Why I always carry a $1 bill and 4 quarters when I teach. (aka ’30 seconds to Oh!-)

For most of my career I’ve taught classes with students with the top test scores in the state and many with the very lowest.  The wide gap between my students experiences and success in mathematics prior to having them in my class is about as wide as is possible in education right now.  This has been my norm.  One I embrace.  I am confident that I have skills that will allow all students to experience both challenge and success in my classroom.

Despite the differences in their test scores, my students have way more in common than their test scores would tell. All arrive with many assets that allow them to grow as mathematicians while they are in my classroom.  My job is to see their assets and use these assets to build off of.  The students with lower test scores have one thing in common  – all have had interruptions to their education due to some reason that means that they were not currently achieving at the level of their age–alike peers on most assessments.

This school year I have the largest proportion of students I’ve ever had that had major interruptions in their mathematical career before arriving in my classroom.  Most of these students are my EL students who may have not been in a math classroom for multiple years for whatever reason.  They are 17 & 18 year olds in Algebra 2.  Some have math skills at the elementary level.  Most have multiple things they were never taught or never had the time to learn well in prior grades.  They are brilliant and the most hard working group of students I have ever taught.  When some question why students so far behind are in Algebra 2 – I fight to keep them in my class (vs putting them in courses as seniors that teach only middle school material).  I believe I can both teach grade level material and also catch them up.   In a month I am doing a session at my state math conference on the 50+ things I do in my classroom to accomplish this goal.  This post though, is a short glimpse into my work with my students with the stories of 2 of my current students and what I learned about them (but mostly about me) in the past week.

In addition to teaching Algebra 2, I spend one hour a day in my schools math center.  A place where students come to take tests or receive tutoring in math.  Each day 5-10 of my own students show up during 1st hour when I am in the math center (along with students from other teachers courses).  I have had the privilege of seeing these students math journey at a much deeper level.  Most days these students are working on homework.  Every day I see growth in these students and also learn about the depths of the holes left to fill in these students.

Fartun’s story

This is the story of ‘Fartun’.  Fartun is in the top 5 of my hardest working students this year. I suspect she spends a minimum of 2-3 hours a day on math homework.  (Note:  this is NOT my expectation of her – I want her to do less)  Her goal is to catch up on everything she is missing.  She is very quiet and smiles with joy frequently.  She arrived in this country a little over 2 years ago and I suspect she talks little due to confidence in her language skills.  She entered class not knowing how to solve simple equations (2x+3=14) and needs a calculator for most simple arithmetic (I am OK with this).  She is a strong mathematician, despite not yet having a flexible understanding of the base 10 number system.

We are currently solving systems of equations.  To support all my students I give them what I call ‘green sheets’.  These are one page (sometimes front and back) of material learned in 8th/9th grade algebra courses that they need to know to be successful in Algebra 2.  In our current unit students have 2 green sheets.  One for solving equations (6a Daily Practice with Green Sheet) and another for graphing inequalities (Unit 6 Inequality Green Sheet).  Everyday in class you hear me say ‘get out your green sheets’ as we work.  In the math center, students know we will not help them until they’ve first looked at their green sheets and/or googled on their phones the topic we are studying.  AS a result, you will always find students with resources out with their homework.  Much of the time they answer their own questions about math without having to ask me or my student teacher for help.  Fartun is able to figure out all kinds of things on her own at this point of the year.  She goes home and watches videos.  She asks her peers for help.  She uses the resources I give her to help her learn the material she missed in the past.

Last week, this was the problem she was working on in my classroom.  

Think of all the skills students need to graph this system.  This was the first system that I had given students with non-integer coefficients.  Fartun was not shaken.  She got to work.  About 10 minutes into the problem she caught my attention with a smile and pointed and quietly said – “this does not make sense.”

Before I tell you how I answered her – check out her work to this point.  Here is a student who struggles with basic arithmetic.  She uses a calculator for everything.  Without any help from me, she has accurately found intercepts – graphed them – found test points and worked to identify the region of the graph that represents the solution.  All without my help.  In many classrooms – high school teachers will shake their heads at students like Fartun as they watch her use her calculator to multiply 8 times 4 or subtract 7 from 10.  Many High School math teachers say – she should not be in an Advanced Algebra classroom with out basic arithmetic skills.  But look at the question Fartun asked me – she has number sense enough to ask a question when something does not make sense to her.  She is not blindly using a calculator to do her work – she demands understanding of what she does.

Back to Fartun’s question…..she asked….”Why is the answer larger than what I started with when there is a decimal?  I don’t get it.

I had heard a similar question from several of my students in the last day.  Their understanding and fluency with decimals is low.  To answer her question I took a $1 bill and 4 quarters out of my pocket and said, “How many quarters are in a dollar?”  She said 4.  I then wrote the following down….

I said, “How many groups of 0.25 are in 1?”  Fartun said, “Oh, I get it!”

Fartun is usually only 30 seconds away from an “OH!” and understanding. In my ideal world I would have Fartun for a 2nd hour of math each day and could slow down and have her and her peers build a fluency with arithmetic and numbers she missed in elementary and middle school.  Unfortunately, I only have 50 minutes with her each day.  For this reason I am always looking for quick ways for her to fill the holes in her prior education. Using money to help make sense of decimals is a go to for me.

Also -I have immersed myself in reading and learning about how to teach elementary mathematics.  Now when I talk about division with students I talk about ‘numbers of GROUPS’, when I talk about subtraction I talk about the ‘distance’ between the numbers.  Us High School teachers to the most complaining about students lack of arithmetic skills, but we are the weakest at knowing how to teach these skills this needs to change….NOW….what are you doing to become smarter as a teacher at not only teaching HS standards but also supporting students lack of numeracy skills?

At the start of the school year Fartun was a student I identified as being the most behind in mathematics.  Notice I did not say ‘low skilled’.  Fartun is a learner.  She is bright and hardworking.  By unit 3 she was turning in tests that looked like this.  Her grade level skills were on par with her peers.  Despite this her pre-grade level math skills & continue to unearth conceptual holes.  We work hard everyday to fill those.  The challenge now is that her brain is saturated and she seemingly at times loses things she’s gained.  She is not losing anything -but a lot of her learning is fragile and need of ongoing reinforcement.  She is amazing.  I hope her next teacher sees this.  I hope she believes this about herself.


Imran’s story

‘Imran’ is outgoing and positive every time I see her.  She is friends with everyone.  When someone enters the math center, she greets them by name (regardless of their grade, race…..).  She grasps new material quickly.  She will forgo doing her own work to tutor anyone in need. She is gregarious, kind and willing to work hard.  I’ve been working hard to convince her to become a teacher – despite the challenges she has of only arriving in this country 2 years ago, learning English and still having lots of holes in her mathematics.  She would be an amazing teacher – I hope she pursues this path.  She is the first to say she was never taught 6-9th grade math.  She has good basic arithmetic skills.  Because she learns new material quickly, I often think she is stronger than she actually is until she asks me a question that reminds me of the holes that remain.

This is the problem that Imran asked me a question about this week….As you look at this – ask yourself – what will Imran struggle with in this problem?  In addition to graphing the system of inequalities, students were asked to identify the corner points in the feasible region.  

What would your students struggle with?  What would a student of yours whose education was interrupted for multiple years ask?  Imran was able to do everything on her own – but she did have one question.

Imran pointed to this and said “Is there a one in front of x?”  I asked her to read the inequality out loud with a 1 and see if it makes sense.  She says “one times x minus 2 times y…..OH!  I get it“.  I said, “could it be a zero?”  She said ‘No it is a one” with confidence and proceeds to write in the invisible 1 and begins working ignoring me. Note:  asking students to read things out loud often allows them to answer their own questions or find their mistakes.

Again, Imran is usually less than 30 seconds from “Oh, I get it“.  I need to be prepared to help her get there that fast.  I could have said ‘Yes, there is a 1”.  Instead I choose to ask questions that hopefully she can answer her own question.  I am training her and all my students to answer questions for themselves when they work at home or anywhere I am not.  I am not going to college with the and it is my goal to empower them to teach themselves if necessary.   Look at her work – Imran is on par with her peers.  All I helped her with was giving her the confidence that there was an invisible one in equation 2.

Imran’s question also shows how important it is for us HS teachers to make the invisible ones and zeros in our work visible.  I ask myself everyday when I am planning – HOW CAN I MAKE THE INVISIBLE, VISIBLE?

One way I do this is by never saying “Cancel the 3’s” in the example above.  Instead I write a big 1 and talk about creating factors of 1.  I also write in the invisible 1’s and zeros into equations often in class -but that is an entirely another blog post.

Some final thoughts

I only have 50 minutes a day  for 1 school year to catch up Fartun & Imran and their peers on 6+ years of mathematics.  This is a seemingly impossible task.  I can either choose to focus on what they still don’t know or I can focus on what they do know and the growth they make each day.  I can either make decisions about what future courses they have access to based on the fact that they need a calculator to add 7 to -3…..OR….I can believe that they have the ability to learn new material at the same rate (if not greater) than their peers who have had interrupted educations and give them access to high levels of mathematics and watch them flourish.

I choose to believe my students can do grade level mathematics no matter how far behind they may appear to be when they arrive in my class.  I believe all students want to do well in math no matter how resistant they may appear.  I choose to see assets.  I believe it is my job to do something different if students are not successful and not blame students.

I am not going to lie – teaching the students I teach is damn hard work.  Fartun and Imran grapple with feeling like the understand very little all the time.  As soon as they mastered one thing in math – 5 other things they’ve yet to master appear.  The seem to never have a chance to feel like they are not behind.  The rarely feel ‘caught up’.  To counteract this we celebrate the mini-moments of Aha’s and Oh, I get its.  I frequently remind them of what they were able to do the first week of school and what they can do now so they can see the huge growth they have made.  If I don’t show them the growth – they only focus on the things they don’t know yet today.  They are now at the point that they are addicted to the struggle of not knowing because they know coming soon is the catharsis of understanding.  They see their hard work resulting in understanding and are empowered with tools of how to learn on their own – even if I don’t follow them to college.

I want to close with one more story about ‘Imran’ followed by a challenge to you.  Right before winter break I wanted my students to solve and write-up a problem that they could not solve easily a few moments.  I selected the classic Painted Cube Task.  I knew many of my students had language issues and interrupted geometry experiences so we started by each holding a cube and recording terms like face, vertices, edges.  We counted the number of each term.  Students explored the task in groups by building cubes, counting and recording information they collected.  In doing this the noticed patterns that emerged and world to generalize rules.

Imran and her peers persisted and worked hard.  When students were convinced they did not understand I would tell them to go back and build cubes again.  I remember overhearing Imran saying to friends on day 2 – “Ms Van was right, if we just build and look for patterns it starts to make sense”.  Imran was confident in a solution.  On day in the math center she was working on writing up her work.  While working she shouts out to me “Ms. Van, what is a vertex?”.  I shouted back my answer “A corner” to her.  Several minutes later she caught my attention again and said “What is a a corner?”

Here was a student that written beautiful rules for the number of cubes with 0, 1, 2 or 3 sides painted.  She understood these rules and had made beautiful diagrams to support her mathematics.  Despite all of this – she is still in need of support.  She had no idea what ‘corner’ meant.  I picked up the kleenex box nearby and pointed at one of the vertices and told Imran ‘this’.  She said “Oh, yeah!  Thanks.”  Again, Imran was 30 seconds away from understanding.

This is her 6 page write-up for the Painted Cube Task.  She did this all on her own.  I hope her future teachers don’t see her deficits and instead see her amazing potential.

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Here is my challenge to you (and as always me) based on what I learned from Fartun and Imran this week:

  • Noticing the small things students ask about more radically changes my teaching that almost anything else.  How can I make space for students to ask questions?  How will I make sure I stop and listen to what students are saying?
  • Will you see students struggling with elementary mathematics as students who are ‘low’ or as students who are powerful problem solvers who can achieve mastery of grade level standards?   How will this view impact what you teach and how you support students?
  • What are you doing as a secondary math teacher to improve your skills of teaching elementary math concepts?
  • How will you help students with holes in the math background feel success with grade level mathematics in your classroom today?  Do the tasks you select have multiple entry points for students?
  • What assets do your students bring to the table?
  • Are you focusing on proficiency or growth?
  • Lastly – most students are only 30 seconds away from “Oh, I get it”.  In a classroom that validates speed in answers (answering the first raised hand for example) – students who are only 5 seconds or even 5 minutes behind their peers often feel like they are years behind and have no hope of catching up.  How can you change the culture of your classroom to eliminate gaps in understanding that are only minutes away for most students?
4.18.18 Follow-up

One of the challenges of writing this post was worrying about talking about ‘low’ vs. ‘high’ students and all the baggage that comes with this.  I hesitated to even push send on this post because like others I hate calling students ‘low’.  There are lots of great posts on why I and others hate the term ‘low’.  Here are a few for further reading.

  • Kristen Gray’s ‘RTI for Adults‘.
  • Brian Bushart’s ‘What we Presume
  • Tracy Zagar talks about this a lot in ‘How not to start class in the fall’.

All of these are great – but from an elementary perspective.  Do you know other posts written by secondary math people?  If so, let me know.

Sara VanDerWerf

I am Sara Van Der Werf, a 24-year mathematics teacher in Minneapolis Public Schools. I have taught math in grades 7-12 as well as spent several years leading mathematics at the district office. I currently teach Advanced Algebra at South High School and I'm also the current President of the Minnesota Council of Teachers of Mathematics (MCTM). I am passionate about encouraging and connecting with mathematics teachers. I'd love to connect via twitter.  Join the community.  Tweet me @saravdwerf.

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