The Science Ed Resource Center at Carleton is one of the places I check in on, and I just saw the page linked above. They're looking for HS teachers who can teach the Climate Detectives module, which has 6 sessions clocking in at about 600 minutes of class time, and 1) have a colleague observe a session, 2) have students take a post-module test in 2 weeks. The aim is to improve the module for others. Look at the webpage for more details -- this is just my quick summary to give you an overview. TERC will also pay HS teachers who can participate a small honorarium to compensate some of your prep time, as well as a materials allowance.

You'll need computer lab space and computer lab resources for your students (Flash, Excel, RealPlayer or other media player). You'll also need some support from your principal and students/their parents. And you'll need to do the whole thing before May 15, 2016.

The Climate Detectives module does look cool: it talks about cores (those long strips of earth or ice drilled out by scientists) and Milankovich cycles (large-scale cycles from the wobble of the Earth, etc., which I mentioned once in a discussion of climate modeling) and geologic time and particle sizes for sand and mud. Very sciency, but with a lot of use of data. Advertise to friends and colleagues! Be a guinea pig for climate science teaching!

*Feel trapped by boring fake word problems in your math textbook? Get intros to real-life issues in the natural world and see math at work.*

So, this post is about using the normal distribution and spreadsheets to deal with real data! November has been very warm, even though I picked the coldest days to go winter camping. How warm is our November here in Minneapolis-St. Paul?

Here is a Google spreadsheet with the minimum and maximum temperatures for every November 8 since 1970:

Google spreadsheet: max, min temps, with precipitation and graphs, Nov. 8

Included are histograms of max and min temperatures, and a scatter plot of max and min against each other. In the worksheet I just wrote, I asked students to create histograms for the maximum and minimum temperatures and look at those histograms to discuss what probability model might be appropriate. I only give the suggestions of uniform and normal, so it's very simplistic. The second page asks students to find some probabilities assuming a normal model.

Simplified spreadsheet: max & min only, Nov. 8

Immediately above is a clean version of the spreadsheet without graphs or additional data, and here below is a version showing what I'm asking students to do to find the standard deviation:

Spreadsheet with standard deviation calculations

You'll need a z-table to find the probabilities, as well.

I think this is a good conversation starter in a class in which you want to get students to work together and argue about whether a normal distribution is a good model, and to get students to talk about what this information means. For instance, in the discussion about global warming/climate change/climate instability, a lot of little anecdotal facts are thrown around: "Last winter was really cold!" or "But it's a really warm fall!" In isolation these facts really don't tell us much about the global patterns in temperature, and we need to make that clear to students on a scientific level. Snow in Iowa doesn't mean global warming is a hoax. At the same time, local changes are really important to the food we're able to grow and the activities we are able to enjoy. Local changes in snowfall and precipitation change our drinking water availability and when we can plant corn. Local changes in heat influence how much milk cows give.

I'm not satisfied with this worksheet yet: it does its tiny part but doesn't go very far or give much of the big picture. Ideal improvements:

- test if a normal distribution is a good model or not
- do some linear regression on max vs min temps
- see if there is a temporal shift in the distribution of maxes and mins

Reasons I haven't done these things:

- hypothesis testing and model validation are complicated topics, and I haven't though much yet how to teach them
- time
- I played around with the temporal shift and can "see" one, but again haven't thought about how to teach the mathematical or statistical analysis of this idea.

Have any of you taught hypothesis testing in a lower-level college class or talked with your high school or college students about how to select models? Anyone else do exploratory data analysis for fun?

Well, there's snow on the ground and the night is pretty cold. Time to eat leftover pumpkin custard and watch movies or something Hope you all had a happy Thanksgiving!

Well, here are a few examples if you're interested!

- Stefan's equation for sea ice thickness: these two posts talk about modeling sea ice thickness with a differential equation but don't ask you to use data to create a model
- Modeling tides on the California coast, with more here: these two posts give worksheets on creating your own model of the tidal patterns at Point Reyes Seashore using actual NOAA data
- Lynx! for cute fuzzy animals with sharp teeth! These two posts have students develop their own trig model for lynx populations, see how bad that model is, and then use a logarithm composed with the trig function to get a model that better fits the sharp population peaks.

Sometimes I feel like teachers who make room for this material are swimming upstream since so many of our high school math curricula don't provide the time for experimental, living mathematics... but every now and then I meet someone who really makes it work. And maybe with modeling as one of the high school Common Core standards there will be some official space for this in high school classes! It is so sad that students can go through 12 years of school and never really see mathematical model-making at work.

Well, the semester is coming to a close again. Two more weeks of class and then finals. My corn has grown pretty tall and survived a squirrel attack, and next weekend I should be able to plant it outside: April 30 is our average last frost date in my area. It's been a nice week weather-wise, though, and so the plants are outside in my thrown-together "cold frame." I'm using soil with built-in compost for the young plants since corn is a heavy feeder, and once I move them to the ground I'll put them into areas where I've pulled up the hairy vetch (which is supposed to fix nitrogen) and added more compost. We will consider manure, but for the moment it seems like we might make it with the contents of our own compost pile this summer. The benefit of working without commercial fertilizer is that we don't have to think very hard about whether we're contributing to the nitrates in the Mississippi river -- even considering that urban areas contribute less than 10% of Minnesota's nitrate runoff.

The Star Tribune (the Minneapolis paper) had an article this weekend about nitrates in well water in the rest of Minnesota. It's costing taxpayers a lot of money -- up to $3,300 per household in some areas to install nitrate-removal equipment and ensure that people can drink the water safely! In fact, nitrates made the news because in Randall, MN, people can't drink the water safely. Nitrate levels are too high and dangerous to human health, especially babies and the elderly. The majority of nitrates in Minnesota water come from agricultural fertilizer runoff, and dealing with the runoff has become a hot local political topic. It's not just a problem in the Gulf of Mexico when it's affecting our local drinking water and lakes.

Nitrates in drinking water would be a great topic for a social studies debate. Farmers need nitrates to grow their corn at a level that provides enough for food products, corn syrup, compostable cups, and ethanol. As taxpayers we subsidize some of these uses, and as taxpayers we pay for the removal of nitrates from the water. Paying for removal of nitrates is hugely expensive: besides the initial up-front cost of the equipment, it looks like it costs 15 to 35 cents per 1000 gallons of water to do the processing and maintenance, while low-nitrate water costs only 5 to 10 cents per 1000 gallons TOTAL. That's a huge percentage increase in water costs! (The figures are from the linked Minnesota Department of Agriculture file.)

Interestingly, I couldn't find nice data online about nitrates in water in Minnesota, so I returned to the tried and true USGS data from the Raccoon River near Des Moines, Iowa. The past month's data provides enough for a numerical integration and estimation worksheet:

Numerical Integration: Nitrates

It is a pretty basic worksheet -- provides enough information for a social-studies debate starting point, but not much more. It would be very cool to make some more sophisticated optimization problems for nitrate use, to help students (and perhaps the public) think about the tradeoffs involved in agriculture and environment.

]]>Let's back up a moment. My garden fever has been ramping up; those of you who live in the great frozen north might understand the hunger to see green things. It's why all the undergrads were in shorts last week (that and we set a record of 84 degrees!). For the past few years my husband & I have tried to start seeds in egg containers, and every year something terrible happens: they blow off the porch, or they all drown in a big rainstorm, or squirrels eat them. No more. This year I marched into Eggplant Urban Farm Supply a few blocks away and made a stand. I spent what felt like an exorbitant amount of money, but it'll be less than a dollar a plant even if we have a few failures. I got a seed-starting tray and a seed heater for our cold house and some hairy vetch and inoculant for nitrogen fixation!

Nitrogen what?

Nitrogen is one of the most important nutrients for plant growth, especially for corn. That's why farmers apply nitrogen-based fertilizers. Last spring I published some worksheets about nitrogen runoff in the Raccoon River: spring melt washes away nitrogen that wasn't absorbed by the soil or plants. Last month I updated that story with some info about Des Moines Waterworks suing local counties for letting too much nitrate wash away: it's really expensive to remove, but you have to, or babies turn blue and die! (Nitrates in our water blocks oxygen transport in our blood.) So, from this blog, I've learned something about nitrogen. And we did a soil test in our back yard and learned our nitrate/nitrite/nitrogen of any kind levels are very low. And this year...

This year we're going to try a Three Sisters garden. Why? I went to North Coast Nosh curated by Sioux chef Sean Sherman and the food was amazing. And I loved the little garden Fort Snelling used to have. And I bought Buffalo Bird Woman's Garden: Agriculture of the Hidatsa Indians , a book of garden wisdom from one woman's memory of her life, first published in 1917, and Original Local: Indigenous Foods, Stories, and Recipes from the Upper Midwest by Heid Erdrich, full of more modern stories, and I was reading the chapter on the three sisters in Braiding Sweetgrass: Indigenous Wisdom, Scientific Knowledge and the Teachings of Plants. So. Three sisters. We'll give it a try.

But for this corn, I need more nitrogen in the soil and I'm afraid compost won't be enough. Remember how in the recent nitrates post I mentioned no-till agriculture? I'm not a farmer, but I started to read about cover crops. I'd planned to try them in the fall, but Eggplant had that hairy vetch, sitting next to the seed-starting supplies, for only $6.25... And that's how I got to be spring flinging hairy vetch covered in inoculant slurry into the garden last weekend. And reading a lot of pdfs on the math of cover crops!

Turns out there is a fair bit of research on nitrogen fixation of cover crops. I've been looking at the CENTURY model of soil organic matter but there are also papers on simpler studies of nitrogen fixation by different plants (clover, vetch, rye, oats). To break up the butterfly madness, I'm prepping an activity on nitrogen fixation by cover crops. It seems like a theme this spring is transitions: transitions between butterfly habitats, between bee life-stages, between garden clippings and fully composted soil. More differential and difference equations to come!

Next week, nitrogen fixation and cover crops. Will this change your fall garden?

*Disclosure:** This website may receive a commission for sales of the books we link to. I do actually own all these books and recommend them!*

**South and north:** Monarchs overwintering Mexico have hit pause on their reproductive lives. Really, it's called reproductive diapause! They get to live 6-9 months down in the highlands of central Mexico, living on the oyamel fir trees in the mountains.

This is not a beach vacation: the monarchs cover the trees high up in the forest. I had the good fortune to visit Cerro Pelon butterfly reserve last January with Joel Moreno of Joel's Butterfly B&B, and these are pictures I took from that trip.

The summer butterflies, up in Minnesota, Canada, and other parts of the northeastern US, only live two to five weeks and reproduce several times during that short lifespan. So in the worksheet activity, students are asked to make a graph leading from immature monarch life -- the egg, larval, and pupal stages -- to either summer butterfly life (short but wild!) or winter butterfly life (a long quiet stay in the mountains, followed by a big migration and then basically one chance at parenthood). In the worksheet I chose to break these up into two different matrices.

**Matrices and graphs: **The worksheet focuses on making the lifecycle graph and then making two matrices, one for summer butterflies and one for winter butterflies. Then what? I ask a conceptual question and quit. That's why I'm calling this a first draft. It's a good start for an activity, but it doesn't have the story arc I want.

I'm basing the math model on the paper by Flockhart, Pichancourt, Norris, and Martin that discusses what is leading to monarch population declines. They conclude that the biggest factor is declining milkweed habitat up here -- up north -- because in Mexico great efforts have been made to protect the butterflies, but up here we are not thinking about how our corn and soybean agriculture is affecting milkweed populations. I'd like to continue the story arc and ask students to do some of that experimentation themselves -- but if we're dealing with multiplying a lot of 5x5 matrices that's got to be a computer experiment. There's some very cool linear algebra and calculus that goes into the model used in the paper, too, so the next step in worksheets is to play with *sensitivity and elasticity* of the matrix model.

If you want a nice intro to population viability analysis, look here for a very straightforward explanation.

If you'd like to learn more about the monarch lifecycle or want more educational activities, check out the Monarch Lab at the University of Minnesota!

]]>First off, if you're familiar with solving systems of differential equations, you might remember that matrix methods are pretty useful in that endeavor!

- Matrices allow you to solve systems of
*linear*differential equations. - Euler's method basically reduces differential equations to difference equations/matrix methods.

Differential equations can be really hard (or currently impossible) to solve. Matrix models are computationally advantageous and let us deal with small populations really concretely. If we can chunk up the life stages of a population, as with the turkeys in last week's post, we can do some pretty slick matrix modeling.

I think it's the structure of the lifecycles and lifestyles of bees vs butterflies that drives the choice. Let's think about this: bees live in hives, the same one for a long time. We can think of a hive as a population whose health we want to model. There are different classes of bees in the beehive, but they all live in the hive at the same time. Butterflies live as individuals rather than in hives or herds, so we can't look at any population smaller than a regional one. Moreover, the migration of monarch butterflies is a really big deal. The winter monarchs -- the ones who fly to Mexico -- have very different lives than summer monarchs. They live a lot longer and in different places. It's almost as if there are two kinds of butterflies separated in time. The time and space dimensions for modeling these populations, then, are pretty different.

So, that's one set of reasons for using different modeling techniques for these different populations. Can you think of others?

Here's a fun fact, though: you can use discrete methods for some bee modeling. In fact, the Fibonacci sequence comes up in bee math! I was too busy this weekend pondering the game theory of pricing books on Amazon (suddenly relevant) to complete the desired insect life worksheet, but I found some really cool resources while reading:

- Fibonacci numbers for rabbits and cows and
*bees*! - A math circle problem set with a picture of Fibonacci's original writing.
- Stepping up in "grade level" and sophistication, a whole book on difference equations and differential equations. I think it would be appropriate for knowledgeable undergrads.
- A bit of a tangent, but a very cool paper about decision-making among beehives. Lots of math, but also good writing and thought-provoking philosophical implications.

Looks like I'm getting drawn toward longer projects here, like the bees and the butterflies... we'll see what happens!

]]>**Raccoon River:** A while ago I wrote some posts about the Raccoon River in Iowa, and the flow of nitrates into the river. The posts talked about how fertilizing our big corn and soybean fields can lead to problems with nitrate runoff, especially in spring when the snow melts and washes leftover fertilizer into rivers and lakes. The associated worksheets were about increasing/decreasing functions, average rate of change, interpretation of graphs, etc.

It's spring again! Nitrates are a topic with continuing relevance even though the worksheet data is from 2013. Recently, the city of Des Moines voted to sue three Iowa counties for not managing nitrate and nitrite runoff: according to the linked National Public Radio report, removing the nitrates in 2013 cost the city $900,000! The New York Times (coincidentally?) recently featured an article about no-till farming, which reduces fertilizer runoff.

So, how much agriculture is practiced in your state? How big an impact does fertilizer runoff have on your ecosystem? Consider asking your students to report on whether their families fertilize their lawns, and find out what your community is doing to deal with runoff into lakes and streams!

]]>rate of change = increase - decrease

and get some pretty cool models for populations, for instance. (My favorite is looking at predator-prey interactions: write two differential equations, one for foxes and one for rabbits, for instance. Foxes eat rabbits, so the populations depend on each other. What happens as one increases and the other decreases? Check out a puma version here.)

However, differential equations can be really hard to solve. Sometimes it's nicer to take a *discrete* rather than continuous approach: use a matrix model! In a matrix model, you divide time up into discrete steps: months or years or stages of life. Then you multiply a population vector that gives population at step n by a matrix that tells you how each population changes. That gives you a new vector that gives population at step n+1.

Here's a non-insect example: wild turkeys. We can classify wild turkeys as poults (ages 0-1), yearlings (ages 1-2), and adults (ages 2+). Every year turkeys get a year older, as we all do! Only yearlings and adults can reproduce. Then you can do some research to find how the population structure works:

- The number of poults each year depends on the reproduction of yearlings and adults. So P(n+1) = F2*Y(n)+F3*A(n): number of poults at time n+1 is a reproductive constant times number of yearlings at time n and a constant times number of adults at time n.
- The number of yearlings at time n+1 is given by how many poults survive! Y(n+1) = Q1*P(n). Q1 is less than one.
- The number of adults at time n+1 is given by how many yearlings survive plus how many adults at time n survive. So that's A(n+1) = Q2*Y(n) + Q3*A(n). Here Q2 and Q3 are also less than 1 (no magical birth of old birds).

It seems like the literature on bees all uses DEs, while the literature on monarch butterfly populations uses mainly matrix models. This might be because of monarchs' special lifecycle: most monarch live, mate, and die up north, in Canada, the eastern US, or the midwest of the US, but some make the long trip to central Mexico to overwinter there. (There's a smaller population that has the same pattern, but with the Rocky Mountains and California replacing the North and Mexico.) The overwintering monarchs live a much longer lifespan and really have a totally different life than the summer monarchs.

I'm working on a worksheet for monarch modeling with a matrix. In the meantime, you can find educational links at Education World and Monarch Watch. Spring is the time to start thinking about butterfly activities, as the monarch migration north starts in April!

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- The book is just about ready. I've been updating it at Leanpub, and it's your last chance to grab it before it goes on sale at Amazon and elsewhere!
- Job changes: as of January I've started a full-time position at the University of Minnesota, working with MCFAM. There I'm involved with teaching, research, and online education. What does this mean for you? While I'm more busy in some ways, there has been a freeing of psychic energy and that may manifest in more posts
- Last, I've gotta expand beyond calculus. There may be some probability or linear algebra mixed in here!

My spring goal is to post regularly (max once a week, though), and to this end I'll start with some links on bees again. More research keeps coming out about possible causes in the decline of the bee populations world-wide, and there are many indications that a lot of factors have come together to contribute to the bee trouble.

Since it's spring, you might consider planting bee and butterfly-friendly plants when you have a chance! It's not too hard to upgrade the insect-friendliness of your yard. In Minnesota, check out the U of MN resources on bee-friendliness. Other places will have different suggested plants. And look out for future Monday posts on EarthCalculus, starting with bees and butterflies!

S

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