“So I’m there on the beach with my friend Ben when we notice a taco cart up the road. Ben wants to walk straight over, but I’m thinking we walk a lot slower in the sand than we do on the street. So I say we walk straight to the street and then down the street to the cart. So we went our separate ways…” Thus begins the first Three-Act math task I ever experienced, courtesy of Dan Meyer.
I was introduced to the work of Dan Meyer in 2010, when I saw his fantastic TED Talk titled, Math Class Needs a Makeover. If you haven’t seen it, it is well-worth the 12 minute investment. In the talk, Dan identifies 5 harmful behaviors in his students that he sees often and that he wants to change. Even though he was a high school math teacher, every adult education math teacher will recognize these behaviors in many of their own math students.
- Lack of initiative
- Lack of perseverance
- Lack of retention
- Aversion to word problems
- Overeagerness for procedures and formulas
One of Dan’s points is that students are conditioned into these behaviors by teachers, textbooks and workbooks. His argument is that students expect simple problems with simple solutions. This “sit-com” mentality of math (simple problems with a resolution/solution in 22 minutes) works against our students because it develops an intolerance for uncertainty and encourages them to shut down as soon as they start to struggle and await further instructions.
To combat this conditioning, Dan sets the following goals for himself, and all of us who teach math:
- Use multimedia (this one poses logistical challenges for many of us in adult education, but it don’t let that diminish the power of the other goals)
- Encourage student intuition
- Ask the shortest question you can
- Let students build the problem
- Be less helpful
The resource I’m reviewing here is something Dan Meyer created to accomplish these five goals and reclaim students’ imagination while encouraging patient problem solving and mathematical reasoning.
The problems are great, almost all involving mathematical modeling. Once you click on a problem, you have all the videos, information and questions you need to get started.
The problems are called Three-Act Math Tasks because they are designed to be rolled out in three stages. I’m going to explain the way it works using a specific problem that I like, so back to the Taco Cart, where we last left Dan and Ben going their separate ways.
Act One: Act One has no numbers. It is about getting students interested, curious and tapping into their experience and intuition. Everyone is on a level playing field. During Act One, students discuss/argue/debate/process the situation. Act One is really nice because it doesn’t feel like a math problem – it is accessible to all and everyone feels entitled to weigh in.
In the case of the Taco Cart, the situation is two friends walking on a beach spot a taco cart. One thinks it’s faster to walk straight to the cart across the sand. The other thinks it would be better to walk to the boardwalk and then down to the taco cart. Students discuss: Who will reach the cart first? What do you think? What do you guess?
Act Two is where students get involved in the formulation of the problem. They decide what information is needed and what matters. Teachers have to be actively less helpful and keep students in control of the problem. Teachers ask students, “What information do you need?” Hold back before giving them all the information. Encourage them to explain their thinking and comment on each other’s thinking. Ask students to justify why they think they need a particular piece of information. Ask other students if they agree that piece of information is needed. After a good discussion, you can share the information which Dan supplies – one piece at a time. In the case of the Taco Cart, Dan gives us the distance that each friend walks and walking speed on the sand and walking speed on the street. The key is that students have asked for each of those pieces of information. If students only think they need the distances, don’t offer up the other piece of information. Let students build the problem. Once they have the information they think they need, let them work on it, share their solutions and discuss their approaches.
Act Three is the answer – usually in the form of a photo or a video (usually under a minute). Then you can have students talk about their initial guesses, their process, their results and the Act Three video.
Many of the Three-Act Tasks also offer “Sequels”, which are extension problems. For example – one sequel suggested after the Taco Cart problem is, “Where would the Taco Cart have to be so that both people reach it at the same time?”
For the sake of folks who have no access to technology, I wanted to start with a few recommended problems that are entirely re-creatable through old fashioned print outs and copies or by acting it out. The key is sticking to the three-acts and not giving out the materials all at once. Note: There are a lot more problems that might be re-created, but this is just to get you started.
Stacking Cups – All teachers need is about 10 styrofoam cups and a ruler for every 2-3 students in their class and they are all set to do this awesome activity, which is an exploration of rate of change and starting amount and is incredibly satisfying and fun.
Dueling Discounts – Act One: You have 2 coupons – $20 off and 20% off. Which coupon should you use? (This problem is great for getting students to analyze proportional relationships and use them in real-world mathematical problems)
Coin Carpet – Act One: If you were going to carpet a room with a single kind of coin, which would be the cheapest option? (This problem is great for students working on area and circumference of circles and proportional reasoning) The sequel is “If you could use any coin in the world, which would be the cheapest option?”
Sugar Packets – Act One: How many sugar packets do you think are in a 20-oz bottle of soda? This problem is re-creatable, but the video is sure to have a positive impact on student engagement – it is a PSA video from NYC.gov and features a man downing packets of sugar while sitting between two people drinking soda. The sequel is nice too – “What kind of food is the equivalent of 50 sugar packets?”
If you are able to show videos to your students, here are a few problems that are good places to get started with the three-act structure of drawing out student observations and intuitions and developing some real math content:
Super Stairs – Act One: Dan stretches at the bottom of a staircase. He runs up one stair and then comes back down. He runs up two stairs and then back down. He runs up three stairs and back down continuing this until he gets all the way to the top of the stairs and comes back down. How many steps will he run on the super stairs? How long will it take him to run all of them?
Lots of ways for students at different levels to work on this and lots of math content connections to be made – using organized charts, recognizing and using patterns, writing generalizations, creating a function, etc.
Coin Counting – A lot of coins are dumped in a bank change collector. What’s a guess for a total that is too big? What’s a guess you know is too small?
The Act Two of this one is modified a bit. We ask students to come up with all the possibilities after we release each piece of information. This is a great problem that has a low entry and a high ceiling. After students formulate the problem and are on their way to solving it, this problem allows for a great range of solution methods, from guess and check to making organized charts to setting up a system of equations.
Nana’s Paint Mix-up – Another good situation for proportional reasoning, this problem involves a series of texts between Dan and his Nana. She wants him to mix red and white paint to make a particular shade. She sends him the wrong combination, but it’s too late, he’s already mixed it. Is there a way to fix the color?
Three-Act Tasks have a powerful impact on adult education students. Even if we just use a few of these tasks over the course of a semester, it would be time very well spent. They can really go a long way in helping students develop curiosity, initiative and perseverance in problem-solving. They help students see how much rich mathematics they can engage in, even if they don’t know a specific formula. And they give students time to learn the mathematics deeply.
The Three-Act Task structure is something we can use to develop rich problem-posing and problem-solving experiences for our students.Dan Meyer’s created the structure, but other teachers are taking it and running with it. Andrew Stadel has developed some great ones – check out his Stacking Cups activity, which is a great follow-up activity to the Dan Meyer’s stacking cup activity mentioned above.
Finally, if you want more ideas for teaching with the Three-Act Tasks, this link will take you to a write-up from Dan Meyer’s blog.
Also, stay tuned for my CollectEdNY review of 101qs, a very easy to use site – also created Dan Meyer – where teachers can create (and share) their own three-act math tasks.
I’m in love with three-act math tasks and really interested to hear what three-acts teachers use with their students and how it goes. Please add a comment below and let me know.