Category Archives: Questions

Is time a variable? Is learning a constant?

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?

Deep Practice, Leveling, and Communication

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 eraseleave 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.

Level Up with Formative Assessment to Improve Communication

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.”


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.”



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

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

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?

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

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?


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.