Implicit Equations for Lines

So, I’ve covered parametric lines already. Another form in which we can write equations for lines using linear algebra is implicit form.

The parameter in the parametric form of a line was a scalar \(\mathtt{k}\). We built the parametric form using a position vector to get us to a starting point on the line. Then we added this to the product of the slope vector and the parameter \(\mathtt{k}\) to get all the other points on the line. The implicitness of the implicit form comes from the fact that we build the equation using the slope vector and a vector perpendicular to the slope vector.

I mentioned back here that perpendicular vectors always have a dot product of 0. So, thinking of \(\mathtt{x-p}\) as the (slope) vector of our line, then \(\mathtt{a \cdot (x-p) = 0}\). With the actual values shown here, we have \[\begin{bmatrix}\mathtt{1}\\\mathtt{3}\end{bmatrix} \cdot \begin{bmatrix}\mathtt{\,\,\,\,3}\\\mathtt{-1}\end{bmatrix}\mathtt{ = (1)(3) + (3)(-1) = 0}\] If we know one point \(\mathtt{x}\) on the line, then the dot product equation is true for any \(\mathtt{p}\) and identifies a unique line. Let’s represent all the parts here as vectors, and more generally. \[\begin{bmatrix}\mathtt{a_1}\\\mathtt{a_2}\end{bmatrix} \cdot (\begin{bmatrix}\mathtt{x_1}\\\mathtt{x_2}\end{bmatrix} \mathtt{- }\begin{bmatrix}\mathtt{p_1}\\\mathtt{p_2}\end{bmatrix}) \mathtt{\,\,= 0 \rightarrow a_1x_1 + a_2x_2 + (-a_1p_1 – a_2p_2) = 0}\]

What’s cool about this equation is that we are all familiar with its form, \(\mathtt{ax_1 + bx_2 + c = 0}\), so long as we let \(\mathtt{a = a_1, b = a_2,}\) and \(\mathtt{c = -a_1p_1 – a_2p_2}\). This is what is called the general form or standard form of a linear equation. Even more interesting is that the coefficients in this form help to describe a vector perpendicular to the line.

Knowing the above and \(\mathtt{y:=x_2}\), we can write the equation for the line at the right as \[\mathtt{4x+3y-(4)(-5)-(3)(6)=0}\] And then, working out that c-value, we get \(\mathtt{4x + 3y + 2 = 0}\). The vector a = (4, 3), which we can rewrite as the ratio –4 : 3, describes the slope of the line.

Now we can easily slide back and forth between linear algebra and plain old current high school algebra with certain linear equations.



Line Segments with Linear Algebra

We saw last time that the parametric equation of a line is given by \(\mathtt{l(k) = p + kv}\), where p is a point on the line (written as a vector), v is a free vector indicating the slope of the line, and k is a scalar value called the parameter. Turning the knob to change k gives you different points on the line. At the right is the line

\[\mathtt{l(k) =} \begin{bmatrix}\mathtt{1}\\\mathtt{3}\end{bmatrix} + \begin{bmatrix}\mathtt{\,\,\,\,1}\\\mathtt{-1}\end{bmatrix}\mathtt{k}\]

Substituting different numbers for k gives us different points on the line. These resolve into position vectors.

This setup makes it fairly easy to make a line segment, and to partition that line segment into any ratio you want (this will be our ‘current high school connection’ for this post).

When \(\mathtt{k = 0}\), we get our position vector back: \(\begin{bmatrix}\mathtt{1}\\\mathtt{3}\end{bmatrix}\). This is the point (1, 3). When \(\mathtt{k = 1}\), we have … \[ \begin{bmatrix}\mathtt{1}\\\mathtt{3}\end{bmatrix} + \begin{bmatrix}\mathtt{\,\,\,\,1 \cdot 1}\\\mathtt{-1 \cdot 1}\end{bmatrix} = \begin{bmatrix}\mathtt{1 + 1}\\\mathtt{3 + (-1)}\end{bmatrix}\]

… which is the point (2, 2). And so on to generate all the points on the line. To generate a line segment from (1, 3) to, say, the point where the line crosses the x-axis, we first have to figure out where the line crosses the x-axis. We can do this by inspection to see that it crosses at (4, 0), but let’s set it up too. We start by setting the line equal to the point (x, 0) and solving the resulting system of equations: \[\begin{bmatrix}\mathtt{1}\\\mathtt{3}\end{bmatrix} + \begin{bmatrix}\mathtt{\,\,\,\,1}\\\mathtt{-1}\end{bmatrix}\mathtt{k} = \begin{bmatrix}\mathtt{x}\\\mathtt{0}\end{bmatrix} \rightarrow \left\{\begin{array}{c}\mathtt{1+k=x}\\\mathtt{3-k=0}\end{array}\right.\]

Adding the equations, we get x = 4, so (4, 0) is indeed where the line crosses the x-axis. To generate points on the line segment from (1, 3) to (4, 0), we use position vectors for both endpoints. Then we can use what’s called a convex combination of k—which is just extremely fancy wording for coefficients that add up to 1. We scale the second position vector, (4, 0) by some k and we scale the first position vector (1, 3) by 1 – k. \[\mathtt{l(k) =} \begin{bmatrix}\mathtt{1}\\\mathtt{3}\end{bmatrix}\mathtt{(1-k)} + \begin{bmatrix}\mathtt{4}\\\mathtt{0}\end{bmatrix}\mathtt{k}\]

Want the line segment divided into fifths? Then just use k values in intervals of fifths, from 0 to 5 fifths. Transpose the k coefficients to get a different “direction” of partitioning of the line segment.

Lines the Linear Algebra Way

Let’s continue with the idea of reinterpreting some high school algebra concepts in the light of linear algebra. For example, we learn even before high school in some cases that a line on a coordinate plane can be defined by two points or it can be defined by a point and the slope of the line.

When we have two points, \(\mathtt{(x_1, y_1)}\) and \(\mathtt{(x_2, y_2)}\), we can determine the slope with \[\mathtt{\frac{y_2 – y_1}{x_2 – x_1}}\]

and then do some substitutions to work out the y-intercept.

The linear algebra way uses vectors, of course. And all we need is a point and a vector to define a line. Or, really, two vectors, since the point can be described as a position vector and the slope is also a vector.

We have the line here defined as a vector plus a scaled vector—scaled by k. (See here for adding vectors and here for scaling them.) \[\color{brown}{\begin{bmatrix}\mathtt{1}\\\mathtt{3}\end{bmatrix}} + \color{blue}{\begin{bmatrix}\mathtt{\,\,\,\,1}\\\mathtt{-1}\end{bmatrix}\mathtt{k}}\] That second, scaled, vector looks like it could do the job of defining the line all by itself, but free vectors like that don’t have a fixed location, so we need a position vector to “fix” that. In general terms, thinking about the free vector as extending from \(\mathtt{(x_1, y_1)}\) to \(\mathtt{(x_2, y_2)}\), we can write the equation for a line as \[\mathtt{l(k) = }\begin{bmatrix}\mathtt{x_1}\\\mathtt{y_1}\end{bmatrix} + \begin{bmatrix}\mathtt{x_2 – x_1}\\\mathtt{y_2 – y_1}\end{bmatrix}\mathtt{k}\] That form is called the parametric form of an equation and can be written as \(\mathtt{l(k) = p + kv}\), where p is a point (or position vector), v is the free vector, and k is a scalar value—the parameter that we change to get different points on the line.

Let’s put this into the context of a (reworded) word problem:

In 2014, County X had 783 miles of paved roads. Starting in 2015, the county has been building 8 miles of new paved roads each year. At this rate, if n is the number of years after 2014, what function gives the number of miles of paved road there will be in County X? (Assume that no paved roads go out of service.)

The equation we’re after is \(\mathtt{f(n) = 783 + 8n}\). As a vector function this can be written as \[\mathtt{f(n) = }\begin{bmatrix}\mathtt{0}\\\mathtt{783}\end{bmatrix} + \begin{bmatrix}\mathtt{1}\\\mathtt{8}\end{bmatrix}\mathtt{n}\] We can see here, perhaps a little more clearly with the vector representation, that our domain is restricted by the situation. Our parameter n is, at the very least, a positive real number, and really a positive integer.

It seems to me that here is at least one other example of a close relationship between linear algebra and current high school algebra instruction that would make absorbing linear algebra into some high school material feasible.

If you’d like to practice with some items related to this post, visit Linear Algebra Exercises I.

How to Use Guzinta Math

  • Adults and students work together to complete modules—at school, at home, or both.
  • Students check in regularly. When their Practice Meter for a lesson is in the red, they should complete one or more modules in that lesson to get their Practice Meter to blue (or at least out of red).
  • Adults check in 1 day, 1 week, and 1 month after first going through a lesson module and require that all practice levels be out of the red or in the blue on that day.
  • PLEASE read the recommendations for adults at the end of this post.
  • Your Practice Meter levels are saved, even if you uninstall and then reinstall the application. To reset a Practice Meter for a lesson, click on the Guzinta Math logo at the center of the lesson homepage.

First things first: head over to the download page and install the application. When you start the application each time, you start at the homepage, which shows all 15 lessons for Grade 6.

After you have completed at least one module in one of the 15 lessons, this will activate your Practice Meter (explained below) for that lesson, and you will see a meter level to the right of the lesson on the homepage. This will allow you to, at a glance, see what concepts need your attention. Above, you can see that I’ve completed at least one module in the Ratio Names lesson (a green bar is shown to the right of the lesson). The Instructor Notes link at the bottom right allows you the ability to download the complete PDF of all the Instructor Notes for the grade level (279 pp).

Lesson Structure

Click on one of the lessons to be taken to the lesson homepage. Every lesson (with the exception of Plotting Ratios) in Grade 6 contains five modules. At the right, the lesson homepage for Ratio Names is shown. Notice the green Practice Meter level in the center icon. This mirrors the level shown on the main home page. Hover over this icon to see the numeric Practice Meter level (the one shown is at 36 currently).

Notice also the Home button in the bottom right corner of the lesson homepage. This can be found on all lesson homepages and will take you back to the main home page.

The 3 main modules, located in the center quad-panel are numbered 1 to 3 (Algebraic Expressions shown at right). The fourth square in the quad-panel is a link to the Instructor Notes for the lesson. Click on that to download a PDF of these notes.

The modules do not necessarily have to completed in any particular order. However, completing them in the order given is recommended.

Adult Interaction Is a Must

Each of the first modules in every lesson is labeled with Guided Practice (Equations and Inequalities shown at left). This means that if it is used in a classroom, it should be used as an activity centerpiece involving teachers and students. If it is used at home, the first module in particular should be the focus of both parent(s) and student.

The Instructor Notes provide an outline for interactive teaching and learning discussions with this material. It is recommended that every module—when first completed—be done together with adult and student.

To use Guzinta Math at home, a teacher may assign a module for homework or practice. Parent and student then discuss and complete the module together and, if it is requested, return the student to school the next day with the completion certificate or email it. (Parents can follow the Instructor Notes for each module as well.) The material can be used also at school or solely at school. In that case, different modules can be completed as a class and others may be assigned for homework. It is not the case that adults should be doing all or even most of the work during these interactions, but they should attend to them, rather than plop students down in front of a monitor to complete these activities alone.

Practice Meter

At 25 and below, the Practice Meter color is red. Between 26 and 79, the color is green. And, a level of 80 or above makes the color blue. Once a module has been completed—either at school with a teacher or at home with a parent or caregiver, the time is recorded for that lesson, along with the Practice Meter level. As time passes, the Practice Meter level decreases to represent a forgetting of the content.

In the first 7 days, the meter decreases at a rate of about 54% each day. That is, it loses a little more than half its value each day. The Practice Meter level of 36 (green) mentioned above would be about 16 or 17 (red) a day later, if no work is done in the lesson. From 7 days to 28 days, the meter only loses about 17% of its value each day. From 28 to 90 days, only 6% is lost each day. And from 90 days on, only 1% is lost each day.

The purpose of the Practice Meter is to provide a visual indication of forgetting and to alert students, teachers, and parents when it is time to revisit a lesson. Forgetting is very useful for learning, so it is important to allow time for the Practice Meter level to decrease before recharging it. A good schedule to keep would be to use a module together as a class or with parents and students as homework and then check in 1 day, then 1 week, and then each month after this first start. Have students get their meters out of the red or in the blue, at least for these check-ins. The goal is to keep this content alive throughout the year—yes, even if students are repeating the same questions. Repetition is excellent for novice learners!

Time-Released Practice Questions

Because the application timestamps the beginning of a student’s work in a lesson, this allows it to reveal new practice questions over time. The table below shows the number of questions (excluding Module 0) asked in each lesson, starting on Day 1 and then the extra questions revealed on Days 4, 9, and 22. These days are measured separately for each lesson, and the timer doesn’t start until after adults and students together complete at least one module in the lesson.

LessonDay 1Day 4+Day 9+Day 22+
Ratio Names31+5+8+6
Ratio Tables28+10+7+5
Comparing Ratios45+0+0+0
Plotting Ratios12+0+0+0
Measure Conversions26+6+12+6
Fraction Division28+6+12+4
Long Division22+6+10+6
GCF and LCM33+6+6+5
Negative Numbers31+0+0+0
Quadrants in the Plane31+7+6+6
Order and Absolute Value34+7+6+3
Numeric Expressions28+8+8+6
Algebraic Expressions46+0+0+0
Equations & Inequalities29+0+0+0

Across the entire Grade 6, students have 424 practice questions on Day 1. Then, over time, this number grows to 651 total practice questions. The zeros in the table show 5 lessons which do not time-release new questions over time (at the moment). The other 10 lessons do.

There are a few reasons for time-releasing new questions: (1) This ratchets up the challenge level for a lesson. The answers for questions revealed on and after Day 4 are not included in the Instructor’s Notes. And, more questions in a lesson means that it becomes slightly more difficult to raise one’s practice meter up to any given level (though the difficulty increase is very minor in most cases). (2) It helps break the repetition a little. (3) Transfer is facilitated when students revisit a previously learned topic in a slightly new context.

Contact and Future Work

I’ve written here about work on Guzinta Math that I’ll be getting to in the near and farther future. If you have any questions or technical issues, please email me at qanda[at]

Some Polite Suggestions

Here are some recommendations for how to think and behave around this material.

  • Ideally, provide guidance in one form or another on EVERY module the first time students go through it. The Instructor Notes provide the answers and some guiding questions for adults for the original practice questions (not the time-released ones). Use those notes to help you guide students. Guidance doesn’t mean you are perched on top of their shoulder, making sure they don’t get anything wrong.
  • Take. Your. Time. Students should live with this content—revisiting it regularly to maintain their Practice Meter levels—throughout the entire year. They don’t need to “get through” the content fast.
  • It’s possible to revisit the same module of a lesson every time to raise one’s Practice Meter level. That’s okay! But encourage students to complete other modules over time as well.
  • Forget whatever label you’ve assigned to your student. “Quick learner”? They still need to work through and revisit this material throughout the year, just like everyone else. Revisiting deepens their understanding and often reveals patches of misunderstanding—where they learned to play the game well but don’t really get it. “Slow learner”? They should be challenged and given high expectations like other students. Again, revisiting over the entire year is key.
  • If your student has memorized the answers to questions, that’s fine, at least once. That’s a good cue to let that module sit and allow forgetting to set in for a while and do other things. Also, if you suspect that a student is moving through stuff without thinking, that’s a clue to SIT DOWN WITH THEM AGAIN and work together. Watch the videos again and discuss. Have them explain things to you. Read the Notes. Have them generalize. Work through the exploratory module even when it doesn’t have questions to answer. In short, make them think and elaborate when otherwise they wouldn’t.

Zukei and Dot Products

Zukei puzzles that ask students to find right triangles seem to rely on an understanding of perpendicularity that is situated more comfortably in linear algebra than in Euclidean geometry. Consider the following, which has a hidden isosceles right triangle in it. Your job is to find the vertices of that isosceles right triangle.


High school students would be expected to look at perpendicularity either intuitively—searching for square corners—or using the slope criteria, that perpendicular lines have slopes which are negative reciprocals of each other. But it seems a bit much to start treating this puzzle as a coordinate plane and determining equations of lines.

The Dot Product

In all fairness, the dot product is a bit much too. Instead, we can operationalize slopes with negative reciprocals by, for example, starting from any point, counting 1, 2, . . . n to the left or right and then 1, 2, . . . n up or down to get to the next point. From that point, we have to count left-rights in the way we previously counted up-downs and up-downs in the way we counted left-rights, and we have to reverse one of those directions. For the puzzle above, we count 1, 2 to the right from a point and then 1, 2 up to the next point. From that second point, it’s 1, 2 right and then 1, 2 down. It’s a little harder to see our counting and direction-switching rule at work when the slopes are 1 and –1, but, in the Zukei context at least, the slopes have to be 1 and –1, I think, to get an isosceles right triangle if we’re not talking about square corners.

This kind of counting is really treating the possible triangle sides as vectors. And, with perpendicular vectors, we can see that we can get something like one of these two pairs (though perpendicular vectors don’t have to look like this): \[\mathtt{\begin{bmatrix} x_1\\x_2 \end{bmatrix} \textrm{and} \begin{bmatrix} -x_2\\ \,\,\,\,x_1 \end{bmatrix} \textrm{or} \begin{bmatrix} x_1\\x_2 \end{bmatrix} \textrm{and} \begin{bmatrix} \,\,\,\,x_2\\-x_1 \end{bmatrix}}\]

The dot product is defined as the sum of the element-wise products of the vector components. In the case of perpendicular vectors, the dot product is 0. Here is the dot product of our vectors: \[\mathtt{(x_1 \cdot -x_2) + (x_2 \cdot x_1) = 0}\]

Some Programming

One reason why this way of defining perpendicularity (with a single value) is helpful is that we avoid nasty zero denominators and, therefore, undefined slopes. With the two vectors at the right, we get \[\mathtt{\begin{bmatrix}0\\5\end{bmatrix} \cdot \begin{bmatrix}4\\0\end{bmatrix} = (0)(4) + (5)(0) = 0}\]

We can take all of the points and run them through a program to find all the connected perpendicular vectors. The result ((0, 1), (1, 0)), ((0, 1), (2, 3)) below means that the vector connecting (0, 1) and (1, 0) and the vector connecting (0, 1) and (2, 3) are perpendicular.

This gives us all the perpendicular vector pairs, though it doesn’t filter out those vectors with unequal magnitudes, which we wanted in order to identify the isosceles right triangle.

There are some Zukei solvers available, though I confess I haven’t looked at any of them. No doubt, one or all of them use linear algebra rather than ordinary coordinate plane geometry to do their magic. It’s about time, I think, we start weaving linear algebra into high school algebra and geometry standards.

Solving Zukei puzzles is not the best justification for bringing linear algebra down into high school, of course. But I hope it can be a salient example of how connected linear algebra can be to a lot of high school content standards.

Guzinta Math 6 (Windows & Mac)

guzinta math windows

This morning, I wrapped up some final work on compiling all 15 Grade 6 Guzinta Math lesson apps into one Windows application (and a few days later into one Mac application). To try it out, visit the download page.

At the moment, since there have been few downloads of the application, you may see a warning from your SmartScreen or other defender, which you can ignore. Once it has been downloaded enough (I don’t know the number) the defender warning will go away.

Modules Zero

guzinta math windows

My main project for the next year is to build a Module 0 for each lesson. At the moment, there is some useful but mostly placeholder material there. Here is a look at the lesson homepage for Fraction by Fraction Division.

What will go in Module 0? Instruction inspired by the principles of variation theory. This Module 0 will not be monitored by the Practice Meter, and will serve to provide students a scaffold into the learning in the other modules as well as provide a differentiation tool if educators should need it.

Grades 7–8 Package and Other Platforms

Before writing the zero modules, however, I’d like to finish updating the lesson apps to Version 5.0. This version includes the zero and fourth modules, and has a bit slicker look. As of this writing, I’ve got 14 lesson apps in Grades 7–8 to update. These will all be updated as individual Chrome apps first and placed on the Chrome Web Store for download to Chromebooks.

Once these are updated to Version 5.0, I can package them together (all 16 lesson apps for Grades 7–8) into one application, just as is done for Grade 6. Done.

Other Enhancements

I’ve got a lot of different enhancements in mind as well. A version for classroom use—where there is a single computer or just a few—with student login would be pretty simple to knock out. That way, individual practice meters could still be tracked while students use just one version of the application. Done. Also, timestamping and individualizing end-of-module certificates is on my mind for the future Done. —as is a repository of practice problems.

What else? A version completely online has been mentioned. I would also like to put together some how-tos and “better” practices documentation for different use cases (home school, school and home, classroom only, etc.). New content for statistics and probability standards, correlations to popular curricula. So many things that will have to wait for another day!


Another task I have on my list to do is to build up the information around the lessons. For example, as mentioned above, each of the lesson apps in this package (and in future packages) has a Module 0 that is not tracked via the Practice Meter, although it does contain questions with correct answers. This will not change—students should have at least one fairly robust module in each lesson that isn’t about being assessed in some way. Even if these modules zero will be used less frequently, it’s important to have them, for reference, for a way to test yourself before diving into the practice meter work, etc.

In some cases, too, the fourth module of an app is not linked to the Practice Meter. These are usually exploratory modules. If a Module 4 has questions whose answers can be checked (by submitting an answer and pressing the check button), then it is linked to the Practice Meter. Otherwise, it is not.

Variation and Example Spaces

I‘ve been thinking a lot about Craig Barton’s wonderful book How I Wish I’d Taught Maths and have been scanning three of his new websites, Variation Theory, Same Surface, Different Deep Problems, and Maths Venns, as well as some research and other books on variation, and a lot of online commentary, in anticipation of starting to implement these ideas in some way.

Writing Algebraic Expressions

As I was reading the last page of Mr Barton’s Book, I was working on instruction around writing algebraic expressions, so this is the topic kind of hovering next to me wherever I go, waiting for when I have time to dig in. This topic is a little more fraught than the purely procedural examples that have been circulating, so it’s worth exploring how variation can be applied to something a little looser.

What does writing algebraic expressions involve (for a beginner)? Well, if I force myself to ignore what other people think writing algebraic expressions involves (essentially ignoring standards and any written material on the topic), then I would say that writing algebraic expressions means to write something like s + 2 or 2 + s when presented with a question like “How old in years will Sam be in exactly two years?”1

This, then, I would call the first example in my example space. Or, rather, an example of an example in the example space—because, if this example is any good, then I will use it as an instructional example to start and leave it out of variation work, which is about PRACTICE, not instruction.2 So, something like this, with the brilliantly simple Silent Teacher method, mentioned in Barton’s book (and a few other places), though without the natural pauses and instructions for students to copy down the correct worked example used during a normal classroom implementation of this.

Try This One

Write an algebraic expression to model the situation.

How old in years was Sam exactly 10 years ago?

I would include a follow-up to this process, here involving a discussion around (a) the idea that the resulting algebraic expression represents an answer to the question of how old Sam will be—it’s just that one part of that expression is not known, (b) asking students to check that the answer makes sense, here by substituting different values for s and comparing the result to the situation, (c) the idea that any letter can be chosen for the variable, and (d) perhaps drawing a visual model of the result (an annotated number line). Some of these could be packaged into the instruction and question above, of course—or perhaps I’ll decide to split this up even more, considering how much “in addition to” I’ve now done about this—but I think that, in general, leaving room for a stepping back step at the end of this is a good idea, to catch the kind of overflow that is difficult to squeeze into expositions like this.

And Now Enters Variation

The paired problem here has opened up a dimension of variation—using addition or subtraction in the expression, so we can play with that during Intelligent Practice (really love that phrase). Technically, the instruction was open to all four operations, but I think it makes sense to focus exclusively on addition and subtraction, leaving multiplication and division expressions for another round.

Here’s what I cooked up.

  1. How much money in dollars did Sam have if he got exactly 10 dollars?
  1. How much money in dollars did Sam have if he got exactly 10 cents?
  1. How much money in dollars did Sam have if he got exactly 2 dollars?
  1. How much money in dollars did Sam have if he lost exactly 10 cents?
  1. How much money in dollars did Sam have if he got exactly 1 dollar?
  1. How much money in dollars did Sam have if he lost exactly 1 dollar?
  1. How much money in dollars did Sam have if he got exactly 50 cents?
  1. How much money in dollars did Sam have if he lost exactly 2 dollars?
  1. How much money in dollars did Sam have if he got exactly 25 cents?
  1. How much money in dollars did Sam have if he didn’t lose or gain any money?

After this, it might be good to have students cut out the strips and place them on a number line.

It’s interesting how much my experience and training rebels against this process. What I want to get to, right away, are the difficult and ambiguous situations. In particular, I started with, and then rejected, a variation sequence involving height: How tall in inches will Sam be if he grows 2 inches? The subtraction variation is bound to confuse: How tall in inches was Sam if he grew 2 inches? That’s tricky.

But knowing about and looking out for those tricky and ambiguous and interesting situations can serve you well creating instructional routines like this. It shows you where you’re going—and your example space can be richer and broader. And if you’re serious about implementing minimally different variation like this, it shows you how far away your knowledge really is from a beginner’s. You just have to learn to have more sympathy for learners who are encountering mathematics for the first time that you’ve seen a gazillion times.

  1. It’s important to me—at the moment, at least—that the examples in this example space should also involve identifying the correct unknown, rather than simply recording the unknown, as would happen with a question like, “Sam is s years old. How old will he be in 2 years?” or with an exercise of the form “2 more than a number.” In both of these cases, the unknown is entirely exposed.
  2. This is an important aspect of variation that I worry will be lost on U.S. teachers. Intelligent practice can’t happen, beneficially, until some acquisition has happened. In 20 years, I haven’t seen a robust public discussion about acquisition. The rhetoric around instruction in the States treats it as just one long assessment, though almost no one realizes that’s what it has become.

Mr Barton’s Book

It wasn’t too long ago—not even three years—that I finished reading David Didau’s terrific book (this one), so I still remember the excitement that I felt reading it, and watching all of the silly certainties of common wisdom in education being dismantled in front of my eyes, making way—I could only hope—for pedagogical practices informed by a real science of learning.

I felt a similar excitement reading Craig Barton’s book How I Wish I’d Taught Maths, because in this book, at long last, are many of those practices in one place, constructed, as readers will see, next to the debris of familiar canards and shallow reasoning that once guided parts of Barton’s teaching.

It is not a book full of proclamations about “best” practice. But you will find in this book a beautiful translation of the science of learning to the classroom. And far from the drudgery that one may imagine this to be, the joy of effective explicit instruction, for both teacher and students, comes through in every chapter of the author’s writing. It is serious, thorough, humble, and humane. And accessible: perhaps the greatest pleasure in reading it is knowing that you could turn around and start to implement many of these practices in short order—or, perhaps, that you already do these things, but don’t know why you should stick with them or how you could improve on them.

I have a lot of underlines and margin notes, but I think these three snippets together, from the chapter on problem solving and independence, are my favorites. The section starts, as they all do, with what the author used to think:

I used to love the sight of my students struggling through problems. Scratching heads, heavy sighs, and even the snap of a pencil thrown down in frustration were the soundtrack to learning. . . .

And then we are introduced to one of these problems, Question 23 from this paper (PDF), along with a deep concern for how novices will handle it. Contrast Barton’s new diagnosis below with common wisdom—that students ask why they are doing math because it is boring, tedious, procedural, or not relevant to their lives.

The task of choosing cards and calculating their totals may prove so cognitively demanding that novices do not have any spare cognitive capacity to recognise patterns. They do not realise that it is not the actual totals that matter, but whether those totals are odd or even. They just carry on regardless. Moreover, students are so consumed with the minutiae of the problem that no cognitive capacity remains to consider the global picture—why are they doing this? The result is that the novices may end up with an assortment of lists and totals, but not actually do anything with it—the fact that this is a probability question was pushed out of working memory long ago when the first set of cards was being processed.

As you might imagine, since the diagnosis is different from that received from common wisdom, the prescribed treatment is different too:

Before I set students off to work independently, I ensure they have enough domain-specific knowledge to solve problems on their own.

Although the snippets above are certainly grist for my mill, How I Wish I’d Taught Maths is not an ideological tome. It is eminently practical, taking the best ideas from all corners of the educational universe, squeezing them through the filter of cognitive science, and setting them in the right proportion to create a firm foundation that any educator—and especially any math educator—can use and build on. I highly highly recommend it to anyone who wants to strive for better in teaching and learning.

Proprioceptive Knowledge

This paper ($) was a nice read, with some fresh (to me) insights about discovery, instruction, and practice. There are many points in it where I don’t see eye to eye with the author, but those parts of the text are, thankfully, brief. I took away some new thoughts, at any rate, the most robust of which was an analogy between learning and proprioception, as the title suggests.

Here are the two main ideas as I see them, involving a very healthy amount of paraphrasing and extrapolation on my part:

  1. An aspect of your learning over any topic that deserves attention from instruction is your subjective, first-person, thinking with the material taught.
  2. Good instruction not only manipulates you into knowing something, but enlists your cooperation in doing so.


Proprioception is the basic human sense of where your body parts are in space and the sense of your own movement in that space (i.e., you don’t use ‘touch’ to know where your left hand is in space; this is proprioception). For learning something abstract like adding fractions with unlike denominators, we might think of proprioceptive knowledge as what it is LIKE to add fractions with unlike denominators—physically, cognitively, etc. Certainly carrying out the computations procedurally is an important part of “what it is like” but there are many others, including “what it is like” to identify situations calling for working out common denominators.

The first paper uses as a candidate for proprioceptive knowledge (although they don’t call it that) an example of working long division to produce repeating decimals. Students are instructed, with an example, that for any number you write as an integer over an integer, the decimal digits will either be repeating zeros or a repeating pattern of some other kind. Students use practice, however, to gain access to the proprioceptive dimension of this instruction—the experience that this is indeed the case; a first-person view of the knowledge. It is not that they are not told why the digits repeat (there are a finite number of remainders that are linked together in what they produce) during the instruction with the example. They are told this. And it’s not necessarily the case that the students don’t understand what they have been told. It’s just that the first-person experience of this is an important node in the constellation of connections that constitute the schema of understanding rational numbers.

Indeed, I would argue that the explicit instruction is absolutely necessary in this example (and almost all other examples). It allows the student to connect his newly gained first-person proprioceptive conceptual knowledge about division and rational numbers with what he understands to be experts’ experience and see them as the same. In other words, the explicit instruction did not take away the student’s ability to discover something for himself, as the idiotic trope goes; on the contrary, it was necessary to facilitate the student’s discovery. Cristina says it well:

Cristina [the teacher in the example] did not consider it a crime to “tell secrets”—like the author, she believed that students have to work to figure them out anyway.

Making Connections

This perspective helps me make some intuitive sense of why, for example, retrieval practice may be so effective. Testing yourself gives you the first-person experience of—the proprioceptive sense of—what it is like to remember something successfully and consistently (and it ain’t as easy as you think it is before you do it). This knowing-what-it-is-like episodic knowledge is a knowing like any other, but it is one that can be easily neglected when dealing with cognitive skills and subject matter.

Or, rather than neglect proprioceptive knowledge, we tend to make it dramatically different from the conceptual, such that students don’t connect the two. It is not necessarily the case that our conceptual instruction is anemic (though it certainly is on occasion) but that our procedural instruction is so narrow, and when it is not narrow it is not procedural. As a result, students gain a strong sense of what it is like to find like denominators, for example, but little sense of what it is like to move around in “higher level” aspects of that topic. When it comes to those higher level aspects, we have done no better in education, really, than to just have students do things in order to learn things or just have students learn things in order to do things.

The episodic knowledge angle can also help make some sense out of the power of narrative, as this is a type of out-to-in instruction which contemporaneously solicits an in-to-out response. (Although, it seems to me to push too hard in the latter direction for an ideally holistic type of instruction.)

How to ‘Do’ Proprioceptive Knowledge

The above suggests some instructional moves. The paper itself suggests building this in-to-out proprioceptive knowledge during practice. Rather than relying on problems which hit only the procedural writing of fractions as terminating or repeating decimals, for example, practice can assist students in “discovering” the explicit conceptual instruction of the lesson from a brand-new, first-person perspective. So, if we want students to understand why rational numbers must have terminating or repeating decimal representations, we need to give them practice that explicitly allows them to ‘feel’ that connection and express it as they work.

For another example, I’m currently working on percents. I certainly want students to be able know what it’s like to determine a percent of a number, purely procedurally. But I also want them to understand a lot of other things: that p% of Q is greater than Q when p is greater than 100 (and why; because it’s multiplying Q by a number greater than 1) and less than Q when p is less than 100 (because it’s multiplying Q by a number less than 1); and so on. I should think about structuring my practice so that these patterns reveal themselves and explicitly point students to them.

Do-It-Ourselves Education

When I lived for a year in Germany in high school as a foreign exchange student, I picked up, among many other things, a great quote from my host father: “Die Vorbereitung ist alles.” When said in my first language, it sounds fairly banal: “(The) preparation is everything.”

In both languages, though, the understood meaning of the phrase has a teleological ring: preparation is all-important for the goal you want to accomplish or some particular end you have in mind to achieve or realize. But I prefer a more extreme interpretation of the quote, in particular for education: that there is no achievement track, only a preparation track (multiple tracks in reality).


How do we get to that achievement above from, say, the middle of the preparation track? We don’t (in general). We move along the preparation track until we are in close enough proximity to the achievement to grab it. We don’t, in fact, keep our eyes on the prize. We keep our eyes on the preparation needed to move us within striking distance of the prize. Indeed, from way back in the middle of the track, the prize may look more tempting than it will appear close up (and it may be a mirage). And we may not be able to grasp it until we’re a little past it in our preparation.

Stay on the Right Track

The goal or achievement can be anything, really. So, for example, just cruise Twitter for a bit to find some quotable goal for education. The Feynman quote on the right is a good example. It is part of a quotation from a letter to a student in 1976, in which Feynman refers to himself in the third person, from The Quotable Feynman:

Just because Feynman says he is pro-nuclear power, isn’t any argument at all worth paying attention to because I can tell you (for I know) that Feynman doesn’t know what he is talking about when he speaks of such things. He knows about other things (maybe). Don’t pay attention to “authorities,” think for yourself.

Okay, great. Sincerely, that’s a great goal. I definitely would like to help students be appropriately mistrustful of authority—to the extent that it stimulates constructive thinking, not just having temper tantrums about authority. Who wouldn’t? So, let’s talk about distrusting authority as a goal for education.

The usefulness of the above interpretation about preparation is that now we must find the image of that goal somewhere along the preparation track and work out how we will connect the beginning and middle of the track to the point where that goal can be attained. Almost instantly we will see that we need to define what we (society) want for students. (For example, students have a lot of authority figures in their lives. Will they interpret the pro-skepticism message in a way that makes them start ignoring what their parents tell them? Does skepticism just mean that they have an ability to say, “I don’t think that’s right” and then never follow up?) But more importantly, we need to think about the steps along the path: What do students need to know first to understand skepticism and how to wield it appropriately? How does that ability progress over time? What knowledge is involved?

I don’t know about you, but when I deliberate on that simple Feynman quote for a while, I think of dozens of different sub-steps I would want to put in place along the preparation track from the goal back to the starting point. And these would probably break down into hundreds of smaller steps. Balancing appropriate skepticism—actionable skepticism, not armchair, consequence-free questioning—with the absolute necessity in modern life of trusting experts and authorities is lifelong work for adults who take on that challenge. If we want it to be an explicit goal for students—and not just a slogan we pass around on social media—then it will require a lot of work and technical planning.

My wish for 2018 and beyond is that, in addition to wanting these kinds of big things for students, we realize the hard, technical, scientific work involved in doing those things ourselves. Let’s leave behind the childish idea that, in order for students to achieve X, they just have to do X. That works for small things, not for anything worth having.

It is not science to know how to change centigrade to Fahrenheit. It’s necessary, but it is not exactly science. In the same sense, if you were discussing what art is, you wouldn’t say art is the knowledge of the fact that a 3-B pencil is softer than a 2-H pencil. It’s a distinct difference. That doesn’t mean an art teacher shouldn’t teach that, or that an artist gets along very well if he doesn’t know that.

–Richard Feynman