I highly recommend you read Anna Blinstein’s account of a math problem that went wrong in one class and right in another. The makeover she applied between classes is available to you no matter what classes or students you teach.
Blinstein notes that “the sheer wordiness and immediate jumping into very abstract ideas was a huge turn-off for many students.”
She describes the makeover as asking students “to try things, engage, take guesses, get a foot in the door, and progress towards increasing abstraction and formality at their own pace.”
Blinstein also notes that she “started with a story.” This is significant! Cognitive scientist Daniel Willingham describes “the privileged status of story.” Stories are often more interesting to people than expository texts and students often learn more from them.
It’s my birthday, but I’m really, really obsessed with all things square. My entire party has a square theme. Of course, I demand a square cake and that all pieces served to guests are perfect squares too.
Of course this isn’t real. No one, not even the spoiled princesses on My Super Sweet 16, has ever asked for such a party. But none of her students cares about that for the same reason that no one cares that the universe of Harry Potter isn’t real:
Blinstein’s students aren’t just reading a story. She’s made them a part of the story.
Crucial to Blinstein’s success here, in my view, is that she has deleted elements of the problem so that she could re-introduce them with her students’ participation. (Also that she has developed an enormous professional community online she could ask for help between classes.)
Her story deepens my conviction that the most productive and interesting problems aren’t assigned on paper, but co-developed by teachers and students in conversation with one another.
Love it! Amazing what a difference a story context makes. #lhcsd https://t.co/cysd5n3LS8— Teresa Lee (@Mrs_TeresaLee) April 10, 2017
Agreed, but it’s interesting to me how few “story problems” contain any of the elements of stories that people enjoy: heroism, conflicts, rising action, resolution, etc.
Kyle PearceApril 10, 2017 - 4:14 pm -
Michael SerraApril 10, 2017 - 4:23 pm -
There is nothing better than having 2-3 identical classes the same day to experiment, develop and refine your curriculum.
Stacie BenderApril 10, 2017 - 6:11 pm -
Your “privileged status of story” resonated with me. I had a similar experience with Geometry students last year. In my first class, they were not interested in the power of trig. Once I moved my real world problem – finding the height of my son’s kite – to the front of the slide deck, suddenly they had a problem they could chew on. Of course there were those who just said “why didn’t you just say ‘really high’?” Posing the “why torpedo mode” problem the next day brought out additional engagement.
Words to teach by.
Chester DrawsApril 10, 2017 - 9:47 pm -
Agreed, but it’s interesting to me how few “story problems” contain any of the elements of stories that people enjoy: heroism, conflicts, rising action, resolution, etc.
English teachers don’t usually complain that their stories contain no elements of algebra or statistics and seek to rectify that. Because, of course, it isn’t a problem.
If you tell a story in Maths that contains any amount of conflict, then your lesson will be about that conflict. Any Maths learnt will be basically zero — it’s hard enough teaching Maths to them when they are focused without them being deliberately sidetracked by extraneous stuff. (And any English lesson which wandered off into algebra would be equally hopeless too.)
No matter how engaging the square cake problem is, I just don’t believe it teaches anything much beyond how to solve the square cake problem. Occasionally that’s OK, because wandering off the reservation isn’t always a bad thing, but what I really want are ways to teach the actual syllabus in better ways. Activities that link, not dead ends, no matter how interesting.
Engagement is not a proxy for learning — if you engage students but don’t teach them anything, then you have failed. I can think of lots of ways to engage my students if I don’t try to teach them very much.
Dan MeyerApril 11, 2017 - 10:47 am -
Click the links, okay? The study participants were engaged and they learned.
Are there studies demonstrating that algebra is a privileged medium for general learning? I’m not aware of any. There are for stories. (Links are for clicking!)
Kyle PearceApril 11, 2017 - 10:54 am -
The word “engagement” is way overused in education and thus I think that leads folks like Chester to misunderstand what you are getting at. I think you have made a significant case over the ten or so years you’ve been blogging that the engagement you reference is that which sparks curiosity and creates a want (or need) for mathematics to address that question that is just bugging you like a pebble in your shoe.
Chester DrawsApril 11, 2017 - 9:37 pm -
My issue wasn’t that they weren’t learning. My issue was that they were learning something that leads nowhere.
Maths is a series of skill that build on each other. We teach factorising, which allows us to later teach solving quadratics. This is the reasons we have a curriculum, which we are obliged to follow.
Why can’t we teach an engaging lesson on factorising?
If I teach an engaging lesson on the Fibonacci numbers, then they have a whale of a time, learn about Fibonacci numbers and have not learnt anything that goes anywhere.
They have been engaged, they have learned, and I have failed in my job. Because my job isn’t to have them learning, it is have them learning the specified material, in the specified order.
Now many Maths teachers argue that we shouldn’t be go guided by what the curriculum says, and we should teach for learning’s own sake. But that is not what we are paid to do. Nor do I ever want to say to some kid that doesn’t pass a required exam that they shouldn’t complain that they can’t become the electrician/doctor/pilot they wanted to be because we put engaging learning above curriculum learning.
I’m sure lots of high school teachers will recognise the situation where kids turn up at your door on day one who love Maths — but can’t actually do any. They’ve had loads and loads of side-tracks, exploring shapes and patterns, collecting and plotting data, playing maths games. Activities like our square cakes. It leads to twelve year olds who think they are good at Maths but who can’t add two negative numbers — because drill is boring, and primary school teachers don’t do boring.
Tracy ZagerApril 13, 2017 - 7:21 pm -
“I’m sure lots of high school teachers will recognise the situation where kids turn up at your door on day one who love Maths – but can’t actually do any. They’ve had loads and loads of side-tracks, exploring shapes and patterns, collecting and plotting data, playing maths games.”
I have never heard this complaint from a teacher. Not once. Ever.
I’m eager for some evidence to support your version of reality. Have any?
Chester DrawsApril 14, 2017 - 4:35 pm -
@Dan: I can’t really have an engaging lesson on factorising, because I pretty much never teach a “lesson on factorising”. It is done mixed it with other things as we make our way through Algebra.
I rarely think much at the lesson level, because I want everything to cohere into related strands, and focusing on lessons devoted to one thing doesn’t do that. Things are taught spread across lessons, easing them in, going back constantly to revise, and sprinting ahead sometimes to where we will be going. I do have “activities”, but they are invariably very short — 15 minutes tops.
One key is there’s always something for them to be doing, and I keep mixing it up. I would argue that the key to useful engagement in Maths is negative — preventing confusion or boredom — rather than finding things that naturally motivate them. My aim is to have a class where kids enjoy themselves and enjoy learning, but enjoying an individual lesson, as such, isn’t part of that.
The bulk of the engagement, of course, comes from the joyful feeling of success when they “get it”. The less time spent getting to that point, the better — and drill gets them there quickest, so I’m happy to have large bursts of drill. (There’s a myth that kids don’t like drill, but there’s drill they don’t mind and there’s drill they hate.)
Hanging out in a High School staff room at the start of the year you might hear different comments compared to a blog like this.
Every year our new school entrants complete an AsTTle test before they start, which includes questions on they judge their ability and attitude to maths. Since they have no teacher that they feel have to be polite to, they can be honest. The correlation between their attitude to Maths and their skills is weak. Worse, their self-assessment of how good they are is also weak — many of them really don’t know that they haven’t been taught any real Maths yet.
JenniferApril 24, 2017 - 10:42 am -
“My issue wasn’t that they weren’t learning. My issue was that they were learning something that leads nowhere.”
Not sure that is possible. Would love to hear your example of when someone learned something that left them “stranded”. (Actually, I wouldn’t love it, that is sad to think about.) In my experience, all engaging learning experiences have increased my critical thinking and problem solving strategies. That skill set can be applied cross-curricular, in turn, opening more pathways than intended.
“But that is not what we are paid to do. Nor do I ever want to say to some kid that doesn’t pass a required exam that they shouldn’t complain that they can’t become the electrician/doctor/pilot they wanted to be because we put engaging learning above curriculum learning.”
Again, need another example. I have failed many exams (not proud of that) in my high school and college days. I could not tell you which exams or courses for that matter. Not because I didn’t learn anything, but because I didn’t/don’t care. No connections were made. My intellectual need was not met. Curriculum is not what we are paid to teach. We have two important parts to our job: Standards and Students. My one goal: Not to produce students who regurgitate. I want students who think critically and question all that I do and tell them. Curriculum will not do that for me. So maybe try something new Chester. Try something you initially don’t agree with. Give it time and watch what grows from it.
Dan MeyerApril 24, 2017 - 11:19 am -
Thanks, everybody! This thread is closed.
Marc PetrieApril 12, 2017 - 5:35 pm -
I like the concept. I actually have started using purely visual introductions, where I show images and have students come up with their own story. I reserve these problems for midway through a series of lessons. I like the ideas put forth by the concept of the 3-act lesson, where I remove vocabulary at the first stage. I also like John Hattie’s work on Visualization.
Johanna LangillApril 12, 2017 - 8:12 pm -
I notice the story is not particularly wordy, at least I it’s written form. I love much about CPM, but I’ve experienced lots of (written) problems where the details about a controversy between ideas felt like someone just slapped some names on it. I don’t know if that’s actually going to capture kids interest.
The craft of teaching.
I wonder how many and which kinds of students are interested in a wordy story problem that they have to read out themselves.
There’s a tension here between storytelling/development with the whole class, and the times when I want students to dig into a text with their team instead.
Joe SchwartzApril 13, 2017 - 4:41 am -
I love Anna’s make-over. Text-heavy word problems pose very high barriers to entry for many of my students. But it’s not just number of words, vocabulary, and the sometimes purposeful obfuscation designed to distract, confuse or trip up the problem-solver; the stories that provide the context for the problems are often poorly written, and make only cursory attempts to engage the reader. I tried to address this issue here:
We talk about pseudo-context, and how the tail wags the (bandana-wearing) dog.
Love this high bar for word problems.
“Those who don’t know how to communicate, or communicate poorly, in some code that only they, or a chosen few, can understand, are destined to unhappiness, and to spread unhappiness around them. If they communicate poorly by intention, they are wicked, or at the least rude, because they have forced upon their readers hard work, despair, or boredom.”
Despair and boredom are fixtures in many math classrooms. And hard work should come from engaging in mathematical tasks that satisfy an intellectual need, not from trying to decipher a poorly written, confusing word problem. I see the freeing of problems from their confinement in convoluted number stories as a big part of the 3-Act project, and of Peter Liljedahl’s work on how to present tasks and problems.
Lilian LeungApril 23, 2017 - 8:27 pm -
This is a very cool makeover; I think one thing about student hating mathematics is word problem; whenever people see word in the math problems, they will tend to just run away. How do you make sure students do not feel like doing a word problem from this makeover?
Dan MeyerApril 24, 2017 - 1:05 pm -
Interesting question, Lilian. I think this storytelling process has more to say about the floor of the task than the ceiling. if you’d like students to be able to persevere through word problems, it may be helpful to give them a way to access those problems like Anna describes here.
Matt JonesApril 25, 2017 - 3:10 pm -
I need some help! It is pretty tangential, but the thread between Chester et al. made me think this might get some good answers here.
I have been unsuccessfully teaching CPM to my community school students (mostly expelled kids) for 3/4ths of the year so far. My biggest hurdles would be getting my challenging population of students to participate in group work and very sporadic student attendance. I read this blog and have read/bought in to Jo Boaler’s books/philosophy. I have not had any training with CPM. Because I have not figured out how to make CPM work for me, my very supportive admin (I’m not being sarcastic, they really want to help) wants me to “I do, we do, you do” my kids and possibly ditch the curriculum. I am not sure what to tell them.
My best approach so far is to ask for some more patience with them and let me try some different strategies for implementing CPM. The analogy that is coming to mind is that its the advent of the car, and because I cant get the engine running right, they want me to ride my bike instead of try to fix the engine. Maybe that’s a crappy analogy, but its all I’ve got at the moment.
Any advice from anyone would be much appreciated.
Dan MeyerApril 26, 2017 - 6:29 am -
Hi Matt, thanks for the question. I suspect it will be difficult for any of us to weigh in on your context while standing so far outside of it. I wouldn’t make group work the hill that good pedagogy dies on, though. I don’t know if CPM’s group work protocols have been tested in situations like yours where the students have vastly different existing prior knowledge and where attendance may be too sporadic to establish the working relationships that group work demands.
Michael Paul GoldenbergApril 26, 2017 - 8:34 am -
I taught my first two years of high school math at a school for “alternative” (i.e., “at risk”) kids. Virtually all tested at a grade 4 level or less in math and literacy. Attendance was sporadic, behavior was a nightmare, getting most of them to do any work was a fool’s errand and I was that designated fool for math (no colleagues in the school, in other words). After one semester of flailing with no set curriculum, I tried CPMP, aka Core-Plus. The first semester was a disaster, though at least I had something to focus on. The second semester (fall of my second year) wasn’t much better. Then, I hit on the idea of skipping to the unit on discrete math, which deal with, among other things, Euler paths and circuits, critical path diagrams (pretty much of a waste) and graph coloring. A few students got the Euler path and circuit stuff reasonably well, but the winner was the graph coloring sub-unit. That got one girl to suddenly write a legitimate A exam after having failed test after test with less than minimal effort. I presented on this at an NCTM annual meeting in 2000 or 2001 in which I urged people interested in using integrated curricular materials with challenging (and challenged) students not to repeat my mistakes (which were legion).
In retrospect, I would be prepared to experiment and try to find a unit that you think would get students excited. The beauty of the graph coloring and other discrete math topics is that they don’t depend so heavily on student success in other topics. Counting is counting, and there is a lot of inherent play in things like graph-coloring problems. Art gallery problems can be a lot of fun and still lead to deep mathematical thinking and learning, done right. Joe Rosenstein at Rutgers has a self-published textbook I’m going to be reviewing soon that has all sorts of material that I believe students could succeed with even if they’ve not done well in math before.
Finally, group work takes enormous seed-planting, cultivation, nurturing, and patience to be successful, even with students who have decent mathematical backgrounds. You should likely not make it the mountain you and your students die on.
Gail CarmichaelApril 27, 2017 - 7:34 am -
Really neat to see this discussion of the use of story in teaching math.
I’m involved with an NSF Pathways grant that’s looking at the use of story in teaching computer science with games (we use games because of where we want to head in a bigger project): http://gailcarmichael.com/research/projects/storycsed
In our research, we felt it was important to distinguish between story context (what you talk about inmost of your post), and story (more what you are talking about in your closing remark, with elements like “heroism, conflicts, rising action, resolution, etc.”). We developed abstract versions of games without any story or context, with story context, and with stories.
We are still working on analyzing the data, but in my earliest pilot I found that the full story was actually a bit of a distraction for the students. The story context worked better. In the pilot, I used activities that were not games strictly speaking.
I hope to delve more deeply into storytelling and whether it actually improves learning outcomes (outside of being engaging). It seems easier to use storytelling effectively in less abstract subjects than math and computer science, but it would be great to figure out how.
Dan MeyerApril 27, 2017 - 10:15 am -
Super interesting, Gail. I think I need to know more about which elements of story are salient here —Â ie. which help with learning and which help with interest and curiosity, and which help with neither.
Gail CarmichaelApril 27, 2017 - 11:15 am -
We won’t know to a very granular level in our study, at least not beyond how we define story context vs. story (see below). I’m not sure anyone knows yet (though I admit I’m behind in the literature since moving to industry)…but I definitely agree it’s a really interesting and useful question.
Here are the definitions we used in our NSF proposal:
> For the purpose of this project, we adopt Bal’s (1997) definition of story as a particular presentation of “a series of logically and chronologically related events that are caused or experienced by actors” (p. 1). A fictional setting, or story context, is defined as the elements of a story world that provide a concrete scenario and vocabulary for the concepts to be learned and applied.
JasonApril 28, 2017 - 11:58 am -
re: your featured quote, I wonder if that’s the right takeaway.
Dan MeyerApril 28, 2017 - 4:49 pm -
Agreed. In my follow-up, I was trying to discern whether the story was just contextual drapery or a more substantial structure.
Mike RuhlNovember 26, 2017 - 12:58 am -
This is Mike Ruhl from the University of Illinois and I am currently a senior preparing to begin my Student Teaching semester. I immediately thought of a speaker at the Illinois Education Association State Conference I just recently attended when I seen this article. In this speakers classroom she would always create a real-world context for her students to solve certain problems and she even had her students create a miniature version of a bus that her students could climb into and “travel” across the country while discovering how mathematics is use when creating maps for certain areas. I truly believe that providing students with a story motivates them to learn more effectively in the classroom!