Known as man’s best friend, dogs have evolved alongside humans for thousands of years and come in many shapes and forms, aiding mankind in a huge range of specialised roles. There are herding and guarding breeds to help manage and protect livestock; hunting dogs used for tracking, killing or retrieving game; dogs used in the workforce for their powerful noses or strong builds; and service dogs for assisting people with disabilities. For centuries, we even had a breed called the turnspit, specialised for turning a wheel, often to power kitchen appliances. But how did we get here?

Turnspit dogs ran in wheels to turn meat, depicted here in a Welsh pub in 1797.

It is well known that we’ve been selectively breeding dogs for these roles for centuries, but how we came into possession of such docile animals in the first place is still up for debate. We know that dogs evolved from prehistoric grey wolves, while grey wolves themselves are still around today. At some point, an evolutionary branch occurred, separating the wolves that remained wild from those that would eventually become our household pets. Although it is not known exactly what caused this distinction, there have been many hypotheses.

The domestication process. (Source)

Some theories suggest that complementary hunting styles led to collaboration between our respective prehistoric species, and the relationship grew from there. Others believe that humans may have hand-reared wolf pups from an early age; the pups would imprint on their carer, forming the initial bond. Aggressive wolves would be culled, whereas successfully socialised wolves would be kept and bred to produce increasingly domesticated companions. The final popular theory is of proto-domestication: the idea that early wolves domesticated themselves in favour of scavenging human scraps, which would’ve been more reliable as a food source than wild prey.

We can model the evolution of these prehistoric wolves into our modern-day pets and their cousins, the grey wolf, using mathematics! In this case we will test the mathematical validity of the proto-domestication theory using a method called adaptive dynamics.

Adaptive dynamics

Adaptive dynamics takes the theory of natural selection and applies it to the parameters defining the dynamical systems that govern a population. This means we assume that evolution will favour traits that are advantageous to the population. In this case, we assume a mutant trait will be successful if it has a faster ‘growth rate’ relative to the trait already resident in the population.

Before we go any further, there are some key simplifications at play in adaptive dynamics that must be stated.

1. Mutations are small: a mutant individual doesn’t differ much from its parent.
2. Mutations are rare: The number of mutants that arise at any one time is very small. This allows us to approximate the total population with the resident population (the one in place before any mutations).
3. The evolutionary timescale is much larger than the dynamical system timescale. The population is allowed to reach equilibrium between successive mutations.

Generally, adaptive dynamics considers the evolution of a single evolutionary trait. Backed up by Dmitri Belyaev’s studies on the domestication of foxes, begun in the 1950s, we can reasonably take the trait controlling the domestication of wolves to be friendliness, tameness, or something similar. The study also found that many traits found in domesticated animals, like ear floppiness, softened features, shorter snouts and smaller bodies, were linked, and often emerged on their own among more docile individuals in the species. This makes the transition from wolves to domestic dogs easier to model, as some characteristics don’t need to be accounted for individually, and instead we can just focus on the single friendliness trait.

Beware of the $f=0$

We can denote the friendliness attribute by the parameter $f\in[0,1]$ in the dynamical system we use to describe the prehistoric wolf population. A friendliness value of $0$ corresponds to completely wild wolves with a high aversion to humans. They are therefore strong hunters who don’t benefit from scavenging leftovers. An $f$ value of 1, however, marks a fully domesticated wolf relative to the standard at the time, before selective breeding. These wolves will be the least fearful of humans and are therefore able to benefit from scavenging scraps from human settlements.

Since we are dealing with wolf populations, it makes sense to use a classic Lotka–Volterra model, often used for predator–prey type relationships, but instead the focus will be on interspecies interactions rather than between wolves and their prey.

For two sub-populations of wolves with different friendliness, we say that the first sub-population, $W$, grows proportionally to its current size, but is limited by competition from itself and the second sub-population, $W’$:
\begin{equation*} \frac{\mathrm{d}{W}}{\mathrm{d}t} = \underbrace{rW}_{{\text{growth}}} – \underbrace{\frac{r}{K}W^2}_{{\substack{\text{self-}\\\text{competition}}}} – \underbrace{cWW’}_{{\text{with }W’}}. \end{equation*}
Here, $r$ is the usual growth rate, encoding information about breeding success and litter sizes; and $K$ is the carrying capacity of the sub-population of wolves $W$, ie their limiting population size in the absence of any other sub-populations.

More generally, for many sub-populations $W(f,t)$ with different values of $f$ this extends to (after some relabelling)
\begin{equation*} \frac{\mathrm{d}W(f,t)}{\mathrm{d}t} = rW(f,t)\left[1\!-\!\sum_{f’}\frac{\alpha(f-f’)W(f’,t)}{K(f)}\right]. \label{pop_density} \end{equation*}
The term $\alpha(f-f’)$ is a Gaussian function describing the competition a wolf of friendliness $f’$ exerts on another with trait value $f$,
\begin{equation*} \alpha(f-f’) = \exp{\left(\frac{-(f-f’)^2}{2\sigma_\alpha^2}\right)}. \label{branch_comp} \end{equation*}
This competition distribution corresponds to individuals with similar traits competing more aggressively, while individuals with sufficiently different trait values compete less strongly, where the spread of the competition is characterised by the width, $\sigma_\alpha$. This is because wolves with the same or similar tameness use similar resources: wild wolves compete for the prey they are hunting, while docile ones compete for human scraps. Two examples can be seen in the graph below.

We can use Gaussian distributions to represent the competition exerted by the trait $f’$ on the trait $f$ by the difference $f-f’$. The two widths, $\sigma_\alpha$, represent different spreads of trait values.

For the carrying capacity $K(f)$, we choose a double Gaussian to capture the potential benefits of both extremes. As seen below, the two resource components are centred at $f=0$ (wild) and $f=1$ (docile) and are described by the function
\begin{equation*} K(f) = \underbrace{A\exp{\left(\frac{-f^2}{2\sigma_\text{w}^2}\right)}}_{{\text{wild resource}}} + \underbrace{B\exp{\left(\frac{-(f-1)^2}{2\sigma_\text{d}^2}\right)}}_{{\text{human resource}}}. \end{equation*}

The carrying capacity of the wolf population varies with trait value $f$ and the niche widths $\sigma_\text{w}$ and $\sigma_\text{d}$. Each hump indicates trait values that promote the highest equilibrium population densities. Shading excludes impossible values of $f$. Here, $A=2.5$ and $B=2$.

When these widths are small enough, the Gaussians representing each food source are more distinct and two separate peaks are visible; in other words, it’s better to be an expert than a jack of all trades. The wildest wolves will be the best at hunting prey: as aggression diminishes, this ability is lessened, so friendlier wolves will have less hunting success. Conversely, the most fearless wolves will get first dibs on human scraps, leaving the more cautious wolves with less.

Similarly, the idea of wild wolves being the best hunters can be flipped: those who have less success with wild prey are more likely to turn to self-domestication.

Larger spreads produce a significant overlap between the two resources, leading to the loss of these peaks and the emergence of some optimum middle ground between wild and human resources.

Who’s a good wolf?

Now that we have our model set up, we can get started on the adaptive dynamics analysis. First off, the resident trait value will always be denoted $f=x$; this is the value of $f$ before it mutates at any given point, while $f=y$ will be used as the placeholder for each mutant friendliness value.

As per our assumptions of adaptive dynamics, mutant rarity tells us that initially we can neglect the mutant population density, ie $W’ = W(y,t)\approx0$, and a large evolutionary timescale lets us set $W(x,t)$ to its equilibrium value, $W^*(x)$. The equilibrium occurs where $\frac{\mathrm{d}W(x,t)}{\mathrm{d}t}=0$; since we assume the population is in equilibrium before mutations arise, our equation for $\frac{\mathrm{d}W}{\mathrm{d}t}$ says this is equivalent to
\begin{align*} rW^*(x)\left[1-\frac{\alpha(x-x)W^*(x)}{K(x)}\right] &= 0 \\ \implies W^*(x) &= K(x). \label{eqn:residentdensity} \end{align*}
The first thing we can do with this model is test the success of a mutant wolf (one which is more or less human resistant than its pack) when the rest of the population all shares the same friendliness value. We do this by finding what The hitchhiker’s guide to adaptive dynamics calls the invasion fitness: the expected per capita growth rate of a rare mutant in the environment set by the resident population.

Denoted $F_x(y)$, where $x$ is the resident trait and $y$ is the mutant trait, this is
\begin{align*} F_x(y) &= \left.\frac{1}{W(y,t)}\frac{\mathrm{d}W(y,t)}{\mathrm{d}t}\right|_{\begin{subarray}{l}W(x,t)=W^*(x)=K(x),\\W(y,t)=0\end{subarray}}\\ &= r\left[1-\frac{\alpha(y-x)K(x)+\alpha(x-x)\cdot 0}{K(y)}\right]\\ &= r\left[1-\frac{\alpha(y-x)K(x)}{K(y)}\right]. \end{align*}
If the invasion fitness of $y$ is positive, the mutant can invade; if it is negative, it will die out.

Given our carrying capacity $K(x)$ above, the plot below shows $F_x(y)$ for $x=0.2$ (pink), $0.5$ (purple) and $0.8$ (blue): resident traits of mostly wild, intermediate and mostly docile.

Notice that $F_x(x)=0$ always: a mutant identical to the resident is neither advantaged nor disadvantaged. So, given the resident trait $x$, which direction for $y$ is evolutionarily more successful? The one that’s positive!

Given the carrying capacity, $K(f)$, shown above with $\sigma_\text{w}=1,\sigma_\text{d}=0.5$ (orange), here are three plots of the invasion fitness, $F_x(y)$, for different resident traits, $x$. Notice that $F_x(x)=0$ in every case: if the mutant is the same as the resident trait, there will be no (dis)advantage. The strength of competition width is $\sigma_\alpha=0.5$.

So the sign of the invasion fitness, $F_x(y)$, matters. This can also be visualised on something called a pairwise invasibility plot (PIP) which shows the sign of $F_x(y)$ on a plot of resident trait value by mutant trait value. These are plotted below.

Plots of mutant trait value by resident trait value. Green regions show where the invasion fitness is positive; white regions are negative. The regions are separated by black contours of $F_x(y)=0$. The left-hand plot shows that the curves of fixed $x$ from the previous graph exhibit the same invasion fitness polarity. The right-hand plot shows how traits will evolve along the diagonal: if a mutant trait is in the green region it will be successful and the population will evolve to adopt this new trait. But if it lies in the white region, the resident trait will remain the same. This evolution sequence is shown for a limited number of mutations but the pattern shows evolution will continue towards the point where the contours intersect. Parameters: $A=2.5$, $B=2$, $\sigma_\text{w} = 1$, $\sigma_\text{d} = 0.5$ and $\sigma_\alpha = 0.6$.

The selection gradient, $s(x)$, for the resident trait, $x$, tells us whether friendlier or less friendly mutants are favoured for each $x$. This plot is based
on the carrying capacity from above with $\sigma_\text{w} = 1$, $\sigma_\text{d} = 0.5$.

We have assumed that mutations are small, so another way to find which direction for $y$ is more successful is to look at the slope of the graph at $y=x$, known as the selection gradient:
\begin{align*} s(x) &= \left.\frac{\partial F_x(y)}{\partial y}\right|_{y=x} \\ &=r\,\Biggl[-\underbrace{\left.\frac{\partial \alpha(y-x)}{\partial y}\right|_{y=x}}_{=\;0} + \frac{\partial K(x)}{\partial x}\frac{1}{K(x)}\Biggr]\\ &=\frac{r}{K(x)}\frac{\partial K(x)}{\partial x}. \end{align*}
The selection gradient tells us information about the speed and direction of evolution: positive $s$ means slightly friendlier mutants are favoured; negative $s$ means slightly less-friendly ones are favoured.

This also tells us whether we move up or down the diagonal of the PIP: if it is green above the diagonal and white below, a friendlier wolf will be more successful so $s(x)>0$ (we move upward and rightward); but the converse means $s(x) < 0$ and we move downward and leftward. To the left you can see $s(x)$ for different values of the resident trait, $x$.

A golden retri-evolutionarily singular strategy

The trait value at which evolution gets interesting is where $s(x^*)=0$: neither friendly nor unfriendly is favoured, so this trait value is called the evolutionarily singular strategy. This can be seen visually as the point at which the $F_x(y)=0$ contours cross in the PIP.

‘Gets interesting’ can mean many things, but in our case, the point of interest we are looking for is where the singular strategy is additionally a branch point. Branch points are defined by their ability to be both convergent and unstable evolutionary waymarks: evolution is attracted to the point but selection can then drive the population to split into distinct branches which diverge away from one another towards different values of the trait, $f$.

Convergence comes from requiring $s'(x^*) < 0$; instability thereafter requires that $F_{x^*}(y)$ has a local minimum at $y=x^*$ so that any mutation away from $x^*$ grows. Some mathematical manipulation can show that meeting these conditions corresponds to satisfying the inequality \begin{alignat*}{3} \left.\frac{\partial^2 F_x(y)}{\partial x^2}\right|_{y=x=x^*} &>\;& \left.\frac{\partial^2 F_x(y)}{\partial y^2}\right|_{y=x=x^*} &>0. \end{alignat*} In our case, this implies
\begin{equation*}\begin{split} r\left(\frac{1}{\sigma_\alpha^2} – \frac{1}{K(x^*)}\left.\frac{\partial^2 K(x)}{\partial x^2}\right|_{x=x^*}\right) \hspace{2cm} \\ > r\left(\frac{1}{\sigma_\alpha^2} + \frac{1}{K(x^*)}\left.\frac{\partial^2 K(x)}{\partial x^2}\right|_{x=x^*}\right) >0. \end{split} \end{equation*}
Let’s test it out on the wolf population. Firstly, our evolutionary singular strategies can be found simply from solving $\frac{\partial K(x)}{\partial x}=0$ (the stationary points of the carrying capacity), which can be done numerically.

Benji’s favourite branch point is under one, in the shade. (Benji, $f=1$, will happily eat human scraps.)

Next, it’s clear from the inequality above that we must have
\begin{equation*} \left.\frac{\partial^2 K(x)}{\partial x^2}\right|_{x=x^*} < 0, \end{equation*} to guarantee convergence, so branch points must be maxima of $K(f)$. The instability condition is then only satisfied for \begin{align*} \frac{K(x^*)}{\sigma_\alpha^2} > &-\left. \frac{\partial^2 K(x)}{\partial x^2}\right|_{x=x^*} = \left| \frac{\partial^2 K(x)}{\partial x^2}\right|_{x=x^*}\\ \implies&\frac{K(x^*)}{\hspace{2em}\left|\frac{\partial^2 K(x)}{\partial x^2}\right|_{x=x^*}} > \sigma_\alpha^2. \end{align*}

Since both sides are positive, this can hold true if the wolf–wolf competition has a narrow enough width! There are two possible scenarios where this can happen — look again at the plot of $K(f)$ above.

Case 1: The yellow plot of $K(f)$:

• The niche widths are narrow.
• The branch point resides near $f=0$.

This is the branch of interest to our domestication story because initially all wolves are wild and use the wild resource.

Case 2: The orange plot of $K(f)$:

• The niche widths are wide.
• The branch point is at a mid-friendliness value.

Both cases can be shown in stochastic simulations under the appropriate constraints derived for branching earlier. You can see these on the next page.

From the plots, it seems as though case 1 is the more realistic choice, although it is worth noting that our example of case 2 is an extreme one. By altering the values of $A$, $B$, $\sigma_\text{w}$ and $\sigma_\text{d}$, the branch point can be pushed closer to $0$ to be more representative of where we would expect the true branch to lie.

In fact, we would expect to find $A$ much larger than $B$ in the early beginnings of wolf proto-domestication (corresponding to the purely wild wolf population having a much higher capacity than the purely ‘domestic’) and when this is true, the branch point is found to be very near zero despite the case we are in.

This is good news since grey wolves are essentially the direct descendants of pre-historic wolves, so it makes sense that the branch point lies right on $f=0$ and dogs have diverged from wolves at that point leaving the wolves unperturbed, rather than all wolves evolving as a whole towards a more docile state before going back on themselves (evolutionarily speaking) after branching.

Stochastic simulations of friendliness evolution in prehistoric wolves over time. Values of $\sigma_\alpha$ have been chosen to satisfy the conditions for a branch point near $0$ in both cases. Each coloured curve represents a different realisation of the simulation to show the pattern is not unique. In both cases, one branch would evolve to consist of purely wild wolves, while the other would grow in friendliness until they eventually become domestic enough for selective breeding to take over (which would require a whole other model). Parameters: $A=2.5$, $B=2$, maximum mutational step $=0.01$. Case 1: $\sigma_\text{w} = 0.5$, $\sigma_\text{d}=0.25$, $\sigma_\alpha = 0.4$. The simulation shows that in the presence of humans, the population would branch at some value of $f$ very close to zero. Case 2: $\sigma_\text{w} = 1$, $\sigma_\text{d}=0.5$, $\sigma_\alpha = 0.6$. The simulation shows that in the presence of humans, the population would grow in friendliness until some critical value is reached, at which point the population splits into increasingly wilder and friendlier wolves. Images: Grey wolf image, Seney Natural History Association, CC BY-SA 2.0.

Give paw

So what have we found out? For the model specified at the beginning of this article, the maths supports the proto-domestication theory that the benefit of human scraps could have driven the evolution of prehistoric wolves into dogs and grey wolves.

What’s most fascinating is not just that self-domestication was attractive enough to cause this change, but that the wild wolf population remained. Evolution was not unidirectional; instead, it supported the creation of two distinct populations within the same geographical area! This is important as speciation is often considered to be driven by geographical separation which consequently causes a species to evolve to adapt to two different environments, which wouldn’t have necessarily been the case for these wolves. Furthermore, this is an important result as it shows the usefulness of mathematics in situations where we can’t know the answer to a question using data or experiments (in this case because it happened so long ago).

Although we still don’t have a concrete answer for the origin of our four-legged friends this maths acts as one more piece of the puzzle in the quest to find out.

The post The domestication of prehistoric wolves appeared first on Chalkdust.

Nursing an identity as a runner for decades despite participation that felt torturous and intermittent, Chris Onstad reveals his on-again, off-again relationship with the sport. In this piece for The Baffler, soaked in sweat and equal parts hyperbole and self-deprecation, Onstad explains how walking became his gateway drug to running during the pandemic.

I maxed out at 43.8 miles on July 25, 2020, which took about fourteen hours of unbroken walking. I limped the last ten miles, the sides of my heels raw like a peeled shallot, in a state I can best describe as “too proud to call an Uber.” My calves above the sock line were satisfyingly gray from soot; below it, a virginal ivory. I had walked the most that could be walked in one summer day’s light: the summit of that restless urge.

I estimate I logged over a thousand miles during that five-month period. Using a browser, I mapped the route from memory after each day’s walk and started posting the daily mileage on Instagram, proving I’d finally committed to the new identity. Walking was a sustainable way for the fitness-minded man to gracefully acknowledge his age, I told myself as my lavender fanny pack and I silently mocked the sprinting ninnies. I even started posting reviews of nonalcoholic beers, like some warped lunatic who doesn’t realize everybody can see right through his midlife crisis.

Writer and comedian Zach Mander took quotes from Karl Marx and paired them with illustrations from books by Richard Scarry like What Do People Do All Day? and Cars and Trucks and Things That Go.

See also Mamdani’s Richard Scarry socialism.

Tags: books · illustration · Karl Marx · politics · remix · Richard Scarry · Zach Mander

about a week before october hit, i saw a post from loren about getting ready for weird web october. this was the third one to be organized (shouts out to jay) but my first time participating. i wish i had written down at the start why i wanted to try it out - i guess i was just in the right headspace at the time it came across my radar - but i'm very glad that i did it because it gave me a morning routine where i got to use and share my brain in a developery way that i hadn't since the late glitch days.

my plan was simple: when i wake up and take my thyroid medicine, i need to wait an hour before i can eat or drink anything that's not water - so while i wait for my coffee time, i open up my laptop and get to work on my weird app for that day. to keep it even simpler, i chose to use p5.js as much as possible to save me time - i'm a slut for html canvas so i knew i'd be using them a lot. and most importantly, if i can't make anything that day, i wasn't allowed to get down on myself about it. this is the same boundary i set when i started my weekly retros and 3-minute vlogs. the most important rule, though, was if i wasn't having fun and learning while doing this, just stop.

it turns out, i was having fun and learning a lot while doing weird web october, so i didn't stop. before i get what i learned, etc, here's the full list of little weird web apps i made every single day in the month of october:

1. reveal
2. spark
3. fake
4. plastic
5. sheet
6. analog
7. organic
8. skeuomorph
9. spicy
10. layers
11. stuck
12. pointy
13. scratch
14. origami
15. wooden
16. instant
17. branching
18. geometric
19. distorted
20. undefined
21. illusion
22. skeletons
23. incognito
24. collage
25. ascending
26. net
27. spiral
28. smooth
29. the end
30. pastel
31. spooky

since this challenge was not centralized to a specific hosting platform, i was able to upload my apps anywhere and still have it linked in an rss feed of submissions thanks to the octothorpes protocol - all it took was adding a link tag in each app to have it indexed. all of the links in the prompt list on the wwo site are feeds of all the submissions by prompt. everyone involved in this is a human genius!

the main thing i learned through doing this challenge is that i'm still capable to do things like this. when i left my corporate job and became a small art business owner, some of my web developer colleagues asked me if i was worried about "losing it" which i interpret as "losing the ability to code". btw it's only men that ever ask that, so i don't believe it's a real idea that applies to me, and yet because it's men asking it my instinct is to prove them wrong - tapping into the superpowers i wielded to stay and succeed in that field as long as i did. i didn't think i couldn't code every day, but i wasn't sure if i'd have the actual appetite to. to me, coding has always been making art through solving puzzles, and i was reminded that this is still the case. not using ai-assisted coding tools absolutely helped with that feeling.

now for some advice i have for others interested in participating next year, whether or not they feel like it's possible:

• i self-host my own site, so finding a place to put these apps was trivial. other participants used (and i recommend, too) neocities which lets you host static pages for free. if you need something full stack, i have this post of glitch alternatives that include some servers
• i don't use ai for anything i code, make, write etc. i encourage anyone who uses ai and/or vibe codes to consider doing an exercise like this without those tools as a way to challenge your brain and you can continue doing your regular work the way you do your regular work.
• if you do use ai, be open about it - you should be proud of your tools like the rest of us are! if you're not proud, then you may want to reflect on why are you are using tools you're not proud of for a hobby daily prompt exercise in the first place.
• i would absolutely consider using a library like p5.js if it will help you work around a specific element of your work quicker and/or easier. for example, p5 mouse events also handle touch events, so it cut the work in half for making my apps work on mobile along with desktop on top of all the sugar it provides for building with html canvas.
• most importantly, if you can't do it every day, don't get down on yourself. again, it's just a hobby daily prompt exercise, the stakes are so low here so take advantage of that by staying chill and relaxed.

next year i'll probably pick a more narrow umbrella for my projects. i'm currently taking a game development course at sfpc, so next year i may try to make a weird web game every day, or perhaps a weird level of a weird web game every day. i've also been talking to the octothorpes protocol folks about how their protocol works and how i could maybe resurrect something like the glitch community code jams with it. i don't have the bandwidth to do that right now, but it's on my mind as i plan my 2027.

whether it's weird web october, inktober, any month or art medium, organized with a community or all on your own, i highly recommend a one-month daily challenge to exercise your brain and time. i'm now getting more done in the morning, feeling more inspired around code, and feeling hopeful and in community with so many others who build weird web things like i do.

xoxo jenn

Here are three mistakes you might see a student make:

These mistakes all have something in common: they’re logical. Each mistake is a generalization of something that students have learned before. The + sign means to add, so we should add across, right? In addition problems the order doesn’t matter, so in subtraction it must not matter either. 8 - 3 = 5, and 3 - 4 = 1, right? When we add, we stack the numbers, shove them to the right, then add, right?

Are These Ready?

One approach is to explain why each of these is a mistake. To explain the place value system, why we add ones to ones and tenths to tenths. To explain why 4 - 3 is different from 3 - 4. I do that. I want my students to understand those things and to see that math is a logical system that makes sense.1

But I’ve always found that approach insufficient. Students aren’t thinking very hard about these problems. They aren’t thinking to themselves, “hmm, 3.2, ok what number is in the ones place and what number is in the tenths place?” The problem is that they’re solving the problem without thinking, relying on the habits they’ve built in the past. My goal isn’t to convince them logically that what they’re doing is wrong. My goal is to get students to slow down and recognize that there’s an extra step they need to take to solve this particular problem.

The approach I’ve taken for each of these three skills is to teach students to ask themselves a quick question. Are these ready? Fractions are ready to be added when they have the same denominator. Digits are ready to be subtracted when the top (or first) number is equal to or more than the bottom (or second) number. Decimals are ready to be added when they have the same number of digits after the decimal. If they aren’t ready, we need to do something else first.

Categorical Atoms

These three skills are examples of what Kris Boulton calls “categorical atoms.”2 A categorical atom is any question that can be answered with a yes or no. There’s a nice approach to teaching skills like these, a sequence of examples called NPPPN. NPPPN means that the sequence begins with a negative example (not ready to be subtracted), then three positive examples, then a negative example. I give students this sequence of examples:

3 - 5: these are not ready to be subtracted

5 - 3: these are ready to be subtracted

9 - 1: these are ready to be subtracted

4 - 4: these are ready to be subtracted

3 - 4: these are not ready to be subtracted

Humans are excellent at recognizing patterns. The goal here is to avoid wordy explanations. Instead, students figure out the pattern through a carefully designed series of examples.

When moving from a negative to a positive example, the goal is to change as little as possible, to highlight the boundary between the non-example and the example. When moving from a positive example to another positive example, the goal is to make a larger change to show the breadth of positive examples.3

Then, I see whether students understood the sequence. I ask students a bunch more questions, in random order. One nice thing about a categorical atom is that it’s easy to check for understanding. Choral response, mini whiteboards, thumbs up/thumbs down, whatever, and I quickly have a sense of whether students can answer the question on their own.

How to Teach It

If I’m going to teach one of these skills, where “are you ready…?” is an atom I want students to understand, I always teach the “are you ready…?” atom a few days before it’s needed.

The NPPPN approach works pretty well, but it’s not perfect. My goal is to teach it, check for understanding, then either teach again or just provide a bit more practice. I do this over a few days, spending as much time as needed. Typically that’s only a few minutes each day. Multiple rounds help to ensure the learning is secure before students need to use this atom as part of a larger procedure. Multiple rounds also give me a chance to help students understand the why: why the ones place and the tenths place need to line up when we add, etc. It’s tough to teach the why and the how at the same time. When I try to teach multiple ideas at the same time, things often get muddled. That’s another benefit of revisiting this idea over multiple days: if I want students to understand the why, revisiting that why multiple times is crucial.

If students are really solid with the “are you ready…?” atom, it just makes everything after it easier. It’s still tough to teach students how to find common denominators, or how to borrow when subtracting, or how to line up the decimals. (If those don’t sound tough to you, please stop by and help me out!) But it’s way tougher to teach those ideas when students keep mindlessly doing something wrong. As I’ve taught this way, I’ve occasionally ended up with frustrated students: they know that the numbers aren’t ready to be added, but they can’t remember what they need to do. “These aren’t ready to be added, help me!” That’s a good problem to have! Students know they need to do something and they’re slowing down to try to figure out what that something is.