Here are a couple of terrifying tweets from my summer.
Twitterverse, I need your help! I am going to be teaching math next year for the first time. 7th grade! Iâ€™ve previously taught ELA & SS. Any teachers/gurus I should follow? Advice/tips for me? Resources you recommend? I know @Desmos is a good resource! #iteachmath #WEAreLakota
— Emily Ladrigan (@EmilyLadrigan) June 28, 2019
Iâ€™ll be teaching 7th grade math next year. Switching from Social Studies. Any tips or helpful websites?? #iteachmath
— Ms. Nesmith (@ms_nesmith) July 24, 2019
I saw those tweets and had to sit back and collect myself.
That’s because I know how well I’m served by my knowledge of mathematics, how that knowledge helps me find value in early student thinking, how that knowledge helps me connect and build on thoughts from different students that, without that knowledge, might seem totally unrelated.
This isn’t a critique of those two newly drafted math teachers at all. Most of my horror here results from the thought of being drafted to teach history after a career teaching math. So what can they do?
You’ll find lots of people in those threads recommending resources and curricula. But resources and curricula are only as good as the teacher using them. A developing teacher can make a good resource bad and an expert teacher can make a bad resource good. (This is why John Mason prefers to talk about “rich teaching” instead of “rich tasks.”)
So my own advice is for these teachers trained in the humanities to focus on their teaching, not the resources or curricula.
Specifically, I hope they’ll resist the idea that math should be taught any differently than the humanities. I hope they’ll resist the idea that only the humanities deal in subjectivity, argumentation, and personal interpretation, while math represents objective, inarguable, abstract truth.
Math is only objective, inarguable, and abstract for questions defined so narrowly they’re almost useless to students, teachers, and the world itself.
In social studies, an analogous question might ask students to recall the date of the Louisiana Purchase or the name of the king who signed the Magna Carta —Â questions that are so abstracted from their context, so narrowly defined, and so objective that they make no contribution to a student’s ability to think historically.
The National Council for the Social Studies describes what’s necessary for students of social studies:
Students learn to assess the merits of competing arguments, and make reasoned decisions that include consideration of the values within alternative policy recommendations. [..] Through discussions, debates, the use of authentic documents, simulations, research, and other occasions for critical thinking and decision making, students learn to apply value-based reasoning when addressing problems and issues.
All of which rhymes perfectly with recommendations from the National Council of Teachers of Mathematics:
Teaching mathematics with high expectations for all students in mathematical reasoning, sense making, and problem solving invites students to learn to identify assumptions, develop arguments, and make connections within mathematical topics and to other contexts and disciplines.
Teaching math like the humanities asks us to:
- Broaden the scope of the problems we assign. We can always narrow the scope in collaboration with students but the opposite isn’t true. Students don’t have the opportunity to “identify assumptions,” for example, if we pre-assume every detail in the problem.
- Focus on mathematical ideas that are big enough to be understood in different ways. Ask students to make claims that demand to be argued and interpreted rather than evaluated by an authority for correctness.
- Celebrate novel student contributions to mathematics. History is made every day and so is mathematics. If our students leave our classes this year without understanding that they have had made unique and original contributions to how humans think mathematically, we have defined “mathematics” too narrowly. (For example, someone just decided to call this shape a “golygon.” If that person has the right to notice and name things, then so do your students.)
Instead of the worksheet above, show your students this video of a pallet of bricks and then immediately hide it.
“Does anybody have a guess about how many bricks we saw up there?”
“Did anybody notice any features about the bricks that might help us figure out exactly how many bricks we saw there?”
“Let’s look at the video again. Okay, what’s the most efficient way you can think to figure out the number of bricks.”
“How were you thinking about the number of bricks you figured out? What assumptions did you make?”
“Someone else got a different answer from you. How do you think they thinking about the number of bricks?”
“Here’s the number of bricks. What’s another question we could ask now?”
These questions rhyme with the kinds of questions you’d hear in a productive, engaging humanities classroom, questions which are no less possible in mathematics!
Humanizing math class means teaching like the humanities. And if you’re joining us from the humanities, please be generous with your pedagogy. We need all of it.
BTW: This is my contribution to the Virtual Conference on Humanizing Mathematics, a fantastic learning opportunity hosted by Hema Khodai and Sam Shah through the month of August 2019.
Daniel PeterAugust 30, 2019 - 8:01 pm -
“The greatest men I have known in my life were Albert Einstein, Niels Bohr, Alfred North Whitehead. They were the freest and most childlike men I have every known. Almost like small puppies dashing into every corner wherever something excited their curiosity, wherever there was something of interest. There were no fences. They were the freest most childlike men I’ve ever known.”
George Wald — Nobel Laureate Biology
I agree we should follow those subjects who actually know what and how to teach like the Humanities. But we don’t have to look just to the humanities – we can look to the greats mathematicians/scientists. How they learned tells us how we need to teach.
Dan MeyerAugust 30, 2019 - 8:06 pm -
No disagreement here, but it’s interesting to me that even though their ideas were born of childlike curiosity, the canonization of those ideas (and indeed the idea of a canon more generally) inhibits so many other students from developing childlike curiosity of their own. Particularly students who don’t share their identity as white western males.
StanAugust 31, 2019 - 3:09 am -
Aren’t you just confusing the cognitive science of how people think about mathematics with mathematics here?
Not that is not an interesting and valuable subject.
But suggesting changing its name to mathematics takes quite some hubris given most mathematicians don’t spend their professional career working in cognitive science and most of the valued achievements in mathematics are notable for their lack of concern for how anyone thinks about them.
William CareyAugust 31, 2019 - 3:51 am -
The idea of restoring math to its place as the queen of the humanities is near and dear to my teaching heart, so I think your advice to new teachers is spot on.
I think, though, that math has even more to offer than you’re giving it credit for. Math deals meaningfully with both the subjective and the objective, the many and the one! For example, a question like “How many numbers are relatively prime to 36,499?” is subjective and argumentative in its exploration (there is an obvious but terrible way to approach this and the argument centers around whether there’s a better way), but objective, inarguable, and abstract in its answer.
But wait! That objective answer points at a profound and beautiful pattern that students apprehend subjectively by argument, and that then broadens out to a scope that includes how secure communication over the internet works.
What’s even more awesome is that we have a wonderful inheritance of what the NCSS calls “authentic documents” that model for students what it looks like to be a working mathematician. It’s easier to believe that you have the right to name things when you know that the *best* mathematicians made up new stuff all the time. In fact, the problem about 36,499 comes from a note by Euler on the totient function.
In the same way that history textbooks aimed at children are worse than history textbooks aimed at fellow historians, math textbooks aimed at children are worse than exposing children to math written for the ages. Nicomachus has more to say about evenness and oddness than any textbook conglomerate! So, just like the humanities rightly point students at authentic sources, so should we!
Dan MeyerSeptember 3, 2019 - 7:50 pm -
Yeah, this is closer to my point than “math is never objective” or that “math is only ever subjective.” The meaning of objective mathematical proofs and axioms is only ever apprehended subjectively by students. Whenever we make meaning together, the humanities offers us a set of tools. PS. Looking forward to re-reading your own treatise now that I’ve had my own go at the subject. Let me know if you decide to post it anywhere so I can link to it.
Jackie PalmquistAugust 31, 2019 - 4:07 am -
Mary LangmyerAugust 31, 2019 - 7:00 am -
Dan MeyerSeptember 3, 2019 - 7:51 pm -
Beautiful story, Jackie. I hope you’ve had a similar effect on your partner.
Joanne WardAugust 31, 2019 - 10:13 am -
Maybe you’ll be interested in this one !?
Humanizing the teaching of mathematics
cheesemonkeysfAugust 31, 2019 - 10:33 am -
As a language, literature, and logic nerd from the humanities side who crossed over into mathematics and math teaching, I appreciated this post. In fact, I’ve pretty much given up on a lot of the current “math ed” trends and gone back to what I **know** to be my best pedagogy that comes from my humanities background.
This year in my teaching, I am noticing a significant shift in my practice from teaching kids math to teaching kids thinking. I’m also more interested in finding ways to get my students to carry their mathematical thinking and reasoning out into the world and into their other classes. I overhear them using language from my class in the hallway or in their other classes. I’m hearing from my non-math colleagues how our shared students are using different and better quantitative or logical reasoning and language in other subjects.
I’ve also started hosting mini-seminars for teachers from other subjects on demystifying math/quantitative reasoning tools and strategies to help them feel more confident in bringing them into their classrooms. You’d be astonished how many science, social studies, and English teachers are still suffering from their own math trauma, which holds them back from integrating math into their teaching and courses. The other day I gave a micro-PD for AP Bio teachers on using Desmos to re-teach standard deviation in their curricula.
So when **I** saw those tweets from other humanities teachers who’d been drafted into teaching math, it didn’t scare me — it made my heart sing. People who love teaching can figure out 7th grade math topics but more importantly, they are open to finding more compelling ways to reach students than they themselves experienced.
Sorry for rambling, but I guess this is me saying I’m all for the humanities invasion of math teaching and learning. And I’m happy to help any other humanists entering the field.
– Elizabeth (@cheesemonkeysf)
AndrewAugust 31, 2019 - 12:05 pm -
I can’t agree that these problems are akin to memorizing historical facts.
In problem two, the area of the base is given instead of the dimensions themselves. For a student who had only memorized V =LxWxH, they would have to extend that knowledge and recognize that LxW is already given and they just multiply by the height.
Also, based on the way the figure is dimensioned, it would be quite easy to assume that B referred to one of the side lengths unless you recognized that the units were given in meters squared.
If the argument is that memorizing historical facts gives students no real experience thinking historically, I agree. But to say that those two problems offer no real experience thinking mathematically would be an overstatement. Even to say that they take the humanities out of mathematics is a stretch. There’s real beauty in how concrete objects can be formally modeled by abstract entities and I see no reason why a push toward the humanities should be in any way absent of that.
Dan MeyerSeptember 3, 2019 - 7:55 pm -
It might be helpful for us to think about the humanity of an educational experience as a continuous variable rather than binary. A binary variable invites us to find exceptions and try to push an experience from one category to the other. What I’m saying is that both the textbook activity on compound prisms and the recitation of facts and dates invites a much narrower range of expression from a student than the counterexamples I’m offering. My goal isn’t to justify the former experience as “not really all that bad,” but to ask “how can I broaden the expression as much as possible here?”
Donovan T BaardaAugust 31, 2019 - 7:21 pm -
As someone with mildly dyslexic and autistic tendencies who excelled at maths and failed at humanities, I always cringe when I see people advocating teaching maths like humanities. There has been a long push towards teaching and assessing maths more like humanities, and things like the physics exam from my days which was “write the number and units in the box” that I got 100% for is now “write a paragraph about this topic” that I would have failed.
I get that this push was an attempt to broaden the appeal and enrollment in these subjects, but it’s not really working, and its been at the price of alienating the students who would have been really good at them (and probably only them).
However, I do get your point that teachers shouldn’t think that maths teaching is about shoving boring facts and formulas down students throats. It’s more like teaching a new language, and with it a whole new way of thinking. Geometry is a visual language in multiple dimensions that can say things in a few diagrams and formulas that you would struggle to say in a million words of english.
The best maths teacher I ever had never showed me how to do anything. She always asked us to figure out how to do things, giving only gentle hints and encouragement.
Anyone who is about to start teaching maths IMHO should do Khan Accademy online, starting right at the beginning with single digit counting and addition, and push through the excerises until you hit your limits. You will probably find something simple you missed in your own education, and maybe even start to love maths… Loving it even a little will make you a much better teacher.
Daniel PeterAugust 31, 2019 - 8:08 pm -
I think your issue is with simplistic and incompetent assessment methods like tests rather than authentic methods. A competent educational system would have assessed you by allowing you to explain your thinking orally – either by listening or allowing you the time to express your ideas orally and have the computer translate to writing. These are called accommodations but really they are how you get your PhD by orally defending your thesis.
SamOctober 27, 2019 - 3:39 pm -
Why do you think teachers that never show you how to do anything and give you gentle hints is so effective? A past teacher I had in college followed that method and all it did was waste time and frustrate everyone especially when she moved on when some people still didn’t understand what was going on.
GarthSeptember 2, 2019 - 12:30 pm -
Another approach to this discussion is teacher certification requirements. Montana is a K-8 non-subject specific certification for elementary and 7-12 for subject specific secondary. A new K-8 teacher will have had one college math course and a math methods course. I remember when my wife went from 5 years as a 3/4 grade teacher to a 7th grade science/math. That math methods course was 6 or 7 years ago. I tutored her for that first year. (I am a high school math teacher.) On the other side as a high school math teacher I would feel totally unqualified to teach 7th grade math. 7th graders think differently than high school kids. I feel for those two teachers in that they are being thrown it the most difficult years for math and for student maturity.
samOctober 27, 2019 - 3:43 pm -
I agree with you so much. Another aspect is to think about those students during their first years. I don’t know about you but I would be an absolute mess. Those students would be their guinea pigs for the year.
Brian RockSeptember 9, 2019 - 4:08 am -
As a veteran social studies teacher, I definitely feel their dread. Even though I love math and I’ve helped students out with their work from time to time, I would feel totally inadequate in terms of creating and sequencing activities to help students learn.
But to your point that math should be taught like the humanities, I find that interesting. We struggle with the same debate in social studies to some extent, and there are plenty of classroom teachers who think that history and historical facts have some kind of intrinsic value. But whether it’s math or social studies, we ought to be teaching those discrete, objective things so that they can use them to think about the world in new ways. This reminds me of one of my favorite ideas from Paulo Freire – you learn to read the word in order to read the world.
Daniel PeterOctober 18, 2019 - 2:18 pm -
Talking about math and humanities. Thought you might find this article of interest: https://www.sciencedirect.com/science/article/pii/S0010027719300927
Philip BrownOctober 25, 2019 - 6:31 am -
This is a compelling article, one that hits close to home. I have a BS in math and have taught for the past 14 years. That degree and knowledge has, as you said Dan, served me well, in developing into an quality teacher. However, once I started to “teach students,” I went from a good teacher to one that really had different outcomes for students. The following isn’t a boast, but a goal. I’ve had 14 of the 23 A-stars awarded for Cambridge IGCSE in the nation over the past 7 years, and only have had 3 students not pass. The average passing rate in the nation is only around 18% of students. I teach in a rural, title 1 school, where most families do not have a single member that attended college. There is absolutely no reason that other teachers couldn’t have results similar to mine.
Over the past few years I have tried to help other teachers, similar to those mentioned in this article, develop into better math teachers. I’ve tried to support their mathematical understanding, and also help them to develop better questioning techniques. I’ve become a Cambridge Math Teacher Trainer, been department head, held tutoring sessions for teachers, and run workshops at MEAD (where you’ll be the keynote speaker this year!).
One of the recurring hurdles these teachers face is confidence. They don’t really understand the math and they often lack the confidence to execute things like you mentioned with the pile of bricks. That lack of confidence isn’t limited to just teachers that don’t know the math well. New teachers have a terribly difficult time because it is often against what is seen as best practice.
The support and focus from administration and school districts continues to be on resources and curriculum, not instructional techniques. So unless these teachers are mighty resources, and perhaps also willing to fight the man (stand up to administrative pressure), it is unlikely their students will realize successful outcomes.
This has perhaps been a tangential rant, I had a lot to get off of my chest, but here’s my conclusion. I think to move forward in the direction you are promoting (and I’m 100% on board), we need to educate the gate-keepers of education about what good math instruction, especially grades 6 – 12, is and what it is not.
Samantha MandzakOctober 27, 2019 - 3:25 pm -
Could you maybe provide more examples. I really loved the brick idea! In class right now we are going over a lesson that I could use in my classroom. We are working on collecting data and finding the best fit line using rubber bands and bungee jumping a barbie from an elevated surface. Would this be an example? Thank you
Dan MeyerOctober 28, 2019 - 3:54 pm -
Great question, Samantha.
It’s interesting to me that the same task can be humanizing or dehumanizing depending on the teacher’s pedagogy.
The Barbie Bungee task can ask students to do routine cognitive work, making measurements, filling out boxes, performing operations, etc.
Does your implemenation of the task instead invite students to make novel contributions to the class’s understanding of mathematics or the context? Does it make room for intuition and early ideas about math? Does it give students a chance to think about how they’d make a confident prediction about the number of bands or does it prescribe a method which students will follow step-by-step? Different teachers of the barbie bungee task will answer this question in very different ways!
Here is a resource I made that might support a humanizing introduction of the problem. I hope you’ll let us know how the task goes for you.