What is the most important question you should ask in mathematics?

For my entire teaching career (getting closer to 30 years) I’ve been asked to focus on working on my questioning of students in the classroom. I’ve been given countless sheets with sentence stems for asking good questions in the mathematics room for both me and students. The amount of questions on these sheets are so many that I, like many of you, never ask any of these questions with regularity. The goal of this post is to give you ONE – and only one (type) – question that I want to propose you should be asking every single day in your classroom – over and over again. Do you have any ideas what this ONE question may be? Keep reading.

During the professional development I lead I often reference the one question I ask more than any others when I teach (though this is NOT the question this blog post is about). What I often say is this question is “What Else? What Else? What Else?” – a lesson I learned years ago from Annie Fetter from her ‘Ever Notice what they Wonder?’ ignite video. In the video she advocates for asking mathematics students “What do you notice? What do you wonder?“. These simple questions are now a common mathematical instructional routine being written into most new curricula and modeled in PD sessions everywhere. WDYN? WDYW? are great questions – but learning to ask, ‘What Else? What Else? What Else?‘ is powerful as the first 3-5 things students say are not that mathy – but when you ask ‘What Else?’ several times – students always start saying mathy things they notice. (Note: Do not tell students to not say the first 3-5 non-mathy things – this will only impact their mathematical identity negatively – let them say these things and chart them with everything else. If they see a llama write it down).

I would love it if all math teachers asked ‘notice/wonder’ and ‘what else….’ questions daily – but those are NOT the most important questions a teacher should ask every class period. So, what is the MOST IMPORTANT question I believe you should ask all the time in mathematics? First a story. You can watch it in this video or scroll past the video to read the written version of what is in the video.

Several months ago I was observing in a alternative High School with my math leader friend Brooke Williams. The class was full of students who were working on everything from pre-algebra to calculus depending on their background and what they needed for graduation. The teacher had a system for students to work independently. As the teacher was working with other students, a student caught Brooke and my eye. We went over and asked him how he was doing. He was working on problem 4 on a sheet where he was being asked to multiply binomials. This was his work.

The student responded to our question of how things were going by saying, “This is my work. I think it’s right.” He had a note of uncertainty in his voice as he said this. As a 6-12 math teacher my mind immediately started diagnosing what I thought maybe he was uncertain of.

I thought, ‘Maybe he is unsure of how to write variables and coefficients (noticing the ‘s7’ vs ‘7s’ we normally write it as) or maybe he does not know what to do next and does not know how to combine like terms.’ In my mind I was jumping to conclusions of how to treat what I thought was wrong with him. As I did this I heard my math leader friend Brooke ask him, “Why did you say ‘I think’?” The student went on to say, “Well, I am not sure about the sign on 14. I am not sure I am multiplying negative Numbers correctly.” As an experienced teacher, I never once predicted this student was struggling with this as the sign on 14 was correct. If I had jumped in and treated what I though he was struggling with – I would have wasted his time. My friend Brooke quickly had a conversation with him around multiplying integers and quickly gave him confidence he understood this. Moments later, without our help, he recorded the correct solution on his paper.

Brooke with her question, “Why did you say, ‘I think’?” had surfaced his reasoning. She was able to support him in what he was unsure with and build is confidence as a mathematician.

The power of the Brooke’s question “Why did you say, ‘I think?‘” is that the young man could have said almost anything. He could have said, “I said ‘I think’ because I have not gotten sleep in several days and I am not sure my tired brain get’s it.” If he said this, he is revealing a part of him I should pay attention to as a teacher. Brooke’s question has the potential of helping us reveal our student’s humaneness, not just their mathematical reasoning. In my experience, using questions with my students similar to Brooke’s has given me insight into students mathematical identity in ways I never been given access before.

I believe Brooke’s question, “Why did you say __________?” is the most important question all math teachers should be asking more than any other question in our classrooms.

I believe the question ‘Why did you say __________?’ – when we as math teachers learn to ask it every class period multiple times – becomes the best formative assessment we will do in our classroom. It provides real-time evidence of student reasoning that we can act on immediately.

Max Ray did an incredible IGNITE talk years ago titled ‘When is 2 greater than 4? A proof by induction.‘. I call it… ‘The amazing 2>4 video‘. If you’ve never seen it – stop what you are doing for 5 minutes and watch it now.

In the video Max talks about how and what we math teachers LISTEN for. He asks, “Are we LISTENING 4 answers or LISTENING 2 students“.

I know for a lot of my career, for most questions I asked in the classroom, I was listening for the answers I wanted to hear and as soon as I heard them I would move on. It created a culture in my classroom that was about answer-getting. The culture in my room was not about surfacing student reasoning. (Sidenote: If you’ve never seen Phil Daro’s video ‘Against Answer getting’ – it’s amazing – check it out HERE.) As I began to transform my classroom and devalue the race to the answer and value student thinking I realized one of the first things I had to work on was to really surface student thinking. I had to change how I questioned students and ask about ideas and not answers. Then I had to listen to students and not for the answers. The question that has transformed my classroom more than any other were versions of the one Brooke asked the student in the story above.

August 2015 was the first year of ‘Math On-a-Stick’ at the Minnesota State Fair. Math On-a-Stick is the creation of Minnesota Math Leader Christopher Danielson. At the 12 days of math fun the first year we watched as families interacted in the space with their children. Some parents would let their children play with the math experiences without interrupting them. We also saw some parents jump in and tell their child how to play with things at the table saying things like “Why don’t you build a patern” or “No, do it this way“. Many times – adults jumping in to interrupt the learning would shut down the child’s natural curiosity and play in the space. Some parents would, as their students as they played, say, “Tell me what you created” and we would hear the child excitedly explain to their parent all about their thinking – and the child would create more. What the parents said and did not say had a powerful impact on how they interacted with the mathematics in the space. Christopher wrote about what he observed that first year in a piece on his blog titled “Let the Children Play”.

What I learned the first year of Math On-a-stick to do even more of in my classroom was to give my students the space for them to make meaning and for me to not try and control every aspect of what that looked like. I learned to ask questions that would surface their reasoning. I started asking more “Tell me about what you ‘ve done so far” “How did you get __________” “Where did the number ______ come from?”…..

Check out his recent tweet from the online math community #MTBoS that connected to me of a visual Tracy Zager uses in PD.

You can hear Tracy’s own words about this picture in a Global Mathematics Department titled “How will we know what they are thinking?”. I love her picture of Heidi Fessenden (skip ahead to the 5 minute mark to see this photo) and what Tracey calls ‘the new posture of teaching mathematics’. This is what I need to look like in my classroom. I need to be listening to what students are saying and not listening for just answers.

As a Secondary Math Teacher – I use the visual pattern each fall in the first week with students. I don’t start with it this way though. I start with just one of the 3 images above in a math talk. I ask “How many white squares do you see?” I follow this up with my favorite question – the question that will help me surface student reasoning, “How do you see that number?“

Check out my nephews above. The spent 1.5 hours like this as my father introduced them to his favorite game and they played with him. I would love it if I could create the conditions in my math classroom so it looked like this every single day. I love using my nephews occasionally to try out mathematics I am doing with students. In order to have them engage in the task above, I got out my square block manipulative. I started by building the first shape in the picture using blue and orange blocks. One at a time I asked my nephews – ages 4, 6 and 7 – how many blue squares there were. I then asked them – How do you know it is 8 blue squares?

I asked my 4-year old nephew, “How many blue squares to you see.” After thinking and interacting with the squares he excitedly said ‘8’. I said, “How do you know it is 8?’ He put his cute chubby fingers on each square and counted ‘one, two, three….eight, nine….”. I’ve learned to not interrupt student thinking (my old self would have jumped in and corrected him double counting) and watched as he caught his mistake and started over and pulled one each blue square away from the inner orange square to count. He looks up and smiled at me when he got to 8. As a Pre-K student – his one-to-one correspondence of number to object is totally where he should be developmentally. (proud aunt).

I then asked my 6-year old nephew in the fall of his Kindergarten year – “How many blue blocks are there?” He looked at the shape and quickly said ‘8‘. Developmentally he should still be one-to-one with number & objects like his brother – but rather than assume I asked, “How do you know it is 8?”. To which he excitedly moved the top 3 blue squares up and said, “because there and 3 here and 3 here and 3 & 3 is 6 and then 2 more makes 8.” Wow! Great thinking for an early Kindergartner.

My 2nd grade, 7 year old nephew, also quickly said, “8” when I asked him how many blue squares there were in this shape. I also asked him, “How do you know it’s 8?“. To which he replied, “Well I know the whole thing is 9 and I took the one away.” After he said this I thought, ‘my 2nd grade nephew is mathematically advanced. It is September and he is beginning to see groups (3 groups of 3) and multiplying’. But instead of assuming this was correct, I asked “How do you know the big square is 9?“. The 2nd grader said, “I just know. I’ve built it so many times I know it is 9″. Through his play he had permanence – memory – that this shape was made of 9 squares.

Later I asked my 2nd grade nephew about how many blue squares were in a larger shape. He did not say, ‘It is 15 and take away the 3“. He did not have this shape memorized and he used similar thinking to his 6 year old brother to determine there were 12 blue squares.

By asking my nephews to not just give me an answer and have me celebrate the correct responses and fix their responses with misconceptions – I’ve instead learned to always ask some version of… “How do you know _________?” of them. This question reveals their mathematical thinking. It reveals their assets. It reveals how they see the world. It reveals not just how they think mathematically, but aspects of their humanity. I am am a better aunt to them, when I listen to them instead of assume or direct. This is the attitude I also need to keep as a daily practice with my students. It is a daily practice I need to bring to my work with adults.

THE ONE QUESTION WE SHOULD ALL BE ASKING EVERY DAY IN MATHEMATICS CLASSROOM IS….

In conclusion, I’d love it if you joined me in working to – in the words of Max Ray – ‘Listen TO our students vs listening for answers’ and to ask questions that surface student reasoning. My favorite question(s) to ask are some form of the ones below. I believe all 4 of these really are the same question.

I ask you to join me in not assuming we know what students are saying by their words. I ask you to daily follow up student responses with “How do you know _______? and to ask questions that will give us a window into how we can support their learning (formative assessment). If you are like me, it will take some time to develop this norm in your classroom – but when we do we will begin to see more of our students assets and take our focus off of what they get wrong.

CHALLENGE YOURSELF: Ask some form of my #1 most important question of your math students (or your own children) for the next 10 days in a row. Then listen to what they say.

I’d love to hear from you! What questions do you think we should be asking in the classroom? Comment below or tweet at me @saravdwerf. How do surface student reasoning and decrease the culture of answer-getting in your classrooms? I’d love to learn from you!

For even more thoughts on LISTENING TO STUDENTS REASONING & the questions we should ask – keep reading to learn from other educators.

Check out some cool threads on twitter about listening to students and surfacing their reasoning. The online community of math teachers has been a huge support for me to deepen my own practice around asking the questions in this blog.

I am so excited to see this resources from Marilyn Burns fall 2020 – Listening to Learn – check out her website to learn more.

“Our job isn’t to fix them (our students who are stuck), it is to understand them.” – Megan Franke

This thread on surfacing student thinking from Dan Meyer recently is chock full of great ideas from him and so many other educators….

A final thought – on surfacing reasoning with adults – not just students. What are the most important questions we can ask of one another?

The questions I am advocating asking students in this blog do not just work with students – I need to use them with adults in my life. For example, I saw this on a teacher Facebook page when I was writing this blog. I immediately started scripting a response to the teacher referenced in this post if she were my nephews teacher. My first thoughts were quite negative. I assumed things about the teacher and what they valued in math education.

If the pictured worksheet were given to my nephew in pre-school and I believe in listening to teachers not just for the answers I assume they will say – My question to the teacher should take a similar form to what i am advocating we ask of our students. For example, I may ask this teacher, “What were the activities that led up to this worksheet?” This teacher may go on and tell me all about these amazing Montessori concrete manipulative experiences my nephew may have engaged in before this sheet were assigned or they may tell me __________….who knows what – but what I know for sure is listening to the teacher will further the conversation more than me listening for the evidence to justify my anger at seeing this assigned to 4 year-olds.

one final note – really, I promise – I’m almost done…

This weekend I was hanging out with my nephews again. I got a text from their mom, my sister-in-law, saying they wanted me to bring my ‘pop-up game’. So of course, I brought my old-school game, Perfection, for my nephews to play with me. After one round my 7-year old nephew, the 2nd grader, exclaimed – “I got 23 pieces!” To which I asked, “How do you know it is 23?” My nephew smiles up at me and says, “Because 5 groups of 5 is 25 and there are 2 pieces left. I took 2 away from 25.” To which I smiled, because – since I’ve learned to listen for my nephews reasoning and not for the correct answers – I know in the last 4 months since I asked him about the number of blue squares he has made growth in his mathematical thinking and has the beginning base for multiplication. So great.

Side Ending Note: In this nation we have great messaging for parents around supporting literacy at home. Most parents will say, “I know we should read 20 minutes every day to our kids.”. Let’s start a campaign that families can support mathematics at home by playing games together for 1 hour a week. As we play games, let’s ask kids “How did you know _______?” and listen to their reasoning. In the words of Christopher Danielson, Let’s let our children play and make meaning of the world they live in without forcing our idea of it on them.