Can We Model Generosity With Mathematics?

four celebrities and their donations — hemsworth $1000000, mariam (hs student) - 75, British Petroleum - $700000, Jeff Bezos — $700000

I posted that image on Twitter last week, asking:

Which of these bushfire relief donors was the most generous? What’s your ranking? What information matters here? What would your students say?

Some teachers quickly identified a connection to ratios.

Meanwhile, Lee Melvin Peralta critiqued ratios as too limited to fully model generosity.

I tend to side with George Box here, who wrote:

All models are wrong, but some are useful.

Anyone who thinks that proportional functions fully describe runners in a race, or that linear functions fully describe the height of a stack of cups, or that quadratic functions fully describe the height of objects under gravity, or that ratios fully describe generosity is, of course, kidding themselves.

But those models are all useful. Ratios are a useful way to think about generosity.

Emily Atkin originally stirred this question up for me in her fantastic climate change newsletter:

Chevron’s donation is paltry, however, given its earnings and relative contribution to the climate crisis. Not only is Chevron the second-largest historical emitter of all the 90 companies, it also earned about $15 billion in 2018. So a $1 million donation amounts to about .00667 of its yearly earnings. To the average American, that donation would amount to about $3.96.

So Atkin is evaluating generosity as ratio of net worth/earnings to donation size. But then she also considers the donor’s contribution to climate change.

The model is complex and grows more complex!

One teacher wanted to add fame and notoriety to our model, something Chris Hemsworth donates that Mariam might not. (Maybe she’s a TikTok teen, though. We can only speculate.) I talked with someone who lives in Australia about this question, and she said Hemsworth is less generous than someone from another country donating the same amount because of his identity as an Australian citizen. Robyn V wondered how to evaluate time donations, and even the donation of one’s life.

So ratios aren’t a perfect model for generosity, but they do offer us an important insight that, under some circumstances, someone who donates $75 is more generous than someone who donates one million dollars, which one teacher noted is a quantity that is really hard for students to fathom!

One teacher preparation program asked the question:

If the ambiguity of the original question strikes you as anything other than a feature, then please don’t risk the conversation.

If you go into the conversation presupposing a model for generosity rather than admitting to yourself in advance that all the models are broken, you’re likely to diminish students who suggest variables you had already excluded.

Okay, yes, um, ‘whether or not someone lives in Australia.’ Okay, that’s one idea, but can I get some other ideas, please? Perhaps ones more related to the math we’ve been studying?

All of these models are complex. All of them are certainly broken. And all of them offer you the opportunity to celebrate and build on your students’ curiosity and contextual knowledge, an experience that is all too rare for students in math class.

BTW: Shout out to Christelle Rocha for her observation that individual generosity is no way to solve the climate crisis.

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I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


    • Btw. I think this answers the question that has been stewing about modeling and defining “real world” vs. “non-real world”
      Obviously in the pure world of mathematics all models are perfect! It isn’t until we try and apply them to the “real world” that they become broken.

  1. Loved the comment, “interestingly fuzzy or annoyingly vague?” – because in many ways that encapsulates the Dan Meyer style of questions!

    Throw up an annoyingly vague question, then let the room straighten it out and tell you what they “should” be answering

  2. If I were solving this, my first step would be discussing how to define generosity more precisely, or rephrase it into a more productive question.

    Are we asking who is most “deserving recognition” or “deserving praise” or “how much they value the well-being of others vs themselves” or “likely to be charitable in the future” or “having the biggest positive impact with their donation” “having the biggest net-positive impact” )

    If this step is missing and it’s left as a vague “who do I like most” type question, I don’t expect it to be a very useful discussion.

    To put it another way, I think it’s important to consider *why* we might care about generosity. Which donation(s) should we write a newspaper article about? Which type of donor should we be marketing charities to? Which donor(s) should we preferentially patronize* due to their donation? Which one deserves the biggest tax write-off? What effect on the world would praising or criticizing each donor have?

    *Buying their oil over competitors. Watching their movies. Using their site. Buying their homemade crafts. Joining their Patreon. Etc.

    One argument for proportional model: If rich people/entities donate a small percentage of their wealth, that means the world is worse off with so much wealth concentrated among these “less generous” entities. Whatever group gives the highest percentage is the group we want to have more money.
    Example: Suppose non-CEOs give 5% of their money to charity, and CEOs give 2% of their money to charity. A world where non-CEOs have 90% of the wealth is more charitable than a world where CEOs have 90% of the wealth.

    One argument for the simple model: Having one more large donor does much more good than having one extra small donor. Thus we should praise/reward large donors when they donate, to encourage them to donate more. Praising poorer donors is perhaps a waste of time.

  3. I don’t believe we really have enough information to answer the question–even stipulating that this is intentionally supposed to be either “an annoyingly vague” or “interestingly fuzzy” question. As Jeremy H. writes above, at some point we will need to define generosity so that we can put each donation into proper context.

    Suppose we define the generosity as the ratio of the gift to each entity’s overall net worth and we see that Jeff Bezos’ donation is the smallest fraction of his net worth. Is that really all we need to know to draw that conclusion or are there other factors to consider? This may not be Bezos’ only charitable donation. Suppose we learn that he has made 18 other charitable donations within the last quarter that when added together, exceed any of the other entity’s donation ratios. Is this question intended to cast a judgement about the values and character of each entity or merely a problem about ratios wrapped in a timely context?

    I think one takeaway for the students is to think more deeply about numbers rather than just taking each of them at face value. But we also want to caution against making a judgement when we think we have all the information (comparing each donation relative to the means of each entity) when there might still be unknown unknowns lurking (failing to consider other donations/contributions the entities have made). As the famous quote says, “it’s not what we don’t know that hurts us the most, it’s what we think we know for sure that just ain’t so.”