I’m very excited to have discovered your blog.

I have recognized three (3) distinct educational methods:
1) Presentation: A single leader lecturing and demonstrating a topic,
2) Master/Apprentice: a one on one hands-on internship with feedback,
3) Performance Interaction: many people seemingly doing their own thing until one of them does something someone else finds worth noting, which often results briefly in #1 or #2, and then a reversion to individual action.

#3 is the type of spontaneous educational experience that happens when someone has a new phone with interesting features, or when basketballers, skateboarders, or jugglers gather and practice in a single location.

The problem with #1 especially, and #2 somewhat is that often the students do not believe that there is ‘real-world’ application for the received information. That is, they have a prejudice about the utility based on how information came to them.

I once tended bar and my barback (assistant who brings ice, washes glasses, etc) could never remember the correct order of the three sinks. He was a chemistry major at University and so I reminded him that the cleanser was acid and the sanitizer base and that if you didn’t rinse the glass between the other two sinks, you would end up with white condensate on the glasses. When I pointed that out, he was amazed. He had never tried to apply what he learned in the classroom to events outside a laboratory.

One great thing about the water-tank video and discussion, is that it helps to bridge that gap.

Currently I’m a student (4th career, I’m almost 46 years old) with the intention of teaching math grades 6-12 by fall of 2011. Having worked as an engineer many years ago, having an aptitude for math, loving education, and feeling a deep passion about students and how public schools operate, I decided to go for this career. As much as I can engross myself in a math problem, I truly struggled with how am I going to answer that infamous question, I’ve heard so many students ask, “Why do we need to know this?” Numerous responses come from the teachers I’ve seen and not a single one of them has managed to satisfy your average student.

To say math helps you learn how to think analytically or will help in a future job (which in some cases is simply not true!) gets eye rolls and even sleeping students–so I’ve seen during my field work.

For the past year +, I’ve been searching for applications that students would really use now or in their future. The WHY has been the driving force–as the online student in this post seemed to be wondering. In some cases the ‘why learn someting’ may only apply to certain careers or just for the sake of education. Not very motivating to students.

Along comes Dan Meyer–who I learned about not even a full week ago on TED– and offers a new focus: ENGAGEMENT.

Such a simple change in focus and yet so powerful! Even if a student buys the WHY, if they’re not ENGAGED it doesn’t really matter. But when we’re truly ENGAGED, we don’t even think to ask WHY.

That’s what’s so great about this water tank; if presented in an ENGAGING fashion, no one cares WHY. Learning for the sake of learning. How cool is that?

Why would anyone even think of having the kids watch the whole video? I cannot watch the whole video at once. I have to take a break for a trip to the bathroom.

I was thinking that with the recent events in the Gulf of Mexico this activity is more relevant. I wonder how students’ estimations compare to those of the oil company and experts with respect to range. Seems like you could run a lot of different ways in trying to determine how to make better estimates with less information. Interesting to student? I don’t know.

I am wrapping up my second year as a Bio teacher and I have been trying to figure out how to do just this kind of teaching in Bio. As it turns out, it will be a moot point because I have been informed that I have been assigned to teach Integrated Physics and Chemistry next year.

I was, as a student, a “why” kid. I wanted to know why I cared. I don’t know that this would give me that answer but it would certainly have helped. I was one of those kids that struggled (to be polite) with algebra and trig and excelled at geometry. I liked your explanation in a different post about why that might have been the case.

When it came time for me to do a masters degree I took a few courses to fill in my weak (took all the field ecology courses I could and avoided the math based science like the plague) science back ground. One of these courses was calculus. I started the calculus course and my immediate reaction was “so that’s</i< the point of algebra, why didn't they just say so?"

Any how, I hope, if I am teaching IPC next year, to use some of these techniques.

I’d like to see some scores comparison on math tests, your kids versus theirs, to see if your methods are resulting in anything positive other than attitudes and classroom involvement. Sure, classroom involvement is fun but does it actually result in these kids being able to solve problems the other kids can? Can they bring their skill levels up to pass state tests, factor, do trig, etc?

I like your enthusiasm and I WANT TO BELIEVE, MAN…but I gotta see the numbers, not just the rhetoric. Maybe you already posted it on your blog and I just haven’t looked for/seen it.

Hmm, I like the sound of the first point. Helping kids score well on tests is always rewarding and ensures they understand the material on some level.

I was hoping for some more data with state tests, though, or PSATs or what have you. So I think tests administered by a third party might be convincing?

I also do agree with you that you aren’t using radical methods, and they certainly do appear engaging.

It just seems that the claims and implications you make about curriculum and teaching methods are very strong and that they should be implemented across the board. To me, that treads into the area of standards, not individual teaching styles. It would radically alter how questions in text books are written! Instead of having all the data thrust into a complex paragraph in front of them and parsing it, they’d be forced to extrapolate, guess, and figure most of it. I mean, I think many teachers like the sound of that, but does it actually WORK?

And it does seem like it would take a lot more time; it’s almost like a science or a physics class instead of a math class. Most traditional math classes aren’t taught in a hands-on way. (Maybe that’s part of the problem!) However, I’d need proof that hands-on math gives better results than hands-off, traditional, abstract ones. Also, you teach remedial students, so there’s no guarantee the methods you use would result in higher learning curves for advanced students. Are you advocating that hands-on, be-less-helpful math is the way you should teach pre-calc, advanced algebra, factoring, quadratic solutions, derivatives?

This actually sounds like it boils down to the experiential/experimental/discovery learning versus…the contrasting philosophy (I haven’t gotten my teaching degree yet :) ), and so I don’t know if there is definitive proof for either side.

Therefore, I’d be happy to view state-administered testing data when considering claims about changing curriculum, text books and question styles.

Off: an awesome WCYDWT just fell into my lap! I included a thermometer picture in a tutorial without looking too closely. The editor caught it, luckily. Check it out:

http://www.shutterstock.com/pic-1334128/stock-vector-thermometer.html

I think he missed the point. That video isn’t supposed to be compelling or engaging on its own. Heck, I haven’t even watched the whole thing (although I did go a little farther to hear the conversation with your neighbor when another poster mentioned it). I wouldn’t expect middle school kids to sit through the whole thing without complaining. That’s the point. They grumble, and as soon as one of them mentions anything involving time, you stop it and then invest them in it. After they’ve guessed, suddenly they’re more interested in seeing how it all comes out.

I can understand the idea that fun discovery learning is very, very important, especially at the early stages of getting them hooked on math.

However, I wonder if it will slow down their progress on the amount of subjects they learn, if they spend more time solving problems.

I guess that boils down to ANOTHER endless education battle, breadth versus depth…

I mean, if we really want those kids to learn those subjects intimately, there’s a chance we will a) alienate those kids who learn at different rates and grasp the subjects before others and b) fall behind on covering what else they need to know.

(Also, I’d be irritated if I were a student and were forced to sit through exploratory learning every single time I came to class. If someone’s already figured out the rules to the game, I want to know about them! I don’t want to waste time inventing a chainsaw every time I want to cut down a tree…)

Hmm, maybe I’m seeing a pattern?

Those who usually score low on traditional tests work better with hands-on practice, that gradually lead up to abstraction. “Just let me do it”, etc.

Those who usually score high on traditional tetst can handle the abstraction sooner, and actually would prefer it to initial hands-on practice. “Just tell me the rules”, etc.

(And of course, the third group, the uninvolved students. “Just let me go to sleep”. ;) )

So, to sum (thanks for the comment, thescamdog), I think discovery learning is fairly important but time-consuming (yeah, I know…) and can serve to alienate those who are ready for abstraction and instruction. (Different approaches for different students, I guess…) I also think it slows down the pace of covering new subjects. However, I think it does offer a key piece of learning something new AND retaining it; I would just prefer more abstraction and instruction over application, but that’s why I’m a math and not a physics major. :)

tl;dr is at the bottom.

Dan: We’re talking about splitting every problem into four or more layers and constructing them separately, in dialogue.

And that sounds like it’s almost focused more on how to solve problems than the actual material. A lot of time is spent breaking down the problem into graspable bits because we want to show them how to abstract the important bits and solve problems. This is actually a separate skill than the material being taught.

Perhaps we could abstract the actual problem solving itself into a class, so they have the instruction on solving problems and then can focus more on digesting word problems. There’s a textbook that focuses on just problem solving tools, “Crossing the River with Dogs”. Maybe that could help them. I know I want to actually create physical representations of tools with words written on them; a paper cut-out of a wrench with the words “What do you HAVE (facts, equations, definitions) and what do you WANT?” written on it, a paper cut-out of a screwdriver with the words “Make a list of results”…something like that to crystallize the fact that these are TOOLS. Then put those tools on the board in a collage format or something. Heavy-handed, but hopefully it gets the message across. Or maybe I’ll have exploratory learning experiences for their daily warm-ups, the end goal being they have discovered and learned how to use a new “tool”…

Anyways, tangent…

tl;dr Problem solving and abstraction skills should be taught in a separate class, and the math word problems left as they are. Maybe.

In addition to “Crossing the River…” I’d recommend this reading on mathematical habits of mind.

Regarding problem solving and abstraction skills, I believe (and this is just my humble opinion) that if you remove problem solving and abstraction skills all you’re left with is the equivalent of getting students to memorize hundreds of digits of pi. Sure, it’s sort of applicable to life and might be fun for some but it’s also pointless. Students who end up liking math like it only because they are good at it. And even those kids won’t retain the information we are teaching them (how many college educated adults do you think remember or can derive the half angle formulas). Instead, the subject is just used as a gatekeeper, allowing “good” students to go on to become STEM majors and future teachers who down the road find themselves in front of their own students wondering why their students aren’t as excited as they were. Even conceptual understanding is somewhat vacuous if you aren’t interested in exploring unfamiliar problems. I see problem solving and abstraction skills as the whole point. What you are calling content is just the medium (unfortunately, it’s a lot easier to assess procedural skills). Our hK-12 math curriculum could just as easily be teaching kids alternate base systems, truth tables, and circuitry logic which some might argue would be more applicable to the “real world”. We haven’t devolved into anarchy because this isn’t being taught in K-12 and we won’t devolve into anarchy if we eliminated or changed most of what is currently taught.

Ok. Now for an argument that is less value based and opinionated. When students are just learning the steps necessary to solve exercises, even the kids who are successful with this have difficulty extending these ideas to new realms and problems involving higher critical thinking (this is backed by research such as Jones & Tarr, 2007). Even if the teacher focuses on why things work, they are still reinforcing the idea that math is a subject where you should know how to solve the problem at the outset (leading to disturbing results such as the belief that someone “smart” should be able to solve any math problem in less than 10 minutes). This leads to a subject that is fragmented, irrelevant, and frankly disliked by too many people.

Exploration takes time…lots of it. It’s also really hard to assess improvement/development in exploration (ie mathematical habits of mind/problem solving skills/heuristics, etc).

“If someone’s already figured out the rules to the game, I want to know about them! I don’t want to waste time inventing a chainsaw every time I want to cut down a tree…” This is back to just my opinion, but I don’t think this is ALWAYS true. People enjoy doing puzzles even though they’ve already been solved by someone. People still watch movies even though others have seen the ending. It’s about my own personal journey. We just need to find a balance so that by the time you’ve invented the chainsaw you still have some interest in cutting down the tree (and make sure you had any interest to begin with).

@Bill: I couldn’t agree more. You are asking students to collect data and explore problems that might be messy.

Reminds me of a GB Shaw quote:

“You see things; and you ask ‘Why?’
But I dream things that never were; and I say, ‘Why not?’”

It’s an oversimplification, but I think of science as the subject of the first line (formulating laws to explain/predict phenomena in the world) and math as the subject of the second (creating a system of rules and exploring the implications).

I think maybe Chris is missing the point a little, kind of going sideways. Maybe?

This is my take on it, about the purpose of math being to discover our environment. (Its original purpose)

It reminds me of a calculus problem, how much work it takes to fill an ice cream cone-shaped tower with water. The curve and maybe the 3 factors make this calculus. To extrapolate downwards, how big must a building be to hold 500,000 shoeboxes? With 18 inches left to the ceiling for the fire marshal? Or even, the amount of shoeboxes Nike makes in one year? Go on down to the elementary, primer level – ‘We have an ice cream tray, and we’re making pops out of orange juice. How many popsicle sticks will me need?

I think the process of starting with nothing, and having to get to a goal, is more creative, more interesting, and, ultimately and more importantly, more likely to cultivate the type of student who can solve the problem, ‘How much oxygen must be created for 2 humans to survive in a spaceship going to Pluto and back?’ ‘How large do pipes have to be to let water flow freely from a sink to the sewer?’ ‘How deep do the girders have to be to support a 12-story building?’

I think that students who can turn problems inside out like that, from concept to detailed numbers rather than from numbers to concept, can’t fail to pass any standardized test, but I really don’t care whether they do or not.

Dan and other commenters,

Great talking with you. Wish we all could do this in person to come to a decision more quickly. I have to get back to my Euler line lesson plan, but I’ll jump back in later tonight.

Sure thing, Dan. :) I always appreciate a good discussion, and your ideas and plans of attack spark just that.

Dan
That post is exactly what I was thinking about when I wrote my comment. I was simply trying to say that the images don’t have to be so elaborate as to scare the average technophobe away. I built that lesson in about 10 minutes. I’m on your side here. The power of that lesson was my ability to hold back info until students saw the need for it. A book can’t do that.

Four words: SHOW ME THE MONEY.

As soon as I see some proof that this works, I’ll be a much happy camper. After all, the currency of math and science themselves is evidence and proof. And to quote Michelle, math is “intended to discover our environment” (although a discussion could be had about whether or not studying water tanks is interesting to everyone; not everyone is a future engineer, and so the theme of the problem does matter).

Everything hinges upon whether it works or not. So, we could cut a bunch of wasted time if it was proven.

Don’t get me wrong, I love enthusiasm, passion and rhetoric as much as the next fellow. Hell, that’s what hooks kids into a subject and teachers into a new pedagogy most times (it seems): a teacher who is passionate about their subject and able to communicate well.

And that, I suspect, could be playing a large role in the “success” this method claims to have. How to control for that, well, that’s part of the challenge of drafting up an experiment.

I know you claim to see the results of it every day. But if YOU weren’t teaching it, if someone else was handed your lesson plan for a term, would the method still hold? And would they retain it?

But seriously, Dan, thanks for your work and theories. As long as it doesn’t become religious doctrine…

Bah! *currencies of math and science themselves are

@ Chris & Maria

That’s why I mentioned scripting, which is used a lot in elementary ed.

I may have a different viewpoint because I am thinking about younger students, elementary and middle school age. That’s why I don’t think about ‘pure’ mathematics, although all math started because of some natural problem, like geometry starting because lands needed to be measured after the Nile River floodings in Egypt, where the Greeks went and learned about it; and even the theory of relativity, etc., started out as an original problem rather than the rote ‘Given that the diameter of the circle is 12 inches…’
Anyway, with scripting, a person thinks about the words they say to get their meaning and emotion across, and WRITE IT ALL DOWN. A teacher using the text then has the option to teach word-for-word, or at least know where the author is coming from.

This is useful in elem. ed. because the pedagogical knowledge of the teachers varies greatly, (some have just barely passed exams after many tries, and some went to Harvard) and is a way of making certain teaching methods universal. It seems to me that math ed. has the same problem, esp. in elem. ed and middle school.

Chris wrote:
After all, the currency of math and science themselves is evidence and proof.

Chris, this isn’t evidence for Dan’s method, and it’s not a math class, but it is evidence based, and it’s physics, which is pretty close to math.

I know I can’t do anything like what Dan does with the media. But I’m a good storyteller, and I hope to use these ideas with my own twist when I get back in the classroom in the fall.

Sue wrote:

Chris, this isn’t evidence for Dan’s method, and it’s not a math class, but it is evidence based, and it’s physics, which is pretty close to math.

I’m not sure what you are referring to by “this” and “it”. The water tank video? I thought he was teaching a math class with that video. And how can the teaching method itself not be used for evidence? Are you saying that the water tank video is not evidence for his method?

Not being snarky, just trying to understand the statement.

Mr. Meyer, Mr. Meyer, can you help us?? *waves hand* ;) :D (How meta is THAT…)

I’ve changed my mind about the howling “PROVE IT” refrain. After all, this is really just a teaching style and I don’t see how it can harm anyone’s learning. Also, it should be more widely adopted to see if other teachers get the same results; THEN it can be the subject of a study if it gets enough support.

Sorry about that. You’re right; it’s better to get new ideas out there and try them out in the field and see if they work.

The reason I changed my opinion is because I was thinking about my own planned styles of teaching and lesson plans, and how counter-productive it would be if I were limited to only given and proven teaching methods. No room for creativity, ya know? How am I going to prove my Clockmaker’s problem solving method(now that has a nice ring to it…the Clockmaker’s Problem…) if I don’t try it out? And having to prove it before it’s fully shaped is just too early in the development cycle.

I’m hesitant to self-promote, especially since the reading/writing of blogs is quite new for me being an old fogy of 33, but I will be addressing some of the research questions that have been raised here in a classroom next year. I’d love for this community to be a part of that and, who knows, maybe I’ll even be able to show you the money.

http://mathteacherorstudent.blogspot.com/

@Sue

That is a brilliant video. I’m excited to try PI.

I’ve heard that quote before and I still like it: “the plural of anecdote is not data”.

Awesome link, thanks.

Dan,
As an ESL algebra teacher my biggest problem is finding ways to get students interested in the discussion. Since videos appeal to visual learners and they have the added benefit of transcending language I believe that they can be of significant value as a teaching tool. Unfortunately sometimes you still have to rely on hard work and repetition to truly master basic skills. Herein lies the rub, math will always require a level of commitment and students must understand that they will only succeed if they truly challenge themselves.

Ethan-

I have to respectfully disagree with the idea that this is not for above average students. What I like about Dan’s ideas is that I believe they would show how math is applied in the world, and the resulting math would be creative, not just the rote ‘I know how to do the math problem very well.’
What if there is something to be measured, somehow, that has never had math applied to it before? How do you create? I believe that minds trained in this manner will be able to ultimately take a situation and design a math problem for it that will ask and answer new questions, not just follow the textbook. And I think that’s what we need right now – a LOT of people who can do that, not just a few.

I agree with a lot of you. We DO fill in so much for the students and do a lot of the “drill and kill” type memorization that can get extremely boring for the kids! I teach lower elementary in a under performing urban school and we are pressured with getting lessons done..almost at the expense of the students understanding the work! It’s all about exposing the kids to more and more and not allowing time for self-discovery. I feel that it is so important for the kids to experience issues, ask questions and come to conclusions on their own. However, I feel each year we get further and further away from this and closer to the scripted lessons that Michelle was talking about above.

Bob,
Yes, That’s a good point.
While videos and other motivational ideas can make good segues to initiating discussion of a problem, but how do we promote the sustained focusd reasoning that’s necessary to solve these problems in a math averse class?

Bob, Ethan, Maria and Amy –

As a student who used to consider myself bad at math, I wonder if the children are really ‘math-averse’ or, rather, ‘failure and confusion-averse’.

When I took a standardized test and realized I had a good score, that I really did know some math, my confidence and interest in it rose. When I understand the math, I really like it, and with the new confidence, when I don’t understand it I think it’s challenging but that I can eventually get it.

Before I used to think it was frustrating and I would never be able to get it, so why try? Looking back, I can see how my brain actually actively disengaged and I distanced myself from the problem.

So, I’m thinking, the students need to really have a good background in mult, div, fract and percents before they can do well in higher math, feel really comfortable with it. So, so many don’t and it ruins math for them.

I’m really worried about the person who said the students move on too fast, because I believe her and I think it’s highly detrimental. Somehow, the students need to get the extra practice, maybe for homework, some incentive if they do extra work (remedial and/or challenger) and turn it in, come in early or stay after school, but it has to be done so that they can understand the material.
I’m thinking that the TED.com lecture about using points like in gaming might be a good idea to try.

After reading the last two posts, it might be helpful for all to read this blog entry. It is long, but very worthwhile:
http://researchinpractice.wordpress.com/2010/07/13/despairing-vs-working-learning-classroom-management-and-learning-math/

Hey, for those who commented on scripts, this is a script, one like what I meant – it tells you what to do, how to do it, things you might say, reponses a student might have – is this the same as the kind of script you’re thinking of?

“Play the water tank video clip in front of a class of students of any age between 12- and 30-years-old. The video is best served cold. No music. No text. No introduction.

Your students will start to inventory their surroundings. They’ll identify the crucial elements of the scene. Again, you shouldn’t say anything.

Twenty seconds into watching the hose dribble water into the tank, ask “how long do you think this is gonna take?” Ask for guesses. Just guesses. Write them on the board next to the guessers’ names. Whenever anyone raises the maximum or lowers the minimum, point it out.

Then turn the clip off. Turn off the projector and proceed to whatever else you had planned for the period.

At least one student will ask what 90% of the class will be thinking:

“Well … who was right?!””