Tag: curriculum

Mathematics Curriculum in the age of AI – What can humans do?

It’s 1976 and I’m in Mrs. Gerber’s second grade classroom. After learning our times tables by heart (I was so proud), I’m sitting at my four-table group and Mrs. Gerber comes over handing out something small and square with buttons that looks really cool. She introduces us to the four-function calculator and while lots of other students are seeing what they can write with numbers upside down on the small screen, I am plugging in the biggest numbers I can think of and seeing what the product would be. When will it be too big for the screen? What will it tell me? Is there a pattern in the numbers?

I was able to come up with some questions that Mrs. Gerber hadn’t even thought of, and it was all because of this new toy that I was able to play with in class. Times have definitely changed since then and the “toys” that students play with in class currently, were probably unimaginable in 1976. However, one thing still rings true: that playing with those toys changes everything.

As a math teacher since 1990, I have seen many phases of technological change in education. I remember learning how to use the TI-83 calculator for the first time and thinking of the wonderful ways that this would change the way I taught. So many questions existed about what was important and what was not to change or retain in the curriculum.

We are once again at a crossroads – one that math education has seen a number of times – where we as a group (or often individuals) decide the importance of specific topics that are “covered” in our curriculum. This autonomy given to math teachers often varies from district to district, even school to school – especially in the independent school world. The eternal question that math teachers do not have enough time to “cover” the contents of the textbook has now come to another moment – What can we leave out and still call ourselves math teachers? Still call our learning outcomes mathematics? The advent of AI should make us all think about what we as mathematicians and educators, think teaching math should be about. AI can serve as tutor, assessor, colleague, student peer and many other roles. What is it we’d want to get out of that “relationship” with the AI?

Personally, I have been contemplating this for a long time and I still come back to the curriculum. There’s so much at this point in education that is extremely intertwined and hard to untangle:

• Working with digital natives for whom technology may not be the best of devices to be interacting with

• The risk-taking and being wrong that students today are mostly lacking

• College admissions dependency on an age-old tradition of racing to Calculus

• The phenomenon of the ‘snowplow’ parent who seeks to remove all obstacles and suffering from a student’ pathway

All these issues affect the way we look at curriculum for mathematics education. Many say that too much access to technology undoes all the skills that may have laid the foundation of students’ mathematical understanding. Others say that students will use the technology as a crutch and not truly “learn” the skills they need or learn from their own mistakes. Teachers will cite the necessity for certain traditional skills based on students’ need to be able to take Calculus by grade 12. Parents will create the obligation of the school to allow students to use AI for homework since the expectations of the school or teacher are “too high.”

It is always important at these crossroads to consider the skills that students learn through the curriculum. What are our goals for teaching factoring? Finding horizontal asymptotes for rational functions? Simplifying complex rational expressions by hand? Memorizing SOHCAHTOA? Why do we teach these skills? If the purpose or answer is so that students can do math “in their head” or “on their own”, my response to this is that eventually this will no longer be required. What we’ve wanted as math teachers as far back as 1990 (and probably further) and what mathematics researchers like Carpenter, Fennema and Schoenfeld have been promoting is “teaching for understanding” or “making sense” of mathematics.

We as a community need to grapple with the question of what is most important to understand and how what we as humans are capable of will be more than what an AI can “understand” for us. Creating a holistic view of teaching mathematics means to dive into the new world of AI and look at the understanding and skills students see as “mathematics” while at the same time considering the new possibilities of what math can be. Instead of factoring, solving, using, “plugging in”, perhaps we change the curriculum so that inquiring, critiquing, deciding and evaluating are the ideas that come to mind when students think of math with AI? Could empathy, discussion, collaboration and abstraction be the main learning outcomes of mathematics education since so much can be done with AI now?

Contemplating these possible futures takes courage, knowledge and experience. I am sure that as a community, we will eventually get there. We will continue to have students that will want to write “hello” upside-down on their calculator (or other words!), but we will also have that students that push the AI to see what they are capable of. We can change what we teach in the mathematics classroom to encourage all students to experiment with curiosity and to help us shape the future.

Creating a Conspiracy in the PBL Classroom

Any Mad Men fans out there?  I just love some of the characters and the struggles they put themselves through.  In one episode from season 5, called “Signal 30,” Lane Pryce needs to take some clients out to dinner and Roger Sterling is giving him some advice on how to woo them to sign a contract with Sterling Cooper Draper Pryce.  Since he is not an account man, Lane is nervous about landing the account.

Roger:  And then it’s kind of like being on a date.

Lane:  Flattery, I suppose…

Roger:  Within reason, but I find it best to smile and sit there like you’ve got no place to go and just let ‘em talk.  Somewhere in the middle of the entrée, they’ll throw out something revealing and you want to wait ‘til dessert to pounce on it. You know, let him know you’ve got the same problem he has.  Whatever it is, and then you’re in a conspiracy – the basis of a quote “friendship”.  Then you whip out the form.

Lane:  What if I don’t have the same problem?

Roger:  It’ll probably be something like he drinks too much, he gambles…I once went on a five minute tear about how my mother loved my father more than me and I can assure you, that’s impossible.

Lane:  Very good then, and if for some reason he’s more reserved?

Roger:  You just reverse it – feed him your own personal morsel.

Lane:  Oh I see.

Roger: (getting up to leave) That’s it, get your answers, be nice to the waiter and don’t let him near the check.

My husband and I watched this episode about the same time I was having a great deal of resistance in my class to PBL.  I was talking to my husband about how to get students to buy into the notion of learning for the sake of learning where everywhere else in their lives what measures their learning are their grades.  Why would I expect anything different from them if this is the culture they were brought up in?  They depend on their grades to get them into a good college and if their grades are not up to a certain standard, they will not “measure up.”

I get this question all the time from other teachers – about  how to motivate students to find the love of learning and the interest in problems when they do not necessary know the solution methods to find them.  I usually tell them the same things – talking about the values of the class, grading class contribution with a viewable rubric,  grading their metacognitive journal writing, rewarding them with an interesting relationship with a great teacher…OK that might be pushing it.

However, this year is different.  I am having the hardest time trying to let them know what I want from them.  They do the homework, try their best, write down notes, but for some reason it feels different.  It’s almost as if there’s this wall between them and me and I don’t know how to get them to see my side.  I have had this problem with students in the past, but usually with a whole class.  Some of them blatantly are interrupting each other and others are obviously ignoring each other.

Then my husband says, “Maybe it’s like the conspiracy.”  I said, “What?” He said,” You know, what Roger was talking about on Mad Men.  Now, Roger Sterling is no saint (those of you who watch the show know this all too well) and I usually take what he says with a grain of salt.  I also would not ever consider taking advice from him, especially about teaching, but I allowed my husband to continue.  He said maybe what I had to do was build up the conspiracy that Roger was talking about.  I had a real problem with that because I am so committed to relational pedagogy that there’s no way I could lie to or mislead a student about their learning.  But that’s not what he really meant.

I suddenly realized that what had happened was I was teaching a curriculum that I didn’t even buy into.  I had just finished teaching them matrices and matrix operations with some problems that I had written, and it went very well.  However, in the end I did have to do Cramer’s Rule and determinants.  I tried motivating the problems about determinants with the area of a parallelogram, which kept them interested for a while, but in the end, with a 3×3 it was just here’s the way to do it.  I’m not sure that I could’ve expected them to have enough prior knowledge to derive the formula for finding a determinant of a 3×3.  As much as I tried to cover it up with problem-based learning, it was still a curriculum that is antiquated and not necessarily what I felt they should be doing and learning.  I couldn’t hide it any longer.

But we’re caught aren’t we?  Do we change the whole system – college prep curriculum, SAT required math, college expectations – and if so how do we do that? (see ahbel.com for a great article on this and a keynote address called Reflections on a 119 year old curricullum!)  Do we move beyond the required standardized testing material and allow our students to see mathematics the way we see it?  Yes, that’s the conspiracy – that’s what my husband was talking about.  When kids complain to me, I will “smile and sit there while they talk” knowing that I’m going to try to get something that we have in common.  “Do you hate solving a system of three equations with three unknowns with a determinant? Oh yeah, I did too in high school.  Wouldn’t it be great if we could do something else?  What else should we do?  Let me find some other problems that might be interesting.”  We have the same problem (literally and figuratively), now we’re on the same playing field having similar motivating factors.

And you know what?  I don’t think it would be the end of the world if they’re not revealing and you reversed it.  We are allowed to say to them that we don’t understand why we are still teaching this and these would be my reasons for taking it out of the curriculum – part of your own personal morsel.  It might actually bring you closer as a class and have you talking about how your hands are tied and we have to get through this “together.”

Yeah, there are little tricks that can be learned and carrots that can be used to get students to do what you want them to do, but in PBL, that’s not the point.  There is very little for them to mimic because it is based on their prior knowledge.  They are the ones who need to move the curriculum forward.  So in a nutshell,

  1. Take action – Get to problems in order for students to start feeling empowered and active in class.  Once they see that they are capable of a great deal on their own, it is amazing what they can accomplish.
  2. Create relationships – be sure that you are being reciprocal in your attempts at problems and valuing theirs.  The concept of Relational Trust and Authority are huge parts of a PBL pedagogy (Boaler, Bingham)
  3. But make sure that you are at least somewhat in control in the end because we are, at least for now, still responsible for making sure that some understanding of what we might consider unnecessary skills, for their next courses or future use.

As Roger said, “Get your answers, be nice to the waiter and don’t let ‘em near the check.”  Create that conspiracy.