The end of the school year is usually an emotional time for me for a multitude of reasons: some students who I never reached, some personal goals unaccomplished, curriculum not covered, rushed lessons, unfinished conversations, the workload doesn’t fit into 24 hours day and “my kids” are about to disappear into the summer. So this post is not about a lesson or a series of lessons, it’s about emotions and perceptions.

**Introduction**

As the school year is coming to an end, so is our Mount Holyoke College Course. During the most recent session with Kaneka Turner, she shared her call to action with us.

- Interview a set of student you serve using the following:

- Are you good at math? How do you know you are? are not?
- Do other people know you are good/not good at math? How do they know this?
- Is there anything about math class that causes you to be good/ not good at math?
- If you could change one thing about math class what would you change and why?

- Consider the results of this interview and the implications for your math classes.
- Plan to or actually make 1 change as a result of the results.

I decided to start with taking two questions and asking my fifty grade 3 students to write an answer. I told my students that I need critical feedback so me and other teachers can be better math teachers, and I trust my kids’ honesty.

**Question #2 (we are moving backwards):
**

**If you could change one thing about math class, what would you change and why?**

There were some interesting ideas, but a found that a lot of the answers were very personal and focused on what students find hard or easy, and then they want things to be either harder or easier.

**Students want:** more problems, more time, more division because multiplication is too easy, more geometry, more multiplication bingo, more Desmos, word problems instead of “just numbers”, learning more strategies, harder, easier, more times tables because I know them, more tests.

**Students don’t want:** subtraction because it is too hard, multiplication because it is too hard, too hard math, too easy math, no standard algorithm, no tests.

This was all alarming, but it wasn’t the worst.

**Question #1**

**Are you good at math? How do you know?**

Most of my students felt they are good at math. And they should. But the reasoning stunned me.

Some responses I hoped to get more of.

- I am good at math because I am good at finding patterns.
- I am good at math because I can find patterns.
- I am good at math because I know good strategies and know how to use them efficiently.
- I re-think what I learned when I get home.
- When I get to solve problems it’s actually kind of fun!

That’s it, five responses, the rest I hoped I wouldn’t get.

- I am not good at math because I don’t understand how standard algorithm works.
- I am good at math because I learned division and multiplication before grade 3.
- I’m ok because I am good at adding but subtraction is confusing.
- I am kind of good because I remember my times tables up to ten.
- I am good at math because I can times big numbers.
- I am good at math because I am fast at multiplying and also correct.
- I am good at math because I can add, subtract, multiply, divide and my brain works fast and smart.
- I am good at math because I get more answers correct.

And the nail in the coffin.

**Emotional moment # 1**

I went through…

**Denial:** They didn’t understand the question. They just tried to give the “right” answer.

**Anger:** Why didn’t they get it? All year was for nothing!

**Bargaining and looking for excuses:** If only program of studies didn’t put that much emphasis on numeracy in elementary. I only have one outcome for geometry and two pages for operations. If only I have asked these questions earlier.

**Depression:** I am a terrible teacher and now it’s too late to do anything. I failed at one thing that I believe really matters. I’ve been leading workshops about math teaching all year and I’m a fraud. I’ve ruined my students’ math education and they will hate math for the rest of their life.

**Acceptance:** Now, what can I do about it? Because I can’t just leave it.

**Continuing the Conversation**

We started by getting together with all fifty kids and my teaching partner and brainstorming what it means to be good at reading. I hoped to get a T-chart and compare reading and math. We did make a T-chart, but a result was surprising. What do you notice if you compare this chart to students’ personal responses?

I noticed that there were more answers that I expected to get in the first place. Were my students just telling me what they knew I wanted to hear now? Did they get more ideas from each other? Whatever it was, my original plan to compare and contrast it with reading was not turning out to be dramatic enough. We noticed similarities.

Then I asked my kids what we did in math this year. They told me about addition and subtraction. I put it on the board and circled it. And then we kept going. We kept going for a while, and when the bell rang I asked the students who still had more ideas to add them on the post-it notes. Here is what we came up with.

And more from the notes: Is 5 closer to 0 or to 10, patterns in a triangle, polydrons, odd and even numbers, egg experiment with weight, paper (cardboard) SOMA cubes, multiplication gummy bears, Euler’s formula and making geometric shapes.

After lunch, I asked students to look at everything we have on the board, on the ideas on the post-its, and put a thumb up if they felt successful at something, did good at it. My students are used to showing the number of strategies they came up with on their fingers, so I saw how more fingers were following the thumbs, kids reminiscing on the year of math and counting their successes. They were beaming. Two girls quietly moved to writing more questions.

**Emotional moment # 2**

I told my students that I am writing their report cards now and no one is getting any 1s, and if they worry about it they shouldn’t. I told that asking good questions is sometimes more important than getting right answers. I told them how I’ve been sharing their work with other teachers who always said what amazing mathematicians they are. I stopped because I realized my rambling might actually spoil the moment. I left all their notes on the board. I want my students to keep seeing it and thinking about their successes. I want them to remember that arithmetic is just one thread in the math tapestry.

**Final thoughts and questions**

I am glad I did not just end my year with the assumption that I know what is going on through my students’ heads as they enter and leave my math classes. I still wonder what I did wrong, what I need to change next year so that my students have this realization about the nature of math as a subject earlier than May. I wonder if something in my words, my lesson design was reinforcing the stereotypes. Do my words and my actions and choices always align? I realized that I want to change how my students feel about mathematics, I need to explicitly design my lessons with this goal in mind. I need to be more careful and more reflective.

I wonder what’s one thing I should change in my math class?

I’m still at the – vicarious – bargaining and looking for excuses stage. Here’s what comes to mind, not terrifically thought-through or anything.

Excuse Number One.

We, children especially, are really good at forgetting, at telescoping experience into almost nothing. The parents I ask, ‘What did you do at school today, Miguel?’ And Miguel says, ‘Oh, I don’t know. I can’t remember.’ Teacher says, ‘What did you do at the weekend, Sara?’ Sara says: ‘I didn’t do anything.’ And yet, lots happened. And somewhere in there, growing up and learning is happening. It’s partly that describing what’s happened, what we’ve learnt is a-whole-nother skill. So we take summaries on their availability: whatever we’ve heard someone say, or springs to mind.

Excuse Number Two

And actually, so much learning happens unconsciously anyway. This is maybe heresy, so, just between you and me. I believe in articulating thoughts and in students actively choosing and thinking things through very consciously. But there’s always plenty that’s going on that isn’t even articulated, or even conscious and that’s fine too. We learnt to walk and to speak without being able to say what was happening at the time. So, if I’m right, then that’s even harder to access when we’re called on to say how we are at learning.

Excuse Number Three

Am I good at reading? is a funny question, isn’t it? People in book groups don’t talk about how good they are at reading. They almost always go straight to the book and talk about that. The reading bit is transparent, invisible almost, like glass through which we view things. And I’m wondering whether there’s a bit of that in maths too…? When it’s going well, we’re interested in the maths itself, not so much our performance.

Further Thought

But I do want students to be better, better than we were, at watching and noticing how they’re doing, what’s going on in their heads.

And Even Further

In terms of ‘Making Thinking Visible’, your board with all the things you’ve done is what was needed. Students won’t have one of those inside to refer to, so they need to have the inventory of what they’ve done there before them. Then their reflections on that are less arbitrary guesses and more based on what actually happened.

Lastly

It’s great that you’re able and willing to share the discomfort and sense of incompleteness of our job, the self-questioning. It helps us all to locate ourselves as teachers. Thank you.

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Re: Excuse Number Two

This idea of, being aware of what we know, is something that I have meant to write about for awhile. It’s a theme I keep getting hit over the head with when learning from some notable thinkers: Charlie Munger, W. Edwards Deming, Peter Drucker, and Elon Musk all harp on some variation of the importance of knowing what you know, whether that is subject matter knowledge, a practice you engage in, or knowing yourself.

Awhile back I caught Kent Haines in the midst of one of his excellent think alouds on Twitter. He had said “If I want to do a good job as a teacher with these kids, I need to know how they think about the big ideas of math.”

Hearing Charlie Munger in my head saying “know the big ideas from different disciplines, use them regularly, and know how they interrelate” I asked Kent a leading question…

“To do a good job as students do they need to know how they themselves think about the big ideas of math?” You can see Kent’s initial reply here https://twitter.com/KentHaines/status/807345310360109057

Connecting this back to Lana’s reflection, I imagine that if I asked all of my 4th grade students from the past two years what the big ideas of 4th grade math are, I bet they would struggle to name even the major content strands. However, I also think we could do an exercise like Lana did with her class and see just how much we had in fact learned (or at least become aware of). I see that as a BIG area of needed improvement in my practice because if the students don’t know what they know, they’ll struggle to use it, as well as to connect it throughout mathematics and across disciplines.

So Lana, I’m taking notes over here. I think your idea to continue the conversation (and how you did it) was perfect as well as your realization that you could do the same exercise earlier in the year. To extend it, I’d be thinking about what practices I, as the teacher am engaging in to continue the conversation, then try to figure out how I make those visible to the students. Why? Because just as I want them to be able to read or problem solve on their own, ultimately I want them to also be able to ‘map’ their own knowledge.

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Thank you for your comments, Mark. This idea in your comment that being aware of our thinking allows us to see how different ideas interrelate is really important for me when I think about teaching. I believed for a while now that teaching in content units without taking time to explore the connections leads to a very fractured learning landscape. But I never considered that even if I try to organize learning experiences in a more recursive fashion, the meta cognitive level of reflection is still not something that will happen naturally. The challenge is (like with all teaching) to not just tell students about the meaning and connection but to help students make these connections, find the meaning.

My students have been drilled on all the growth mindset lingo for years now. They can tell that mistakes are opportunities for learning, challenges are exciting and the journey is more important than the destination. They watched videos, they drew pictures, they listened and talked. Most don’t really buy it though. So I’m trying to be careful with my math agenda and not push it too hard so students start saying superficial things because they know I want to hear them. I want them to learn to “map their knowledge” not mine.

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Thank you Simon, you made me think about the difference between unconscious conceptions and the ideas that are put into words. As an adult, I often find it difficult to express myself when the emotional aspect is involved. And judging yourself as being good or not good at something would definitely bring up some emotions. A lot of kids responses were either “I am good” or “I see how I am improving”, but the justification of it is the rational part. There are marks in their report cards, there are worksheets-bases tutoring services, there is parents assigned homework, there is a pressure to perform to make sure that you “can do it better than others”. I wonder if thinking about “being good at something” shifted it towards thinking about skill performance, because it is measurable and quantifiable.

I was thinking if next year I would want to have a space for this “math mind map”, just like I have wonders and claims wall. Kids refer to the wall a lot. They are proud to note how someone’s “new question really connects to my conjecture”. Maybe building a map throughout the year and explicitly noticing/wondering and reflecting on it might help students to watch and to notice.

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Thanks for this post, which I’m finding really thought provoking.

I have a group of high school geometry students that has had a really hard time seeing proof as something valuable to learn and improve at. Some of these kids have really had success in their past math classes, and that success has mostly come in developing their arithmetic and algebra skills. Skill development, more than proof, explanation, or even problem solving, seems to me what they think of as the point of math.

I’ve been thinking about this group — and now I’m thinking about your post — and I’m wondering if there’s a progression of learning about what it means to be good at math.

The point of thinking of this as a progression of learning about what ‘being good at math’ means is that the attitudes of kids are natural results of the experiences they’ve had so far. If it’s a progression, that means it might take time to develop their thoughts about what ‘being good at math’ means, and it might take years. That doesn’t mean that we shouldn’t nudge them along that progression, but we also shouldn’t expect kids to have ‘expert-level’ ideas about the nature of math and mathematical skill either.

Here’s what I’m thinking. When kids say that being good at math has to do with speed, or with mastering operations, or with understanding subtraction…well, this is not what a mathematician would say. But the math that kids are studying is very different from the math that a mathematician studies. Kid math focuses A LOT on computation, EVEN IF we listen to strategies, allow kids to make sense of problems on their own, do mental math, number talks, justify our reasons. Kids aren’t exactly wrong that speed or computational proficiency has something to do with mathematical skill with this material.

With time, we want kids to have a richer vision of math. But there are also decent reasons, I think, to focus on number operations in these early years. It is the bread and butter of the rest of school math, and it would be a shame to not help kids develop these skills.

At the same time, this is going to have a natural consequence: kids overemphasize certain mathematical qualities over others.

Now, maybe we can do some things to help kids move along the learning progression. Many students won’t learn a more sophisticated story about what math skill is like until later grades (a lot of people talk about college math as the brand new world).

I suppose my point in this too-long comment is that we should just as much expect kids to have sophisticated ideas about the nature of math as we would for them to have sophisticated ideas about the math itself. Which is to say, they’ll have some incohate thoughts that we can help them develop. A lot of their ideas will be undeveloped, because they’re just starting to experience math. Their prior experiences are going to bias them towards a particular vision of math; that’s ok and natural. Our job can be to help them improve these ideas towards more sophisticated, grander visions of what math is and can be.

Thank you for the fantastic, open, and thought-provoking post!

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Thank you Michael for your comment. I think some of my worries come from my own negative school experiences and the fact that my own understanding of the subject was that math is mainly about of nonsensical boring routines that I need to perform quickly.

I agree that the understanding of mathematics and the nature of it might evolve and become more sophisticated with age and experience but I want to make sure that I support this development.

I hope that my students will eventually start recognizing more of their ideas and their learning as mathematical without needing my validation for it. But you are right, that would probably not happen over one school year.

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Lana, your end-of-year tweets have a much needed context now. This is a beautiful reflection.

What’s unspoken but obvious is how much work you put into improving your practice. Full time teacher, the Mt. Holyoke program, taking higher-level math courses, and still finding time to share and reflect with all of us. Thank you for that.

This isn’t just a wonderful reflection on learning and improving your practice, but an important reminder that learning means we’re not perfect, and all of the attendant emotions that come along for the ride with that reminder.

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After reading through your post a number of times I can see that you explored the disconnect between how your students thought about being good at math and how they thought about being successful with the math activities that you wrote on the board.

Still, I can’t help but think about what I would say if someone asked me if I was good at math. I admit that the first thing that comes to my mind is answering in terms of how well I do calculations. I make mistakes constantly. This morning I was fretting over getting a “wrong answer, ” and it took me many minutes for me to realize that I was adding a 2 instead of subtracting it (in the discriminant of quadratic formula, doing it in my head). But, oh yeah, I enjoyed fretting over it. Really. So the mistakes make me bad at math? The fact that I enjoy it makes me good? Just not a fair question…would not want to even answer it.

But ask if I like trying to understand things that I didn’t know before. Or does getting a wrong answer discourage me or pique my curiosity about what I’m missing? How about applying what learn? Even now, even though I should know that I should consider these qualities to be math thinking, they are not what I, on a gut level, consider to be math. So how can I expect a child, who likely has a perspective even narrower than mine, consider these things to be math?

I think that the only way they/we/ I begin to think of things such as soma puzzles, patterns in triangles, stories about shapes, and building & drawing models as math is if there is a constant reminder of this fact. They are not going to get it by osmosis, because what we get by osmosis is that math is calculations.

This week I am working with first graders, (eighty of them.which tires me out and renders me mostly brain dead at the end of the day, so you’re not hearing much from me this week..), and we are doing what I do with students, which is making books. But, this week, for the first time, partially because of your post, I am telling them explicitly that we are also doing math. They look shocked. And I say it’s not the kind of math that uses numbers. Through folding, cutting, assembling we are finding shapes within other shapes. We are finding small triangle in big triangles. We are making small rectangle out of big rectangles. We transformed a square into a pentagon to form a pocket to hold the pieces of our book-in-progress. They won’t know this is math unless I tell them, and keep telling them, which now I am trying to do more.

So for next year, not only do I hope you have this talk with your student sooner but I hope you can find some wall space in your classroom that is labeled something like Playing With Math Ideas, and every time you do your egg weighing experiment, or talk about infinity, or explore patterns, or even just get more fluent with number facts, that you somehow add reference to all of these activities to you math ideas board. Hopefully, doing this kind of math PR will remind them that math thinking takes many many forms, and chances are they will find that there is much about math that they connect to.

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Thank you Paula. I actually transferred all students ideas on the poster, and there is still enough space to be going for the last 5 weeks of school. I love your idea of starting in the beginning of the year and filling this space as we go. I can think about some interesting questions and conversations that can come out of it too. We can pause a few times a year to notice and wonder about things we learned and about things that we want to learn. We can look for connections and see how building models can help us find new calculations strategies, or calculations can help us to do geometric art.

My students are finishing their ink drawings of the famous landmarks, and I am thinking to do a reflection to remind them how much math went into this work: identifying shapes and symmetry, learning about shapes in architecture, functionality and aesthetics.

I also thought how earlier this year my students insisted that only regular hexagons are hexagons because this was based on their common “shape encounters” experiences. So if we start using the word “math” to identify a broader range of experiences than arithmetic, the shift should eventually occur and hopefully lead to happier math lives.

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