The story of 2 words & one simple tweak to get all students talking.

I currently have a really great student teacher.  Two things I love about her is that she is reflective and a quick learner.  Everyday she is working on the craft of teaching mathematics.  Every time I have a student teacher I learn new things.  Having a student teacher also reminds me of things I do unconsciously in my own teaching.  Having a student teacher allows me to name some things in my own practice that have an impact on student engagement and achievement.  Today’s post is the story of one of those things I now do so well, I don’t even know I am doing it anymore.  5-8 years ago, this ‘thing’ was something I had to intentionally choose to do each day.  Here is the stories of why I made this change to my teaching and a challenge for you to join me.

True Story #1 – 20 years ago

About 5 years into my teaching career I was assigned to teach 3 sections of math to Juniors and Seniors who had not passed our states ‘Basic Standards Math Test’, a test of 8th grade mathematics in an era where Algebra was usually first introduced in HS.  Currently the material on this test is taught to students in grades 4-7.  On the first day of class I asked 80 students this question:


80 out of 80 students got this problem incorrect.  It was a multiple choice problem and not a single student answered correctly. NOT.A.SINGLE.STUDENT.  These students were Juniors and Seniors in HS and not a single one answered correctly.  Lucky or me I was in a reflective mood that day and rather than assume that my students struggled with basic math operations (it turned out most did not)  and teach a month long unit on multiplication, I dug deeper.  I followed the question above with the one below and….

in a year

…EVERY student answered correctly!  EVERY.SINGE.STUDENT.  Wow.   MY.STUDENTS.ANNUAL DID.NOT.KNOW.THE.WORD.ANNUAL.  My student’s were simply struggling with an academic vocabulary that blocked them from showing what they did know.  Subconsciously, I had know this and like many teachers teaching in a school of high poverty (mine was around 90% at the time) and large number of ELL students, I had adopted a savior mentality (I now recognize how dangerous this is) and had simplified the language I had been using in the classroom.  For example instead of always using the term ‘coefficient’, I would say, “What is the number in front of the letter x?”.  I did this with the best intentions – I wanted to teach in such a way that I would not leave anyone out.  My students were participating and were able to perform on my classroom assessments, but I was handicapping them for future math courses.

This day in my classroom was one of the most pivotal of my career.  it sent me on a journey of learning.  I knew that I did not have the skills to teach the students I had in front of me.  The word ‘ANNUAL‘ became the motivation to change what I do.   I read everything I could get my hands on at the time related to math and best practices in vocabulary and reading strategies.  This did not take long as there was not much out there on this topic at the time and the things that were out there where shallow and lacking in more than the same 5 things repeated in every resource.  I became a bit frustrated, so I changed tactics.  I started making friends with ELL teachers and picking their brains.  I learned a lot.  20 years later I have dozens and dozens of simple tweaks (this post is about just one of them – I’ll blog about others when I have time to do so – if you like)  I’ve made to my teaching to engage students who struggle with academic vocabulary.  (and let’s face it, most of our students struggle with this).

True Story #2 – 8 months ago.

This year was my first year back teaching in a HS after an 11 year hiatus in MS and leadership positions.  As part of my teaching assignment I spend 2 hours a day in my schools Math Center tutoring students.  During the first week of school a student I did not know walked in and asked for help with his Pre-Calculus homework.  He opened up his book to the problem set shown below. I asked “Read the directions out loud, what are you being asked to do?”  He read the directions, shrugged and said “I don’t know how to do any of this”.  I looked at the problems and started thinking silently to myself, “Yikes, this young man is NOT ready at all to be in Pre-Calc if he can’t even do #1”.


Luckily, i did not go with my first fleeting thought, I reread the directions with a simple tweak, “In problems 1-6, find the real solutions,  or the value of ‘x’ that makes each equation true, of each equation.”  The student’s face lit up with a smile and he said “Oh, that is all I need to do?” and proceeded to relatively quickly solve each equation.  The words ‘Real solution‘ had tripped him up and made him believe what he needed to do was much more difficult.  Studies say that many white educators will underestimate his ability to be successful in math just by his skin color. When this student shrugged & said “I don’t know how to do any of this”, his actions supported the unconscious biases many of us have of students not traditionally enrolled in higher level math courses.  At one time in my career I probably would have diminished this students potential because of his actions, but I am thankful for the work I’ve done to change how I interact with students so I can see and promote their amazing potential.

I snapped a photo of this students textbook on this day and posted it by my desk to remind me to dig deeper than my first gut reaction to a students words or actions.  The words ‘REAL SOLUTIONS‘ became the words I used to challenge myself to continue learning and using best practices with students.

A simple tweak to engage students.

I learned this strategy from an ELL teacher about 8 or 9 years ago.  I don’t know if this strategy has a name, but I call it the ‘Pairing words/definitions in same sentence‘ strategy.  This ELL teachers said to me, “Sara what you should be doing multiple times each class period is to define mathematical terms immediately after using them in the same sentence. Students will learn and use the mathematical terms you want them to use if you simply always pair the term/definition.  Try it.  Watch your ELL students and students in poverty start answering more questions in class.”  So I tried it for 21 days in a row (’cause somewhere in my life I heard I need to do something for 21 days in a row to make it habit) and she was right.  This simple tweak resulted in more and more students understanding what I was saying.

My first attempts to do this were choppy and rough.  After doing it multiple times per class for 21 days it became natural.  Now days I notice myself doing this anytime I ask a question and most students stare blankly at me.  I quickly restate the question and pair the the mathematical term with a definition in the same sentence and all of sudden many more students are engaged and have a access point to what we are taking about.

I get this strategy may not make sense to you without some examples so….Below are some examples of what this looks like in my classroom.  My goal is to seamlessly pair mathematical terms with a quick definition all in the same sentence in an effort to invite more students to engage and also build their mathematical vocabulary.

EXAMPLE #1: What is the leading coefficient, the number in quad equation.front of x squared, in the expression?  (note:  This is one example of what I could do with this sentence.  I may have to define ‘expression’ or give more specifics to the word ‘leading’)

EXAMPLE #2: Which 2 integers, numbers that are not decimals or fractions, have a sum, add up to, 10?   (Note: If you want to see if any of your students struggle with mathematical terminology, write the sentence ‘Find two integers whose product is 12.’ and watch what your students do.  If even one or 2 struggle with this prompt for more than 15 seconds, you need to incorporate ELL best practices into your classroom)

EXAMPLE #3: I occasionally use this technique when responding to what students say.  To increase the chances of all students talking in class I use low risk questiowhich one doesn't belongns like ‘Which One Doesn’t Belong?’ from Mary Bourassa’s site.   I encourage students to use rich mathematical vocabulary, but don’t demand it when we are noticing and wondering.  I see it as my role to take students where they are at in terms of vocabulary and give them
supports to use more and more math vocabulary as they build their confidence, so it may look something like this….

  • Me:  Which one doesn’t belong?
  • S#1: The red one because it is the only one with a ‘t’ in it.
  • Me:  You selected the red expression because it is the only one that uses the variable ‘t’.
  • S#2:  The orange expression doesn’t belong because it is the only one with an ‘x two’in it.
  • Me:  You selected the orange expression because it has x two, pronounced’x-squared’ as one of its terms……..and so on….

Note: There is a second ELL best practice in this Example.  Do you notice what else I did to help students enter a conversation?  More on this in a later post.quad line graph

EXAMPLE #4: “What is y-intercept, where the graph meets the y-axis,  of the linear equation, the equation whose graph is a line?”

EXAMPLE #5: “The shape of the quadratic equation, the equation with an x squared term, is a parabola, or a U shape.

EXAMPLE #6: What are the coordinates, the x & y values, of the vertex, the minimum or maximum point of the curve, of the quadratic function?

A couple of notes on the examples above;

Note #1:  I don’t pair a vocab word and definition with EVERY math term I use each day.  I choose to do it with the terms I think students struggle the most with.  If I don’t know which terms they struggle with, then it is my job to use formative assessment information to learn which terms I should be pairing with definitions.

Note #2:  Though my examples show me being OK with students using casual math descriptions I very much demand that all students use appropriate math terms when I assess their mastery of learning.  I have lots of other techniques I’ve learned from ELL staff to assure ALL students are using appropriate math vocabulary.  By the way, I have found my lowest performing students have a sense of empowerment once they have a vocabulary that allows them to understand what they have previously quietly let float over their heads.

Note #3: I almost never pair more than one word with its definition in the same sentence like my examples above.  I guess I did this above just to give you an idea of what I do in my classroom.  It was surprisingly difficult for me to think of examples of what I do on the spot.  It is so much easier to do while I am teaching a live group of students.

I challenge you to make friends with the ELL staff in your building.  Go and ask them for their tips in engaging the students in your classroom (ELL, those in poverty….) in building a rich mathematical academic vocabulary.  If you are like me and still have a handful of days to teach this spring, try pairing words with their definitions in your classroom before you go home for the summer.  It gets easier I promise you.  In the fall, I also promise you that if you choose to use this strategy 21 days in a row, you will begin to see yourself doing it without thinking AND more importantly you will see some students you’ve never seen raise their hands before engaging in class conversations.

If you liked my thoughts on this strategy, let me know and I will write about the dozen other things I do to helps students engage in classroom discourse using a rich mathematical vocabulary.  Until then, enjoy the amazing weather outside (well, it is amazing in MN, I am not sure what it is like in your end of the world).


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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