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Archive for the ‘Learning Through Problem Solving’ Category

I was in a meeting this morning, and we were discussing how to connect literacy across the curricular areas. I flashed back to high school, and a great short story we read. I started wondering whether I could use Shirley Jackson’s “The Lottery” in a math class. Then I began to wonder if a 3773 short story would fit with Dan Meyer’s 3 Act Mathematical Story Telling.  Here’s what I would try with this story.

Act I

Have students read The Lottery, by Shirley Jackson. Ask them what they wonder about. They will probably wonder about lots of things non-mathematical. Eventually they might wonder (Spoiler Alert!) what Tessie Hutchinson’s chances of winning the lottery were.

Act II

Ask the students what information they require to be able to answer the question. If they wonder how many families were in the first draw, you can have them look back through the story and count, or tell them that there were 16. They will also need to know that there are five members in the Hutchinson family in the second draw.

Act III

Students work it out. I still need to come up with a better way to reveal the answer, which is that Tessie had a 1 in 80 chance of winning the lottery.

Sequel

If this lottery has been going on all of Old Man Warner’s life, what is the probability that he survived to age 77?

Edit

Kendall reminded me that I started with connections to English class, and I meant to close with connections to English class. I would totally do this in collaboration with my school’s English teacher.

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I support mostly high school math teachers. I work with colleagues who support K-9 teachers. Last week, I eavesdropped on two of them as they tried to come up with a 3 Act Math Story in style of Dan Meyer that would apply to division 1 students. This week’s Parks and Recreation may have provided us with one. You be the judge.

Act One

Click on Andy to play the movie.

Act Two

Find out what the students wonder about and what information they will need to answer their questions. I suspect they will wonder whether it will really be a billion nickels. Depending on how young a group you give this to, they may need to know that nickels are worth $0.05 or that there are 20 of them in a dollar. Canadian kids may need to be told that those wacky Americans use paper for $1 instead of coins.

Act Three

The good folks over at Parks and Recreation didn’t film the right answer for us. If anybody wants to withdraw 20 000 nickels, stack them up in some way, film it or photograph it, and send it my way, I would appreciate it. Otherwise, this is the best I can do. Give them a photo and some information.

$1000 = 20 000 Nickels

Sequels

Could Andy hold 20 000 nickels? How much would they weigh? What size container would he need? Would they fit in his trunk? If he piled them all in a giant stack, how high would they reach? What about a billion nickels? How much would they weigh? How high would they reach if all stacked up?

Edit (June 16, 2013) The story about Samsung paying off an Apple lawsuit using truckloads of nickels is a really nice sequel to this one. Some conversation on Twitter last night led me back here and I realized I never updated the sequels to include links to it. I have been using screenshots of this site, which I believe this is the origin of the story. The Humor>Satire also clearly indicates it’s a fake story. Timon Piccini sent links to this story last night, which isn’t as obviously fake. This morning, while updating, I found the following YouTube video purporting to be the 30 trucks delivering the nickels. I’m no world traveler, but it looks awfully European to me. It still might be fun to run it by a class full of kids.

 

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While in the Pennsylvania Convention Center for ISTE 2011, I came across this intriguing piece of art wedged under an escalator.

I have since learned that the piece of art is by Mei-Ling Hom, and is called “China Wedge”. It  is composed of many Chinese cups, bowls and spoons wedged into a space under an escalator. It appears that the space is not entirely filled with cups, bowls and spoons, but they go about 2.5 deep all around. The sculpture, which was commissioned for the Center’s Arch Street Concourse, pays homage to Philadelphia Chinatown, which is adjacent to the Convention Center. (Source (Spoiler Alert):  http://philadelphia.about.com/cs/artmuseums/a/paconvention.htm)

I took some photos, because I wanted to turn this into a math story. I didn’t get all the shots I needed, so I have to give a big thank-you to Max Ray, who went back and took care of the Act II photos with his girlfriend, who happens to be a photographer and got great shots. The Act II photos containing referents and shots of Max are all the work of Kaytee.

Here is my attempt at a three act math story as described by Dan Meyer here.

Act I – The Video

Act II – Some More Information in Photos

Act III – The Answer
  • Word document containing all the information I could find about the China Wedge.
Sequels
I struggle with this one.  Any feedback would be greatly appreciated. I’ve only one idea so far.
  • If the entire area under the escalator was filled with cups, bowls and spoons, how many more would have been needed?
In his last one of these, Dan asked whether this broad outline is enough for teachers to go by. I know it is enough for lots of us to run with. If you need more details about how to make this work in a classroom, contact me and I will spell it out a bit more. I would present it in much the same way I discuss in my learning through problem solving explanation.

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Note: This post is part 7 of 7.  Click here for Part 1 2 3 4 5 6.

This post will describe my afternoon at a local high school, trying learning through problem solving lessons all day.  The morning is described in part 6.

Class #4 – Math 10-4 Math 10-4 is a Knowledge and Employability math course.  It is intended for students with low IQ’s, who are unable to complete our regular math program.  Students who go this route in High School receive a certificate of achievement, rather than a high school diploma.

LTPS – Bucky the Badger

I got the Bucky the Badger video from Dan Meyer when he did a session in Edmonton in March.  It’s a funny video, and the math involved is not complicated.  I thought it would be perfect for this class.

I followed the same process, whereby I played my altered question video.  I asked them what they wondered about.  They were all over the place.  They wondered why he had such poor pushup form. They wondered why that mascot costume was so ugly. It was taking every ounce of patience I had for wait time, because I was starting to think they were never going to get to any kind of mathematical question. Eventually, they decided they might be a bit interested in how many pushups the mascot did. Several were sure it was 83, so I explained the process of doing the pushups again to them.

If Wisconsin scores a touchdown, he does 7 push-ups because their score is now 7.  If they follow that up with a field goal, he does 10 push-ups, because their score is now 10.  At this point he has done 17 push-ups in total.  Then if they score a touchdown, Bucky does 17 push-ups, because their score is now 17, bringing his push-up total to 34. Then I asked them again how many they thought he did.  Guesses ranged from 83 to 300. I couldn’t get them over 300 as a guess, so I let them loose, thinking I’d explain it later.

I asked them what further information they required.  They said they needed to know how the Badgers got to 83 points.  I gave them TD, FG, TD, TD, TD, TD, TD, TD, TD, FG, TD, TD, TD.

They needed much help.  This was my first experience in a Math 10-4 class and I have no special education training. They were, however, engaged and trying hard with it. They were neat kids. Some started by building the chart from the video on the board. Many began to get answers in the ballpark of the correct answer.

One group was done, and had it right.  I gave them the extension: Does it matter where the two field goals occur?  They concluded it did.  So I asked them to figure out where to put the two field goals so that Bucky ended up doing the most possible pushups.  They concluded that the two field goals should come at the end. I thought they were wrong, not because I had done the question before, but because a group of teachers told me that when I gave them the same extension in a session I did. I told the kids they were wrong for that very reason.  They proved to me that they were right, and were pretty proud to have bested some math teachers. I showed the class the answer video, and they were pleased to have been right.

The biggest success here was a girl who initially asked, “who cares?” when we came up with the question. She appeared totally disengaged. During the activity, however, she noticed that a pair of boys had made an error in a chart they were producing on the whiteboard. She came up, grabbed the marker from their hands, fixed the error, and went back to her desk. She accidentally got engaged in the problem. It was a major success.

Curricular Fit: 100% – Pattern Extension is in Math 10-4.  This activity could also be used as an introductory activity in Math 20-1 Sequences and Series.

Class #5 – Math 20 Honors This class was an honors level grade 11 math course.  Most of these kids will go on to AP Calculus next year. They are currently on our old curriculum, so they are following our Math 20 Pure course.

LTPS – Spider and Grasshopper

I used the spider and grasshopper question I found on Andrew Shores’ blog.  I started by giving them this image.

This group was astute enough to point out that I was manipulating them because there really was only one question they could ask here. They humored me, though, and got to work on figuring out when the grasshopper would meet his demise.

They worked hard on it. I had groups making charts, others trying a logical proof. Some groups were trying to find and solve a system of equations.  What they all realized eventually, of course, is that the grasshopper never hits a spider web. I was quietly telling some groups to prove this. Then I made a mistake. I got the whole class’s attention and pointed out (without allowing them to discuss it) that there was no answer, and therefore the question had shifted from being “find out when they meet” to being “prove that they never meet”.  This was far too teacher directed, and some quit on me after hearing it.

What went well: Several students were working hard on a general proof, and enjoying it. A few of them had it generalized nicely.

What I could have done better: Lots. I should have stopped them and let them have a discussion whereby they could have discovered that they were all finding no intersection.  That discussion would have led to the next question, rather than me giving it to them.  I also wrote my general proof on the board at the end, rather than having one of them do it. That’s one place where this whole process tends to fall apart for me. I fall back on my old habits of wanting to do too much that is teacher directed. I need to remember to turn more over to the students.

Curricular Fit: 100% – Logic, reasoning, and proofs are in Math 20 Pure. This will also have good fit in Math 20-2 under the new curriculum, where logic and reasoning are major topics.

Class #6 – Math 20 Pure This was a regular Math 20 Pure class.  These are our pre-calculus students at the 11th grade level.  Most will go on and take calculus in grade 12.

LTPS – Mozart’s Dice Game

I used the Mozart’s Dice Game, as described in an earlier post on my blog. We created and listened to a minuet using the site linked in that previous post. Then I told them that I was pretty sure that no one had ever heard that minuet before. I asked them why I was so confident. They suggested that there must be a lot of them. I asked them how many. They didn’t know, so I suggested they figure it out. We had a hard time with the part of the process whereby they are to come up with a guess, because the number is unfathomable. The correct answer is 129 octillion, 629 septillion, 238 sextillion, 163 quintillion, 50 quadrillion, 258 trillion, 624 billion, 287 million, 932 thousand and 416 (Thanks Wolfram). In the end, their guess of “lots” was pretty accurate.

They got to work, and most made really nice progress and ended up inventing the fundamental counting principle themselves.

What went well: This class enjoyed the problem, and made really nice progress with it.

What I could have done better:  One student had answered the question before the rest of the class had even had time to figure out what the question was.  Some quit because they knew someone else had already done it.  I should have given him an extension (how long would it take for the whole world to listen to these, if we divide it up equally) to keep him quiet and busy.

Curricular Fit: 0% – this does not fit in Math 20 Pure.  This can be used in our curriculum, however, since the fundamental counting principle is in Math 30 Pure, Math 30-1, Math 30 Applied and Math 30-2.

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This one didn’t get much love when I threw it out on Twitter under the #anyqs hashtag.  I still think it has nice use in a Math 10C or 10-3 class on measurement.

It’s a bit of a shaky video, so I paused it at the crucial parts.  If it’s still hard to tell, I’m sitting on the side of the road.  I have about a half tank of gas.  My truck tells me I can go 163 miles until empty.  The sign I’m stopped at tells me it’s 16 km to Leduc, 137 km to Red Deer, and 275 km to Calgary.

Let’s fit this into the 7 steps of LTPS.

  1. Play the video. 
  2. Ask the students what they wonder.  They will likely wonder where I can get to before running out of gas.  I was wondering if I could make Calgary.
  3. Ask them to guess how far I can get.  Near Calgary?  Past Calgary?
  4. Ask them what other information they require to solve the problem.  They will likely need some conversion factors.
  5. Students solve.  Teacher circulates and offers support and/or extensions.  Extensions could involve litres per 100 km or miles per gallon based on the fact that I have a 100 gallon tank in that truck.
  6. Students share answers. The teacher can’t play an answer video, because I didn’t actually drive until I ran out of gas.  Sorry I didn’t take that one for the team for the sake of math education.
  7. Teacher summarizes learning with a brief wrap up on metric to imperial conversions.

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Since I was filling my daughter’s sandbox anyhow, I decided to film it and turn it into a math problem.  I’ll fit this into the 7 steps of the Learning Through Problem Solving approach discussed previously on this blog.  Here’s how to make this work in your classroom.

  • Play the question video.
  • Ask students what question they want to explore.  They will likely come up with “Does he have enough bags of sand?” or “How many bags of sand is he going to need?”
  • Elicit student guesses.  Students may assume the answer is 20, because that’s how many bags are stacked up.  You should tell them that the guy in the video is a notoriously bad measurer, and he could have way too many or way too few. As a class, agree on a range of reasonable answers.
  •  Ask the students what further information they need to answer their question.  Provide them the measurements of the sandbox, and the information from the bag of Play Sand as shown on this handout.
  • Allow students to work on the problem.  Students who finish could be given an extension like this Google image of a local playground.  Tell them that the sand was put in at a uniform depth of 15 inches.  Ask them how many bags that would take. I would use a park near their school that they might remember playing in as a child.
  • Share student solutions.  Have students share solutions with other students, or with the whole class using a document camera or chart paper.
  • Play the answer video.  Discuss sources of error.
  • Summarize what was learned about volume.

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I attended a presentation by Nelson today.  Nelson is the only approved publisher for our new Math 20-2 course.  One of the things the presenter shared with us is a perfect fit for the learning through problem solving I have been discussing on this blog recently.  I don’t know why, but I feel like I should compliment a publisher who gets it right. Here’s the goods.

Course: Math 20-2

Unit: Inductive and Deductive Reasoning

LTPS: The Mary Celeste

1.  Play the first 2:42 of this video.

2.  Ask the students what they wonder about.  They will most certainly wonder about what happened to the crew of the Mary Celeste.

3. Ask them to think of a theory on their own, and write it down.

4. Ask them what further information they require.  They will most likely suggest that the video didn’t give them all the details.  Provide the following facts, which I have taken from Page 5 of Principles of Mathematics 11 (Nelson, 2011).

  • The ship’s hull was not damaged.
  • No crew or passengers were on board.
  • No boats were on board.
  • Ropes were dangling over the sides of the ship.
  • Only one of the two pumps was working.
  • The forward and stern hatches were open.
  • Water was found between the decks.
  • The only dry clothing was found in a watertight chest.
  • Kitchenware was scattered and loose in the galley.
  • The galley stove was out of place.
  • No chronometer or sextant was found on board.  Both of these instruments are used for navigation.
  • The ship’s clock and compass were not working.
  • The ship’s register was missing. The ship’s register is a document that notes home port and country of registration.
  • The ship’s papers were missing.  These papers could have included a bill of sale, ownership information, crew manifest and cargo information.
  • The cargo, 1701 barrels of commercial alcohol, had not shifted.  When unloaded in Genoa, 9 barrels wer found to be empty.
  • The alcohol was not safe to drink, but it could have been burned.

5. Let the students work in pairs to formulate a conjecture about what happened.

6. Have students share their theories with other groups. Normally the answer video would be played here.  Unfortunately, the answer to this one is unknown.  The best we can give them is this more detailed video. You could play the one below in its entirety.

7. Teacher summarizes what a conjecture is.

Have fun with this one.

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Note: This post is part 6 of 7.  Click here for Part 1 2 3 4 5 7.

On May 5, I spent the day at a local high school. The Math Department Head asked me to come to her school and model learning through problem solving in various classes throughout the day.  In the end, I did 6 LTPS examples in 4 periods.  It was a lot of fun for me to be back in front of kids, and I enjoyed having the opportunity to try out some of the things I’ve been playing around with.  The request to visit the school, and the subsequent planning I did was the inspiration for this series on my blog.

Here is what I did, and how it went.  It wasn’t perfect. This post will describe my morning. The next one will describe my afternoon.

These will end up being long enough posts without me explaining the lessons in detail.  I will eventually put full lesson packages for all of these on this blog.

Class #1 – Math 24 For those readers not from Alberta, Math 24 is a non-academic math class for 11th graders.  It is the minimum requirement for graduation.  Topics include ratios, consumer math (mortgages, loans, budgeting), earning an income, tax, scale diagrams, and statistics. It is the last year for this course, since our new curriculum reaches grade 11 next year.  This course will be replaced by Math 20-3.  For more information on our revised curriculum, click here.

LTPS – Which liquid is most expensive?

I started the day with a problem I’ve used before.  This picture is all over the internet if you google “Price of HP Ink 45”.

Since I couldn’t find prices for the 3M PF-5030, I modified it slightly. I started by giving pairs of students the first page of this handout (printed in colour).

I asked them to cut out the pictures.  Then I asked them what they wondered about. They wondered what some of the pictures were. They wondered if they were all liquids, and then they wondered how much liquid was in each. Very shortly, they began to wonder which cost more.  That’s where I was hoping they would go.  I asked them to sort them in order from least expensive to most expensive.  Then I got them to get up and take a look at how all the other groups had ranked the liquids.  Most had the water being the cheapest, and the oil being the most expensive, but there were variations.

Then I asked them how we could tell for sure which was the most expensive. They told me they’d need to know how much each one cost, and how much liquid was in each one.  I gave them the following information. As I was giving them the price of oil, it was in the process of dropping 10% in one day, but I digress.

  • Blood $200 for 500 ml
  • Perfume $30.29 for 50.3 ml
  • Oil $112.54 for a barrel (158 987.3 ml)
  • Penicillin $16 for 100 ml
  • HP Ink $35.99 for 42 ml
  • Red Bull $3.29 for 59 ml
  • Vodka $20.99 for 750 ml
  • Water $1.99 for 500 ml

They got right to work on the math.  The classroom teacher and I circulated and helped keep them going in the right direction.  In the end, almost all were able to re-rank the liquids correctly.  Many were surprised at the results.

What went well:  The students were great.  They were engaged, enthusiastic, and had fun with it.  This was a great class.

What I could have done better: I may have helped some of them too much when they got to the calculation part.  They were struggling with the unit rate.  I probably should have let them struggle a little longer.

Curricular Fit: 100%.  Unit rates are in Math 24. It could also be used in Math 10-3.

Class #2 – Math 10-3 Math 10-3 is a class for non-academic 10th grade students.  It is intended for students who failed Math 9, or students who know they want to pursue a trade.  It is designed as a trades and workplace math course.  Topics covered include measurement, geometry, ratios and earning an income.

LTPS – The Water Tank

I borrowed Dan Meyer’s water tank problem for this class.

I began with the video.  It didn’t go long before they were complaining about being bored, and starting to be disengaged.  I stopped the video and asked them what they wondered about.  The wondered what it was.  They wondered why some guy was filling it up and recording it.  They wondered how much water it would hold. It didn’t take long for them to wonder how long it would take to fill up.  I asked them for guesses.  The range was between 5 minutes and 30 minutes.  Then I asked them what they would need to figure out the answer.  They asked for what shape it was, how tall it was, how long one side was, what the area of the base was (which I gave them, because that’s a real tough calculation in Math 10-3, with no trig covered yet) and how fast the water was going into it.

They got to work.  I’ve been out of the classroom way too long, because one fellow figured it out and he was so excited I almost hugged him.  That would have been weird and creepy, so I restrained myself.  Many groups needed help, and I helped as little as I could.  In the end, we had answers ranging from 7 minutes to 9 minutes (8.23 minutes being the correct answer based on the measurements I gave them).  One girl was insistent that the answer was 8 minutes and 23 seconds, which led to a good conversation after we watched the answer video.  We sped the video ahead to 7 minutes, and a class of non-academic 10th graders watched intently.  They were disappointed that they were wrong until we managed to figure out that 8.23 minutes was actually right, and equivalent to 8:13.

What went well:  Kids who were initially disengaged got invested in it once they had a guess registered.  The teacher told me that it was nice to see his class so engaged.

What I could have done better: The were so engaged, I should have had them attach their names to guesses on the board.

Surprising:  I had a little more time left, so I showed them the Bucky video (see Math 10-4 lesson).  Then I left.  One girl was so into it, she came and found me in the classroom I went to next to show me her answer to the Bucky problem. This is a non-academic kid, remember, going out of her way to solve a problem, and proud enough of it to come and find me and share it.  Hook, line, and sinker.

Curricular Fit: 100% – Volume of solids is in Math 10-3. Could also be used in Math 10C.

Class #3 – Math 10C Math 10C is our combined grade 10 math course. It is the entry to pre-calculus 11 and 12 or foundations 11 and 12.  Students who pass math 9 take this course.  It is our most academic grade 10 level math course.

LTPS – The Giant Gummy Bear

I used the Giant Gummy Bear problem, exactly as described in an earlier post.

After watching the question video (in which I edited out the number of small gummy bears and the dimensions of the giant one), I asked the students what they wondered about. They wondered how long it would take to eat the giant gummy bear. They wondered how many small gummy bears were equivalent to the giant gummy bear. They wondered how tall the giant gummy bear was. They wondered if it was real, and if they could really get giant gummy bears. I asked them where they wanted to go with it, and they decided to figure out how tall the giant gummy bear is. I explained that the number of small gummy bears equivalent to the giant one was a necessary step along the way, so we elicited guesses for both the number of small gummy bears, and the height of the giant one.

I asked them what they needed to know to figure out the answers. They said they needed to know the size of a small one.  They were excited when I pulled out bags of gummy bears for each group from my lunch kit. They were convinced I had a giant one in there (I didn’t).  I gave them the gummy bears, and they got to work. All groups got very reasonable answers for the number of small gummy bears equivalent to the giant one (I think the range was from 900 to 1600).  Many used the calories, and some used the mass. Interestingly, those that used the calories were closest to the answer Vat19 alleges is correct.

It fell apart a bit at the point where they had to figure out the height of the giant one.  Going from a 1 to 1400 ratio from calories proved to be a difficult thing to scale up to the giant one.  They weren’t considering the volume (3 dimensional) of the bears, and were getting unreasonable answers.  Fortunately, they knew their answers were unreasonable. Unfortunately, they didn’t know how to fix them.

I played the answer video.  Most were satisfied with their answers to the number of small gummy bears, but very few had anything reasonable (other than their guess) for the height of the giant one.

We debriefed the problem. We talked about why their answers were so varied for the number of gummy bears.  They discovered that some had used the calories, some had worked with mass.  They discovered that there was a variation of between 24 and 30 gummy bears in the bags I handed out. No one thought to question the claim made by Vat19 in the video. Are our kids too trusting of advertising? This error analysis discussion was rich.

What went well:  The video and process really had the class engaged.

What I could have done better:  When they struggled with figuring out the height of the giant one, I didn’t help them properly.  They would have gotten there if I had reminded them of their three-dimensional geometry from an earlier unit of study.  Instead, most left this part frustrated.

Curricular Fit: 70% – Proportional reasoning is not in Math 10C (It’s in our Math 20-2). The measurement component of this exercise fits Math 10C.  The entire thing could be used with 100% fit in Math 20-2.

After this class it was lunch time.  I’ll describe the afternoon in the next post.

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Note:  This post is part 5 of 7.  Click here for part: 1 2 3 4 6 7

Recently, I was asked to model learning through problem solving in a school.  I will post about that experience in part 6, but I thought I should provide the teachers there with an outline of a learning through problem solving process prior to my visit to their school.  With credit to Dan Meyer, who gave me many of these ideas, this is what I gave them. What follows is a step-by-step description of the process I use.

Learning Through Problem Solving Process

1. Present the problem.

  • The problem is best presented using a multimedia artifact like an article, video, picture, story, song or any other multimedia artifact.
  • It is best if the question the teacher wants the students to explore is not explicitly stated in the artifact.

2. Have students come up with the question they want to answer.

  • Ask students what perplexes them in the artifact.  What questions do they have?  What do they wonder about?
  • Let this discussion go on long enough for them to come up with the question that you want them to answer.  This is the hook.  They feel like the question came from them, rather than their teacher.

3. Ask them to intuitively answer the question by providing a guess, a lowest reasonable answer, and a highest reasonable answer.

  • This is one of the most important steps, and is easily overlooked.
  • Ask students to make a guess.  No mathematics allowed.  They can use only their intuition.  Allow them to discuss and debate what they think is a reasonable answer.
  • By the end of this discussion, the teacher should have recorded on the board a range of reasonable answers.
  • Students could be asked to attach their names to guesses within that range of reasonable answers. I like to have them put their name somewhere on a continuum between the two answers.
  • This process makes it safe for students to be wrong, and allows them to recognize wrong answers later on if the answer they come up with doesn’t fit in the range.

4. Provide them with clarification and any information they think they require in advance of beginning to work on the problem.

  • Ask the students if they need any clarification on the question before they get to work.
  • Ask them what further information they require (if applicable)

5. Students work on the problem.

  • The teacher’s role here is to circulate and make sure that the groups (or pairs, or individuals) are on task.
  • Some groups will require help to get started.  Don’t let them opt out.
  • Some groups will finish quickly and ask if they are right.  If they are wrong, ask them a question to steer them in the right direction.  If they have the right answer, don’t tell them because as soon as they know they are right, their thinking will stop. Instead, give them an extension.  Extensions are challenging to create.  They can’t be the same question with different numbers, because that’s just more of the same work.  Instead, extensions must truly extend the student’s thinking.

6. Share student solutions.

  • The amount of time required to finish will vary based on the problem.  Some will take only a few minutes, and others might take a whole period.
  • Do not interrupt the group until they have all gotten an answer. Nothing is more frustrating than being truly engaged in a problem, and having your thinking stopped.
  • The teacher should not give the solution or the answer.  Have students present their solutions in one of the following ways:
  • Use a document camera for students to share with the entire class. (stressful for some)
  • Have groups share with another group. (safer)
  •  Some teachers have students working on boards around the classroom.  In this situation, the class can sit down and look at all the solutions simultaneously.

7. Teacher summarizes the learning.

  • The teacher should spend a few minutes summarizing what mathematics was learned.
  • This is not time for the teacher to show his own method of solving the problem.  It is simply time to consolidate the learning.

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Note: This post is part 4 of 7.  Click here for part: 1 2 3 5 6 7

There is much talk in our district about student engagement and 21st century literacy. The learning through problem solving approach discussed in the previous posts fits nicely within these topics. When a learning through problem solving approach is used, I believe that in addition to math content, students learn the following critical skills.

  • Perseverance and Persistence
  • An ability to approach a problem in a variety of ways
  • Collaboration
  • Communication
  • Presentation
  • Peer Review
  • Self Correction

These things, folks, are 21st century skills.

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