And now, finally, let’s get to the formula for combinations. The math in my last post got a little tricky toward the end, with the strange exclamation mark notation floating around. So let’s recap permutations without that notation.

4

× 3

× 3 × 2

× 3 × 2 × 1

÷ (1 × 2 × 3)

÷ (1 × 2)

÷ 1

You should see that, to traverse a tree diagram, we multiply by the tree branches, cumulatively, to move right, and then divide by those branches—again, cumulatively—to move left. The formula for the number of permutations of 2 cards chose from 4, \(\mathtt{\frac{4!}{(4-2)!}}\), tells us to multiply all the way to the right, to get 24 in the numerator and then divide two steps to the left (divide by \(\mathtt{(4-2)!}\) or 2) to get 12 permutations of 2 cards chosen from 4.

Combinations

An important point about the above is that the number of permutations of \(\mathtt{r}\) cards chosen from \(\mathtt{n}\) cards, \(\mathtt{_{n}P_r}\), is a subset of the number of permutations of \(\mathtt{n}\) cards, \(\mathtt{n!}\) The tree diagram shows \(\mathtt{n!}\) and contained within it are \(\mathtt{_{n}P_r}\).

Combinations of \(\mathtt{r}\) items chosen from \(\mathtt{n}\), denoted as \(\mathtt{_{n}C_r}\), are a further subset. That is, \(\mathtt{_{n}C_r}\) are a subset of \(\mathtt{_{n}P_r}\). In our example of 2 cards chosen from 4, \(\mathtt{_{n}P_r}\) represents the first two columns of the tree diagram combined. In those columns, we have, for example, the permutations JQ and QJ. But these two permutations represent just one combination. The same goes for the other pairs in those columns. Thus, we can see that to get the number of combinations of 2 cards chosen from 4, we take \(\mathtt{_{n}P_r}\) and divide by 2. So, \[\mathtt{\frac{4!}{(4-2)!}\div 2=\frac{4!}{(4-2)!\cdot2}}\]

What about combinations of 3 cards chosen from 4? That’s the first 3 columns combined. Now the repeats are, for example, JQK, JKQ, QJK, QKJ, KJQ, KQJ. Which is 6. Noticing the pattern? For \(\mathtt{_{4}C_2}\), we divide \(\mathtt{_{4}P_2}\) further by 2! For \(\mathtt{_{4}C_3}\), we divide \(\mathtt{_{4}P_3}\) further by 3! We’re dividing (further) by \(\mathtt{r!}\)

When you think about it, this makes sense. We need to collapse every permutation of \(\mathtt{r}\) cards down to 1 combination. So we divide by \(\mathtt{r!}\) Here, finally then, is the formula for combinations: \[\mathtt{_{n}C_r=\frac{n!}{(n-r)!r!}}\]

So, did you come up with a working rule to describe the pattern we looked at last time? Here’s what I came up with:

As we saw last time, the “root” of the tree diagram (the first column) shows \(\mathtt{_{4}P_1}\), which is the number of permutations of 1 card chosen from 4. The first and second columns combined show \(\mathtt{_{4}P_2}\), the number of permutations of 2 cards chosen from 4. So, to determine \(\mathtt{_{n}P_r}\), according to this pattern, we start with \(\mathtt{n}\) and then multiply \(\mathtt{(n-1)(n-2)}\) and so on until we reach \(\mathtt{n-(r-1)}\).

The number of permutations of, say, 3 items chosen from 5, then, would be \[\mathtt{_{5}P_3=5\cdot (5-1)(5-2)=60}\]

This is a nice rule that works every time for permutations of \(\mathtt{r}\) things chosen from \(\mathtt{n}\) things. It can even be represented a little more ‘mathily’ as \[\mathtt{_{n}P_r=\prod_{k=0}^{r-1}(n-k)}\]

So let’s move on to the “legal” formula for \(\mathtt{_{n}P_r}\). A quick sidebar on notation, though, which we’ll need in a moment.

When we count the number of permutations at the end of a tree diagram, what we get is actually \(\mathtt{_{n}P_n}\). In our example, that’s \(\mathtt{_{4}P_4}\). The way we write this amount is with an exclamation mark: \(\mathtt{n!}\), or, in our case, \(\mathtt{4!}\) What \(\mathtt{4!}\) means is \(\mathtt{4\times(4-1)\times(4-2)\times(4-3)}\) according to our rule above, or just \(\mathtt{4\times3\times2\times1}\). And \(\mathtt{3!}\) is \(\mathtt{3\times(3-1)\times(3-2)}\), or just \(\mathtt{3\times2\times1}\).

In general, we can say that \(\mathtt{n!=n\times(n-1)!}\) So, for example, \(\mathtt{4!=4\times3!}\) etc. And since this means that \(\mathtt{1!=1\times(1-1)!}\), that means that \(\mathtt{0!=1}\).

So, for the tree diagram, \(\mathtt{_{4}P_4}\) means multiplying all the way to the right by \(\mathtt{n!}\). But if we’re interested in the number of arrangements of \(\mathtt{r}\) cards chosen from \(\mathtt{n}\) cards, then we need to come back to the left by \(\mathtt{(n-r)!}\) And since moving right is multiplying, moving left is dividing.

4

4 × 3

4 × 3 × 2

4 × 3 × 2 × 1

÷ (4 – 1)!

÷ (4 – 2)!

÷ (4 – 3)!

÷ (4 – 4)!

The division we need is not immediately obvious, but if you study the tree diagram above, I think it’ll make sense. This gives us, finally, the “legal” formula for the number of permutations of \(\mathtt{r}\) items from \(\mathtt{n}\) items: \[\mathtt{_{n}P_r=\frac{n!}{(n-r)!}}\]

Last time, we saw that combinations are a subset of permutations, and we wondered what the relationship between the two is. Before we get there, though, let’s look at another possible relationship—one we only hinted at last time. And to examine this relationship, we’ll use a tree diagram.

Tree Diagram

This tree diagram shows the number of permutations of the 4 cards J, Q, K, A—the number of ways we can arrange the 4 cards. The topmost branch shows the result JQKA. And you can see all 24 results from our list last time here in the tree diagram.

4

4 × 3

4 × 3 × 2

4 × 3 × 2 × 1

÷ 3!

÷ 2!

÷ 1!

÷ 0!

Here’s where, normally, people would talk about the multiplication 4 × 3 × 2 × 1 and tell you that another way to write that is with an exclamation mark: 4! But that’s skipping over something important.

And that something important is this: Notice that the first column of the tree diagram—the root of the tree—shows 4 items. This is the number of different permutations you can make of just 1 card, chosen from 4 different cards. And the first and second columns combined show the number of permutations you can make of 2 cards, chosen from 4 cards (JQ, JK, JA, etc.).

And so on. You might think that to go from “permutations of 1 card chosen from 4″ to “permutations of 2 cards chosen from 4″ you would multiply by 2. But of course that’s not right (and the tree diagram tells us so). You actually multiply 4 by 4 – 1. And to go from “permutations of 1 card chosen from 4″ to “permutations of 3 cards chosen from 4″ you multiply 4 • (4 – 1) • (4 – 2).

We’re on the verge of being able to describe the relationship, which I’ll put in question form (and mix in some notation to):

What is the relationship between the number of permutations of \(\mathtt{n}\) things, \(\mathtt{P(n)}\), and the number of permutations of \(\mathtt{r}\) things chosen from \(\mathtt{n}\) things, \(\mathtt{_{n}P_r}\)?

We can see from our example above that \(\mathtt{P(4)=24}\). That is, the number of permutations of 4 things is 24. But we also noticed these three results: $$\begin{aligned}_{\mathtt{4}}\mathtt{P}_{\mathtt{1}}&= \mathtt{\,\,4}\cdot \mathtt{1} \\ _{\mathtt{4}}\mathtt{P}_{\mathtt{2}}&= (\mathtt{4}\cdot \mathtt{1})(\mathtt{4}-\mathtt{1}) \\ _{\mathtt{4}}\mathtt{P}_{\mathtt{3}}&= (\mathtt{4}\cdot \mathtt{1})(\mathtt{4}-\mathtt{1})(\mathtt{4}-\mathtt{2})\end{aligned}$$

A New Formula?

Study the pattern above and see if you can write a rule that will get you the correct result for any \(\mathtt{_{n}P_r}\). Check your results here (for example, for \(\mathtt{_{16}P_{12}}\), you can just enter 16P12 and press Enter).

The rule you write, if you get it right, won’t be an algorithm. But it’ll work every time! This is the step we always skip when teaching about permutations! The next step is to think hard about why it works. We’ll get to the “legal” formula for permutations next time.

I have now been blogging for 16 years, and my very first post (long gone) was on combinations and permutations. So, it’s fun to come back to the idea now. In 2004, my experience with the two concepts was limited to how textbooks often used the awkward “care about order” (permutations) or “don’t care about order” (combinations) language to introduce the ideas. So, that’s what I wrote about then. Now I want to talk about how the two concepts are related.

What They Are

When you count permutations, you count how many different ways you can sequentially arrange some things. When you count combinations, you count how many ways you can have some things. So, given 2 cards, there are 2 different ways you can sequentially arrange 2 cards, but given 2 cards, there’s just one way to have 2 cards.

Right off the bat, the language is weird, and it’s hard to see why combinations should ever be a thing (there’s always just 1 way to have a set of things). But combinations make better sense when you are not choosing from all the elements you are given.

So, for example, how many permutations and combinations can I make of 2 cards, chosen from a total of 3 cards?

Now having the two categories of permutation and combination makes a little more sense. There are 6 permutations of 2 cards chosen from 3 cards and there are 3 combinations of 2 cards chosen from 3 cards. That is, there are 6 different ways to sequentially arrange 2 cards chosen from 3 and just 3 different ways to have 2 cards chosen from 3. And you can see, by the way, that the combinations are a subset of the permutations.

In fact, let’s do an example with 4 cards to show the actual relationship between permutations and combinations. Here we’ll just use letters to save space. The permutations of JQKA if we choose 3 cards are:

That’s 24 permutations. For combinations, we get JQK, JQA, QKA, KAJ. That’s 4 combinations. What’s the relationship? We’ll come back on that next time.

I‘ve just started with Six Not-So-Easy Pieces, based on Feynman’s famous lectures, and already there’s some decently juicy stuff. In the beginning, Feynman discusses the symmetry of physical laws—that is, the invariance of physical laws under certain transformations (like rotations):

If we build a piece of equipment in some place and watch it operate, and nearby we build the same kind of apparatus but put it up on an angle, will it operate in the same way?

He goes on to explain that, of course, a grandfather clock will not operate in the same way under specific rotations. Assuming the invariance of physical laws under rotations, this change in operation tells us something interesting: that the operation of the clock is dependent on something outside of the “system” that is the clock itself.

The theorem is then false in the case of the pendulum clock, unless we include the earth, which is pulling on the pendulum. Therefore we can make a prediction about pendulum clocks if we believe in the symmetry of physical law for rotation: something else is involved in the operation of a pendulum clock besides the machinery of the clock, something outside it that we should look for.

Rotation Coordinates

Feynman then proceeds with a brief mathematical analysis of forces under rotations. A somewhat confusing prelude to this is a presentation that involves expressing the coordinates of a rotated system in terms of the original system. He uses the diagram below to derive those coordinates (except for the blue highlighting, which I use to show what (x’, y’) looks like in Moe’s system). What we want is to express (x’, y’) in terms of x and y—to describe Moe’s point P in terms of Joe’s point P.

“We first drop perpendiculars from P to all four axes and draw AB perpendicular to PQ.”

The first confusion that is not dealt with (because Feynman makes the assumption that his audience is advanced students) is what angles in the diagram are congruent to θ shown. And here again we see the value of the easy-to-forget art of eyeballing and common sense in geometric reasoning.

The y’ axis is displaced just as much as the x’ axis by rotation, and “displaced just as much by rotation” is a perfectly good definition of angle congruence that we tend to forget after hundreds of hours of deriving work. The same reasoning applies to the rotational displacement from AP to AB. If we imagine rotating AP to AB, we see that we are starting perpendicular to the x-axis and ending perpendicular to the x’-axis. The y-to-y’ rotation does the same thing, so the displacement angle must be the same. So let’s put in those new thetas, only one of which we’ll need.

Inspection of the figure shows that x’ can be written as the sum of two lengths along the x’-axis, and y’ as the difference of two lengths along AB.

Here is x’ as the sum of two lengths (red and orange): \[\mathtt{x’=OA\cdot\color{red}{\frac{OC}{OA}}+AP\cdot\color{orange}{\frac{BP}{AP}}\quad\rightarrow\quad x\cdot\color{red}{cos\,θ}+y\cdot\color{orange}{sin\,θ}}\]

And here is y’ as the difference of two lengths (green – purple): \[\mathtt{y’=AP\cdot\color{green}{\frac{AB}{AP}}-OA\cdot\color{purple}{\frac{AC}{OA}}\quad\rightarrow\quad y\cdot\color{green}{cos\,θ}-x\cdot\color{purple}{sin\,θ}}\]

So, if Joe describes the location of point P to Moe, and the rotational displacement between Moe and Joe’s systems is known (and it is known that the two systems share an origin), Moe can use the manipulations above to determine the location of point P in his system.

Another, exactly equal, way of saying this—the way we said it when we talked about rotation matrices—is that, if we represent point P in Joe’s system as a position vector (x, y), then Moe’s point P vector is \[\small{\begin{bmatrix}\mathtt{x’}\\\mathtt{y’}\end{bmatrix}=\begin{bmatrix}\mathtt{\,\,\,\,\,cos\,θ} & \mathtt{sin\,θ}\\\mathtt{-sin\,θ} & \mathtt{cos\,θ}\end{bmatrix}\begin{bmatrix}\mathtt{x}\\\mathtt{y}\end{bmatrix}=\mathtt{x}\begin{bmatrix}\mathtt{\,\,\,\,\,cos\,θ}\\\mathtt{-sin\,θ}\end{bmatrix}+\mathtt{y}\begin{bmatrix}\mathtt{sin\,θ}\\\mathtt{cos\,θ}\end{bmatrix}=\begin{bmatrix}\mathtt{\,\,\,\,\,\,x\cdot \color{red}{cos\,θ}\,\,\,+y\cdot \color{orange}{sin\,θ}}\\\mathtt{-(x\cdot \color{purple}{sin\,θ})+y\cdot \color{green}{cos\,θ}}\end{bmatrix}}\]

The first rotation matrix above actually describes a clockwise rotation, which is both different from what we discussed at the link above (our final matrix there was for counterclockwise rotations) and unexpected, since we know that Moe’s system is a counterclockwise rotation of Joe’s system.

The resolution to that unexpectedness can again be found after a little eyeballing. The position vector for point P in Joe’s system is at a steep angle, whereas in Moe’s system, it is at a shallow angle. Only a clockwise rotation will change the coordinates in the appropriate way.

It has been now just two years since I reviewed Mr Barton’s stellar first book. I say “just,” in part because the last three weeks during this pandemic have felt like five years, and in part because Barton packs so much into his second book, it is a little surprising he did it in just two years.

The central theme of Reflect, Expect, Check, Explain is using and constructing ‘intelligent’ sequences of mathematics exercises, “providing opportunities to think mathematically.” The intelligence behind these sequences is the way we order and arrange them, allowing for comparison (reflection) between two or more exercises, the anticipation of what the answer or solution method will be (expectation) based on what the previous answer or solution method was, determination of the answer (check), and then an explanation of the connection between the exercises (explain).

Consider, for example, the sequence at left, from early in the book. During reflect, for the first pair of exercises, I can notice that the lower and upper bounds have stayed the same, and the second number line has minor ticks for every second minor tick of the first number line. I can also notice that the sought-after decimal value is at the same location on both number lines. This noticing can lead me to expect that since I identified the missing value for the first number line as 2.6, my answer should be the same for the second number line. It’s possible, though, that I won’t come up with an expectation. In the check phase, I fill in the values for the equal intervals on the second number line, coming up with the value for the question mark. Finally, when I explain, I either have a chance to talk about my earlier expectation and explain why I was off or why my expectation was correct or, if I couldn’t formulate an expectation, I can explain why the question-marked values are the same even though the tick marks are different.

As I move through the sequence, there are really interesting thoughts to have.

Why did the question-marked values line up when moving from 10 to 5 equal intervals (between Questions 1 and 2) but not when moving from 5 to 4 equal intervals (between Questions 3 and 4)?

Why does “lining up” fail me in Questions 4, 5, and 6 when it worked between Questions 1 and 2?

I can’t rely on inspection every time to figure out the intervals. Is there something I can do to make that task simpler?

Is the question-marked value in Question 9 just the question-marked value in Question 8, divided by 10?

Can I extend my interval calculator method to decimals?

If this were the entire book, that would be enough for me, to be honest. But Mr Barton spends an exemplary amount of effort addressing possible questions and misconceptions about such sequences (the FAQ chapter is excellent) and explaining how these sequences can both fit into more extensive learning episodes and can function in different ways from practice. All the while, the sequences remain the stars of the show.

I highly recommend (again) Mr Barton’s book, especially to math teachers. He outlines in brilliant detail how you can turn a set of boring exercises into a powerful method for soliciting students’ mathematical thinking. No revolution required.

Choice Quotes

Below are just a few snips from the book that I added to my notebook while reading. These are not necessarily reflective of the entire argument. But after a long day of educhatter, which more often than not reads like an ancient scroll from some monist cult, it is comforting to read these thoughts and know that there is still a place for practical, technical, dispassionate thinking about teaching and learning in the 21st century—a place for waging the cerebral battle, rather than constantly leading with our chin or our hearts.

Teaching a method in isolation and practising it in isolation is important to develop confidence and competence with that method, and indeed, students can get pretty good pretty quickly. But if we do not then challenge them to decide when they should use that method – and crucially when they should not – we deny them the opportunity to identify the strategy needed to solve the problem.

There are two main arguments in favour of teaching a particular method before delving into why it works.

The path to flexible knowledge The key point that Willingham makes is that acquiring inflexible knowledge is a necessary step on the path to developing flexible knowledge. There is no short cut. The ‘why’ is conceptual and abstract. We understand concepts through examples. The ‘how’ generates our students’ experience of examples. In other words, often we have to do things several times to appreciate exactly how and why they work.

Motivation As Garon-Carrier et al. (2015) conclude, motivation is likely to be built on a foundation of success, and not the other way around.

The mistake I made for much of my career was trying to fast track my students to this [problem solving] stage. This was partly due to my obsession with differentiation – heaven forbid a child should be in their comfort zone for more than a few seconds – but also based on my belief that problem solving offered some sort of incredible 2-for-1 deal. I thought it would enable my students to practice the basics, whilst at the same time allowing them to develop that magic problem solving skill.

I will again quote John Mason: “It is the ways of thinking that are rich, not the task itself.”

Check out Barton’s online courses, which now includes a stellar course on Intelligent Practice.

A projection is like the shadow of a vector, say \(\mathtt{u}\), on another vector, say \(\mathtt{v}\), if light rays were coming in across \(\mathtt{u}\) and perpendicular to \(\mathtt{v}\). For the vectors at the right, imagine a light source above and to the left of the illustration, perpendicular to the vector \(\mathtt{v}\). The projection, which we’ll call \(\mathtt{p}\), will be a vector that will extend from the point shown to where the end of the shadow of \(\mathtt{u}\) touches \(\mathtt{v}\).

If you take a moment maybe to read that description twice (because it’s kind of dense), you may be able to tell that the vector \(\mathtt{p}\) that we’re looking for will be some scalar multiple of vector \(\mathtt{v}\), since it will lie exactly on top of \(\mathtt{v}\). In fact, given the picture, the projection vector \(\mathtt{p}\) will have the opposite sign as \(\mathtt{v}\) and will have a scale factor pretty close to 0, since the projection vector looks like it will be much smaller than \(\mathtt{v}\).

That information is shown at the right. Helpfully, the vector \(\mathtt{u-p}\) is perpendicular to \(\mathtt{v}\), so we know that \(\mathtt{\left(u-p\right)\cdot v=0}\). And, using the Distributive Property, we get \(\mathtt{u\cdot v-p\cdot v=0}\).

Since we know that our sought-after vector \(\mathtt{p}\) will be some scalar multiple of \(\mathtt{v}\), we can substitute, say, \(\mathtt{cv}\) for \(\mathtt{p}\) in the above to get \(\mathtt{u\cdot v-cv\cdot v=0}\). And a property of dot product multiplication allows us to rewrite that as \(\mathtt{u\cdot v-c\left(v\cdot v\right)=0}\).

This means that \(\mathtt{u\cdot v=c\left(v\cdot v\right)}\), which means that the scale factor \(\mathtt{c}\) that we’re after—the factor we can multiply by \(\mathtt{v}\) to produce \(\mathtt{p}\)—is \[\mathtt{c=\frac{u\cdot v}{v\cdot v}\quad\rightarrow\quad p=\left(\frac{u\cdot v}{v\cdot v}\right)v}\]

I‘ve started a writing project recently that I’m having a good time working on so far. I’ve called it Scala Math (and on Twitter here) for now, because its central focus is deconstructing concepts and procedures into steps, and la scala is Italian for ‘staircase’. You can see the word at work in ‘escalator’, ‘scale’, etc. Scala is also the name of a programming language. Here are some reasons for that I found online.

Most of the projects I’ve worked on over the past few years have also been ways for me to learn new software languages or libraries. For Geometry Theorems, it was d3. For Scala, it was React—as well as the beautiful, amazing database that a normal person can actually look at and edit and it’s still a database: Airtable.

How It Works: Learn

Every Scala has a display window—where images and videos are shown—and a steps window, where you find the text of the steps, or ‘parts’. These areas are divided by a brain, which I’ll talk about below. When you land on a Scala (this one is Solving Arithmetic Sequences), the first thing shown in the display window is an image presenting a quick snippet of what will be covered. The image shows an essential question at the top. The use-case for the snippet was a student wanting a quick reminder about something they are working on, perhaps for homework, without having to search online and wade through tons of stuff that sorta-kinda matches what you want but not really.

The remainder of the section shown at left (called ‘Learn’ mode) is a series of steps (in this case, six), explained with text, audio narration, and the accompanying images that you can see appearing when clicking on each step. The dot navigation at the top shows us that we are on the first screen of this Scala.

Each step card has a button to replay the step, which can be pressed at any time while the step is active, and a button (up arrow) to go to the preceding step.

How It Works: Reflect

As you can see at the end of the video above, there is a Reflection question which calls for a short or extended text response. This is where the audio input on my cell phone comes in handy. Students’ responses are, at the moment, compared to a few ‘correct’ responses that I have written, and others have conributed to. The response which has the highest numerical match on a scale from 0 to 100 is presented as your score, and the pre-written response is presented as a suggested answer.

How It Works: Try

After the Learn phase is the Try phase, which consists of example-problem pairs (usually; for a very few cases, so far, stepped-out problems only). Or, more specifically, stepped-out problems followed by not-stepped-out problems. These look a little different from what I typically see as example-problem pairs, where the example and the problem are set side by side. Here, the problem follows the example, and the example is not provided when solving the problem. The typical sequence is shown below.

For the Try and Test phases, it’s always multiple choice, although it’s in the plan to look at other response inputs. When students are logged in, they build up (not earn; see below) points for every question. Right now, it’s just 50 points for each, though that gets cut in half and rounded up to the nearest integer for every incorrect answer. For an item with 3 choices, the lowest point total possible is 13. For an item with 4 choices, the lowest is 7.

On desktop, students can have the question read aloud via text-to-speech. As far as I know, that hasn’t yet come to mobile as a built-in feature, but I’ll keep my ears open for when it does.

How It Works: Test

Finally, there’s the Test phase. This is typically 4 to 6 questions that are of the same form as the ‘problems’ in the example-problem-pair Try phase. I’m just showing one such question in the video at the right.

When students are logged in, they can earn points by taking the test. The points are built up in both the Learn and Try phases. I have described how the points work for the Try phase above. The Learn phase is simpler: just clicking on a step builds up 100 points. At the moment, no points are tied to the Reflect question.

Once a student reaches the Test phase, the greatest number of points he or she can ‘bank’ is the number he or she has built up over the course of the Learn and Try phases. And the Test phase is fairly high stakes, in that each incorrect answer divides the total possible points to earn in half.

The stars shown on the score modal are awarded based on percent of total points earned. For the lesson shown in this post, the total that can be earned is 1700. So, approximately 560 points is 1 star (33%), 1130 points is 2 stars (66%), and 1360 points is 3 stars (80%).

Finally, to make sure this product connects knowledgeable people with students (whether they be parents or teachers or both) and guards against mindlessly pressing buttons to earn points, there is a final front-and-back activity, wherein students solve a different problem by listing the steps themselves and showing all their work.

Pay attention to your thought process and how you use expert knowledge as you answer the question below. How do you think very young students would think about it?

Here are some birds and here are some worms. How many more birds than worms are there?

Hudson (1983) found that, among a small group of first-grade children (mean age of 7.0), just 64% completed this type of task correctly. However, when the task was rephrased as follows, all of the students answered correctly.

Here are some birds and here are some worms. Suppose the birds all race over, and each one tries to get a worm. Will every bird get a worm? How many birds won’t get a worm?

This is consistent with adults’ intuitions about the two tasks as well.

Interpret the Results

Still, what can we say about these results? Is it the case that 100% of the students used “their knowledge of correspondence to determine exact numerical differences between disjoint sets”? That is how Hudson describes students’ unanimous success in the second task. The idea seems to be that the knowledge exists; it’s just that a certain magical turn of phrase unlocks and releases this otherwise submerged expertise.

But that expert knowledge is given in the second task: “each one tries to get a worm.” The question paints the picture of one-to-one correspondence, and gives away the procedure to use to determine the difference. So, “their knowledge” is a bit of a stretch, and “used their knowledge” is even more of a stretch, since the task not only sets up a structure but animates its moving parts as well (“suppose the birds all race over”).

Further, questions about whether or not students are using knowledge they possess raise questions about whether or not students are, in fact, determining “exact numerical differences between disjoint sets.” On the contrary, it can be argued that students are simply watching almost all of a movie in their heads (a mental simulation)—a movie for which we have provided the screenplay—and then telling us how it ends (spoiler: 2 birds don’t get a worm). The deeper equivalence between the solution “2” and the response “2” to the question “How many birds won’t get a worm?” is evident only to a knowledgeable onlooker.

Experiment 3

Hudson anticipates some of the skepticism on display above when he introduces the third and last experiment in the series.

It might be argued, success in the Won’t Get task does not require a deep level of mathematical understanding; the children could have obtained the exact numerical differences by mimicking by rote the actions described by the problem context . . . In order to determine more fully the level of children’s understanding of correspondences and numerical differences, a third experiment was carried out that permitted a detailed analysis of children’s strategies for establishing correspondences between disjoint sets.

The wording in the Numerical Differences task of this third experiment, however, did not change. The “won’t get” locutions were still used. Yet, in this experiment, when paying attention to students’ strategies, Hudson observed that most children did not mentally simulate in the way directly suggested by the wording (pairing up the items in a one-to-one correspondence).

This does not defeat the complaint above, though. The fact that a text does not effectively compel the use of a procedure does not mean that it is not the primary influence on correct answers. It still seems more likely than not that participants who failed the “how many more” task simply didn’t have stable, abstract, transferable notions about mathematical difference. And the reformulation represented by the “won’t get” task influenced students to provide a response that was correct.

But this was a correct response to a different question. As adults with expert knowledge, we see the logical and mathematical similarities between the “how many more” and “won’t get” situations, and, thus we are easily fooled into believing that applying skills and knowledge in one task is equivalent to doing so in the other.

The term ‘entia successiva’ means ‘successive entities.’ And, as you may guess, it is a term one might come across in a philosophy class, in particular when discussing metaphysical questions about personhood. For instance, is a person a single thing throughout its entire life or a succession of different things—an ‘ens successivum’? Though there is no right answer to this question, becoming familiar with the latter perspective can, I think, help people to be more skeptical and knowledgeable consumers of education research.

Richard Taylor provides an example of a symphony (in here) that is, depending on your perspective, both a successive and a permanent entity:

Let us imagine [a symphony orchestra] and give it a name—say, the Boston Symphony. One might write a history of this orchestra, beginning with its birth one hundred years ago, chronicling its many tours and triumphs and the fame of some of its musical directors, and so on. But are we talking about one orchestra?

In one sense we are, but in another sense we are not. The orchestra persists through time, is incorporated, receives gifts and funding, holds property, has a bank account, returns always to the same city, and rehearses, year in and year out, in the same hall. Yet its membership constantly changes, so that no member of fifty years ago is still a member today. So in that sense it is an entirely different orchestra. We are in this sense not talking about one orchestra, but many. There is a succession of orchestras going under the same name. Each, in [Roderick] Chisholm’s apt phrase, does duty for what we are calling the Boston Symphony.

The Boston Symphony is thus an ens successivum.

People are entia successiva, too. Or, at least their bodies are. Just about every cell in your body has been replaced from only 10 years ago. So, if you’re a 40-year-old Boston Symphony like me, almost all of your musicians and directors have been swapped out from when you were a 30-year-old symphony. People still call you the Boston Symphony of course (because you still are), but an almost entirely different set of parts is doing duty for “you” under the banner of this name. You are, in a sense, an almost completely different person—one who is, incidentally, made up of at least as many bacterial cells as human ones.

What’s worse (if you think of the above as bad news), the fact of evolution by natural selection tells us that humanity itself is an ens successivum. If you could line up your ancestors—your mother or father, his or her mother or father, and so on—it would be a very short trip down this line before you reached a person with whom you could not communicate at all, save through gestures. Between 30 and 40 people in would be a person who had almost no real knowledge about the physical universe. And there’s a good chance that perhaps the four thousandth person in your row of ancestors would not even be human.

The ‘Successive’ Perspective

Needless to say, seeing people as entia successiva does not come naturally to anyone. Nor should it, ever. We couldn’t go about out our daily lives seeing things this way. But the general invisibility of this ‘successiveness’ is not due to its only being operational at the very macro or very micro levels. It can be seen at the psychological level too. Trouble is, our brains are so good at constructing singular narratives out of even absolute gibberish, we sometimes have to place people in unnatural or extreme situations to get a good look at how much we can delude ourselves.

An Air Force doctor’s experiences investigating the blackouts of pilots in centrifuge training provides a nice example (from here). It’s definitely worth quoting at length:

Over time, he has found striking similarities to the same sorts of things reported by patients who lost consciousness on operating tables, in car crashes, and after returning from other nonbreathing states. The tunnel, the white light, friends and family coming to greet you, memories zooming around—the pilots experienced all of this. In addition, the centrifuge was pretty good at creating out-of-body experiences. Pilots would float over themselves, or hover nearby, looking on as their heads lurched and waggled about . . . the near-death and out-of-body phenomena are both actually the subjective experience of a brain owner watching as his brain tries desperately to figure out what is happening and to orient itself amid its systems going haywire due to oxygen deprivation. Without the ability to map out its borders, the brain often places consciousness outside the head, in a field, swimming in a lake, fighting a dragon—whatever it can connect together as the walls crumble. What the deoxygenated pilots don’t experience is a smeared mess of random images and thoughts. Even as the brain is dying, it refuses to stop generating a narrative . . . Narrative is so important to survival that it is literally the last thing you give up before becoming a sack of meat.

You’ll note, I hope, that not only does the report above disclose how our very mental lives are entia successiva—thoughts and emotions that arise and pass away—but the report assumes this perspective in its own narrative. That’s because the report is written from a scientific point of view. And from that vantage point, people are assumed (correctly) to have parts that “do duty” for them and may even be at odds with each other, as they were with the pilots (a perception part fighting against a powerful narrative-generating part). The unit of analysis in the report is not an entire pilot, but the various mechanisms of her mind. Allowing for these parts allows for functional explanations like the one we see.

An un-scientific analysis, on the other hand, is entirely possible. But it would stop at the pilot. He or she is, after all, an indivisible, permanent entity. There is nothing else “doing duty” for him, so there are really only two choices: his experience was an illusion or it was real. End of analysis. Interpret it as an illusion and you don’t really have much to say; interpret it as real, and you can make a lot of money.

Entia Permanentia

Good scientific research in education will adopt an entia successiva perspective about the people it studies. This does not guarantee that its conclusions are correct. But it makes it more likely that, over time, it will get to the bottom of things.

This is not to say that an alternative perspective is without scientific merit. If we want to know how to improve the performance of the Boston Symphony, we can make some headway with ‘entia permanentia’—seeing the symphony as a whole stable unit rather than a collection of successive parts. We could increase its funding, perhaps try to make sure “it” is treated as well as other symphonies around the world. We could try to change the music, maybe include some movie scores instead of that stuffy old classical music. That would make it more exciting for audiences (and more inclusive), which is certainly one interpretation of “improvement.” But to whatever extent improvement means improving the functioning of the parts of the symphony—the musicians, the director, etc.—we can do nothing, because with entia permanentia these tiny creatures do not exist. Even raising the question about improving the parts would be beyond the scope of our imagination.

Further, seeing students as entia permanentia rather than entia successiva stops us from being appropriately skeptical about both ‘scientific’ and ‘un-scientific’ ideas. Do students learn best when matched to their learning style? What parts of their neurophysiology and psychology could possibly make something like that true? Why would it have evolved, if it did? In what other aspects of our lives might this present itself? Adopting the entia successiva perspective would have slowed the adoption of this myth (even if were not a myth) to a crawl and would have eventually killed it. Instead, entia permanentia, a person-level analysis, holds sway: students benefit from learning-style matching because we see them respond differently to different representations. End of analysis.

A different but similar perspective on this, from a recurring theme in the book Switch:

In a pioneering study of organizational change, described in the book The Critical Path to Corporate Renewal, researchers divided the change efforts they’d studied into three groups: the most successful (the top third), the average (the middle third), and the least successful (the bottom third). They found that, across the spectrum, almost everyone set goals: 89 percent of the top third and 86 percent of the bottom third . . . But the more successful change transformations were more likely to set behavioral goals: 89 percent of the top third versus only 33 percent of the bottom third.

Why do “behavioral” goals work when just “goals” don’t? Behavioral goals are, after all, telling you what to do, forcing you to behave in a certain way. Do you like to be told what to do? Probably not.

But the “you” that responds to behavioral goals isn’t the same “you” whose in-the-moment “likes” are important. You are more than just one solid indivisible self. You are many selves, and the self that can start checking stuff off the to-do list is often pulling the other selves behind it. And when it does, you get to think that “you” are determined, “you” take initiative, “you” have willpower. But in truth, your environment—both immediate and distant, both internal and external—has simply made it possible for that determined self to take the lead. Behavioral goals often create this exact environment.