions.  I wander around and document how many answers are correct and how many they had to do in deep practice.  Here’s an example of a student’s work on a traditional Algebra I homework assignment.  (I just asked for a random homework page that showed deep practice from the students my last class on Friday.)

Student example of deep practice.

The diagnosis for the student above is really simple since there is evidence of her work and thinking. In problem number 4, there was a simple division error; she reduced 12/15 incorrectly the first time.  In questions 5 and 16 (and other problems on the page) we can identify a pattern in her work.  Her fundamental misunderstanding is about Integers.  Subtracting is the same as adding a negative.  The negative is missing in step two for this child every time the variable term is subtracted from a constant.  A simple-to-fix error IF it can be diagnosed.  (The correction for number 5 is also not correct…sigh…there is still so much work to do.)  We should make the time to coach our students to value proof-reading their work and analyzing their mistakes for learning.    

As fate would have it, I ran into the mother of the child whose work is displayed above the same day that I collected this example.  Mom stopped me to tell me a story about her child, M.  According to M and her mother, M has not felt successful in previous math classes.  M’s mom says math is now M’s “favorite class”.  Her confidence has increased significantly and is still increasing. M also feels comfortable asking you questions and knowing you will willingly answer them.  Mom also says the biggest change in M is her willingness to make a mistake.   She quoted her child “Mom, Ms. Gough has taught me that it is okay to make a mistake.  We learn from them.  It’s how we learn.” 

Worth repeating   

“…it is okay to make a mistake.  We learn from them.  It’s how we learn.” 

Let me pause here and say that I do not grade homework; I do not give credit for homework; I do not count homework in any way.  We do, however, document homework; we also document attendance in Office Hours.  My students help me with this documentation.  If and when a student is disappointed or frustrated with their progress, we can look at their rate of homework completion and how much deep practice is attempted compared to the number of times they come for addition coaching during Office Hours.  Often the data is very revealing to a student.  I struggle with my homework, and I have questions, but I never come to work on these questions.  Hmm… what steps could be taken to begin to improve?

One more thing about the above work…We have had to work really hard to correct a misunderstanding with kids that you can see in the above work.  She has documented that she had 6 correct answers and 6 deep practice. From my point of view, she has 11 correct, and 1 that still needs work.  It has taken about 3 months to get kids to the understanding that their homework is “correct” if they arrived at answer with good work no matter how many times they had to try. 

We are trying to move learners from “If at first you don’t succeed, hide all evidence that you tried!” to M’s attitude of “…it is okay to make a mistake.  We learn from them.  It’s how we learn.” 

Assessment, Learning, Questions, Time

Is time a variable? Is learning a constant?

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I think that I need some counseling – a reality check, if you will – about the meaning of: 

Time is the variable and learning is the constant. 

Is it true all of the time, some of the time, regularly with some exceptions, or only applicable to small children learning to walk and talk?

There are deadlines – we have an exam coming up that signifies the end of first semester.  So let’s take the idea of unlimited time off the table.  What, then, does “Time is the variable and learning is the constant” look like in practice, realistic on-the-ground, in-the-trenches, day-to-day practice?

A child performs poorly on a relatively high-stakes assessment; an assessment that carries a grade that some use to “define” this child.  Do we offer this child a mulligan?  Do we offer every child the same opportunity for said mulligan?  Do we require the “do-over” of this child?  Do we require the “do-over” of every child?

Let’s break these questions down with some details and more questions.  If we embrace “Time is the variable and learning is the constant”, then YES to the 2nd chance… or…are there more questions to be asked?

  • AG was sick the entire week prior to the assessment…mulligan?
  • DP is so nice, sweet and tries so hard…mulligan?
  • SM was irresponsible having no homework showing no effort…mulligan?
  • JH is disrespectful and disagreeable…mulligan?
  • RT is misplaced in this course (shh, she just can’t cut it here)…mulligan?  

If yes is the answer to any (or all) of the above, is the do-over optional or required?

What about an assessment?  Is time a variable?  Should AG, DD, SM, JH, and RT be able to (have to) express their learning, understanding, and growth in the same 55 minute time frame? 

  • AG is a visual leaner; she needs to draw and outline her thinking before documenting it formally.
  • DP is a memorizer; as long as there are no curve-balls on the assessment, she is great.
  • SM is unprepared but SMART; he can pull the information back, but it is slow going due to lack of practice.
  • JH checks his work on every question; he is determined not to make a mistake.
  • RT is carrying a heavy personal load at home and can’t concentrate; everything is a struggle.

Do we provide these students additional time to show and document their current level of ability, work, and understanding, or at the end of the period, is time up?  Does it matter which child you are? Does knowing more about why the additional time is needed change your feelings about providing additional time?  (What if you can’t know every child’s story?)

The two latest arguments being discussed within my team are that we were not preparing our students to take the SAT if we give them additional time and that something is wrong if you can’t do the work in the given time-frame.  I don’t wake up every morning planning to improve the future SAT scores of my 8th grade students; should I?  How will I know what is wrong if I can’t see this child’s work?  How will this child know what is wrong if they are not given the opportunity to “fall down”? What am I unintentionally teaching by providing or denying additional time?

Assessment, Questions

Deep Practice, Leveling, and Communication

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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 am intrigued by what deep practice might look like in other classrooms and/or disciplines.  In ¡Inglés fatal!, TSadtler is starting to write about deep practice and what it might look like with his students.  (The 13-year prologue post that includes his first mention of deep practice including the powerful questions “Can I do this and will it result in meaningful, ‘well-myelinated’ learning for my kids?”)

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.

Now is the time to give them voice, confidence (and trust), and a safe place to struggle.

Assessment does not always have to carry a grade. Learning should not be punitive. If the struggle causes a student to learn, that struggle should be rewarded.

Assessment, Questions

Level Up with Formative Assessment to Improve Communication

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How often does a student come for extra help and say “I don’t get it”?  And, how many times have I replied, “That is not a question”?  I need to help them diagnose what they “don’t get.”  More importantly, they need help diagnosing what they “don’t get.”

We’ve spent the last year reading and attending conferences on assessment.  We formed a study group that meets after school to work together to learn to better assess learning.  In Algebra I, we have modified our assessment plan to include 2nd chance tests, test analysis, and self-reflection.  New as of August is an attempt to use formal non-graded formative assessment based on a 4-point rubric to help kids level up.  An important note here is that the target level is Level 3; the level 4 questions are there to provide enrichment for students that always “get it.”

Our formative assessment on slope would be an example of this work.  Students are given this assessment to help them chart their progress and understanding on the basics of finding slope.  They worked independently to attempt to answer all questions; this took about 30 minutes.  At the end of the “testing period”, the final answers were displayed (page 9 of the formative assessment on slope), and students worked in teams to assess, correct, and reflect on their work.  (Pages 4-5 of the formative assessment on slope.)  Homework was assigned based on the individual student’s level.

How do students confidently determine their level?  For some students it is obvious; for other students, they have to ask for someone else’s opinion?   Let me say how really great it is when a struggling student asks for my opinion on his or her level.  It is even better when the same student can tell me what they can do well and where they need help.  To be invited by a child to be part of their learning team is very motivating and removes all the frustration and irritation.  The request for help or advice sends a strong message of interest.

An unexpected by-product of this type of formative assessment is the leveling up of their vocabulary.  Rarely does a student now say “I don’t get it.”  Much more often a child will come by after school and say “I need help writing the equation of a line when you give me a point and the slope.”

Progress.

One of my students, MR, says “I think that the formative assesments are great!!  They really help me to study and they help me to know what will be on the tests and what I need to further study! Knowing that level three is the target level, always giving us a goal to strive for and to study for is great!”

CH writes “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.”

Worth repeating: 

“makes me believe it is okay if you do not know the material at first.”

Progress!

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“What a child can do today with assistance, she will be able to do by herself tomorrow.” ~ Lev Vygotsky.

Questions, Technology

Learning, liking what we do, bright spots, and literacy

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I do not know how to find the square root of a whole number without a calculator. I have number sense; I can estimate that the square root of 21 is between 4 and 5, closer to 5. I wonder if my students’ work with fractions and decimals falls into this category.

From Powerful Learning What We Know About Teaching for Understanding: “Students do not routinely develop the ability to analyze, think critically, write and speak effectively, or solve complex problems from working on constrained tasks that emphasize memorization and elicit responses that merely demonstrate recall or application of simple algorithms (Bransford, Brown, & Cocking, 1999; Bransford & Donovan 2005).”

I worry that our students never get to the solve-complex-problems stage of learning in math. I’m afraid that we assume if they can’t do the “basics” then they are not qualified to attempt sophisticated interesting problems. If we would dare to start with the complex problem, would we interest more students in learning – even learning the “basics”? If we allowed technology to crunch the numbers, would students experience more engagement and attempt more interesting, complex, elegant problems? Would they ask to learn to improve the “basics” that they deem necessary or important? Would they use technology to aid in their learning? Are we brave enough to test this hypothesis?

In Bo’s It’s about Learning blog from June 25, 2010 I read: “We should be recreating more of the moments when things work well, when our strengths are revealed and engaged, when our efforts are at our best.”

Imagine you are sitting in Algebra I looking at one of your papers where every problem is wrong; you do not have one right answer on the entire page. You know that you have done everything the way you were taught. You know and can express that you have used the correct inverse operations to solve the equations? Or, you know the quadratic formula and can correctly interpret the results IF the results are correct? How frustrating!

Think of Gillian from The Element. Gillian did not perform well on tests; her work was difficult to read, and it was often turned in late. She was “a problem” in class. Sound familiar? In Gillian’s case, she needed movement. What if you need technology? What if we could reveal your strengths in algebra by simply allowing you to leverage technology to show your work and effort at its best? Would you find the motivation to work on your deficits if you find your strengths first?

Questions, Technology

Calculator is to Arithmetic as Spell Checker is to Spelling???

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Is using a calculator for math comparable to using a word processor for English? Is the calculator an arithmetic checker like the word processor is a spell or grammar checker?

My friend Jeff makes a good point about technology integration (advances) in English. Students that use a word processor must still proof their writing. Do we worry that kids won’t learn to write because of spell check or grammar check? Or, do we think that because of these tools they are free to concentrate on ideas, organization, voice, word choice, sentence fluency, conventions, and presentation? (Okay, conventions have to do with grammar and spelling, but I’m making a point here.)

The grammar or spell checker does not always catch “there” when I mean “their”. A calculator will not catch that I meant (-2)^2 when I entered -2^2. The calculator does not know that I mean 1+(6+4)/2 when I enter 1+6+4/2.

As Peyton pointed out in the Writing Workshop meeting, MS Word will not alert you to your error in writing “warmest retards” when you meant to write “warmest regards”. Your calculator will not alert you to an entry error; it will not know that you entered -3.75 when you wanted -3.57.

The spell checker automatically corrects some of my incorrect spelling. When I type “recieve” it automatically changes it to “receive”. When I type “calcualtor”, the word automatically changes to “calculator”. When I write “I never here anyone…”, the grammar checker alerts me to check my spelling or word choice. The Nspire calculator will autocorrect a little bit, but it assumes what you mean. For example, if you open parentheses, it will close them. However, you must make sure that it closes where you intend to end the grouping.

This is an interesting place for me in my thinking. I know that it has to do with age appropriate learning. I believe that young children should learn their numbers and arithmetic just like they learn their letters and words. I believe that junior high students should learn how to graph and solve equations by hand, graphically, with tables and spreadsheets, and with technology.

How much more could we learn about algebra, calculus, and statistics if we used technology to accommodate 8th graders that struggle to compute? Don’t our students need to spend more time on data gathering, mathematical modeling, and interpreting graphs and less time on mechanics.

Have you seen Conrad Wolfram’s Ted talk Teaching kids real math with computers?  What do you think?

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Just look at any TED talk by Hans Rosling to see examples of how critical the analysis and synthesis of mathematical information is to our future.