Watch Dan Meyer's video and answer the following question?
(a) Can the ideas in this video help you to be a better teacher? Please explain why or why not?
(b) Can you come up with a more challenging interesting type of question like the "Tank Filling" one presented by Dan? Give an example
This video gives teachers the insight that giving more information about a particular problem or breaking the question into steps is not helping are students. Real life problems have very few givens, usually you have to assess the problem and see what you know about it or what you could find out. Teachers or textbooks are eliminating those important skills. Real world problems rarely have step by step instructions to solve, so why do are textbooks have so much guidance to the answer. Setting up are students to failure if they can’t breakdown a word problem for themselves. Most questions in mathematics are too closed in the curriculum. I feel students at first will be lost with all that freedom to the question since there use to receiving all the parameters of the problem. At the same time more engaged since there making the problem. Giving the students that freedom allows them to be more involved and engaged in the problem at hand. Soon there will be able to make logical decisions and be able to build the problem. Students are too orientated to what the answer is. Always asking is this the right answer? Which isn’t the important part of the word problem in the first place. It is the students thought process and intuition to the problem. So by allowing the students to mold the problem to how they see it in the real world eliminates that eagerness to have the “right answer.” As most students will have different answers depending on how they see the problem and what parameters they give. I agree with this approach and plan to use multimedia, encourage student intuition, ask the shortest questions you can, let students build the problem and be less helpful.
ReplyDeleteHere are some question examples that are more open ended. How many drywall sheets would you need to drywall your bedroom? How much would it cost? How much diesel would a bus burn from the school to your stop?
Nicely thought out enty. I wonder if other issues with word problem are they (a) have no relevance to the student's world, (b) are not authentic, and (c) are not clearly articulated or presented in a context the student understands. I especially like your diesel burning question.
DeleteI think that too often we teach students how to do things step-by-step and what formulas to use, instead of letting students develop some intuition about what the problem is asking and think about how to solve the problem. I think the ideas in this video are a good reminder that we need to let the students think for themselves and develop problem solving skills instead of just turning them into calculators. We have to teach the students how to do mathematical operations and processes, but we should also let them think about how to use them and which situations to apply them to in order to solve problems.
ReplyDeleteAt my placement I noticed that when students are given word problems, they just jump into the math and do not think about what the numbers represent. We should get to the point where they understand what the numbers represent and be able to rationalize which processes they will need to follow to get to the right answer. Too often students are focused on getting an answer and don’t even think about what the question is asking them to find, so I really like how in the video he reduces the word problems to one simple question. This would help the students to understand what they are trying to do.
For a sample problem, I like the classic how much will it cost to paint a room question. You could show a picture of a room before and after it is painted and have the students think about what they will need to know (surface area of walls and ceiling, how much paint will cover what area of wall/ceiling, cost of paint can, how much paint is in a can). Then have them come up with and follow the steps needed to solve the problem.
Why is it that we ask student to do things step-by-step? And why do students focus on the answer only? It certainly is a prevalent culture and I wonder if you have any thoughts about that?
DeleteI think that many students want to be told exactly what to do and memorize the steps instead of thinking about what they are trying to do. Our society seems to be used to "instant gratification" so I think students just care about getting the question done and finishing as fast as possible, and don't care as much about learning how to do the math.
DeleteAs a student teacher, I have learned that word problems are a great challenge for many students. I understand that the wording and context of the problems are irrelevant to the vast majority of elementary and secondary students and thus are difficult for them to comprehend. Nonetheless, it was after watching this video that I saw how much the word problems students are given can be improved.
ReplyDeleteIt is ironic to think that providing students with more information can hinder their learning. The majority of people would think that students would perform to the best of their abilities (on a word problem) if they were given all of the necessary information. However, if all information is provided, students begin to depend on procedure and thus can succeed simply because they memorized the steps to achieve the answer (rather than understanding why the steps they took make logical sense).
This video showed me how giving students very little will actually encourage their critical thinking skills and promote their success. By giving students seemingly no guidance, they are able to identify what information they need and what approach they believe is most appropriate. This teaching strategy gives students the ability to explore and succeed in a much more autonomous learning approach.
Ultimately, this method is far more valuable and effective in comparison to the sub steps approach which ‘spoon feeds’ students the answers.
In addition, providing real life visuals (such as the bucket filling with water) gives value and relevance to the word problem. Many students see very little connection between math and their lives and incorporating these links as much as possible will (hopefully and ideally) encourage student engagement and motivation.
I undoubtedly agree with Dan Meyer and his philosophy of teaching math, however, this method may not grasp students’ attention as much as he may make it seem. Some students simply don’t like math and almost nothing will encourage them to get over their ‘math phobia.’ In addition, depending on the students, some classes will be difficult to control and thus classroom management becomes an issue. As a result, ‘fun’ activities with little guidance nor instruction may be very difficult (or simply impossible) to use in these types of learning environments. With that said, this is undoubtedly the most exceptional approach to word problems I have seen thus far. Consequently, I will put this method into practice during my future placements and my years of teaching.
Very thoughtful entry. And yes I wonder how much certain problems will intrigue students. BUt I wonder (a) if it is a genuine puzzle or problem whether students will be naturally curious and (b) if we fundamentally change the culture of the math classroom from obtaining answers to asking, puzzling, and thinking whether math would be hated as much.
DeleteIn addition, a few examples of ‘challenging’ and ‘interesting’ questions are found below:
ReplyDelete1. What curve(s) is being represented in this photograph? What is the axis of symmetry? What is the vertex? What is the maximum? What are the coordinates of the vertex? Etc. https://www.google.ca/search?q=water+parabola&source=lnms&tbm=isch&sa=X&ei=xVN4Uum8BqiiyAHtgoGwCQ&ved=0CAcQ_AUoAQ&biw=1366&bih=577#q=water+parabola+hair+flip&tbm=isch&facrc=_&imgdii=lAEc-JcRoF6-tM%3A%3BR8wCxcUuSZzIuM%3BlAEc-JcRoF6-tM%3A&imgrc=lAEc-JcRoF6-tM%3A%3BYHQFMquGGSDW4M%3Bhttp%253A%252F%252Ffarm5.static.flickr.com%252F4082%252F4775260501_a354ef6f1f_m.jpg%3Bhttp%253A%252F%252Fflickrhivemind.net%252FTags%252Fhair%252Cwhip%252FInteresting%3B240%3B160
Or
http://www.flickr.com/photos/100564167@N05/9832188734/in/photolist-fYQxaq-bwEuRZ-hbAUxL-VAdwG-5dnJGk-dnjoBQ-5SRKXL-4jz8K1-6mPake-8Yd1vf-a9EtCZ-5SHvhR-5SMQNq-cdDYm7-ctB8Qy-8Ezqv4-bUEswA-bUGqBG-bUDq7b-bUGqR1-bUDqsA-bUEsJN-bUDpGJ-bUDqPd-8AQedf-gKjpr-aAfqqG-4juzvr-6HGRDE-E7ZBT-6qXEzV-c9vdw5-bFp2az-9NGKrb-6HrVUA-5hxStX-2nfiH8-575TpQ-6uHM1f-BnG5H-hgAMDc-5SK1sv-MyU37-54Y4Xd-dSdGsR-7Bgjwp-6W3y3x-6W7B7U-sH6Y9-9gnsNf-8f2gT
2. How much water is required to fill up one of these balcony pools? How much will it cost? How long will it take to fill up?
https://www.google.ca/search?q=middle+east+crazy+balcony+pools&source=lnms&tbm=isch&sa=X&ei=3VF4UoXfAqLcyQGzpIDQDg&ved=0CAcQ_AUoAQ&biw=1366&bih=577#q=middle+east+balcony+pools&tbm=isch&facrc=_&imgdii=_&imgrc=y6OtcgRBuVXaxM%3A%3BHeTvq8tEfNVE3M%3Bhttp%253A%252F%252Fhelablog.com%252Fwp-content%252Fuploads%252F2012%252F02%252Fswimming_pool_balcony_mumbai_1.jpg%3Bhttp%253A%252F%252Fhelablog.com%252F2012%252F02%252Ftheres-a-swimming-pool-on-your-balcony%252F%3B550%3B550
3. Teach probability (data management) using a game of blackjack
4. How long will it take to empty the following container of water? http://www.flickr.com/photos/85328548@N00/2405163109/in/photolist-4Ex6ct-6CiikS-92KyQS-7fh3sT-6hSNuj-bsSyiT-4VrWwT-37Awq-d8MQG3-4atGRS-9zPz6u-8vCZPK-8BFLtp-do1Fza-cifXyA-8M9BWu-8M9C3o-8M9C7Q-8M6yRn-8M9BRN-7Fto6V-4Roh3Z-5KqeP1-eUn3ZG-aHDBor-xJg1Y-8WqXK5-44uWmx-7gHY8N-ccZWS-ccZWR-aneg1J-aneh9N-Aq6a8-6RZ28G-dUxRvi-ccZWT-ccZWW-9t6TY3-8WqsQ1-8Wqs2G-8WnkEX-5ZuZBC-b9aanR-Aq6id-9zCWYF-ugChx-5VN7Rt-anioPK-anmJB9-ccZWU
5. How can we make a better airplane than this one? http://www.flickr.com/photos/65638020@N00/896794325/in/photolist-2nfiH8-7Bgjwp-6W3y3x-6W7B7U-575TpQ-sH6Y9-9gnsNf-6uHM1f-BnG5H-8f2gT-8RciXR-7pjh5s-aFRZCT-6xr4Ka-9suSk4-hgAMDc-9DwLxm-8f2id-9Qb4J-CTXRz-62Wn5a-4V36Nf-aSEKvk-7E3kuj-9H3Jsg-5SK1sv-69Uy7j-6FjjYw-b4tXW2-eYRC1N-cKsT5-e1zm7V-7g9aSg-4LJtjf-fArP8e-cKrGj-cKt4c-d77PN9-MJ5iM-83z5bb-4PNkpw-MyP1j-84j6sH-M4iz6-DLd9B-4nwPqX-5yQKEB-84j6fF-MyU37-5yQEJg-3jP5TZ
I especially like question 5 - it is so way out there that students would be intrigued for sure.
Deletea). I believe that the ideas in this video can help me to be a better math teacher because students will be engaged in deriving the solutions to problems using inductive and deductive reasoning and discussing the solutions to problems. One of the reasons students have difficulty with finding the solutions to word problems is that they have not been taught how to think. They have only been taught what to think. In my opinion, I believe that inductive and deductive reasoning should be taught as an introductory topic in any mathematics or physics course.
ReplyDeleteb). What will be the length and the thickness of the shadow of a tree at 12:00 P.M.? 3:00 P.M.? 6:00 P.M.? What is the relationship between the length and the thickness of the tree's shadow?
I would encourage you to watch the video again. I don't believe DM was pushing inductive or deductive reasoning and I am pretty sure he would not support the notion of this being a subject to be taught. I think he might have been saying, you get students to think bu giving them genuine problems/puzzles to think about and NOT giving them too much structure or steps. I'm curious - how you you teach deductive and inductive reasoning?
DeleteI was puzzled by your question - what do you mean by thickness (width?) Shadows, in my mind, are 2 dimensional. I'm curious about your second question? To me the position of the sun is the key here but maybe I need to think about it more?
I thought this was an excellent video. Having taught math and science during my first placement, I found myself enjoying the science more because of the endless opportunities to create student-driven investigations. I did strive to incorporate similar investigations in my math classes but I found it more difficult to teach the curriculum this way. I suppose I just needed a little bit more inspiration. This video gave me that. It provided me with some insight as to the opportunities we have as math teachers to encourage student thinking and inquiry. I love the idea of replacing generic word problems that give step by step procedures to follow with more realistic questions that would actually be useful in the real world. Questions the students could relate to and appreciate, without feeling like they need to be mathematicians to comprehend. The video of the container being filled with water for example was excellent. Why pose questions using neat, labelled sketches, when we can bring in (or show on a video) the actual object in question?
ReplyDeleteIn one of the math investigations that I used during my placement, I had the students discuss the number of pizzas that could be ordered if there were eight toppings to choose from. We did a little bit of role playing as students acted out a scene from a pizza parlour where one of the customers was taking too long to decide what topping(s) they should choose. I had the students try to problem solve, and then we explored Pascal's triangle as a way to identify patterns to solve a simpler problem. We then used the triangle to answer the more complex pizza topping question. It was amazing to see everyone make the connection, and then use Pascal's triangle to solve all types of problems. The students really enjoyed the lesson and I loved hearing them all say how cool it was!
In secondary school, one of the most relevant math units to real life is the finance unit, where students learn how to calculate simple and complex interest, present and future values, and monthly payments of annuities. These concepts are so important and yet, students are often taught to simply plug in numbers into formulas and find the unknown value. All too often, there is no real understanding of the math concepts. Instead, we could give students some Monopoly money and have them invest it, or use it to mortgage a home. We could even do this with real money where students purchase things they need or want like calculators (so many students lose their calculators, so having them finance their own would be a good way to have them take some responsibility for their belongings while having them learn the math behind annuities). We could have students purchase or sell baked goods or any other items, and be required to calculate the sale price, or the final price including tax.
We should be teaching math in a way that is relevant to our students and their real lives. I always hear my students say that they hate word problems. This is because there is a total disconnect due to the manner in which the questions in textbooks are posed. Why would anyone for example care about how long someone would take to drive on a highway and a country road to get to their destination? Rather, we could pose the problem as an investigation where students are required to travel by foot or by hands and knees to a particular destination and determine the fastest possible route. Real world + real fun = real learning.
Lots of good reflection here. I wonder if we let student ask the questions whether that might work even better. There is a technique called Think-Puzzle-Explore. You ask students what they think about a concept/situation, say time. They write all their thought down. Then you ask them what to they have questions about - what do they find puzzling. They generate so many interesting questions. Then you set them to the task of discovering and exploring the questions they have asked.
DeleteThrough my personal experience as a high school student (more specifically grades 11 and 12), I often found myself just looking at the numbers provided in many word problems and then scrambling through my notes and textbook for an equation that could incorporate all my known values. This became an issue because I was just trying to memorize a bunch of equations rather than understanding their purpose. It wasn't until university where we had to apply the concepts to real world situations and I found myself struggling with understanding everything that was going on. As time passed, I started to realize that understanding the problem lead to more success. The courses I struggled in most were the ones where I had “impatient” problem solving skills.
ReplyDeleteDan Meyer’s brings up good points in this video. I really like how he breaks down one of the word problems and explains how the four separate layers compressed into one will lead into impatient problem solving. I didn't realize that when we create word problems with the visual, structure, sub steps, and question, we are just paving a smooth path from start to finish and having students fill in the gaps. Perhaps the reason why many students struggle with word problems is because they aren't actually determining the steps to solve problem. By giving students an opportunity to think, they will be able to use their intuition and be aware of understand their thought process. I do think his ideas can help me become a better teacher, but more specifically, it will allow our students; our future become better patient problem solvers. Though, I do think it will be hard to teach in this mind set at first. The reason why I say this is because in my experience, I am used to having a word problem with a bunch of sub steps that lead me to the answer. It’s important to remember that many students will not choose a career in math when they are done high school. However, if we can teach then the foundation of being a patient problem solver, they will be able to take these skills with them wherever they go in their future. So yes, I do think his ideas will help me become a better teacher.
An example question:
How long would it take a maintenance worker to cut the grass in Horton Park on a sitting lawnmower?
The teacher can show a video of someone cutting a large field on a sitting lawnmower. This will get them to see how much grass gets cut as the mower moves. It will show that the speed will determine how long it could take. Also, they would realize that the larger the field, the longer it will take.
The students would have to think about the dimensions of the park, the dimensions on the deck of the lawn mower, and the speed at which the lawnmower travels. Students would then need to know how much grass gets cut per hour (or minute). They would then be able to put together the steps needed to solve this problem. Once that is established, the teacher can provide them with numbers to solve.
This is an excellent entry - You really get at the heart of what the video was saying and you pose a very good questions about the difficulty in applying this technique. I wonder even if a student was not specializing in math, whether they would prefer the Dan Meyer type problems - I think they might and th more students get exposure to these types of problems perhaps the more they will be better able to deal with open-ended problems in their future.
DeleteThank you. I agree. It is important to think beyond the "math" and look at what skills the student will gain from learning how to be a patient problem solver.
Deletea) I agree quite strongly with many of the ideas in this video and I think they are important for good math teaching. The most important, in my opinion, is the idea of not giving students so many intermediate steps. I think the biggest danger of this is that it gives the impression that there is only one way to solve a problem. It's also useful to combine the art of estimating and making assumptions with other elements of the math curriculum by removing all the numbers from word problems.
ReplyDeleteb) Suppose you're trying to walk from one corner of a long rectangular field to the opposite corner in the middle of winter. The shortest distance is across the diagonal, but because there is ice and snow in the field you must walk slower through the middle of field than you can walk on the sidewalk that goes around the outside. What path should be taken to get to the opposite corner in the shortest time?
For this problem students would have to make a number of assumptions. How fast can a person walk on the sidewalk and through the snow? What are the dimensions of the rectangle? Do the dimensions of the rectangle affect the answer? Students would find that for some combinations of walking speeds and rectangle shapes it would be fastest to walk on the sidewalk the whole way; for others it would be fastest to walk straight along the diagonal, but for some situations a more complicated path combining the sidewalk and the field might be faster.
Warren - that's a great problem. I do challenge you to questions something Meyer's may have said - There must be some reason why this type of activity is not main stream in the schools?
DeleteI can think of several reasons:
Delete-These are not the type of problems commonly found in textbooks
-These problems are harder to mark
-Teachers are afraid of giving so little structure and instructions
-These are not the type of problems found on standardized tests such as the EQAO or the SAT
-Especially in elementary school, teachers who are weak in math themselves may be uncomfortable with these types of problems
This video helps me become a better teacher because he explains how to use hooks related to real life examples. People already having some form of knowledge about a situation will be better able to answer questions as they are able to predict the outcome. Through use of actual experiences already encountered it allows individuals to more easily understand and apply new theoretical concepts. Through his illustration of filling a tank we can all relate to it as we have all done it before.
ReplyDeleteA similar type of question is how many times does your dryer rotate per hour? As it is tedious to count the number of rotations that occurs within an hour it would be easier to calculate it based on a known rotations per minute. Similarly, in order to figure out how many sheets of shingle are required to cover roof with a 200 meter area, it would be more efficient to calculate this based on how much a single shingle would cover. - (Sylvia)
Sylvia, Do you think that having knowledge about a situation is the critical feature here? I understood that the key issue was (a) open ended questions that are authentic/real and (b) not giving students to much structure or guidance so they can become true problem solvers. Your second example seems more like what DM was talking about - the first seems a little less challenging or open-ended.
DeleteYes, it is crucial to have knowledge about a situation. This will help you visiualize the problem and be able to break down small steps to solve that problem. Students are able to structure an answer as they have prior knowledge. Without prior knowledge, they would be lost and unable to even solve a problem and students will just simply use the formula and plug in numbers. By not given students enough structure or guidance will allow students to struggle and try to resolve a solution by using what ever knowledge they have. They will build upon that prior knowledge and research more information.
DeleteI really liked this video, it really exemplifies everything that I observed on placement as well as what I remember Math to be like when I was in elementary/high school. Textbooks teach us to be like machines or robots, where we memorize the equations and/or steps and we just plug the numbers into the proper places because the textbook says so. I love the idea of getting rid of all of the steps and presenting ONLY the problem itself to the students. Let the students fill in the blanks and lead class discussions about ways in which to proceed. Also taking it to another level of bringing in real pictures and videos instead of a clipart from the textbook is a way to bring connections and real-life contexts to the class, otherwise students will be disengaged because they won’t see the relevance. These ideas can definitely help me to be a better teacher because they will guide me to teach in ways that are students-led and discovery based. As a teacher I cannot give the students all of the steps and expect them to understand the reasoning, they must find out for themselves WHY these are the steps.
ReplyDeleteSome other great open-ended questions that could be asked, similar to the ‘tank filling question’, could be…
How much paint will you need to paint your bedroom? (Given dimensions)
How much would it cost to do so?
Clear, concise reply. Your questions at the end is reasonable although I wouldn't give the dimensions (let them figure out this step in the problem too). Now I challenge you do find a problem in this video? If it is so simple, why do teachers give all the steps? Is it easy to come up with these kind of problems for all concepts? How do you eventually consolidate the numerous possible solutions that students come up with?
Deletea) I believe that the concerns that Dan Meyer expresses in his video about student’s inability to reason through math is valid. Dan’s ideas for presenting application problems is very innovative (especially when he presents them with multimedia) is very inspiring and will help me be a better teacher. My current teaching style consists of showing a real world example of where the math I am about to introduce can be applied. Then throughout my instruction I make reference to that example. I would incorporate Dan’s ideas in my style of teaching, by bringing the example I introduced in the beginning back to the forefront at the end, and having the class generate a process in which they would solve a problem related to the example. In classes that I am not introducing a new topic, I would like to present real world problems as described in Dan’s video, however I don’t think this would happen more than two times a week, since I can see that this style of teaching would take a lot of time, and I do believe that some students need to be given the chance to practice the computation as well as reasoning.
ReplyDeleteb) I would present the question to the class, then provide the background with an image of the school’s flagpole. I would then pose leading questions to help students to generate their solution. Once we have a method or process, we would actually test the method and make the measurements. After the class finds their answer I would compare it to an actual value (if available), and have the class generate some factors that could have affected how accurate our solution could be.
Question: How much rope do we need for a flag pole?
Background: The rope on the school flag pole got shredded and portions of it are lost. We need to replace the rope, but don’t know how long it needs to be. We don’t know the height of the flag pole, nor can we take the flag pole down to measure its height.
What factors do we need to think of? What is available to us?
- Pulley system on flag pole means the length will have to be doubled
- Pulley system goes from the top of the pole to some point that is reachable from the ground
- Sun and shadow of the flagpole on the ground
Using what we are given, how do we find a solution?
- Relative triangles and ratios to find the height
- Measure the shadow that a meter stick gives off, and measure the shadow of the flagpole
- Measure the distance from the ground to the bottom of the pulley system, and subtract that value from the height of the pole found
- Double the length of the pulley system to find the amount of rope needed.
~Adele Hedrick
Interesting posting especially the part when you say you might not have time for students to problem solve because they need to calculate the actual answers. It's funny how old methods keep pulling us back. Why do students need to practice computation? That seems to be about the lowest level of learning. Also, I am assuming you will make your questions as short s possible and NOT list any of the background questions you posted. The idea is for students to work on these kind of open ended problems themselves. They need to be open enough so that there will be a variety of solutions and ideas. The "right" answer is not that important - maybe "rightish" answers .
DeleteI believe the video helps teachers out there (not just math teachers, although concepts like the dreaded "word problem" fall into math-like courses) become better teachers.
ReplyDeleteIt shows that overteaching ideas and concepts can actually regress student learning. Having students find they way to those ideas and concepts on their own can not help them formulate their own solutions (solution that they themselves understand, not just regurgitate) to answer word problems, but they can use that critical thinking they've developed to answer more real-world problems that they will encounter throughout their lives; problems that aren't just plug-and-solve.
Out of the 5 issues that were causing students to struggle with word problems, I think the one that stuck out the most was "Lack of perseverance" - students want to be able to throw the answer out there and move on to the next problem. But sometimes they do it and they won't know why they're doing it. Thus, when they encounter a problem where they can't just do that, they lack motivation to answering. Thus, creating an enviroment where the students can take a step back and really analyse what they see and deduce strategies for solving said problem can help them do the same for tests. Although, some may fear teaching like this with a curriculum so large will hinder how much time you really have. I disagree - if you're helping them form their thoughts properly, you should have enough time to do everything and more. Sure, it may be tough at first, but it will have its rewards later on.\
A question such as "You have the following materials. Each of these materials cost a certain amount per quantity. You create a house using all the materials you have and go and sell it for the following amount. Is there a profit?" suffices.
Students would need to take a step back and just look at the last part - the question. "Is there a profit?" From that, they would need to think about what a profit it? Next thought would be how much they're selling for and how much they spent in the first place. From there, they have to figure out how much they spent. Finally, they can solve the problem.
Rather than have students just calculate numbers for the sake of, teaching your students to make these connections throughout solving will get them to figure out what they need to do to solve.
You make a good point about extending what Meyer's said to other subjects - maybe not all other subjects, but science-related ones certainly come to mind. In the real-world there are very few plug and solve problems, so why teach them in school? Why do you think there is "lack of perseverance"? My guess, and it is just a guess, is the quality of the problem given. For example, your problem is very open-ended, probably way too big. It would take a team of experts days to come up with possible solutions, so students might get discouraged and stop trying. Maybe get them to build something they would like to build like a product they could sell (e.g., clothing, games, tech gadgets). So scope ans focus of problem are important. Finally, while making connections is critically important, the actual process of brainstorming and coming up with a plan and working in teams may be even more critical in the long run. Decent post
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ReplyDeleteThis video got me thinking back to a question that was posed to us at the beginning of the term, and that is “Why do we learn math?”. We were also asked to think about what answer we would give to a student who is posing that question. If a student asked me why we learn math I would answer back with a broader question, namely “why do we learn anything?”. A lot of students might answer things like “to get a job” or “to make mom and dad happy”. However, the answer I would be guiding the student to is that the point of learning anything is to gain general knowledge and insight about a broad spectrum of subjects in order to make connections. It might be hard to determine how biology is connected to math and philosophy. However, a medical doctor for example needs ample knowledge from all three fields of study in order to do his job. Everything in the world is connected, and finding these connections requires knowledge as well as critical thinking skills. Math in particular prepares students to think critically about complex issues, analyse and adapt to new situations, solve problems of various kinds, and communicate their thinking effectively. Once mastered, these skills can be applied to any career one chooses to pursue.
We now run into a different problem... From my experience over the years, as well as the recent placement, I have met countless students, some in grade 11 and 12 (close to the end of high school) who have yet to develop these critical thinking and problem-solving abilities. I have always wondered why. Moreover, when they mentioned that they would want to become doctors and lawyers and engineers, I could not help but wonder how they were going to reach their goals without these important skills. Watching the Dan Meyer video made me realize that the problem is that in schools, students are nowadays being spoon-fed information, especially in math classes. By giving students guidelines for every step in a problem, for example, what we are doing is we are training lazy brains. Lazy brains are growing to be unmotivated towards problem-solving on their own.
I do believe that watching this video and understanding why students are having such a hard time with word problems will help me become a better teacher. Educating by spoon-feeding information to students is teaching students to learn in a mechanical way. It’s like saying “follow these steps, and don’t worry about why you have to do it in this order”. Students are being trained to memorize equations and plug in values to solve, because the textbook or the teacher says so. Then you give your students a word problem and expect them to understand why they need to think about it in a certain way, and all of a sudden you find that they are lost. It should be no surprise that students hate solving word problems. Word problems require critical thinking. How can students like something that requires critical thinking when we are emphasizing a mechanical way of learning?
Wow. You have set the bar high on posts - This is excellent - quite a high level of thinking. My only comment (and it is not just for you).- Students keep referring to "word problems" - I think Meyer was pointing to genuine problems - yes you communicate those problems with words - but I "personally" took him to mean problems that could be used to introduce concepts. Maybe I took it to far, but it seems more like a philosophy, as you suggest, than referring to traditional word problems in math". Well done.
DeleteHere is an interesting video I found about math and "being lazy"... While I do not agree with a some of the things mentioned in this video, I do think that it's an interesting idea for everyone to think about...
Deletehttp://www.youtube.com/watch?v=tg0Z--pmPog
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ReplyDeleteThe video makes a solid point about getting rid of all the steps, and instead presenting only the word problem and allowing students to develop their own way of solving it. As teachers we can pose open-ended questions, but we will still run into the trouble of changing this lazy brain culture, and this isn’t something that can happen overnight… Changing the way students learn takes time and effort, and it’s especially difficult at the high school level where students have already gone through school for a good number of years and have been taught to learn in a mechanical way all along. Let’s assume for example that these students have a need for steps in a word problem. We could still give them some guidance to the way they can solve the problem. Instead of explicitly paving a smooth path for them to reach the answer (ex. What is the vertical change? What is the horizontal change? What is the ratio between them? and Which section is the steepest?), why not let the problem have the following guiding sub-questions: What do I know? Is there is any information missing, and if yes how am I going to find it? What should I assume? What do I need to find out?
By guiding students through what they need to think about, instead of guiding them to the steps and to the answer, you are helping them understand how to think critically and why they are following those steps. Students should slowly become more comfortable and confident in their ability to think critically and problem-solve on their own, and then we can present the problem without any guiding sub-questions. The goal is to spark students’ interest in problem-solving by bringing real-life open-ended questions to the class and allowing students to make connections. An example of one such open-ended question similar to the tank-filling question from the video could be “How many tiles will you need to tile the entire floor in your kitchen?”.
Now I see in your first paragraph a tendency to slip back to supporting the lazy brain - have enough courage to let students work on interesting, truly open-ended problems in groups - See what they come up with, then provide the scaffolding. If you provide hints during the process, you are enabling what you refer to as "laziness". I think what might be harder is change in the teacher, not the students (smile).
DeleteThat said, you have some good prompts listed in terms of focussing students to think and sometimes scaffolding will be needed.
I completely agree with letting students work on open-ended problems and seeing what they come up with before providing the scaffolding. And if as teachers we all suddenly changed the way we taught, and supported this way of teaching from the primary grades, this would make for an ideal situation...
DeleteHowever, what I was referring to is, for example, teaching a grade 8 math class that has done things differently with respect to word problems for the past 7 years (and even harder if you are teaching grade 12). I do believe that as teachers we tend to teach how we were taught. So changing the way we teach, and the way students learn, is not something that can happen overnight. Therefore, my suggestion was to wean this way of learning mechanically by still providing "some" guidance for students, without actually guiding them to the answer. Instead, we can guide them to what they should be thinking critically thinking about until that guidance is no longer needed. An analogy would be, for example, using floats for a while until you learn how to swim and you no loner need them (it's mostly encouraging knowing that you won't drown)...
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ReplyDeleteYes, this video can definitely make me a better teacher, especially with respect to math. I really liked the way Dan broke down his ideas about problems in math curriculum into five points. These ideas were relevant and important, and I think that if math teachers consider them critically and try to combat them in their lessons, a significant improvement to their teaching abilities would be made. In my opinion, the most important of the problems that Dan identified was students' aversion to word problems. In my experiences both tutoring and teaching (I was fortunate enough to get to teach mathematics during my first placement) I have found that students do in fact have a serious aversion to word problems. In fact, I remember having a serious aversion to word problems myself when I was learning mathematics. I really liked the way Dan combated this aversion to word problems by getting students involved and engaged. Really, he kills two birds with one stone in his teaching strategy. Students resenting math and being afraid of it are common problems in mathematics education. By Dan's involvement of the students in word problems, he is not only engaging them and making math interesting, but also providing them with a tool to combat their fear of word problems: relevance! Making math real and tangible makes it less scary for students, and I think that his use of student involvement in word problems is a great strategy which I really want to incorporate into my lessons.
Another interesting point that Dan makes in his video is student's eagerness for a formula in math. When tutoring my high school math students, I am constantly asked "but how do I know which one to plug the numbers into?", and "is my teacher going to give me the formula?". These questions often come with a tone of thinly veiled panic. I always respond the same way... "does it really matter? Let's find a way to make the formula on our own so you don't need to worry about it!". This is often met with a weird look, and apprehensiveness. However, after I explain it to them further, and teach them how to go about building a formula, they are often impressed with how easy it can be! This is a great feeling as a teacher, and after watching Dan's video, I see how significant these skills are and feel proud that I have the ability to teach this to a student. Making a student less eager for a formula and more eager to actually understand what is going on in the problem is a really important part of mathematics education, and teachers who incorporate this into their methods are doing well for the teaching of mathematics.
Very through and thoughtful entry. You explain your reasoning well. I am wondering if there is anything that you disagreed with - I find these videos very convincing and I nod my head a lot and it is hard to find something that I disagree with, I think because my guard is down a little. The first few points are very convincing.
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ReplyDeleteFinally, I really liked the small part of Dan's video which relates "sitcom problems" to mathematics. I like how he used this point to touch upon the importance and relevance of mathematics to real life. After all, a significant portion of our lives are spent problem solving. We combat our real-world problems and decision making by weighing out variables, testing things out, and seeing what works to get us to the right answer. This is really what math is all about. When teachers make this connection and apply it to their lessons, they are making math interesting to their students, and giving them a purpose.
While watching the video and attempting to come up with an interesting problem like Dan's water tank one, my mind snapped to a problem that I have encountered during various tutoring sessions. The question gives a student the dimensions of a sheet of cardboard, and asks them to find the maximum volume of the box that they can make with it. Rather than having the students simply look up the formula from the sample problem that is two pages previous to the question, I would give each group in the class a sheet of cardboard of that size and a bunch of plastic balls, or packing foam. I would then ask them to make the biggest box they could, and see which group could fit the most material into their box. This is an interactive and engaging activity for students. Also, an element of competition among groups would be incorporated, which I think would engage students and make them more eager to get involved.
Again, another thoughtful piece. How hard to you think it is to create questions like Dan's - ones that truly engage students. I wonder.
Deletea) For starter I really like his humour and also his perspective. I think sometime we spoon feed too much, and his method of “reducing” the problem is an interesting way of teaching. I think this perspective could definitely help me become a better teacher. I think by using the reduced method, i.e. ‘how long does it take to fill the tank?’ rather than the full problem, it opens up for more discussion, and more possibilities. Using this idea it takes away the boring directed learning of just setting up the problems and giving a formula. Any time you break away from directed learning I think you become a better teacher. This pedagogy can shift the balance in the classroom – the teacher isn't dominating the lesson and ultimately the teaching.
ReplyDeleteHowever, I think this way of teaching would be difficult if the classroom community wasn’t very developed or inclusive. I also would have liked more examples in the video. I am also questioning how you would use this method at the very beginning of a unit or topic. How would students know how to solve the problem if they had no idea of rates or volume?
b) During my placement my class needed to learn how to measure reflex angles. Instead of showing them the method I put a reflex angle on the board and posed the question “how would you measure this angle?”. The class then engaged in a discussion on method they could use. One group said that you could use two protractors at the same time. Another group decided that they could measure the interior angle and subtract it from 360° (since the full circle would be 360°). Another group came up with a method of extended on of the lines and measuring twice. As a class we used each method and decided which answers were reasonable and which were not. I guess this is somewhat like Dan’s method in that I turned the lesson around and had the student come up with possible solutions. I didn't spoon feed the class and it was a collective decision when we decided upon a solution.
Solid points AND I wonder if you could flesh out your objections a little more. Do classes naturally have no community? Can't the teacher help create the community? Couldn't questions like Dan's help? Also, I wonder if you would struggle with this approach (as would other teachers) because you believe you have to teach the problem components before a student does the problem or provide more scaffolding, While you don't want to put in too hard a problem, I think Dan's point is that genuine struggle is critical. Finally, your sample problem seems to be a good start, but there is no real world connection. I wonder if there needs to be a real-wrold connection? What do you think?
DeleteThe ideas in this video cannot possibly make a better Teacher before those same ideas inspire a better system. The ideas in the video hinge on the distinction between a didactic and dialectic form of teaching. Our world, our society, our education system has formed an impression that the current didactic method was preferable to the old dialectic. Teachers provide education, students devour then regurgitate that information and the Teacher assesses and evaluates the Student’s presentation of the information. The system requires learning goals and curriculum expectations. Students are expected to perform to standardizations that facilitate hundreds of comparisons. Teachers are given A number of students to complete Q specific expectations in E years all intended to lead perfectly into the problems on the EQAO testing. We expect patient problem solving to develop in a mechanical system.
ReplyDeleteDan Myers presents great ideas. Pointing out the problems above will always be simpler than suggesting a solution. However, they are presented in the same system he suggests discourages patient problem solving. His solution appears to be a formula to get the information from the problem (disengaged students) to the solution (critically aware students) through a less didactic process. Everyone has pointed out that we as high school teachers will compete against 8 – 11 years of learned mechanicalness. As well it is clear that the mechanical system is leaving children behind. The important factor is that the system will still be in place no matter how I teach. That system says that when a student struggles I am to provide more scaffolding and support. Should that student continue to struggle there are checks in place to ensure his or her teacher has provided those supports. If, for some reason, a student were to struggle with the fewer layers provided for answering Dan’s real life problems the system would not appreciate a teacher whose defense is: “it is okay I saw it on a ted talk”.
We were all able to disagree with a suggested rule-of-thumb that was to be used as an estimate for teacher led time in a class. Yet, we are willing to accept that the ideas in this video could make better teachers. Do not get me wrong here; I love the idea of molding a generation of thinkers that are critically aware rather than mechanically stunted. If only for the purely selfish reason that; “I will retire in a world run by these students”. There are great benefits in having citizens at every standard who think through the reasoning of their actions rather than weighing the consequences. But the systems I will teach in will not allow for these ideas to make me a better teacher. There are rewards to being able to ask a class if they can think of a way that will guarantee a profit from the Lotto 649. Remember 649 tickets are only $2 and there are only 14 million tickets. The simple math behind the combinations of numbers and the plausibility of purchasing 14 million tickets can lead to amazing discussions like thinking about the fact that 14 million seconds from now is Christmas Eve (roughly 44 days).
But I digress; my important conclusion is that Dan has come up with another brilliant solution to how Charlie and Alan can get Jake out of trouble this episode and completely ignored the serious problems like Charlie’s alcoholism. But the only way to get Charlie to stop drinking was to have the character get hit by a train because as a society that is the only solution we could accept; and as a teacher in a system I cannot gamble on an interesting idea when the system tells me success is ensured by a scaffolding method.
Andrew
DeleteSimple calculation error 14 million seconds is closer to 140 days not 44 days.
DeleteInteresting and unique point. But if I follow your logic, nothing will ever change. In the classroom, the teacher is the ruler - they have a lot of autonomy. I would argue that one of the "only" ways to change is to do it at the grassroots level. For the students in Dan Meyer's little classroom world, there is change. A number of teachers use problem-based learning successfully within "the system". How would you or anyone begin to change "the system"? There have been numerous attempts, but I think it is a cultural shift that needs to happen. The system gives a lot of leeway to do what you want (inspite of itself) simply because teachers essentially have tenure. Good reflection though.
Deletea)This video brings up a point that many teachers probably have not thought about. I personally never thought that giving students steps or a specific method for solving a problem could be damaging to their overall math learning. Watching this video has caused me to question the use of the optional textbook scaffolding activity sheets that were used by my associate teacher. Not only was the original question in the textbook broken up into parts, but the scaffolding sheet provided even more guidance to the solution. To me, it seemed like scaffolding upon scaffolding. For most of the Grade 7 students in the class, some kind of guidance was necessary. Many of them struggle with math and are not accustomed to doing math problems any other way. But the scaffolding activity sheet was given automatically and even to those students who perhaps would be able to do without it. I noticed that many of them didn’t even give the problems a good attempt before they came to ask for help. If this is the way students are taught throughout elementary school, how can we help them to eventually become less dependent on step-by-step instruction and questions? How do we help them develop their thinking skills? This can be quite a challenge. I think that for the best results, a different approach needs to be started in the early grades. But even if we start later, the students could benefit greatly and become more successful in math. After watching this video, I will be very careful about using the activity sheets provided by the textbooks. I definitely will not use them automatically. Students need to be encouraged to think and work on the problem instead of just giving up and expecting guidance. This will be difficult since students have learned to expect and receive guidance since they were very young.
ReplyDeleteb)An example question:
You are preparing a party. Your guests include adults and children. Determine how big your cake should be so that everyone gets one appropriate sized piece and there is no leftover cake. How would the size of your cake change if you plan for the children to have a second piece of cake?
Most students love parties and especially cake. This problem would connect math to a real life scenario. When preparing a party, we have to decide how much food will be enough for our guests. When purchasing a cake, we need to decide which size to buy. Usually, people try to minimize cost and buy the right amount of food.
A solid reflection - I wonder is there anything you might challenge about what Dan Meter said? How difficult would it be to run a class the way he is suggesting? What are the road blocks? You sample problem is a reasonable start - I wonder if it is too easy. It would be intresting to see what the students would do
DeleteI agree with Dan’s views wholeheartedly, and I absolutely think that this video can help you become a better teacher. As he talks about in the video, the question ultimately comes down to “what are we trying to teach students in math class?” Is the goal for students to be able to memorize formulas, or memorize steps and other procedures for solving problems which – as far as most students are concerned – are artificial in nature and don’t have much relevance to real life? Is the goal for them to get a good grade on a test because they’ve learned to plug in numbers into a calculator? Or is the goal for students to develop an intuitive understanding of mathematics concepts and how they relate to real life? I would argue for the latter.
ReplyDeleteThere are a number of reasons why I think this approach is superior. First of all, when a student learns a concept, and I mean truly understands it inside and out, then they are far more likely to remember it in the future (and even if they forget, they will probably be able to re-learn it more quickly). Focusing on understanding of concepts rather than memorization of steps is also more interesting, and thus more likely to get students engaged in their work. Finally, this method allows them to see the applications of math to real life far more easily, especially if they are the ones coming up with the steps and the solutions and then comparing them to experimental evidence (such as watching the tank-filling video). When you truly understand the concept behind a problem, then you are able to apply that knowledge to any similar problem, and are able to immediately diagnose which strategies will work and which ones won’t.
Another point that was made in the video, which I also agree with, is that of students having very little patience for problems which are not quickly resolved, and I think that this is a bad trend for society. The most worthwhile problems are indeed the ones that take more time to solve, that require critical thinking, and that entail some potential errors and misjudgments. Focusing on these types of problems is a good way to counteract the impatient mindset of students.
An example of a challenging and interesting question that I have to share is something similar to what I came across when I was in Grade 12. This is a problem that you would do for a Grade 12 Calculus and Vectors class.
The problem revolves around the concept of optimization. The following scenario is described to the students:
Let’s say a man is trying to get from point A to point B as quickly as possible. Point B is 1 kilometer east of point A and 200m north across a river. The question is: what path (or what direction) should the man take to get there as quickly as possible?
In order to answer this students first need to identify what is being asked (minimize the time required to get to point B), and then come up with factors that may affect the outcome. In this case the key factors are the rate at which the man can travel on land (walking/running) and the rate at which he can get through water (swimming, or perhaps a boat, etc…). They would also need to realize that in order to get somewhere as quickly as possible, you essentially want to do two things: take the shortest possible path, and travel as quickly as possible.
There are a lot of factors to consider here, but that is precisely what we want students to think about. They all have the intuitive ability to at least figure out what is necessary as long as they are given sufficient time to think about it.
Once they have determined what is required, then they would move on to figuring out how to make the necessary calculations. They obviously might not be able to solve everything on their own, but as long as they are actively engaged in thinking then they will be learning something and benefiting from the class. The teacher can help guide them along the way.
A very through reflection - Is there anything you disagree with? Also, your problem seems a bit like most textbook problems - I think it might need a littel richer context? Still the river/ land choice is a good one.
DeleteI have been thinking about a similar idea to the one presented in this movie by Dan (more in the context of physics, but still the same concept), but he really put it into words for me. Pardon my divergence (no pun intended) but this kind of problem is being done weekly by my favourite math/physics web-comic, xkcd; readers submit questions, and the author picks interesting ones, then solves them by using creative problem-solving, reasonable assumptions, and a bit of research. http://what-if.xkcd.com/1/ for anyone who is interested in seeing what I think would be an engaging model for how this creative problem-solving could be done.
ReplyDeleteBack to the matter at hand, I think that this form of problem-solving is a great idea to promote critical thinking and problem-solving skills, but I wouldn't pose every single problem I give my students like this. I do not see how we could evaluate students by giving them questions like this on a test or exam, especially if it is inconsistent with other schools, or even other teachers and classes within your school. It seems to me that evaluation via this system would only work in the long-term if every school in your district is playing by the same rules.
In terms of the virtue of the strategy, I agree with everything that Dan is saying; students desperately need the critical thinking skills that define mathematical problem-solving. The methodology he proposes is really well-defined too. As long as a teacher could generate interest from their class, this technique would push students to a better mind-set for approaching mathematics and science.
In conclusion, I like Dan's technique, I would definitely use it, but in moderation. Here is my sample problem:
1) How many cars do you think the GM plant can make in a week?
2) How do you think GM could optimize their car-making process to produce more cars in a week?
-Bryan Gatehouse
A solid reflection. I am not so sure about the consistency issue. Quite frankly there is no consistency among schools in terms of assessment. The teachers need to address the curriculum in a general way. I think Dan is talking about using this strategy for teaching and I suppose it could be used in a formal test f the students had practice. Ultimately, you as the teacher, have to decide what you feel is important base don the Ontario Curriculum. I think the final question could be the same, it's just that students who have not had to memorize procedures, might do better. Good sample questions.
Deletea)
ReplyDeleteThis video was a good choice to end the term with (even though you did attach a link to it in an email early on in the course), primarily because it encompasses many of our in-class discussions – it touches on key questions such as “How can we make math relevant?”, “What is good direct instruction?”, and “How can we use technology/multimedia in our lessons?”. It also focuses on improving the traditional system of questioning and problem-solving implemented by teachers (which we also touched on in class).
As I watched and listened to Dan Meyer explain the root of the problem with the types of questions I grew up answering in elementary/high school (with all of the variables labelled, and all of the steps mapped to parts a, b, c, and d), I began to feel as if I had been cheated out of a quality education – after all, now that I think about it, I’m much better at using the given knowns and unknowns and inserting into a formula than I am at looking at an open-ended problem and narrowing down the key variables on my own. Realizing this, I also feel like I have been cheating the students that I have tutored over the past few years; for example, if I was walking a student through textbook/homework questions, I would always encourage a process of reading the question, writing out the knowns, and using the formula to solve for the unknown. I did not realize that I was inhibiting some very important problem-solving skills (i.e. formulating your own approach without having your hand held the entire way).
As a teacher candidate, I don’t want to ever cheat students in the same way again, so I must try to make a conscious effort to improve the way I ask questions in the classroom (in class discussion, during a lesson, in homework, or on a test). As I consider my next placement and the lesson plans I will make for math and science classes, I will think hard about the types of questions I will ask, and within such questions I will try to eliminate ‘substeps’ and ‘distractors’ from problems so that students will be challenged to determine what variables matter, and what procedure will get them to the solution.
Near the end of the video, Dan had five points of advice for teachers to improve instruction and questioning; apart from his first point (use multimedia), the points he made were in direct contrast to my typical instructional approach (and to how I was taught). For example, I would have never thought to ask the shortest questions possible in instruction/assessment and be more specific in class discussion – on the contrary, I always thought that being able to take given information from longer word problems was a very necessary skill, and I would normally encourage students to follow the same process by which I was taught (as I explained previously). Yet, I see how asking shorter problems (such as, “how long will it take you to fill the water tank?”) would serve as a much better catalyst for development of real-life problem-solving ability (because in real life, the variables are not all spelled out for you, and the problem itself can usually be presented in a short sentence or two).
Overall, for the aforementioned reasons, I think that if I could learn to effectively utilize some of Dan’s strategies I would definitely be a more effective teacher.
Very through and solid reflection. Is there anything you disagree with in the video? How challenging would it be to teacher this way?
DeleteI essentially agree with everything Dan said, but I do wonder if it would be entirely practical to teach this way for every lesson. His suggestions are quite the departure from the traditional teaching formula of (lesson -> examples/discussion -> class work), and I'm really not envisioning a whole lot of 'structure' in these types of lessons - if we only taught this way, we would have to rethink the whole notion of traditional assessment. Essentially, we'd be isolating our classroom from the rest of the school, where teachers use the traditional formula with tests and quizzes and homework (and long word problems) – as a teacher, the greatest challenge would be the fact that you’d be doing away with so many conventions. Personally, this would cause me to constantly question my methods; I’d no longer be validated by the fact that I am following the same path as other teachers in the school. Overall, this style of teaching seems fantastic, but I’m not sure if it would be ideal for practical reasons, especially for a starting teacher (where you are under the watching eye of the principal, parents, students, and other teachers).
DeleteTo incorporate some aspects of Dan’s method into your lessons would be helpful, however; for example, I have used his blog website (http://www.101qs.com/) for my lesson plans during FEII – some have been very successful, others less so.
b)
ReplyDeleteHere are some interesting types of questions like Dan’s water tank question:
1) What could I do to make this bouncing ball bounce higher?
Video: http://www.youtube.com/watch?v=3jI57WMOzbU
2) How big could a person be before this bath tub overfills when they get in?
Image: https://www.google.ca/search?q=bathtub+diagram&safe=off&espv=210&es_sm=93&source=lnms&tbm=isch&sa=X&ei=4CiIUpWSN-TB2QXBzIGIAQ&ved=0CAcQ_AUoAQ&biw=944&bih=951#es_sm=93&espv=210&q=bath+tub+with+water&safe=off&tbm=isch&facrc=_&imgdii=bQcz1izXp83UyM%3A%3BW2WqCPwc1MHYsM%3BbQcz1izXp83UyM%3A&imgrc=bQcz1izXp83UyM%3A%3BEguiOjICgW0CXM%3Bhttp%253A%252F%252Fthedirty30sclub.com%252Fblog%252Fwp-content%252Fuploads%252F2012%252F11%252Fbathtub.jpg%3Bhttp%253A%252F%252Fthedirty30sclub.com%252Fblog%252F2012%252F11%252Fthe-mental-asylum-test%252F%3B540%3B360
3) How much time could Columbus have saved if he took the most direct route?
Image: https://www.google.ca/search?q=columbus+trip&safe=off&espv=210&es_sm=93&source=lnms&tbm=isch&sa=X&ei=hiuIUoX9FcqU2AXRpICYDg&ved=0CAkQ_AUoAQ&biw=944&bih=951#facrc=_&imgdii=_&imgrc=o1b6kklNd4BmCM%3A%3B8X-IkrN852cs9M%3Bhttp%253A%252F%252Fer.jsc.nasa.gov%252Fseh%252Fneatmap.gif%3Bhttp%253A%252F%252Fer.jsc.nasa.gov%252Fseh%252Fvoyages.htm%3B546%3B244
Interesting examples!
DeleteThe ideas in this video can definitely help people be better teachers. It's good to see someone taking a 'misunderstood' (dull) subject like math and re-branding it as entertaining, relevant, and doable by anyone who is capable of analyzing a situation around them.
ReplyDeleteDan Meyer states that students today are "impatient with things that don't resolve quickly" and many textbooks lead students through a problems with easy to follow steps and plenty of information so right from the start not much thinking is required. This problem becomes worse once students understand how to find answers in the textbook without doing any critical thinking; in effect making them experts at "decoding a textbook", a skill that may never be used in the real world. Heavy reliance on formulas and "molded" questions is also to blame.
During my previous placement I had the opportunity to teach students how to solve some "molded" word problems (math patterning). It was obvious that most of them were intimidated by too many words and given data. I had to walk them through how to disassemble a problem and figure out which data was immediately relevant and which could be ignored. Once the overall problem was simplified into just the first required step they had less trouble starting it. So in effect I had to do what Dan is talking about and provide a problem that's general and has limited information. Students had to imagine a situation, imagine what needs to be done initially, and think about what we need to do the fist step only. Once they got that far they could start evaluating where they are now, what else they need, and where they can find the next bit of data. I didn't have the opportunity to create videos and multimedia presentations of how the problem looks in real life, but the examples I used did consist of normal people 'trying to do something' and I did get students to provide lots of ideas and solutions. Just making them consider how they would react when faced with some real-world problem got many of them involved and interested.
I did scaffold the steps initially so students could easily follow along and do similar problems on their own. Dan says this makes students good at solving textbook problems but not real-life problems but I think this is not entirely accurate. Sure his way of teaching sounds better but I think I had some success in teaching my students how to dissect a huge paragraph problem appropriately. Should they be bombarded with data in the future I hope they will be sufficiently skilled to step back, break the problem up, and tackle it step by step. While there are times when a problem is first conceived and can be summed up in one sentence (and include no data) there might also be times when you jump into the middle of a project where the problem is already somewhat developed. I don't think it's entirely useless to be good at taking lots of information and knowing how to sort it before starting.
While I do like his method of posing problems it might not always be practical. Creating videos, resources, and using multimedia is time consuming. Mastery of various software might also be required. Finding techniques that your particular students will enjoy might also be challenging. For a beginning teacher all of this may be intimidating, especially when following established curriculum is hard enough. I'd love to rebuild textbook questions but I feel like this will be exhausting if done regularly (and by someone with limited experience).
Another challenging question which student can understand might be: During winter break you start shoveling driveways. On the 1st day you only shovel one driveway, but each day you get better. By the 7th day you're shoveling four driveways. How much better are you getting each day? (side note: having trouble making a interesting question that doesn't involve physics...)
Thoughtful detailed reflection. I get the sense that your heard Dan but you are not really buying the approach and I wonder what you think the value of current words problems are as they are approached by teachers in a step-by-step fashion. I suppose the key is how much scaffolding your provide and whether you ever remove the scaffolding. Solving open-ended, messy problems is probably a more important skill today than it was in the past.
DeleteI really enjoyed the ideas presented by Dan Meyer in this video. I agree completely that too often math is viewed by students as “textbook interpretation” rather than seeing it as “the vocabulary of your own intuition”. The way the education system is now, people only come to this realization after studying math to at least the post-secondary level. That means pretty much 99% of people never get to appreciate math in this way. His speech was thought provoking and the points he raises are useful for math teachers to think about. It reiterates the idea that students should really be learning through self-discovery, not the spoon-fed methods that are used in today’s classrooms. The molds of traditional math teachers should be broken and instead embrace (even if it is just partially) these new learning techniques suggested by Meyer. To truly relate math to real life, the question can’t already be established. In the real world the problem must first be formulated before a solution can be attempted, and students should be taught to do exactly that.
ReplyDeleteIt’s important to recognize that as amazing as this technique sounds it may not be entirely practical. To let the students learn through the formulation of the problems will often be time consuming, and obviously there are curriculum expectations that need to be met. That isn’t to say that it would be impossible though. I think teachers should try to encourage student learning through these methods as often as possible. Hopefully in the future more of the expectations will be specifically tailored to mathematical exploration in this way. Also, the fact that every student’s response will be different will make assessment even longer than before (as talked about in the last blog post it was already at 3+ hours per night). Due simply to the time constraints with regard to marking this method could not be done on a daily basis. With time and a changing education system we may be able to accommodate for this, but in my opinion it just wouldn’t be possible on a daily basis as of now.
A more challenging/interesting problem, like Dan’s “Tank Filling” one, could be “When will my ride get here?” Students often ask this question and would have to establish known/relevant information (how far the car needs to travel, its average speed, when it leaves, does it make other stops) and then through detailed thought they could figure out how to calculate the answer.
A thoughtful, balanced post. I was wondering about your time issue, though. Do you just move on and "cover" curriculum regardless of whether the students learn it? It's a risky game to play and I am not clear why assessment should take longer? Good sample question BTW.
DeleteDan Meyer makes some important points that I will definitely take into account as I teach math. I agree with this notion that the world needs more patient problem solvers. Students often try to find a formula to use so they can plug in the given numbers and get an answer. I also noticed that when some students come across a question in which the content is new to them because they haven’t been given a similar example, they freeze and they’re not willing to attempt it. I have heard students say that they can’t do a question because the teacher didn’t show them how to do it, even though they have been taught everything they need to know to solve the question and just need apply that knowledge. Maybe teachers have been giving students too much so students haven’t learned to use their intuition when approaching problems. It also doesn’t help that textbooks refer to examples to help a student solve a practise problem because it makes them think they need to be given instructions and a set of steps to solve problems.
ReplyDeleteI have never before considered asking the students the shortest question I can. My goal has been to have students understand the problem so they can determine what to do in order to solve it. His approach seems more interesting and one I will definitely try. I have students break down the problem by writing down what is given in the question and what they need to find. If I ask the shortest question by asking just what they need to find, it gives students the opportunity to first think at ways in which they can solve the question and then determine what other information they need to solve it. This lets them build the problem, which I would think gives the students more understanding of the problem.
Being less helpful is a difficult thing because I think students just expect answers and to be shown how to answer a question. When I help students, I try to guide them in the right direction by using what they’ve learned but not telling them what to do. It’s usually a hit or miss and I think it’s because they haven’t learned this way. I definitely want to implement this approach of teaching math and creating patient problem solvers but I think it might take more time for students to develop these habits.
Examples of problems:
How much carpet would cover the whole floor in an apartment? (Provide a map of the apartment and the dimensions the carpet could be purchased)
How long will is take a container of water to empty if water is dripping out of a small hole? (A video can be shown of water dripping out at a constant rate).
Joe Anne
Good balanced reflection AND can you see anything that might be challenging with respect to Meyer's video? Good first example, although you might make it even more open ended - Tiling the school for example. The second problem is really like Meyers (Smile)
DeleteThe idea behind this question is two folds. Can the video help me be a better teacher? The short answer is absolutely, but we also have to consider if the current teaching environment and curriculum would allow one to maximize the idea behind this video and the answer to this is no. There is no hiding that we leave in the 22 minute sitcom age. The fact that today’s students cannot wait for 1 minute while a computer is loading is a great example of our reality. I am sure that all teacher candidates would love to become the best teachers they can be and we have grand ideas about how we will change the life of students we will touch. However the reality that we deal with is not the same. The video uses 7 minutes and 30 seconds to show a water tank filing up, this is great in many perspectives the most important of which is to start a conversation about application of mathematics. But we have to ask ourselves at what expense did we arrive at our destination? Do we teach everything that is required, but at best with half effort, or do we put full effort with no child behind and then guarantee that some of the materials will not be covered. As much as this video excites me about the potential a teacher can have, the sheer reality is that it disappoints me that I cannot implement to its full benefit.
ReplyDeleteOther examples that can be applied is measuring the distance of a thunderstorm as it happens
Shime - it seems like you have given up before you have even started teaching. You don't really delve into how Meyer's ideas can help and you make the bold (and unsupported ) claim that the the current teaching environment will not permit Meyer's methods to work. And yet many teachers have followed this approach. Keep in mind that most high school teacher have complete autonomy and tenure to do what they want. If you really beleive you can't make a difference, I would reconsider the profession you are in. Teaching is WAY to hard to maintain the status quo.
DeleteMath is a subject that a majority of people dislike, if they don’t understand what is going on immediately, then many check out quite quickly after they stop understanding. Through observation as well as personal experience I notice that this happens more often than not with word problems. When they are worded even slightly different than the practice/sample problem then this can cause panic in a lot of people. By using Dan Meyer’s technique of patient problem solving, you are using different parts
ReplyDeleteDan Meyer definitely made me rethink textbook problems and how students can be interested in word problems instead of fearing them! He was able to use simple, open ended questions to start getting all of his students to have an input. Within this problems, he gets his students gears going and makes them real think about real life questions. He lets them guess the answer, he asks them how they came up with the answer, he ensures that every students answer is valuable, by writing it on the board. He is able to interest every student in the class by using real life examples and with his water tank question, he takes real life example to a new level by video taping the activity actually occurring.
In conclusions, I would say that his teaching techniques would make me a better teacher. Just by watching his video, I kept thinking to myself, "Man, I wish my teachers taught me like this." He is literally able to bring material some may find boring, to life. He makes it easy and fun, and this is something that I want to try and bring to my students. To get them excited to come to math and want to participate on a regular basis.
The interesting question that I found, is very similar to Dan's but also involves a real life situation. The textbook question is: "A goldfish need 1000 cubic inches of water to live in. An aquarium is 20 inches in diameter. How many fish can live in an aquarium?" This can simply be shortened to "How many fish can live in an aquarium?"
This link can be shown: http://www.youtube.com/watch?v=sI9OTeTGcrw
Or you can show them a live feed of an aquarium.
Some students will try to count all of the fish, some may figure out how many fish are in one section and multiple by the different sections. But at this point, everyone should be giving you answers and invested in the lesson.
Good post - decent overview and good sample question. Is there anything about the Meyer video that you would find challenging?
DeleteInitially I thought of many words to describe what Dan Meyer was preaching in this video. Blasphemous, crazy, illogical, impossible. I suppose having these preconceived notions of what a math problem should be was so drilled into my head that I couldn't accept how ridiculously flawed they are. I agree with almost everything he said in the video. The way students are essentially taught to do nothing but memorize the patterns in math problems is doing them a disservice. We need to be asking high-order questions that ask students to reflect, collaborate, and create.
ReplyDeleteDan's example of the water tank problem is ingenious in that he engages students curiosity in order to motivate them to solve the problem. By providing minimal assistance it is up to the students to come up with both the problem and the solution to the problem. This is excellent training for the realities of adult life, regardless of whether they pursue math or sciences after school. In the real world there is no set way to solve a problem, indeed many times the problem needs to be understood or deciphered before it can be solved.
One example of a problem similar to the water tank example would be the following:
An airplane is flying when all of a sudden the engine fails and the plane is left without power. How long do the crew have to parachute out of the plane before it hits the ground?
This question would require students to think about the initial speed and altitude of the plane, as the time required is dependent on these two factors. Students would also need to be able to recognize that the fall of the plane is governed by two vectors, that of the forward motion and that of gravity.
Love your question and your post is reasonable solid, although you appear to agree with everything Meyer said - you had doubts before. What would be challenging about Meyer's approach?
DeleteIn high school, specifically in the senior grades, I was always taught how to solve a word problem by reading it and picking out the key points. I found that teachers used this method in Math as well as Science and while I was learning, I thought it was a great way. All through high school and at the beginning of university, I was constantly just trying to plug different numbers into the corresponding formulas but not really knowing why. Students are trained to become robots with certain types of math and science problems and are just learning how to plug the numbers in and solve the equation but not really how it applies to the problem itself. Listening to Dan Meyer speak made me realize that we need to teach students how to solve these problems step by step and just give them the basic information which, in the end will be more valuable for them.
ReplyDeleteWord problems that are being created in the textbooks lay out all of the steps for the students and give them all of the information that they will need. We are taking out the skill of really having students think about the answer and how it relates to the question being asked. Many of the problems given in textbooks are also very irrelevant to students, which needs to change. If we start giving our students word problems that actually relate to their lives and cause them to think about real life problems, there is a better chance that they will engage themselves in the lesson. Not only do we need to create real life word problems but we also need to ensure that we have not been paving a path for the students from the start to the finish of the problem.
Overall, I think that in order to get students to actually understand the questions they are doing, we need to take away all of the sub steps that are given and force students to figure these steps out on their own. I think that this method of teaching will make me a much better teacher but it will take time to adapt to. Seeing as I grew up just filling in numbers into a formula, I think that it will take me some time to think of different questions to ask and figure out how I can relate them all to the course. In the long run though, I think that it will be well worth it and will hopefully prove to be very beneficial for the students.
Some of the more challenging and interesting questions could be:
At Christmas time, how many strands of lights are needed to decorate the outside of your house?
Some of the factors that the students might think about are; How big is the roof? Are there different levels on the roof? Do you decorate the front door? Are there columns in front of the house that get decorated?
You want to have a paper airplane competition in the classroom. What is the best way to make a paper airplane so that it efficient as possible?
Students first need to determine what the word efficient means in the problem. Does it mean that it flies the farthest, it flies the fastest, it flies the straightest or something completely different? This question is up to the students’ interpretation and imagination.
Solid, balanced, detailed post. Two interesting examples at the end. Is there anything about Meyer's video that makes you feel a bit uneasy? Like it might not work?
DeleteWow. Dan Meyer really gives the insight of what NOT to do as effective teachers. As a highshcool student, I've had MANY great teachers who made my life simple by repeatedly giving questions that were just plug and play or followed a certain process that we were taught, or easily figured it out. Those courses always got me the high marks I needed to get into university. But when university hit, I really wished I had learned WHY we did things the way we did, and why we used certain formulas or process to solve certain questions. There I was, in UofT, left to learn most of highschool math on my own so I could pull through with decent marks in my first year math courses.
ReplyDeleteI believe the different strategies Dan Meyer talked about would defnitely help me in becoming a better teacher. I cut out all the 'extra' wordings of a question from a math word problem and gave it to my cousin. It was amazing to see how creative he got with the question, he was asking questions he would have never thought of if he was given the entire word problem where he had to pick out the numbers and find a formula they fit in.
I think a great question would be getting the students to think about how they can create an enclosed figure with the maximum volume. I would get the students to work in groups and give them each certain sized Bristol-boards. The idea is for them to work together to make a figure, which ofcourse has a side that I can open to put stuff in. The goal is to fit the most amount of chocolate/candy I'll have to test their figure. They would then be required to estimate the volume and find ways to calculate the actual volume of their figure. Ofcourse, the group with the maximized volume would get to keep the goodies!
Good post - good reasoning. I wonder how interested students would be in your question though. Are there any point Meyer made that might be challenging? Hard to apply?
DeleteAlisa (posted by Robin)
ReplyDelete"I do not think that this video will change my teaching style as the content of the video are ideas I have already been considering since I made the choice to go into the teaching profession. I think that if the lesson is not relevant to students and cannot jump off the page and come alive a teacher can just accept that their students will not pay close attention. Students have the mentality that they only want the formula and to be shown how to use it, this is what they believe math is. With all the knowledge in the world available at the finger tips of students why would they not believe that knowing a formula and how to use it is enough for them to be able to learn the next concept. So no, my teaching style will not change, I will continue to work to make my lessons come alive for students and seeing this video reassures me that I am on the right path to connecting my students to the material.
I think that a similar idea to the filing of the tank could be how long it takes their iPhone battery to charge from an empty state. I will give them the capacity of the battery and the rate at which it will charge and have them calculate the time it will take. Then we will actually charge the phone battery from an empty state and see how accurate the calculations were."
Alisa - Yes but Meyer is saying much more - Minimum words in questions, less scaffolding, less step-by-step. You have zeroed in on the "relevance" part but I think Meyer was focussing on the challenge part and the open-endedness of questions. Like your iPhone question - I wonder how much math they would use for that question?
DeleteI do believe so. It showed me another method of teaching. However, most importantly, not only does this video help me become a better teacher, but it’ll also benefit the students. This method of teaching allows students to become problem solvers, which is not just used in math, but everywhere. The students are able to apply this method of thinking to any other problem, whether you use equation A or equation B. With the “plug and chug” method, it won’t allow students to think beyond the problem. How can we create strategic and analytical students with this method?
ReplyDeleteI really like Dan Meyer’s idea of “Let students build the problem”. We aren’t supposed to spoon feed them and tell them to think in certain steps. My first placement AT gave me an advice that I feel relates to Dan Meyer’s idea. He said that we should spend more time lesson planning and less time teaching. He would give the question to class, and then give them some time to converse to figure out how to build the problem. This creates thinkers of “should we do this?”, “what happens if this happens”, “are there other factors that might affect this problem?”, “do we really understand the problem?”
If the students really understand the problem…then they can build the problem. They don’t really understand the problem when it’s all just plugging information into a formula.
However, I wouldn’t use this idea all the time. I would probably use Dan Meyer’s ideas in the middle or the end of finishing the topic/strand. I believe this because you do need to teach the fundamentals first and guide them a bit. The students need to understand the basic concepts and background first. You can’t ask “Kew’s age is a multiple of 6, and Paulo’s age is a multiple of 9. They are the same age and both coach their daughter’s baseball teams. How old might they be?” (took from a math textbook from my first placement. I forget which textbook it is, sorry.) and expect the students to build the problem on that and know that finding the LCM is the most efficient way to solve this problem if they don’t even know what a multiple is or how to find the LCM. In saying all of this, I would teach LCM first and guide them, then throw in a question like this and use all of Dan Meyer’s steps.
A challenging question: You give a cashier $30. How much change will you get back?
The students have to build on the problem. What did you buy? Is $30 enough, or do you need to pay more? Is there tax included? Is there a discount?