The Director of Innovative Teaching and Learning Director at my school sent me a Twitter message the other day about PBL and math. His honest question got me thinking and writing, and although I started using the response box on Twitter, my response got so long that I began a document so I would not lose it all if something happened. Much of my influence on the topic comes from Dan Meyer, though the #MTBoS and Twitter have certainly contributed far more to my understanding than I could ever imagine or cite here. (I simply cannot remember all of the places I have read on the topic).
His question from Twitter:
I’ve been wrestling with this question, but I don’t teach math so I can’t answer it. It’s easy for me to take anything I teach in video production and apply it to a real life task. I basically give the kids the role of video producer, cinematographer, or writer, and they’re off to the races. I teach them the skills necessary to create a product and we’re good. Is the same thing possible with math classes if students were given the role of engineer, builder, or analyst? PBL isn’t always popular in math classes b/c teachers say it just doesn’t fit, but it boggles my mind why this is the case. Aren’t there a host of professions that rely heavily on math? My kids aren’t Steven Spielberg, but they’re getting the simple version of what it’s like to be a real life director. Is there potential for math classes to offer the same thing? Thinking outside the box.
My Lengthy response that wouldn’t fit into the Twitter message box:
I completely see where you are coming from to give you context on what is going on in the math teaching world right now, here is how I see it from my research over the last couple of years. There is a big debate/push/investigation going on in math right now (see #MTBoS) about what it means to incorporate “real world” into the math classroom. On one hand, you have word problems that deal with real world situations and has Ss talk about and solve problems that have “real life data” in them. This type of situation is dependent on providing interesting/relevant information and has students working to solve a certain type of problem using “real world” situations. The problem is that it is often “fake world”, that is, stilted to interject a mathematical concept that either defies logical problem solving reasoning or forces mathematical reasoning in a way that does not scaffold student understanding. On another hand, you have a fostering of mathematical thinking that comes from the analysis of created situations to have students deduce mathematical thought processes gradually. It emphasizes a 3-Act structure to “hook” students and have them make conjectures about data, later evaluating their thinking which is followed by a shift toward an algorithm or specific mathematical approach. One blogger used the analogy “if ‘Algebra Skill X’ is the Aspirin, how do you create the headache?” – an emphasis on leveraging intellectual need to use math to solve a certain situation, artificially creating a need for math using relatively “real-world” contexts.
To execute these “real world problems” there is a need to have a certain amount of skills development in order to execute the mathematical procedures needed to solve the problem. Word problems are often a supplement to the actual goal of learning “skill X”, and are, therefore, usually an afterthought (it has traditionally been in my classroom). Teaching word problems is always tough because it is messy, so the simpler option is to teach the algorithm and be done with it. Once students can do the skill, we tend to stop teaching it because it seems to be a step backward to go from “being able to do 25 problems and get the right answer” to “not being able to solve a word problem”. There is a disconnect between the skill and the need to solve a “real world” problem.
A PBL approach ( like your producer example) would give students a problem/project to work on that could, in theory, use all of the skills that we have learned in a given unit. Certainly, there are many examples of professions that use math all day every day to do what they do, but these professionals have an entire bag of skills from which to draw, whereas we are building a repertoire of skills for students to draw from when they get there. It seems to be a cop out to even say this, because it sounds like I am saying that our approach negates all relevance of math in the real world up to the point of an engineering degree. To say that there is no place for PBL in the math classroom up to the point of designing a bridge is preposterous, but there are issues in thinking that PBL could replace what we currently do. There is such a vast number of skills that comprise “Algebra 1” that taking time to do a project to prove proficiency in a certain set of skills would take a very specific project and a lot of time that I don’t think would be feasible in many classrooms. Not impossible, but difficult.
My uneasiness as a math teacher is that if I give them a project to work on that COULD possibly use the skills that I am aiming at, the one project would not give them enough practice with that skill for me to feel comfortable with them being proficient to build upon it for later classes OR to allow them to pass a test like the ACT or SAT. At the very least, how do I determine if a project is worth taking 1-2 weeks when that time is lost to teaching other skills? There is uncertainty in taking that risk and I think many of us are nervous about leaving huge gaps in understanding for later teachers to fill in (often to the detriment of their content). Alas, the aims of the current math classroom are more of a list of skills to be mastered than a contribution to a larger knowledge base. Solving a “big” problem is not a priority, it is the ability to solve all of the little problems that prove proficiency in skills that is the priority. Fostering mathematical thinking along the way is my answer to the question of “where are we going to use this in the real world” because frankly, many of the skills that they learn are somewhat irrelevant, save for a select few people that end up taking a given class in college or end up teaching it to high school students. Our emphasis should be on making them think like a mathematician and seeing the interconnectedness of the skills as part of a larger body of work, not just doing a bunch of problems, but I have students to prepare for the next class and they have to have certain skills. To teach them perseverance, attention to detail and a proactive, positive approach to problem solving is my “real-world contribution”.
The short answer to your question is this: it is really messy to do PBL in the math classroom and the projects detract from the efficiency of learning skills, of which we have a lot in the math realm. Typically, the shortest route to the right answer is the best way because although we (should) value alternate solutions, chances are that we are not going to be inventing anything new in the math world, so the skills are taught as-is. The application of skills to situations is fine, but to take time to do a project and solve a problem inhibits the process of learning the concrete skills necessary to “be successful” in math, at this level or the next. PBL would/could be great, but the lack of practice to learn all of the algorithms necessary to pass many math classes and be successful in subsequent ones is problematic and is a huge damper on PBL in the math classroom. Not to diminish the aims/objectives of a science class- but doing a PBL to learn the Scientific Method is a huge plus, and one that lends itself to many of the aims of PBL (student voice/choice, group work, problem solving), however, the aim of learning the quadratic formula iis far less suited to PBL and the investigation needed to understand the depth of the material you would be studying would be a significant tangent for a skill that should take about 2 days to “completely learn”.
Much of this sounds jaded and negative, but simply put, the math classroom is in a state of flux, and I don’t think it is going toward PBL. My feeling is that much of the “rote practice” or “drill and kill” model is being supplemented with activities that are trying to teach the concept of “mathematical thinking”. This type of shift should create learners that think about situations mathematically and connect what they have learned with application to a “real life” situation, but I don’t know that math classes will go the way of Project Based Learning, if anything is a fit, I think it would be Problem Based Learning, but even this is a hard sell for many that value efficiency and memorizing processes to the model of deducing processes using patterns or discussion. While this fosters deeper learning, it is a process that requires patience (and time) which few teachers feel they have given the sheer breadth of material that they need to cover in a year.
Common Core is trying to help this problem by shrinking the number of skills that are learned, but until a student has gone through CC for many years, the fruits of this system will not be realized and it must survive the scrutiny of teachers that have students with vast knowledge gaps in the meantime. All the while, ACT and SAT still rely on a form of math test that requires the ability to see math as many skills that can be done as algorithms, and though they are trying to shift, they cannot go too far away from where math has come from and remains in colleges today – a set of problems that require certain processes to solve. The most efficient way to learn how to wire a light switch is to wire 15 of them at a workbench, not build a house, though the house is what we would desire as our end goal. While it would be fun for a kid to build a birdhouse to learn how to measure distances, a worksheet with 12 segments on it that they have to measure and write the length next to is just more efficient, and math teachers are all about efficiency.
If I assume for a second to try to create a math scenario like the producer scenario that you give would be to have a student be an Accountant. All the skills that go with being an accountant involve math, but we as an Educational community have a class for this – Accounting. You just have to be in college to take it, so you have to go through years of math classes to get there. With the goal of teaching Algebra 1, I suppose a good goal would be to be an Engineer – teach the skills of parabolas and lines with the goal of contributing to a bridge project that the class is making. Along the way, you may use the skills that you have learned, but not all of the skills necessary for building a bridge fall under Algebra 1, nor are all the skills of Algebra 1 going to fall under the umbrella of bridge building. This sentiment is true for video production as well (you will not use the Ken Burns effect in all movies), however, the design of the bridge matters little in light of the question “how do I use factoring quadratics to solve this?” because you just won’t, so we won’t put them in a situation to build a bridge that doesn’t use the skills necessary to master Algebra 1. The process of making a movie, however, could involve the Ken Burns effect, and having that tool in your bag could be useful later, options are good and they could contribute to a masterful project somewhere along the line, factoring, however, is really only useful for one thing – breaking down bigger problems that could show up later in your math career, making them easier to solve. Many of the skills of math are self-serving – they “help you solve problems later in other math classes” (a response that is classic when answering the question “when will I use this in ‘real life’?”). Truly the PBL for Algebra 1 is passing the AP Calculus exam – to conquer that, you have to be a master of all things Algebra. If you mastered all of the skills of building a house, the test would be building a house. High School math is a house of Calculus, and mastering all of the skills along the way is extremely self-serving in this way, the math is a means to another means in our short 4 years.
In closing (sorry this is so long), I love the Engineering class that we have for seniors. It allows them the opportunity to use math and science to build and do things, however, it would take a brilliant design for a course to teach all of Algebra 1 with projects. The practice and demonstration of skills is necessary for the math classroom, especially in foundational courses, so the structure of teach, practice, evaluate, repeat is too easy and too efficient to abandon at this point. Even if the design were there, it would be a hard to implement as it is more work with less tangible progress. The less formulaic teaching of mathematical thinking that can come through the math classroom is a more realistic goal at this point and it leaves less to chance in terms of what is taught, and I think it is the uncertainty of leaving the teaching of skills to chance that terrifies me (and other math teachers). I suppose this essay (which is what it has turned into) is a mix of my experience and my frustration with the situation. You hear people say all the time “I use math in my job all day”, but I just don’t know what that looks like for a class, but I am sure that there is no person whose job title is “expert Algebraist” (except if they are a teacher). There are, however, writers, Biologists, movie producers and historians. The skills of math seem somewhat separated from the application of math in the classroom for a variety of reasons, and I think it boils down to the uncertainty of a guarantee that the skills will be better taught using the project as opposed to a “traditional method”. The teaching of intangible skills like attention to detail, perseverance and communicating strategies can be effectively taught alongside of the skills and are not in any way a question mark for the math teacher. PBL could certainly have a place for certain sets of skills, especially in the lower levels of math (elementary), but high school math is so specified (self-serving) and broad (the number of skills), that it provides a unique challenge to find a balance of gimmicky and helpful when it comes to projects. It is a hard sell because it “doesn’t fit” but I think that is because math teachers like things neat and orderly (we have to, after all, it is math!) and PBL seems very open-ended for the specific skills that we are teaching given the sheer number of them. Again, it seems to be a cop-out, but if I had to guess, I think “doesn’t fit” is a kind way of saying “we don’t want to spend time on something that won’t as effectively accomplish our aims as what we are currently doing”. I guess that’s a little more than $0.02, but it is something that I have been wrestling with for a while now as well.