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I have read and heard from some other instructors that they attempt to encourage students who find math hard by saying "math is hard for me too, in fact it's hard for everyone!" I have tried this a few times, but it always rings false to me, because it seems clear to me that math is not hard for me and never was, at least not in the same way that it's hard for the students in question. And it seems to me that the students must realize that too and conclude that I'm lying to them.

Is there a way that I can understand and explain "math is hard for everyone" in such a way that both the students and I can agree and believe that it applies to me as well as to them? Or, failing that, is there something better I can say?

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    $\begingroup$ Not an answer: 1. As a teacher you should be authentic, so yes, you shouldn't tell anything about you that is not true. 2. I'd compare the situation to athletes: Probably PE was never hard for people who are professional athletes now, but it is still hard to be very good at something. So, there is still a lot a professional athlete can tell about hard work to be good in PE. $\endgroup$ – Dirk Dec 6 '17 at 8:47
  • $\begingroup$ While I don't teach math, I always admit to the students when the subject is hard for me to understand (and it happens often enough). Then I present them the same information and sources I have on it and aks them to make their own conclusions. After some discussion, the topic usually becomes more clear to me and to the students as well. On the other hand, if I do understand something well, I never say that I don't, but still ask the students what they think and if they agree with my reasoning $\endgroup$ – Yuriy S Feb 6 '18 at 13:34
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I've long read - perhaps misread - the following comment from Einstein as lacking empathy:

enter image description here

(What use is: Oh? You have a headache? I can assure you, mine is worse!)

I feel as if sometimes teachers' phrasings around this shared difficulty in mathematics suffer similarly. The route that I prefer is an attempt to shift students' views about what "doing mathematics" consists of, and, in particular, emphasizing that doing mathematics is about being stuck.

This idea is well-illustrated in Ben Orlin's MathWithBadDrawings ("The State of Being Stuck") recollection of an interaction with Andrew Wiles. The first time that I, personally, recall thinking of mathematics in this way came from it being framed as such by Neil Grabois. Students sometimes harbor a conception of mathematicians as sitting down to work, proving their theorems for the day, and then moving on; whereas, the way that mathematics actually proceeds is somewhat different. Discussing mathematics as an area in which practitioners are, by the very nature of the discipline, often stuck can (I think) help students to reconsider what it means to find the subject "hard."

Moreover, I believe that the expectation of getting stuck, eventually unstuck, and then stuck again can reduce anxiety among students. After all, this is a natural part of the process of doing mathematics, and we can discuss - without needing to stake claims over whose "problems are greater" - the frustration (or other feelings) that comes with getting stuck, feeling as if we are making headway and realizing we are not, or even figuring something out only to move on and immediately get stuck again.

(To hear me literally voice this feeling on mathematics and being stuck, my school has a video up in which I speak about exactly this for a blitzed 10 seconds, between 2:18 and 2:28!)

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    $\begingroup$ I like this answer a lot, and I even sent Ben Orlin's "The State of Being Stuck" to all of my courses this semester as an announcement with my comments about how much it accurately describes my own experiences in doing math. I believe this kind of thinking is especially important in today's culture where it seems like students view themselves as either completely successful or a total failure, with no in-between state of "just not a complete success yet". Talking about being stuck as a natural part of the process should help them to avoid that false dichotomy. $\endgroup$ – Brendan W. Sullivan Dec 6 '17 at 17:35
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    $\begingroup$ @brendansullivan07 The italicized yet in your comment leads me to recall the wonderful, appropriate-for-all-ages video: The Power of Yet $\endgroup$ – Benjamin Dickman Dec 6 '17 at 22:40
  • $\begingroup$ @BenjaminDickman, I hoped maybe your "appropriate-for-all-ages video" suggestion would help my college students. But I'm thinking they might not appreciate Sesame Street. I googled 'pwer of yet' to see if anything else good came up. Nothing yet. $\endgroup$ – Sue VanHattum Dec 9 '17 at 15:11
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    $\begingroup$ @SueVanHattum Maybe start by asking if they are fans of the featured singer, Janelle Monáe (they may also recognize/know her from the movie Hidden Figures!). $\endgroup$ – Benjamin Dickman Dec 9 '17 at 15:28
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In the framework of tutoring at a community college (Remedial Math to Differential Equations) and teaching College Algebra and Physics at a Liberal Arts College. Most of my interactions are with people who think they can not do math or where not made to do math. My philosophy is that it is time spent working on problems, the use of that time that make the difference. I also distinguish between arithmetic and mathematics. Arithmetic you can buy a $15 calculator that can do for you, mathematics is the language and patterns we use arithmetic in.

I tell my students, math is hard.

I have a t-shirt that says "math is hard, so is life, get over it." I wear that occasionally in class.

I also tell the story of why I think math is "easier" for me. I had older aunts and uncles. My Uncle John taught me the times tables, but not memorizing he said if you had 3+3 that is adding 3 twice, this is the same as 2 times three. In kindergarten class we were counting and I recited this like a parrot. The teacher noticed and complimented me on my math. I liked the pat on the back, so every time math came up I tried hard. I put more time into it. And I put my time into when I was young. I exercised the muscle often, in my free time. In doing that the progression in mathematics was "easier" for me. But if we add up the time and effort spent studying and learning it wasn't "easier" I just put that time in at a different place in my life and liked it.

I explain to my students that we are learning algebra. This is a language. A language that takes the idea of counting and removes it from the object we were counting and looks at the pattern of quantity. When you learn a language you can think in a language. The goal of algebra is that you have the language of quantity as part of your skill set.

If you think about how you learn language, you spend the first 7 to 8 years of your life learning your native language in bits and pieces. Imitating the words, phrases and sentences around you. Learning how to attach them to objects, actions, things you want, things that don't exist yet. By the time you are in elementary school you begin to study the language itself and how you can use it to express more. This is hard, it takes lots of work and practice, but we do it early and it surrounds us, so we do it. With this skill we can now rap, sing, speak, debate, question, write poetry, write technical manuals, fiction, document life and so on.

Math is similar, but we already have language and we are adding to the tool set, so we borrow some of the knowledge from spoken language and start learning this new language. For some parts of it we are like an infant repeating sounds and trying to attach them to objects. This frustrates our older selves, that do not expect to be infant like in learning.

This is the hard part:

Mathematics talks about the pattern of quantity, but quantity does not directly exist. The number three is a noun, but I can not give you a three, I can only give you a three of something. This abstraction we have to form a place for in our thoughts.

This is the hard part.

The big secret is everything is hard at some point in learning it, it is new and different and there is a risk. Recognizing that, analyzing the risk, understanding our feelings and managing them is a skill that work for any part of life that is hard.

And with any skill practice, making mistakes, correcting mistakes and learning from the mistakes is part of making that particular skill less hard.

I have a friend that uses a video on juggling. Juggling is hard. But you can watch a video on juggling. Now you know how to juggle. You still need to practice the skill, make the mistakes and correct them. The practice for some will be easier than others, but this is about you not them. Practice.

Juggling 3 Balls

I also give the students these documents with the syllabus:

Math Myths

I talk honestly about the parts of mathematics I found easy. I talk about the parts I found hard, and how I got past the harder parts.

And sometimes I found things other people thought were really hard easy - Tensor Calc, why because I have developed more geometrical thinking by the time I took it and it made sense in the framework of what I was studying.

Compared to statistics, I had a philosophical bias against the idea of chance. To get past that I had to do the entry level problem more often, I had to develop a sense of random when I was studying it.

I think being human and explaining mathematics as a human invention we use to model and make predictions is a better reference point also then the philosophy that mathematics is the truth. By making it a human invention, looking at the rise from 1 a few, many, too much, to counting, to the creation of sets as human answers to the questions they had at the time a great framing tool in the discussion.

In the end I tell them it comes down to time spent doing mathematics, a willingness to make mistake and not expecting a perfect answer immediately and correcting mistakes when made. If this mindset is added to the habits, the hard part can be dealt with.

So what you want to do is look back at when you spent time learning it. Integrate that time, add it up, that is where it was hard for you. Hard is effort, you just did it at a different time and with a different attitude. The difficulty doesn't change, but the perception of that difficulty does.

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  • $\begingroup$ I don't agree that "spending time" is the same as "being hard". It seems to me that part of the reason you and I spent time on math and got better at it is that it wasn't as hard for us to begin with. (Of course there are other factors too.) $\endgroup$ – Mike Shulman Dec 5 '17 at 16:53
  • $\begingroup$ It was hard, it didn't seem hard. And that is where the word hard lets us down. There are people that love to landscape and garden. The work is hard, back breaking hard, rough on the body, physical, the energy consumption is high. But they enjoy the result and have found how to enjoy the work. The work is still hard. $\endgroup$ – Jeffery Thompson Dec 5 '17 at 18:27
  • $\begingroup$ I still disagree. It's not just about enjoyment or lack thereof; some things really do literally take less work for some people than for other people. $\endgroup$ – Mike Shulman Dec 5 '17 at 22:50
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    $\begingroup$ Yes but I think the effect is over estimated and the amount of time spent, just in contemplation or learned at different ages is greatly underestimated. $\endgroup$ – Jeffery Thompson Dec 5 '17 at 22:57
  • $\begingroup$ That may be true (although it would be all the better for evidence), but in this case I think it's the perception that matters rather than the reality, at least for the students' reaction. $\endgroup$ – Mike Shulman Dec 6 '17 at 17:14
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First, I'd want to figure out why math is so much easier for you than it is for them. Often it's simply background knowledge and work habits.

Second, you can model the type of struggle you expect from your students. What do you expect them to do when stuck? Do you actually make mistakes in front of them and then catch them with a convincing check? Do you ever solve a problem very inefficiently and then look back and ask "Hey, is there a better way we could have done this? How early on in the process could we possibly have known this? Or that?" Do you explain not only how you knew to do X do, but how you knew to do X at this exact point in time, and how you knew W, Y, and Z were not relevant?

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Two quick thoughts that I started writing as a comment, but maybe they're better as an answer.

First, I'd avoid saying anything to the effect that math is hard for you, as doing this will likely come across as contrived, and even if true in some other sense such as having difficulties with research, it's likely to sound hollow (like a multimillionaire complaining about tax shelter difficulties to someone homeless).

Second, if there's something you're really bad at (and I mean truly pathetic), then you might want to share those difficulties one-on-one (or in a very small group meeting). However, you'll want to avoid something that completely defeated you to the extent that there's no way to spin it as something you managed to persevere through. In my case, it was Foreign languages. I was extremely lucky to eventually land in (after 3 previous programs) a graduate program that required only one Foreign language and lucky again that I managed to (quite surprisingly, and not just to myself) pass the language exam (in French, the only language my high school offered and which in high school I barely scraped by in). I also had great difficulties as an undergraduate, when my problems caused two delays in graduation, the second being when I wound up transferring to a less selective university (which at that time had no university-wide Foreign language requirement) after several of my attempts at the (supposedly entrance level) language competency test were so far from acceptable and after a couple of failures of the first semester French class, that I didn't see any other alternative. (Those who know of my interest in math history, and wonder how I manage when I’m so bad with Foreign languages, may be interested in my posts in the March 2005 sci.math thread On the cardinal of the Cantor set.)

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  • $\begingroup$ Your "first" is exactly how I feel. Your "second" is a good idea; but why would you restrict it to one-on-one or small groups? $\endgroup$ – Mike Shulman Dec 5 '17 at 16:51
  • $\begingroup$ I have mentioned it in classes, but the danger there is that too much time could be spent on a tangent if you're not careful, and there's always going to be a few students who think you're wasting their time. But if done quickly as an aside, no more than 20 or 30 seconds, I guess it's not going to be a problem. I've certainly had a few teachers confess in class to being really, really bad at something, besides handwriting or drawing pictures (which I thought was so overdone that I tried my best to never say this in class, but instead let my lousy stick figures of people speak for themselves). $\endgroup$ – Dave L Renfro Dec 5 '17 at 17:13
  • $\begingroup$ I talk about being a slow learner with music, and how my love for music makes me want to keep practicing. $\endgroup$ – Sue VanHattum Dec 9 '17 at 15:17
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Do you think this is a good idea to share with students? That is, "I have read and heard from some other instructors" could be reason enough to consider an idea, but why do you want students to hear this message that "math is hard"?

If it's because you want to sympathize with them that learning math is work, then maybe talking about the merits of hard work in general would free you up from having to fib about your own math abilities. You could discuss how you have seen others (students, colleagues) find success learning hard topics in math. Have you seen students struggle and have an "a ha" moment? Maybe ask your current students if they've pushed through a hard topic and how they made it to the other side. Do they find that particular topic "hard" anymore? I suspect they'd say "well, it was until I realized (such and such)." The point is, an individual may not always find math difficult, though math may always remain hard for people on average.

I'll assume you don't want to tell them it's alright to just be bad at math, but I know many students already carry that message with them. For some students, if their parents were bad at math, and told them so, then they feel resigned to a similar fate. If you, the instructor, are telling them that math is "hard", the message is probably more like "math is hard for many people, which is why we have so many different math classes. These classes build on each other, so I routinely see students who come in with low skills and eventually reach the higher levels. It can be a very hard subject, and some students won't make it to the end. But I have been through it, and I want you to succeed", etc.

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