problem solving mathematical investigation and modelling module

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Problem Solving, Mathematical Investigation, and Modeling (6081)

This course intends to enhance the students’ knowledge and skills in dealing with real-life and/or non-routine applications of mathematics. Students will have the opportunity to explore the use of problem-solving strategies or heuristics as they engage in mathematical investigations, formulate and justify conjectures, make generalizations, and communicate mathematical ideas.

• Teacher: MELANIE GURAT
• Enrolled students: 2

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Math 15: problem solving, mathematical investigation, and modeling.

• Teacher: Christian Cabral

Mathematics for Teaching

This site is NOT about making mathematics easy because it isn't. It is about making it make sense because it does.

What is mathematical investigation?

Mathematical investigation refers to the sustained exploration of a mathematical situation. It distinguishes itself from problem solving because it is open-ended.

I first heard about math investigations in 1990 when I attended a postgraduate course in Australia.  I love it right away and it has since become one of my favorite mathematical activity for my students who were so proud of themselves when they finished their first investigation.

Problem solving is a convergent activity. It has definite goal – the solution of the problem. Mathematical investigation on the other hand is more of a divergent activity. In mathematical investigations, students are expected to pose their own problems after initial exploration of the mathematical situation. The exploration of the situation, the formulation of problems and its solution give opportunity for the development of independent mathematical thinking and in engaging in mathematical processes such as organizing and recording data, pattern searching, conjecturing, inferring, justifying and explaining conjectures and generalizations. It is these thinking processes which enable an individual to learn more mathematics, apply mathematics in other discipline and in everyday situation and to solve mathematical (and non-mathematical) problems.

Teaching through mathematical investigation allows  for students to learn about mathematics, especially the nature of mathematical activity and thinking. It also make them realize that learning mathematics involves intuition, systematic exploration, conjecturing and reasoning, etc and not about memorizing and following existing procedures. The ultimate aim of mathematical investigation is develop students’ mathematical habits of mind .

Although  students may do the same mathematical investigation, it is not expected that all of them will consider the same problem from a particular starting point.  The “open-endedness” of many investigation also means that students may not completely cover the entire situation. However, at least for a student’s own satisfaction, the achievement of some specific results for an investigation is desirable. What is essential is that the students will experience the following mathematical processes which are the emphasis of mathematical investigation:

• systematic exploration of the given situation
• formulating problems and conjectures
• attempting to provide mathematical justifications for the conjectures.

In this kind of activity and teaching, students are given more opportunity to direct their own learning experiences. Note that a problem solving task can be turned into an investigation task by extending the problem by varying for example one of the conditions. To know more about problem solving and how they differ with math investigation read my post on Exercises, Problem Solving and Math Investigation .

Some parents and even teachers complain that students are not learning mathematics in this kind of activity. Indeed they won’t if the teacher will not discuss the results of the investigation, highlight and correct the misconceptions, synthesize students’ findings and help students make connection among the math concepts covered in the investigation. This goes without saying that teachers should try the investigation first before giving it to the students.

I think mathematical investigation is constructivist teaching at its finest. For a sample lesson, read Polygons and algebraic expressions .

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23 thoughts on “ What is mathematical investigation? ”

Can anyone give me an example of it?

I think there’s a need for students to conduct investigations, especially in math, in order for them to really learn the language of mathematics. We also have to note that those young ones have their own way of doing things; which in some ways may simplify processes. Naci John

Is it possible to teach all math topics using investigation method?

No. Only those topics in math that involves generalizing and conjecturing using the mathematics students at that grade level know may be taught by math investigation. However, you can always start any topic by math investigation but it doesn’t have to be an investigation all the way. The students may see patterns and be able to state a conjecture but they will not always have the capability to explain or justify it mathematically. Doing math investigation is more for developing disposition to think mathematically. Don’t overdo it.

I find this article very informative. I hope UP NISMED can organize a workshop for teachers and students for this essential undertaking in Math. I think reconciling LS and MI will be a great boost in the teaching strats of teachers and mathematical abilities of the students.

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Thank you for posting it.

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In elementary school I had to take investigations. In all 956 lessons standard addition is mentionedONCE and not truly taught. same sor addition and multiplication. long division is never properly taught. they never even mentioned how to do it. they teach a guess and check method leaving you with a remainder- no decimals. This lleaves students with lower grades on timed tests because they are rushed due to the innefficient method they first learned. The board of education deemed the program “sub standard”. Many students who used the program in elementary school needed help with math in middle school. So answer this, why do we continually use investigations?

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please, pki investigate naman to: How many other rectangles can be made with twelve tiles?

To me, Math investigation is an ally of constructivism. Students are engaged in tasks that allow them to put in more on what they know as they tinker on a math problem. They do exploration and research to be able to shed more light to the problem. Creative and critical thinking come to the fore. The investigation may take time. But, at the end of the process, students defend what they got, defend it, and share it.

Math investigation takes time. It is quality time. The teacher stands by for consultation and extends assistance, as needed by the students.

My book “High School Geometry with Practical Work and Portfolio Making” is rich on math investigations. My other book “Science Research and Statistics” has much more on investigations particularly on Science Investigatory projects.

hello! i am trying to teach my students to appreciate the process of mathematical investigation and to develop the skills needed. however, i am having a hard time organizing my lessons. i want to them to develop the thinking skills necessary for them to create their own math IPs or math research papers later on. 1. do you have any suggestions or activities or lesson plans that i can refer to so that the skills can be developed gradually? 2.can you recommend any textbook that can be used for this purpose alone including different mathematical model?

i hope that you could answer my queries. thank you very much.

It’s best to expose students to problems with many solutions/answers first before giving them mathematical investigation tasks. You can also convert some of your exercises to problems and mathematical investigation. Check out my post on exercises, problems and math investigations .

I think our use of the Connected Math series really falls in line with this post. Curriclum planning needs to become as strignent as lessons planning that we as teachers work extremely hard on.

• Pingback: Making connections through math investigation: polygons and operations with algebraic expressions- Part 1 « assessing and teaching K-10 mathematics

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Problem Solving and Mathematical Modeling

• First Online: 11 January 2023

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• Parikshit Narendra Mahalle 7 ,
• Nancy Ambritta P. 8 ,
• Sachin R. Sakhare 9 &
• Atul P. Kulkarni 10  

Part of the book series: Studies in Autonomic, Data-driven and Industrial Computing ((SADIC))

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A problem is a puzzle or task that requires a logical thought process or fundamental mathematical steps to solve it. The puzzle generally represents the set of questions on the underlined use case which also consist of complete description of the use case along with the set of constraints about the use case. Logic is very importantly used to solve the puzzle or problem. Logic is defined as a method of human thought that involves thinking in a linear, step-by-step manner about how a problem can be solved. Logic is a subjective matter and varies from person to person. Logic is directly linked to the natural intelligence of human being and is a language of reasoning. Logic also represents the set of rules we use when we do reasoning. It is observed that majority of the employment of fresh engineering graduates across the globe is in software and information technology sector.

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Mahalle, P.N., Ambritta P., N., Sakhare, S.R., Kulkarni, A.P. (2023). Problem Solving and Mathematical Modeling. In: Foundations of Mathematical Modelling for Engineering Problem Solving. Studies in Autonomic, Data-driven and Industrial Computing. Springer, Singapore. https://doi.org/10.1007/978-981-19-8828-8_2

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The Possible Collapse of the U.S. Home Insurance System

A times investigation found climate change may now be a concern for every homeowner in the country..

This transcript was created using speech recognition software. While it has been reviewed by human transcribers, it may contain errors. Please review the episode audio before quoting from this transcript and email [email protected] with any questions.

From “The New York Times,” I’m Sabrina Tavernise. And this is “The Daily.”

[MUSIC PLAYING]

Today, my colleague, Christopher Flavelle, on a “Times” investigation into one of the least known and most consequential effects of climate change — insurance — and why it may now be a concern for every homeowner in the country.

It’s Wednesday, May 15.

So, Chris, you and I talked a while ago about how climate change was really wreaking havoc in the insurance market in Florida. You’ve just done an investigation that takes a look into the insurance markets more broadly and more deeply. Tell us about it.

Yeah, so I cover climate change, in particular the way climate shocks affect different parts of American life. And insurance has become a really big part of that coverage. And Florida is a great example. As hurricanes have gotten worse and more frequent, insurers are paying out more and more money to rebuild people’s homes. And that’s driving up insurance costs and ultimately driving up the cost of owning a home in Florida.

So we’re already seeing that climate impact on the housing market in Florida. My colleagues and I started to think, well, could it be that that kind of disruption is also happening in other states, not just in the obvious coastal states but maybe even through the middle of the US? So we set out to find out just how much it is happening, how much that Florida turmoil has, in fact, become really a contagion that is spreading across the country.

So how did you go about reporting this? I mean, where did you start?

All we knew at the start of this was that there was reason to think this might be a problem. If you just look at how the federal government tracks disasters around the country, there’s been a big increase almost every year in the number and severity of all kinds of disasters around the country. So we thought, OK, it’s worth trying to find out, what does that mean for insurers?

The problem is getting data on the insurance industry is actually really hard. There’s no federal regulation. There’s no government agency you can go to that holds this data. If you talk to the insurers directly, they tend to be a little reluctant to share information about what they’re going through. So we weren’t sure where to go until, finally, we realized the best people to ask are the people whose job it is to gauge the financial health of insurance companies.

Those are rating agencies. In particular, there’s one rating company called AM Best, whose whole purpose is to tell investors how healthy an insurance company is.

Whoa. So this is way down in the nuts and bolts of the US insurance industry.

Right. This is a part of the broader economy that most people would never experience. But we asked them to do something special for us. We said, hey, can you help us find the one number that would tell us reporters just how healthy or unhealthy this insurance market is state by state over time? And it turns out, there is just such a number. It’s called a combined ratio.

OK, plain English?

Plain English, it is the ratio of revenue to costs, how much money these guys take in for homeowner’s insurance and how much they pay out in costs and losses. You want your revenue to be higher than your costs. If not, you’re in trouble.

So what did you find out?

Well, we got that number for every state, going back more than a decade. And what it showed us was our suspicions were right. This market turmoil that we were seeing in Florida and California has indeed been spreading across the country. And in fact, it turns out that in 18 states, last year, the homeowner’s insurance market lost money. And that’s a big jump from 5 or 10 years ago and spells real trouble for insurance and for homeowners and for almost every part of the economy.

So the contagion was real.

Right. This is our first window showing us just how far that contagion had spread. And one of the really striking things about this data was it showed the contagion had spread to places that I wouldn’t have thought of as especially prone to climate shocks — for example, a lot of the Midwest, a lot of the Southeast. In fact, if you think of a map of the country, there was no state between Pennsylvania and the Dakotas that didn’t lose money on homeowner’s insurance last year.

So just huge parts of the middle of the US have become unprofitable for homeowner’s insurance. This market is starting to buckle under the cost of climate change.

And this is all happening really fast. When we did the Florida episode two years ago, it was a completely new phenomenon and really only in Florida. And now it’s everywhere.

Yeah. And that’s exactly what’s so striking here. The rate at which this is becoming, again, a contagion and spreading across the country is just demolishing the expectations of anyone I’ve spoken to. No one thought that this problem would affect so much of the US so quickly.

So in these states, these new places that the contagion has spread to, what exactly is happening that’s causing the insurance companies to fold up shop?

Yeah. Something really particular is happening in a lot of these states. And it’s worth noting how it’s surprised everyone. And what that is, is formally unimportant weather events, like hailstorms or windstorms, those didn’t used to be the kind of thing that would scare insurance companies. Obviously, a big problem if it destroys your home or damages your home. But for insurers, it wasn’t going to wipe them out financially.

Right. It wasn’t just a complete and utter wipeout that the company would then have to pony up a lot of money for.

Exactly. And insurers call them secondary perils, sort of a belittling term, something other than a big deal, like a hurricane.

These minor league weather events.

Right. But those are becoming so frequent and so much more intense that they can cause existential threats for insurance companies. And insurers are now fleeing states not because of hurricanes but because those former things that were small are now big. Hailstorms, wildfires in some places, previous annoyances are becoming real threats to insurers.

Chris, what’s the big picture on what insurers are actually facing? What’s happening out there numbers-wise?

This is a huge threat. In terms of the number of states where this industry is losing money, it’s more than doubled from 10 years ago to basically a third of the country. The amount they’re losing is enormous. In some states, insurers are paying out $1.25 or even $1.50 for every dollar they bring in, in revenue, which is totally unsustainable.

And the result is insurers are making changes. They are pulling back from these markets. They’re hiking premiums. And often, they’re just dropping customers. And that’s where this becomes real, not just for people who surf balance sheets and trade in the stock market. This is becoming real for homeowners around the country, who all of a sudden increasingly can’t get insurance.

So, Chris, what’s the actual implication? I mean, what happens when people in a state can’t get insurance for their homes?

Getting insurance for a home is crucial if you want to sell or buy a home. Most people can’t buy a home without a mortgage. And banks won’t issue a mortgage without home insurance. So if you’ve got a home that insurance company doesn’t want to cover, you got a real problem. You need to find insurance, or that home becomes very close to unsellable.

And as you get fewer buyers, the price goes down. So this doesn’t just hurt people who are paying for these insurance premiums. It hurts people who want to sell their homes. It even could hurt, at some point, whole local economies. If home values fall, governments take in less tax revenue. That means less money for schools and police. It also means people who get hit by disasters and have to rebuild their homes all of a sudden can’t, because their insurance isn’t available anymore. It’s hard to overstate just how big a deal this is.

And is that actually happening, Chris? I mean, are housing markets being dragged down because of this problem with the insurance markets right now?

Anecdotally, we’ve got reports that in places like Florida and Louisiana and maybe in parts of California, the difficulty of getting insurance, the crazy high cost of insurance is starting to depress demand because not everyone can afford to pay these really high costs, even if they have insurance. But what we wanted to focus on with this story was also, OK, we know where this goes eventually. But where is it beginning? What are the places that are just starting to feel these shocks from the insurance market?

And so I called around and asked insurance agents, who are the front lines of this. They’re the ones who are struggling to find insurance for homeowners. And I said, hey, is there one place that I should go if I want to understand what it looks like to homeowners when all of a sudden insurance becomes really expensive or you can’t even find it? And those insurance agents told me, if you want to see what this looks like in real life, go to a little town called Marshalltown in the middle of Iowa.

We’ll be right back.

So, Chris, you went to Marshalltown, Iowa. What did you find?

Even before I got to Marshalltown, I had some idea I was in the right spot. When I landed in Des Moines and went to rent a car, the nice woman at the desk who rented me a car, she said, what are you doing here? I said, I’m here to write a story about people in Iowa who can’t get insurance because of storms. She said, oh, yeah, I know all about that. That’s a big problem here.

Even the rental car lady.

Even the rental car lady knew something was going on. And so I got into my rental car and drove about an hour northeast of Des Moines, through some rolling hills, to this lovely little town of Marshalltown. Marshalltown is a really cute, little Midwestern town with old homes and a beautiful courthouse in the town square. And when I drove through, I couldn’t help noticing all the roofs looked new.

What does that tell you?

Turns out Marshalltown, despite being a pastoral image of Midwestern easy living, was hit by two really bad disasters in recent years — first, a devastating tornado in 2018 and then, in 2020, what’s called a derecho, a straight-line wind event that’s also just enormously damaging. And the result was lots of homes in this small town got severely damaged in a short period of time. And so when you drive down, you see all these new roofs that give you the sense that something’s going on.

So climate had come to Marshalltown?

Exactly. A place that had previously seemed maybe safe from climate change, if there is such a thing, all of a sudden was not. So I found an insurance agent in Marshalltown —

We talked to other agents but haven’t talked to many homeowners.

— named Bobby Shomo. And he invited me to his office early one morning and said, come meet some people. And so I parked on a quiet street outside of his office, across the street from the courthouse, which also had a new roof, and went into his conference room and met a procession of clients who all had versions of the same horror story.

It was more — well more of double.

A huge reduction in coverage with a huge price increase.

Some people had faced big premium hikes.

I’m just a little, small business owner. So every little bit I do feel.

They had so much trouble with their insurance company.

I was with IMT Insurance forever. And then when I moved in 2020, Bobby said they won’t insure a pool.

Some people had gotten dropped.

Where we used to see carriers canceling someone for frequency of three or four or five claims, it’s one or two now.

Some people couldn’t get the coverage they needed. But it was versions of the same tale, which is all of a sudden, having homeowner’s insurance in Marshalltown was really difficult. But I wanted to see if it was bigger than just Marshalltown. So the next day, I got back in my car and drove east to Cedar Rapids, where I met another person having a version of the same problem, a guy named Dave Langston.

Tell me about Dave.

Dave lives in a handsome, modest, little townhouse on a quiet cul-de-sac on a hill at the edge of Cedar Rapids. He’s the president of his homeowners association. There’s 17 homes on this little street. And this is just as far as you could get from a danger zone. It looks as safe as could be. But in January, they got a letter from the company that insures him and his neighbors, saying his policy was being canceled, even though it wasn’t as though they’d just been hit by some giant storm.

So then what was the reason they gave?

They didn’t give a reason. And I think people might not realize, insurers don’t have to give a reason. Insurance policies are year to year. And if your insurance company decides that you’re too much of a risk or your neighborhood is too much of a risk or your state is too much of a risk, they can just leave. They can send you a letter saying, forget it. We’re canceling your insurance. There’s almost no protection people have.

And in this case, the reason was that this insurance company was losing too much money in Iowa and didn’t want to keep on writing homeowner’s insurance in the state. That was the situation that Dave shared with tens of thousands of people across the state that were all getting similar letters.

What made Dave’s situation a little more challenging was that he couldn’t get new insurance. He tried for months through agent after agent after agent. And every company told him the same thing. We won’t cover you. Even though these homes are perfectly safe in a safe part of the state, nobody would say yes. And it took them until basically two days before their insurance policy was going to run out until they finally found new coverage that was far more expensive and far more bare-bones than what they’d had.

But at least it was something.

It was something. But the problem was it wasn’t that good. Under this new policy, if Dave’s street got hit by another big windstorm, the damage from that storm and fixing that damage would wipe out all the savings set aside by these homeowners. The deductible would be crushingly high — $120,000 — to replace those roofs if the worst happened because the insurance money just wouldn’t cover anywhere close to the cost of rebuilding.

He said to me, we didn’t do anything wrong. This is just what insurance looks like today. And today, it’s us in Cedar Rapids. Everyone, though, is going to face a situation like this eventually. And Dave is right. I talked to insurance agents around the country. And they confirmed for me that this kind of a shift towards a new type of insurance, insurance that’s more expensive and doesn’t cover as much and makes it harder to rebuild after a big disaster, it’s becoming more and more common around the country.

So, Chris, if Dave and the people you spoke to in Iowa were really evidence that your hunch was right, that the problem is spreading and rapidly, what are the possible fixes here?

The fix that people seem most hopeful about is this idea that, what if you could reduce the risk and cause there to be less damage in the first place? So what some states are doing is they’re trying to encourage homeowners to spend more money on hardening their home or adding a new roof or, if it’s a wildfire zone, cut back the vegetation, things that can reduce your risk of having really serious losses. And to help pay for that, they’re telling insurers, you’ve got to offer a discount to people who do that.

And everyone who works in this field says, in theory, that’s the right approach. The problem is, number one, hardening a home costs a fantastic amount of money. So doing this at scale is hugely expensive. Number two, it takes a long time to actually get enough homes hardened in this way that you can make a real dent for insurance companies. We’re talking about years or probably decades before that has a real effect, if it ever works.

OK. So that sounds not particularly realistic, given the urgency and the timeline we’re on here. So what else are people looking at?

Option number two is the government gets involved. And instead of most Americans buying home insurance from a private company, they start buying it from government programs that are designed to make sure that people, even in risky places, can still buy insurance. That would be just a gargantuan undertaking. The idea of the government providing homeowner’s insurance because private companies can’t or won’t would lead to one of the biggest government programs that exists, if we could even do it.

So huge change, like the federal government actually trying to write these markets by itself by providing homeowner’s insurance. But is that really feasible?

Well, in some areas, we’re actually already doing it. The government already provides flood insurance because for decades, most private insurers have not wanted to cover flood. It’s too risky. It’s too expensive. But that change, with governments taking over that role, creates a new problem of its own because the government providing flood insurance that you otherwise couldn’t get means people have been building and building in flood-prone areas because they know they can get that guaranteed flood insurance.

Interesting. So that’s a huge new downside. The government would be incentivizing people to move to places that they shouldn’t be.

That’s right. But there’s even one more problem with that approach of using the government to try to solve this problem, which is these costs keep growing. The number of billion-dollar disasters the US experiences every year keeps going up. And at some point, even if the government pays the cost through some sort of subsidized insurance, what happens when that cost is so great that we can no longer afford to pay it? That’s the really hard question that no official can answer.

So that’s pretty doomsday, Chris. Are we looking at the end of insurance?

I think it’s fair to say that we’re looking at the end of insurance as we know it, the end of insurance that means most Americans can rest assured that if they get hit by a disaster, their insurance company will provide enough money they can rebuild. That idea might be going away. And what it shows is maybe the threat of climate change isn’t quite what we thought.

Maybe instead of climate change wrecking communities in the form of a big storm or a wildfire or a flood, maybe even before those things happen, climate change can wreck communities by something as seemingly mundane and even boring as insurance. Maybe the harbinger of doom is not a giant storm but an anodyne letter from your insurance company, saying, we’re sorry to inform you we can no longer cover your home.

Maybe the future of climate change is best seen not by poring over weather data from NOAA but by poring over spreadsheets from rating firms, showing the profitability from insurance companies, and how bit by bit, that money that they’re losing around the country tells its own story. And the story is these shocks are actually already here.

Chris, as always, terrifying to talk to you.

Always a pleasure, Sabrina.

Here’s what else you should know today. On Tuesday, the United Nations has reclassified the number of women and children killed in Gaza, saying that it does not have enough identifying information to know exactly how many of the total dead are women and children. The UN now estimates that about 5,000 women and about 8,000 children have been killed, figures that are about half of what it was previously citing. The UN says the numbers dropped because it is using a more conservative estimate while waiting for information on about 10,000 other dead Gazans who have not yet been identified.

And Mike Johnson, the Speaker of the House, gave a press conference outside the court in Lower Manhattan, where Michael Cohen, the former fixer for Donald Trump, was testifying for a second day, answering questions from Trump’s lawyers. Trump is bound by a gag order. So Johnson joined other stand-ins for the former president to discredit the proceedings. Johnson, one of the most important Republicans in the country, attacked Cohen but also the trial itself, calling it a sham and political theater.

Today’s episode was produced by Nina Feldman, Shannon Lin, and Jessica Cheung. It was edited by MJ Davis Lin, with help from Michael Benoist, contains original music by Dan Powell, Marion Lozano, and Rowan Niemisto, and was engineered by Alyssa Moxley. Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly.

That’s it for “The Daily.” I’m Sabrina Tavernise. See you tomorrow.

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Hosted by Sabrina Tavernise

Featuring Christopher Flavelle

Produced by Nina Feldman ,  Shannon M. Lin and Jessica Cheung

Edited by MJ Davis Lin

With Michael Benoist

Original music by Dan Powell ,  Marion Lozano and Rowan Niemisto

Engineered by Alyssa Moxley

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Across the United States, more frequent extreme weather is starting to cause the home insurance market to buckle, even for those who have paid their premiums dutifully year after year.

Christopher Flavelle, a climate reporter, discusses a Times investigation into one of the most consequential effects of the changes.

On today’s episode

Christopher Flavelle , a climate change reporter for The New York Times.

Background reading

As American insurers bleed cash from climate shocks , homeowners lose.

See how the home insurance crunch affects the market in each state .

Here are four takeaways from The Times’s investigation.

There are a lot of ways to listen to The Daily. Here’s how.

We aim to make transcripts available the next workday after an episode’s publication. You can find them at the top of the page.

Christopher Flavelle contributed reporting.

The Daily is made by Rachel Quester, Lynsea Garrison, Clare Toeniskoetter, Paige Cowett, Michael Simon Johnson, Brad Fisher, Chris Wood, Jessica Cheung, Stella Tan, Alexandra Leigh Young, Lisa Chow, Eric Krupke, Marc Georges, Luke Vander Ploeg, M.J. Davis Lin, Dan Powell, Sydney Harper, Mike Benoist, Liz O. Baylen, Asthaa Chaturvedi, Rachelle Bonja, Diana Nguyen, Marion Lozano, Corey Schreppel, Rob Szypko, Elisheba Ittoop, Mooj Zadie, Patricia Willens, Rowan Niemisto, Jody Becker, Rikki Novetsky, John Ketchum, Nina Feldman, Will Reid, Carlos Prieto, Ben Calhoun, Susan Lee, Lexie Diao, Mary Wilson, Alex Stern, Dan Farrell, Sophia Lanman, Shannon Lin, Diane Wong, Devon Taylor, Alyssa Moxley, Summer Thomad, Olivia Natt, Daniel Ramirez and Brendan Klinkenberg.

Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly. Special thanks to Sam Dolnick, Paula Szuchman, Lisa Tobin, Larissa Anderson, Julia Simon, Sofia Milan, Mahima Chablani, Elizabeth Davis-Moorer, Jeffrey Miranda, Renan Borelli, Maddy Masiello, Isabella Anderson and Nina Lassam.

Christopher Flavelle is a Times reporter who writes about how the United States is trying to adapt to the effects of climate change. More about Christopher Flavelle

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