This is a talk I gave awhile ago looking at why students hate word problems, posing five ways to improve them, and introducing this thing called “three-act math.”

Hi Dan.
I haven’t watched the video, but this is an opportunity for me to put forward some radical ideas, so here goes:

We teach the math, then we wrap it up in more, or less sometimes, artificial scenarios called word problems. The students know that this is a disguise for some sums, and will do anything to dig out the numbers, guess a procedure and get “the answer”.

This is a cart before the horse situation. The sensible way would be first of all to look at situations familiar to the students, pose interesting, realistic problems, numerical or otherwise, and see what happens. Common sense will mostly get one to the end, and if not then a bit of “guidance”. Then an approach to the problems which looks for similarities, irrelevant details, and some prior knowledge will lead to a natural development of “theory”. This can be done at almost any level, from K to Bachelor degree, and is how a meaningful treatment of for example, abstract algebra, or vectors, or fractions can be created.
I do like the three act approach, but it would be good to do this from the start. Then there would not be any “word problems”, only problems.

I did watch the video, which reminded me that learning persistence can be a significant benefit of applied, 3-act problems. They do take longer than standard word problems, but students need to understand that many real problems do indeed require extended period of effort. Fortunately, good applied tasks (with an immediate and clear goal, student engagement in assembling information and tools, a satisfying and authentic conclusion and a challenging sequel) CAN keep students engaged long enough to reflect and learn.
As a physics teacher (a discipline that relies on word problems even more heavily than mathematics), it is clear to me that including just a few well-developed authentic tasks during the year can build credibility about the value of both math and science—and it can change the way students approach the “routine” classroom work. A common mistake physics teachers make (especially now that “engineering design” has been added as a new responsibility) is to save their big authentic tasks (toothpick bridges, egg drop, etc.) until the end as capstone projects. That is too late to reap the full benefits.

Hi, while watching this video a question is coming to me over and over: should we show these videos to our students BEFORE teaching them the necessary skills for solve them, like howardat58 says?
Last year I showed some three act videos to my students and they felt so bad not knowing even how to pose a question or an objective to solve related to them…..and when we finally got to pose some questions to solve they felt pretty bad not knowing even which information they had to know or not having enough math skills to solve them….Maybe did I show the videos too early? Or is this a “normal” reaction when we challenge students with “real” math?

For example, in the video of the basket ball throwing, how can they solve it if they have never heard about parabolic movement? if they don’t know anything about parabolas? When and how should we introduce these concepts to students, “before” or “after”? Thanks!!

To Maria Guerrero
But they do know about parabolas, they don’t necessarily know that the name is “parabola”. They have all thrown balls, or rocks(!). They can be prodded into seeing that since the path is a curve it is not going to have a y=mx+c equation. So what could it have? This is a chance to consider symmetry, and thus good reasons for placing any vertical axis through the apex, and there are other good reasons for placing the horizontal axis touching the apex. I’m sure that good reasons can be found for flipping the curve around the x axis, and we have a nice looking curve which gets bigger quicker than the x value, at least after a while. the question then is
If it isn’t y=mx, what could it be? What could we have which is a bit more complex than x itself? OK, at this point, when silence has lasted for a minute or two some suggestions are in order. “How about y=x^2 or y=x^3 or y=x^99. Go on, plot them.”.

@Maria, if a students wants to know whether the ball goes in, and has an intuition about its path, we’re in great shape. They want instruction. I try to use these tasks to motivate the need for explanation.

For that particular lesson, students need a tool like Geogebra for modeling the parabola over the basketballs. Without that tool, the lesson would be quite frustrating, I’m sure.

couldn’t disagree more.
text books serve a purpose. particularly that of honing one’s practice with a view to competency that serves larger questions of interest to the learner.
they ARE NOT meant to hook a student’s interest – that is incumbent upon the teacher and in fact upon the student’s own input.

my very brightest and most curious students thrive most most vividly when they have text questions with which to hone their understanding. in fact they beg me to give them more to practice – and with good reason. Those very staid questions serve an entirely different purpose – to support their practical and natural inclinations of curiosity with rigour and raw skill acquisition.

every resource has its place, and the professional teacher makes the most of every resource they have.

Textbooks should supplement learning, not function as the primary catalyst to spark inquiry and I have to say I feel sorry for the students whose teacher relies on the text to function as such.

Thanks for your answers! I do use Geogebra when I work with parabolic throwing (I don’t use Dan’s video, but I film my students using catapults they’ve built themselves), but I do all this work AFTER explaining how parabolic movement works, and how to draw a parabola in Geogebra. Should I be more daring and let them discover the concept by themselves? Do you achieve that? maybe your students are older than mines…. (15)

Another terrific talk. Yes word problems are artificial and contain many assumptions. Your method of eliciting everything – question, information req’d and technique req’d sets up a student to independently become maths problem solvers to any problem, not just school/exam based. It’s very empowering. School at the moment does the opposite and kids believe it’s about passing an exam. I had a new student yesterday who told me that she ‘didn’t want to learn anything new’. A sad indictment of current methods at school. Love your passion Dan.

I am NOT a math-teacher at any level, so I do not regularly follow your blog (though I am most keenly interested in enhancing the ‘effectiveness of math ‘learning+teaching’). I have developed a powerful ‘systems aid’ to problem solving and decision making, which I am promoting. This tool is called the ‘One Page Management System’ (OPMS). More information about the OPMS is available at the attachments to my post heading the thread “Democracy: how to achieve it?” – see http://mathforum.org/kb/thread.jspa?threadID=2419536.

I have seen a fair number of your blog-posts, kindly brought to attention to participants at the forum ‘math-teach @ drexel’ by Richard Strausz (who IS a math teacher and who is also a regular follower [and admirer] of your blog).

My comments on your blog: I think it is useful indeed – though it probably could be enhanced/improved quite significantly in many ways. This is based on my own judgement , as well as on the opinions of several of your followers, whose remarks and comments I have seen at your blog.

I do believe you might like to take note of and respond to the serial and continuing put-downs by one Robert Hansen at the above-noted Math Forum of what I believe you are trying to accomplish vis-a-vis the ‘learning and teaching of math’. Robert Hansen’s latest quite vicious remarks appear at a thread he has titled “Dy/Scam’s Latest” (dt. Oct 21, 2014 6:01 AM, http://mathforum.org/kb/message.jspa?messageID=9625018).

I do believe you should take note of his attacks on you and your blog. (If you’re willing to do a [tiny] bit of learning, alongside a fair bit of ‘unlearning’, the OPMS process would help you develop the most effective possible response to Robert Hansen’s rather vicious and damaging lies).

Mr. Meyer,
I must say that you have inspired many math teachers and I am one of them. I love the concept of 3-act math and only wish we had some more examples of you putting it to use in your classroom. I’ve tried it rather unsuccessfully in the past and am a bit stymied about why it hasn’t worked like I thought it would.
I would like to comment on the criticisms I read here and elsewhere about your approach. I think it is obvious that teaching is an art form where the medium (students) is inconsistently delivered to the artist (teacher). It makes no sense whatsoever to me why someone who teaches one type of student at one level would criticize the techniques used by another teacher who teaches a different type of student at another level. For example, in reaction to this blog post “mike” comments that his best and brightest students beg him for more practice. Perhaps that has more to do with the way he practices his art, but I suspect it has just as much to do with the medium he is dealing with.
As a teacher, I have never in my years of teaching, met a student who begged me for more word problems. If I had that problem, I certainly would fail to understand what you are getting at here. Since I don’t, I relate to you entirely.
Thanks for your contributions to the improvement of math education.

Movie’s stalling out on me… but I was just noticing earlier today that one of the strengths of our completely re-worked “Math Literacy” course is that we do, now, start with a “word problem” and then generate the math from the real situation, instead of teaching a math procedure and then artificially twisting some “real” situations. It works a lot better…

(and meant to say that I got the video going in the meantime — we don’t exactly manage “three acts” but do achieve many of the same ends. And one of the guys in the first pilot section of it did, honestly, really want more word problems. )

To be fair, not all my students beg for text book questions, but those seeking refinement certainly do. I spend weeks at a time not touching a text book in all my classes, including those who would shy at begging for more practice questions.

The more my lessons are not my lessons at all, but rather my students’ lessons, the better they are. I try to get out of the way of their learning as much as possible. They’re not the medium, they’re the artists, to paraphrase your analogy.

But speaking to my criticism of the other Dan’s commentary, early in the video he goes out of his way to say that he’s not picking and choosing text questions as Straw Man exemplars of poor practice and yet proceeds to do exactly that.
Has *any* teacher ever thought that blindly pulling out a random question from a text as a lesson plan was likely to result in a quality lesson? I think not. But they serve a definite purpose for skill refinement and the text questions Dan selects as particularly poor actually serve that purpose as well as any.

The 3-act lesson is a nice way of thinking about lessons, but not really new. What makes it effective is the passion, energy, and approach to the structure itself.

We have been trying to incorporate more of the 3 act problems with our students. What is nice is the incorporation of multimedia for students to see how math correlates to life. But the biggest struggle that we are still having with our students when they start to work out the problem is the UNDERSTANDING. We have students that struggle too often with reading and then have issues trying to decide what they are truly suppose to be answering. To help them out I go through how to dissect a word problem with them first, you can see my poster at http://www.teacherspayteachers.com/Product/How-to-dissect-a-math-word-problem-1565684.
What I really feel is the underlying issue for math word problems is one the application which the 3 act questions help solve but two do the students know what they are reading in order to solve.

Yesterday at a conference in Australia a speaker commented on learning being visual – this is a big part of why what you share works.
I explain making meaning as follows: you need both sides of the brain to make meaning.

Left side of the brain Right side of brain
If I say.. you see a picture of:
black cat a black cat
pink hippopotamus a pink hippopotamus (as
ridiculous as it seems!)
love ? (see how difficult this is as it
has many meanings to each
person let alone trying to get
two people to have the same
meaning!
I say the word splunge which I know would have multiple spellings but what sort of picture do you have? A person doing the splits with a lunge, a long piece of sponge…

I think this is a big part of why the visuals work so well – it allows us to create a shared meaning and then learn mathematics.

## 18 Comments

## howardat58

October 20, 2014 - 3:41 pm -Hi Dan.

I haven’t watched the video, but this is an opportunity for me to put forward some radical ideas, so here goes:

We teach the math, then we wrap it up in more, or less sometimes, artificial scenarios called word problems. The students know that this is a disguise for some sums, and will do anything to dig out the numbers, guess a procedure and get “the answer”.

This is a cart before the horse situation. The sensible way would be first of all to look at situations familiar to the students, pose interesting, realistic problems, numerical or otherwise, and see what happens. Common sense will mostly get one to the end, and if not then a bit of “guidance”. Then an approach to the problems which looks for similarities, irrelevant details, and some prior knowledge will lead to a natural development of “theory”. This can be done at almost any level, from K to Bachelor degree, and is how a meaningful treatment of for example, abstract algebra, or vectors, or fractions can be created.

I do like the three act approach, but it would be good to do this from the start. Then there would not be any “word problems”, only problems.

## Fred Thomas

October 21, 2014 - 5:06 am -I did watch the video, which reminded me that learning persistence can be a significant benefit of applied, 3-act problems. They do take longer than standard word problems, but students need to understand that many real problems do indeed require extended period of effort. Fortunately, good applied tasks (with an immediate and clear goal, student engagement in assembling information and tools, a satisfying and authentic conclusion and a challenging sequel) CAN keep students engaged long enough to reflect and learn.

As a physics teacher (a discipline that relies on word problems even more heavily than mathematics), it is clear to me that including just a few well-developed authentic tasks during the year can build credibility about the value of both math and science—and it can change the way students approach the “routine” classroom work. A common mistake physics teachers make (especially now that “engineering design” has been added as a new responsibility) is to save their big authentic tasks (toothpick bridges, egg drop, etc.) until the end as capstone projects. That is too late to reap the full benefits.

## Maria Guerrero

October 21, 2014 - 9:48 am -Hi, while watching this video a question is coming to me over and over: should we show these videos to our students BEFORE teaching them the necessary skills for solve them, like howardat58 says?

Last year I showed some three act videos to my students and they felt so bad not knowing even how to pose a question or an objective to solve related to them…..and when we finally got to pose some questions to solve they felt pretty bad not knowing even which information they had to know or not having enough math skills to solve them….Maybe did I show the videos too early? Or is this a “normal” reaction when we challenge students with “real” math?

For example, in the video of the basket ball throwing, how can they solve it if they have never heard about parabolic movement? if they don’t know anything about parabolas? When and how should we introduce these concepts to students, “before” or “after”? Thanks!!

## howardat58

October 21, 2014 - 10:58 am -To Maria Guerrero

But they do know about parabolas, they don’t necessarily know that the name is “parabola”. They have all thrown balls, or rocks(!). They can be prodded into seeing that since the path is a curve it is not going to have a y=mx+c equation. So what could it have? This is a chance to consider symmetry, and thus good reasons for placing any vertical axis through the apex, and there are other good reasons for placing the horizontal axis touching the apex. I’m sure that good reasons can be found for flipping the curve around the x axis, and we have a nice looking curve which gets bigger quicker than the x value, at least after a while. the question then is

If it isn’t y=mx, what could it be? What could we have which is a bit more complex than x itself? OK, at this point, when silence has lasted for a minute or two some suggestions are in order. “How about y=x^2 or y=x^3 or y=x^99. Go on, plot them.”.

## Dan Meyer

October 21, 2014 - 12:28 pm -@

Maria, if a students wants to know whether the ball goes in, and has an intuition about its path, we’re in great shape. They want instruction. I try to use these tasks to motivate the need for explanation.For that particular lesson, students need a tool like Geogebra for modeling the parabola over the basketballs. Without that tool, the lesson would be quite frustrating, I’m sure.

## mike

October 22, 2014 - 5:52 pm -couldn’t disagree more.

text books serve a purpose. particularly that of honing one’s practice with a view to competency that serves larger questions of interest to the learner.

they ARE NOT meant to hook a student’s interest – that is incumbent upon the teacher and in fact upon the student’s own input.

my very brightest and most curious students thrive most most vividly when they have text questions with which to hone their understanding. in fact they beg me to give them more to practice – and with good reason. Those very staid questions serve an entirely different purpose – to support their practical and natural inclinations of curiosity with rigour and raw skill acquisition.

every resource has its place, and the professional teacher makes the most of every resource they have.

## Dan Meyer

October 22, 2014 - 6:04 pm -mike:Says who?

## mike

October 23, 2014 - 2:43 am -Says me.

Textbooks should supplement learning, not function as the primary catalyst to spark inquiry and I have to say I feel sorry for the students whose teacher relies on the text to function as such.

Again, every resource has its place.

## Maria

October 23, 2014 - 7:19 am -@Dan and howardat58:

Thanks for your answers! I do use Geogebra when I work with parabolic throwing (I don’t use Dan’s video, but I film my students using catapults they’ve built themselves), but I do all this work AFTER explaining how parabolic movement works, and how to draw a parabola in Geogebra. Should I be more daring and let them discover the concept by themselves? Do you achieve that? maybe your students are older than mines…. (15)

If you get to do that I’ll try, for sure!

## Paul Carson

October 23, 2014 - 7:30 am -Another terrific talk. Yes word problems are artificial and contain many assumptions. Your method of eliciting everything – question, information req’d and technique req’d sets up a student to independently become maths problem solvers to any problem, not just school/exam based. It’s very empowering. School at the moment does the opposite and kids believe it’s about passing an exam. I had a new student yesterday who told me that she ‘didn’t want to learn anything new’. A sad indictment of current methods at school. Love your passion Dan.

## GS Chandy

October 23, 2014 - 11:50 pm -Mr Meyer:

I am NOT a math-teacher at any level, so I do not regularly follow your blog (though I am most keenly interested in enhancing the ‘effectiveness of math ‘learning+teaching’). I have developed a powerful ‘systems aid’ to problem solving and decision making, which I am promoting. This tool is called the ‘One Page Management System’ (OPMS). More information about the OPMS is available at the attachments to my post heading the thread “Democracy: how to achieve it?” – see http://mathforum.org/kb/thread.jspa?threadID=2419536.

I have seen a fair number of your blog-posts, kindly brought to attention to participants at the forum ‘math-teach @ drexel’ by Richard Strausz (who IS a math teacher and who is also a regular follower [and admirer] of your blog).

My comments on your blog: I think it is useful indeed – though it probably could be enhanced/improved quite significantly in many ways. This is based on my own judgement , as well as on the opinions of several of your followers, whose remarks and comments I have seen at your blog.

I do believe you might like to take note of and respond to the serial and continuing put-downs by one Robert Hansen at the above-noted Math Forum of what I believe you are trying to accomplish vis-a-vis the ‘learning and teaching of math’. Robert Hansen’s latest quite vicious remarks appear at a thread he has titled “Dy/Scam’s Latest” (dt. Oct 21, 2014 6:01 AM, http://mathforum.org/kb/message.jspa?messageID=9625018).

I do believe you should take note of his attacks on you and your blog. (If you’re willing to do a [tiny] bit of learning, alongside a fair bit of ‘unlearning’, the OPMS process would help you develop the most effective possible response to Robert Hansen’s rather vicious and damaging lies).

Best wishes

GS Chandy

## Dan Thomander

October 24, 2014 - 5:39 am -Mr. Meyer,

I must say that you have inspired many math teachers and I am one of them. I love the concept of 3-act math and only wish we had some more examples of you putting it to use in your classroom. I’ve tried it rather unsuccessfully in the past and am a bit stymied about why it hasn’t worked like I thought it would.

I would like to comment on the criticisms I read here and elsewhere about your approach. I think it is obvious that teaching is an art form where the medium (students) is inconsistently delivered to the artist (teacher). It makes no sense whatsoever to me why someone who teaches one type of student at one level would criticize the techniques used by another teacher who teaches a different type of student at another level. For example, in reaction to this blog post “mike” comments that his best and brightest students beg him for more practice. Perhaps that has more to do with the way he practices his art, but I suspect it has just as much to do with the medium he is dealing with.

As a teacher, I have never in my years of teaching, met a student who begged me for more word problems. If I had that problem, I certainly would fail to understand what you are getting at here. Since I don’t, I relate to you entirely.

Thanks for your contributions to the improvement of math education.

## Geonz

October 24, 2014 - 12:03 pm -Movie’s stalling out on me… but I was just noticing earlier today that one of the strengths of our completely re-worked “Math Literacy” course is that we do, now, start with a “word problem” and then generate the math from the real situation, instead of teaching a math procedure and then artificially twisting some “real” situations. It works a lot better…

## Geonz

October 24, 2014 - 12:05 pm -(and meant to say that I got the video going in the meantime — we don’t exactly manage “three acts” but do achieve many of the same ends. And one of the guys in the first pilot section of it did, honestly, really want more word problems. )

## mike

October 25, 2014 - 7:03 am -@Dan Thomander

To be fair, not all my students beg for text book questions, but those seeking refinement certainly do. I spend weeks at a time not touching a text book in all my classes, including those who would shy at begging for more practice questions.

The more my lessons are not my lessons at all, but rather my students’ lessons, the better they are. I try to get out of the way of their learning as much as possible. They’re not the medium, they’re the artists, to paraphrase your analogy.

But speaking to my criticism of the other Dan’s commentary, early in the video he goes out of his way to say that he’s not picking and choosing text questions as Straw Man exemplars of poor practice and yet proceeds to do exactly that.

Has *any* teacher ever thought that blindly pulling out a random question from a text as a lesson plan was likely to result in a quality lesson? I think not. But they serve a definite purpose for skill refinement and the text questions Dan selects as particularly poor actually serve that purpose as well as any.

The 3-act lesson is a nice way of thinking about lessons, but not really new. What makes it effective is the passion, energy, and approach to the structure itself.

## Van

November 18, 2014 - 8:13 am -We have been trying to incorporate more of the 3 act problems with our students. What is nice is the incorporation of multimedia for students to see how math correlates to life. But the biggest struggle that we are still having with our students when they start to work out the problem is the UNDERSTANDING. We have students that struggle too often with reading and then have issues trying to decide what they are truly suppose to be answering. To help them out I go through how to dissect a word problem with them first, you can see my poster at http://www.teacherspayteachers.com/Product/How-to-dissect-a-math-word-problem-1565684.

What I really feel is the underlying issue for math word problems is one the application which the 3 act questions help solve but two do the students know what they are reading in order to solve.

## Christine Lenghaus

December 5, 2014 - 2:42 pm -Yesterday at a conference in Australia a speaker commented on learning being visual – this is a big part of why what you share works.

I explain making meaning as follows: you need both sides of the brain to make meaning.

Left side of the brain Right side of brain

If I say.. you see a picture of:

black cat a black cat

pink hippopotamus a pink hippopotamus (as

ridiculous as it seems!)

love ? (see how difficult this is as it

has many meanings to each

person let alone trying to get

two people to have the same

meaning!

I say the word splunge which I know would have multiple spellings but what sort of picture do you have? A person doing the splits with a lunge, a long piece of sponge…

I think this is a big part of why the visuals work so well – it allows us to create a shared meaning and then learn mathematics.