The Reflective Educator

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Math is Not (Strictly) Hierarchical

My students drew me into another political argument. Eh. It happens. Lately, political debates bother me. They just show how good smart people are at rationalizing. The world is so complicated - the more I learn, the less clear anything gets. There are too many ideas and arguments to pick and choose from. How can I trust myself to know the truth about anything? And if everything I know is so shaky, what on earth am I doing teaching? I guess you just do your best. No one can impart perfect universal truths to their students. * AHEM * ... Except math teachers. Thank you.

There is a widely accepted truth that learning in math is strictly hierarchical. Before children can learn how to factor polynomials, they have to first learn how to factor whole numbers. Before they can learn how to classify polygons, they have to learn what all the different types of triangles are. But this idea is false.

It is true that one will find factoring the polynomial x2+8x + 12 easier and faster to do if one knows the factors of 12 are 1, 2, 3, 4, 6, and 12. We might be able to say that knowing how to factor 12 is a prerequisite for factoring the polynomial x2+8x + 12. This is a much narrower version of hierarchical knowledge structure where with careful curriculum design one can deliberately select numbers for factoring polynomials that are known (or at least likely to be known) to students. It is well established that there are some multiplication facts that students find easier than others; if we restrict ourselves to only using the factorizations that children are more likely to be familiar with, then at least when introducing the idea of factoring polynomials, we offer a greater variety of children access to the ideas.

But it is also possible to continue to teach students how factor numbers while teaching them how to factor polynomials. First, we remember that it is likely that if a child reaches our classroom and does not know some idea from a previous year, they have unfinished or incomplete learning, not unstarted learning. We don’t need to introduce the idea from scratch; we can probably rely on relearning opportunities instead. Next, we can and should always embed opportunities for learning prior ideas in our teaching since students always come to us with varied understanding. This does not mean having the different goals for different students, rather that we know our aim is to reduce variance in student understanding and build a community of learners in doing so. (Aside: An excellent tool for supporting incomplete learning are the instructional routines described by Routines for Reasoning and the fine folks at the Teacher Education by Design project.)

The Difference Between Formative Assessments and Formative Assessment

Formative assessments are tasks selected by educators intended to help them gain insight into what children can do, know, or believe. Formative assessment is “the process used by teachers and students to recognise and respond to student learning in order to enhance that learning, during the learning” (Cowie & Bell, 1999 p. 32).

Formative assessments are tasks done by students. Formative assessment, the process, has educators clarify the learning intentions with students, activate students as owners of their own learning, activate students as resources for each other, elicit evidence of student learning, and provide opportunities for feedback that move the learning forward (aka. The five formative assessment strategies from Wiliam, D., 2011, Embedded Formative Assessment).

Formative assessments require a careful distinction to be made between the purpose of the assessment; if an educator uses the assessment to evaluate the student and creates grades from the assessment it’s actually summative. If they use the information to modify their instruction, the assessments are formative. Formative assessment, the process, requires no such distinction.

Formative assessment, the process, has a rich set of research that supports how it improves the learning conditions for students. Formative assessments, by contrast, sometimes end up with teachers having their vitality sucked from them as they pour over spreadsheets full of numbers but no insights trying to figure out how to improve student learning.

Formative assessment can be embedded in everyday instruction so that action can be taken immediately whereas formative assessments nearly always require educators to set aside time to give the assessments and time later to look at the results.

Formative assessments are an important dimension of formative assessment (since they do provide an opportunity to elicit evidence of student learning), but the full set of formative assessment practices have a far richer impact on student learning.

Assessment Changes the Learner

In physics, there is a limitation on the measurement of particles called the Heisenberg Uncertainty principle. The principle says, “the position and the velocity of an object cannot both be measured exactly, at the same time” (source). The reason why is that the act of measuring the position of a particle changes the uncertainty in its velocity and the act of measuring the velocity of a particle changes the uncertainty in its position. The more precisely we measure either quantity, the greater the uncertainty in the other. The critical idea is that by observing a particle, we introduce uncertainty in what we can know about the particle because the act of measurement changes the particle.

In a similar way, assessing a learner changes that learner. There are three ways learners can change during the process of assessment.

  1. Every time we retrieve ideas from our memory, our ability to retrieve those ideas again is strengthened, even if we are unsuccessful in remembering the idea. Our memories are not like computers, they tend to strengthen as we revisit ideas and diminish otherwise.
  2. As we attempt tasks and are successful or not successful on a task within a particular domain, we adjust our self-image within that domain. When we repeatedly fail to accomplish a task, we tend to think of ourselves as unable to be successful. When we accomplish tasks too easily, we tend to put less effort into more challenging tasks.
  3. Some tasks and supports for accomplishing those tasks give us the opportunity to learn new ideas. As a result, every time we solve a new problem, we are changed a little by the experience.

I’ve often heard it said that during assessment learning stops, but it’s possible that lots of learning can be taking place during these times, depending, of course, on the nature of the assessment.

Leaving Students Behind

Unfortunately, this happens all too often.

An airplane taking off while leaving most of the passengers behind.

There are two main ways teachers can avoid this happening in their class.

The first is through participatory engagement where activities are designed so that all students have a role in the activity. Sometimes this looks like students answering questions on individual mini-whiteboards so that for every question every student answers and at other times this looks like all students working in small groups on group-sized whiteboards (see non-permanent vertical surfaces).

The second is by utilizing instructional routines that embed formative assessment and strategies for supporting the learning and engagement of all students.

In both cases, the key idea is that in order to prevent some students from not getting on the plane, the teacher needs insight into what every student is learning.

Engaging Students Remotely

Many teachers have told me that they have difficulty feeling like their students are engaged in their remote classes. They open up their synchronous Zoom sessions and see 20 black squares with the video off. They see 30% of the students completing homework. Attendance in their classes is way down.

Here are some theories about what might be happening.

  1. School is about compliance, when we remove the relational aspects of school, students stop complying.
  2. Students are uncomfortable sharing themselves on camera. After all, almost none of them have experienced remote teaching via Zoom and there is strong peer pressure to conform to what the rest of the group does — would you be the only student with your camera on?
  3. Students are unable to turn their cameras and microphone on because the conditions at their home don’t allow it. Maybe they have siblings who are also on Zoom sessions and/or they don’t have private space to attend class?
  4. Students might not know how to turn on their cameras. In the past 8 months, I’ve helped at least a dozen people learn how to use Zoom for the first time and in almost all of these cases, I spoke with the person on the phone and coached them through doing things like starting a Zoom session and turning on their cameras. It stands to reason that at least some children don’t know how to use this technology either.
  5. Students feel uncomfortable sharing their thinking and emotions during mathematics class because they feel anxious to perform. Math class is already challenging for many students, learning remotely is certain to be more difficult, so whatever emotions students feel about math class are likely to be heightened.

Which of these issues is the main issue? I don’t know! I suspect that a variety of different issues impact student engagement and participation in math class. Rather than offer a neat solution, I’ll suggest a process we can go through to find a solution.

Using the network improvement framework developed by the Carnegie foundation, We start by defining the problem as multi-faceted and potentially involving multiple different causes or drivers.

The proposed drivers of student engagement in math class.

Each of these primary drivers of student engagement/participation potentially can be broken down further into secondary drivers. For example, if our primary driver is “Home conditions” then this might be broken down into secondary drivers of “No Internet”, “No private space”, and possibly “No time”.

For each secondary driver, we propose a change. What can we do differently in order to impact this aspect of the challenge? For example, if students do not have Internet at home, then we find out if we can provide home WiFi hotspots, much like this report suggests many rural school districts have done. For some of these drivers, we may not yet have a change idea and that’s fine but as we work together to solve this problem, our collective efforts may yet yield some strategies we can try.

Ideally, instead of everyone trying to tackle this problem independently, we work together to find solutions that appear to work in our varied contexts and then report back these proposed solutions for other people to test. When faced with a common problem, we are more likely to find robust and replicable solutions if we work together on the problem.

I have some wonderings though that I think our hive mind might be able to answer?

  • Are these the right primary drivers? Are these reasons above realistic reasons why students might have trouble engaging in even the most basic sense in math class?
  • Can we break these primary drivers down into secondary drivers? Can we determine what potential issues might exist in each of these categories?
  • What change ideas do you have? If you have identified and solved a problem that relates to student engagement, please share it here! I know of many teachers who are desperate for ideas to make their classes feel a bit more normal.

Improving Multiplication Practice

Here’s an activity that lets students practice multiplication facts. It’s basically a flashcard application.

The issue here is that if a student does not know their multiplication facts, they have no way of figuring them out. The feedback is extremely simple, too simple to be useful.


Here’s another activity.

This activity is slightly better. Students still do not directly know what the correct answer is, but they at least now have a mechanism for determining it. Unintuitively, feedback that requires a student to think is more effective than simpler feedback, so it’s possible that this level of feedback is just right — students may not actually benefit from just being given the answer.


This next activity is more introductory by design.

The benefit of this activity is that students are more able to connect something they know to something they may not know. For example, if students know that 2 × 4 = 8, they may be able to use this fact to derive that 3 × 4 = 12.


This activity is an extremely traditional activity where students fill in an entire hundreds chart.

The key benefit to this hundreds chart versus a pencil and paper activity is that students can request feedback at any time by clicking submit. This means that students who are using patterns to complete the table are able to double-check those patterns before having a mistake propagate into other rows and columns of the chart.

What none of these applications do is give feedback to a student based on their thinking. How would one go about designing such an application?

In this 2016 paper presented at ICCM 2016, the authors analyzed common errors made by students to single-digit arithmetic problems to see if they could rationales for these errors. Knowing why a child might write that 3 × 5 = 12 would be useful when engineering feedback for that child! In this case, the authors note that the child may have been skip-counting or using repeated addition and lost track of how many times they added 3 together. So instead of 3, 6, 9, 12, 15, the child thought 3, 6, 9, 12. The feedback here might be to skip count with the child and count the number of counts with them. If a child enters 3 × 5 = 14, then it might not be the number of skip counts that is the issue, but the addition from 12 to 14 instead of 12 to 15. Our feedback would have to be different!

Unfortunately, I have not yet designed an activity that does this, but the idea of feedback matching the thinking students do is a key component of the work I do.

The point is that with some small design decisions, we can modify activities that essentially only assess learning or build recall and turn them into activities that students can learn from.

Precision In Language

Alice’s Adventures in Wonderland

“Then you should say what you mean,” the March Hare went on.
“I do,” Alice hastily replied; “at least-at least I mean what I say-that’s the same thing, you know.”
“Not the same thing a bit!” said the Hatter. “Why, you might just as well say that ‘I see what I eat’ is the same thing as ‘I eat what I see’!”
“You might just as well say,” added the March Hare, “that ‘I like what I get’ is the same thing as ‘I get what I like’!”
“You might just as well say,” added the Dormouse, which seemed to be talking in its sleep, “that ‘I breathe when I sleep’ is the same thing as ‘I sleep when I breathe’!”

― Lewis Carroll, Alice in Wonderland


When I was early in my career and teaching algebra to 9th and 10th graders, I saw that they often wrote things I did not understand. Here’s one example:

-10 + -10 = 20

I asked my students where they came up with the idea that a negative plus a negative equals a positive and they told me their teacher told them “a negative and a negative equals a positive.” Notice that a literal word for word translation of what the student said into algebraic symbols is equivalent to the generalization above. Students listened to their teacher, learned the idea the teacher presented and applied that idea appropriately. The problem is that the idea is a shortcut for a more verbose and more precise mathematical statement.

This issue can happen at any time during a student’s time in school. A teacher might notice that a student, in the early stages of learning subtraction, has written that 2 – 7 = 5. At this point, the teacher might say, “Make sure you put the larger number first because when we subtract, we always take-away the smaller number from the larger number.” Later on, when students are learning about negative numbers, this early learning on subtraction can be sticky, and make understanding why 2 – 7 = -5 more challenging.

A critical idea in mathematics is that how we define mathematical objects has consequences for what the properties of the defined objects are. For example, there is a debate between whether trapezoids should be defined as quadrilaterals with at least one set of parallel lines or exactly one set of parallel lines. The first definition means that parallelograms, with two sets of parallel lines, would be trapezoids. The second definition means that trapezoids are distinct objects from parallelograms. Exploring the consequences of different definitions is critical work in mathematics but this work should be done collectively where possible so that definitions can be refined and concepts clarified.

When we use simplified language in order to help students understand a concept, students will often do this mathematical work on their own, so if we want students to understand the boundaries of the mathematical ideas and not over-generalize, we need to be careful that the language we use is precise and that say what we mean, and not more. It is also critical that we regularly check how students actually understand ideas, not how we think they understand them.

That’s Mathematics

Ed Southhall invited me to sing a line from the song, “That’s Mathematics” by Tom Lehrer and I happily obliged, along with a few other friends of Ed’s.



One thing I really like about this song is the variety of different things that Tom calls out as being mathematics. Mathematics is no one thing, it is a bunch of different things to different people.

Online Learning Recommendations

Given that many schools (and entire school districts) may be closed down during the coronavirus outbreak, I decided to write this post with recommendations for schools that may attempt to implement online learning during this time.

I read through this review of the research on online learning, which contains these high-level recommendations. There are some caveats with this research, especially given that most research on online learning has been done with older students and that the sample sizes with k to 12 students are relatively small. That being said, some evidence for effectiveness is better than no evidence at all. A further caveat: these recommendations are based on effect sizes, which I have not included since they are notoriously unreliable to compare.

  1. Instruction combining online and face-to-face elements had a larger advantage relative to purely face-to-face instruction than did purely online instruction.

    If your school is closed completely, then this may be impossible. That being said, online programs such as Zoom or Big Marker may allow for some “face to face” interaction to occur. These programs will also help with the next recommendation.

  2. Effect sizes were larger for studies in which the online instruction was collaborative or instructor-directed than in those studies where online learners worked independently.

    This basically means that you should design activities that are either led by a teacher or activities that have students work together in small groups. Resources like Google Docs and Skype will be helpful for students working together but given the high possibility that some students will engage in off-task and/or anti-social behaviour (such as teasing or bullying), having some moderation and oversight of these online spaces will be helpful.

  3. Elements such as video or online quizzes do not appear to influence the amount that students learn in online classes.

    Creating a bunch of video lessons of a talking head working through some math problems and then quizzing students on what they have learned afterwards is not supported by the existing evidence on online learning. Given that educator planning time is in short supply, it’s probably best to plan other types of activities.

  4. Online learning can be enhanced by giving learners control of their interactions with media and prompting learner reflection.

    A good example of a program that allows for this is Geogebra. See for example this set of constructions puzzles that require students to think and make decisions while they work through the problems.

    Another example is the DreamBox Learning math program, which also requires students to actively engage with mathematics. Disclaimer: I work for DreamBox Learning as a mathematician and senior curriculum designer.

  5. Providing guidance for learning for groups of students appears less successful than using such mechanisms with individual learners.

    This recommendation suggests that feedback and support for students should be individualized for online learning, rather than given to the entire group. This does not necessarily mean that one should avoid providing scaffolds (such as guiding questions) to the entire group or that teachers necessarily need to work with individual students, only that whatever guidance and feedback is provided, it should be directed where possible to individual students.

Based on my experience as a parent to my sons, who have both engaged with online learning and are in elementary school and high school right now, I have some further recommendations.

  1. Actively engage the learning guardians of students in the process of learning.

    My sons’ experiences have been far more productive when we have sat down with them while they work through the online course material. This does not mean that we do the work for our children, but rather than we are there to support, encourage, and nurture their development as learners.

    It will be helpful to offer explicit advice for how learning guardians can support their learners, especially given the range of knowledge and experience those learning guardians will bring to the task. You may even want to include videos of what class looks like and descriptions of instructional routines that learning guardians can use with their learners.

    Also, offer suggestions of activities to learning guardians that they can do with their children in their care that are not on a computer and do not require the learning guardians to be experts in any particular subject matter.

  2. Skip watch-the-video-then-fill-in-the-blank-spaces activities.

    I can say from experience that these types of activities are ubiquitous in online learning and result in nearly no learning. I watched my son listen to a video in one tab while dutifully recording the answers in his worksheet in another. I quizzed him 5 minutes later and he could remember literally nothing at all from the worksheet or the video.

    Given that completing these particular courses was a requirement at his school, I taught him a much more productive learning strategy. First, attempt the worksheet and fill in every blank, even if one has to guess. Next, watch the entire video without writing or doing anything else. Now go back to the worksheet and change as many of the answers as one can without going back to the video. Rewatch or listen to the video with the worksheet and change answers as necessary. This is still a terrible experience but it at least has the possibility to result in some learning.

  3. Provide devices for students to work if at all possible or at least ensure that any online learning activities can be completed with a smartphone.

    While access to computers and the Internet keeps increasing, there are still households that do not have access and so providing equitable access to resources to all families is a key responsibility of schools, particularly when expecting students to engage in online learning.

  4. Where possible, engage students in synchronous activities rather than asynchronous activities.

    One of the more successful online classes my son took was with the Art of Problem Solving. Each week my son met with the entire class in an online chat program where the teacher mostly posed questions and occasionally told the students information, while the students responded to the questions in the online platform. He also had a physical textbook, a bank of unlimited practice problems to work on, and challenging problems to complete each week. The chat program was nothing amazing, but it mostly kept my son engaged for the full 90-minute sessions.
  5. Use simple assignments that do not require students to navigate complex instructions.

    Even with assignments with simple instructions, there is a lot of potential for student learning. Given that your students will be working remotely and with limited direct support, you don’t want students spending too much of their time figuring out what they are trying to accomplish.


What other recommendations for teachers and schools who may be suddenly engaged in online learning do you have? What question do you have that I have not yet answered?

Why Is a Negative Times a Negative Positive?

There are different possible answers to this question, depending on the standard of proof one needs and the background knowledge one brings to the question.


Mathematical consistency and patterns

Try solving each of these problems, paying attention to the previous set of problems as you do so. Look for patterns to make solving the problems easier.

3 × 3 = ?
3 × 2 = ?
3 × 1 = ?
3 × 0 = ?
3 × -1 = ?
3 × -2 = ?
3 × -3 = ?
2 × -3 = ?
1 × -3 = ?
0 × -3 = ?
-1 × -3 = ?
-2 × -3 = ?
-3 × -3 = ?

The answers to these problems are below but I really do recommend taking the time to solve the problems above on your own first, so you get the sense of how students might think through this set of problems.

3 × 3 = 9
3 × 2 = 6
3 × 1 = 3
3 × 0 = 0

At this stage, many people will notice the answers are 3 smaller each time and the number being multiplied by 3 is one smaller each time, so they continue that pattern to answer the following questions.

3 × -1 = -3
3 × -2 = -6
3 × -3 = -9

Now, we decrease the first number in the pattern by 3 and one has to make some deductions about what the answer should be.

2 × -3 = -6
1 × -3 = -3
0 × -3 = 0

One might now notice that the answers are going up by 3 each time as we increase the first number, and so it is reasonable to continue this pattern.

-1 × -3 = 3
-2 × -3 = 6
-3 × -3 = 9

While to some this pattern may seem obvious, when someone is still in the middle of learning this concept, they have less cognitive capacity available to accomplish the task at hand (multiplying numbers together) and accomplish the additional task of looking for patterns in their answers, so this is where someone else prompting them to stop and look for patterns in their work so far will be very useful.

Prerequisite knowledge: One has to know what these symbols mean, what is meant by finding one number times another, and how negative numbers work in terms of counting down and subtraction.


Mathematical consistency and mathematical properties

Let’s look at a problem that we can do in more than one way, borrowed from the Khan Academy.

5 × (3 + -3) = ?

If we add the numbers inside the parenthesis first, then this is 5 times 0 which is 0, since 3 + -3 = 0.

5 × (3 + -3) = 0

But what if we distribute 5 through both terms first?

5 × 3 + 5 × -3 = ?

Since distributing the 5 across the addition does not change the value of the expression, we know this is still equal to 0.

5 × 3 + 5 × -3 = 0

But this means that 5 × 3 and 5 × -3 are opposite signs, so since 5 × 3 = 15, then 5 × -3 is -15. Let’s look at another example.

-5 × (3 + -3) = ?

We know that this is the same as -5 times 0, so this has a value of 0.

-5 × (3 + -3) = 0

Similar to before, we distribute -5 through both terms.

-5 × 3 + -5 × -3 = ?

Again, the distribution of terms does not change the value of the expression on the left-hand side of the equation, so the result is still 0.

-5 × 3 + -5 × -3 = 0

We know from before that -5 × 3 is -15 so we can substitute that value for -5 × 3 in the left-hand side of the equation.

-15 + -5 × -3 = 0

Therefore -15 and -5 × -3 are opposites since they add to 0, so -5 × -3 must be positive.

Nothing in what we did for the two examples above is specific to the value of 5 × 3, so we can repeat this argument for every other multiplication fact we want to derive, so these two ideas can be generalized.

Prerequisite knowledge: One has to know what these symbols mean, what is meant by finding one number times another, how the distributive property works, and how negative numbers can be defined as the opposites of positive numbers.


Representation on a number line

Imagine we represent multiplication as jumps on a number line.

3 times 3 on the number line

For 3 × 3, we draw 3 groups of 3 moving to the right. Both the number of groups and the direction of each group are to the right.

But what about 3 × -3? Now we have 3 groups of the number still, but the number is negative.

3 times -3 on the number line

If we find -3 × 3, the size and direction of the number we multiply are the same, but now we are finding -3 groups of that number. One way to think of this is to think of taking 3 groups of the number away. Another is to think of -3 times a number as being a reflection of 3 times the same number.

-3 times 3 on the number line

So -3 × -3 is, therefore, a reflection of 3 × -3 across the number line.

-3 times -3 on the number line

In one sense though, this visual argument is just mathematical consistency represented using a number line. If multiplication by a negative is a reflection across 0 on the number line, and we think of negative numbers as being reflections across 0 of the number line, then multiplication of a negative number times a negative number is a double-reflection.



Karen Lew has this analogy.

Multiplying by a negative is repeated subtraction. When we multiply a negative number times a negative number, we are getting less negative.

This analogy between multiplication and addition and subtraction helps students nicely connect the two concepts.

Joseph Rourke shared this context.

A gambler loses $10 per day. How much more money did they have 5 days ago?

Here, the loss per day is one negative and going backwards in time is another.

@M_Teacher_w_T shared this analogy:

“An enemy of my enemy is my friend.”

This aims not at the algebraic or arithmetic properties of numbers but more at the oppositeness of negative numbers.

Prerequisite knowledge: All contexts that build new understanding require students to understand the pieces of the context fairly well, so it is especially important to probe how students understand an idea when it is presented contextually.


Algebraic proof from first principles

From Dr. Alex Eustis, we have this algebraic proof that a negative times a negative is a positive.

First, he states a set of axioms that apply to any ring with unity. A ring is basically a number system with two operations. Each operation is closed, which means that using these operations (such as addition and multiplication on the real numbers) leads to another number within the number system. Each operation also has an identity element or an element that does not change another element in the system when applied to it. For example, under addition, 0 is the additive identity. Under multiplication, 1 is the multiplicative identity. The full set of axioms required is below.

Axiom 1: a + b = b + a(Additive commutivity)
Axiom 2: (a + b) + c = a + (b + c)(Additive associativity)
Axiom 3: 0 + a = a(Additive identity)
Axiom 4: There exists −a satisfying a + (−a) = 0(Additive inverse)
Axiom 5: 1 × a = a × 1 = a(Multiplicative identity)
Axiom 6: (a × b) × c = a × (b × c)(Multiplicative associativity)
Axiom 7: a × (b + c) = a × b + a × c(Left multiplicative distribution)
Axiom 8: (b + c) × a = b × a + c × a(Right multiplication distribution)

From these axioms, we can prove that a negative times a negative is a positive. I’ll reproduce Dr. Eustis’s proof below and include the reference to the axioms used. First, we prove that a = −(−a).

Corrolary 1
a = a + 0(Axiom 3 and Axiom 1)
a = a + (−a + −(−a))(Axiom 4 applied to −a )
a = (a + (−a)) + (−(−a))(Axiom 2 – the associative property)
a = 0 + (−(−a))(Axiom 4)
a = −(−a)(Axiom 3)

So now we know that if we introduce negative numbers a is equal to −(−?).

Corrolary 2
0 = a + (−a)(Axiom 4)
0 = (0 + 1) × a + (−a)(Axiom 3 and Axiom 5)
0 = 0 × a + 1 × a + (−a)(Axiom 8)
0 = 0 × a + (a + (−a))(Axiom 5 and Axiom 2)
0 = 0 × a + 0(Axiom 4)
0 = 0 × a(Axiom 3 and Axiom 1)

Proving that 0 = 0 × a is the kind of painfully obvious idea that hardly requires proof but it establishes a relationship between multiplication and the additive identity in the real numbers, which is not yet included in the axioms above.

Next, we prove that (−1) × a = −a.

Corrollary 3
a = −a + 0 × a(Corrolary 2 and Axiom 3)
a = −a + (1 + (−1)) × a(Axiom 4)
a = −a + 1 × a + (−1) × a(Axiom 8)
a = (−a + a) + (−1) × a(Axiom 5 and Axiom 2)
a = 0 + (−1) × a(Axiom 4)
a = 0 + (−1) × a(Axiom 3)

Now, finally, we can prove that (−a) × (−b) = ab.

(−a) × (−b) = (a × (−1)) × (−b)(Corrolary 3)
(−a) × (−b) = a × ((−1) × (−b))(Axiom 6)
(−a) × (−b) = a × (−(−b))(Corrolary 3)
(−a) × (−b) = a × b(Corrolary 1)

This last “proof” though is unlikely to justify that a negative times a negative is a positive for any students though. It’s the kind of thing which is a required level of justification for a mathematician interested in rigorous proof who would likely consider the other justifications “patterning” and not sufficient.

A critical idea of proof though is that the intended audience of a proof is left convinced that an idea is true, and so I posit that the algebraic “proof” presented here is no proof at all for almost everyone.

Prerequisite knowledge: While I went through and added the justification for each step of the proof that was missing, I needed a fair bit of fluency with the original set of axioms. I also needed to not lose sight of the overall goal and to be able to recognize the structure of each part of the argument and match that structure to the axioms.


A simpler algebraic proof

This algebraic proof from Benjamin Dickman is much simpler than going back to a proof based on the axioms of arithmetic.

a + (−a) = 0
a × b + (−a) × b = 0 × b
ab + (−ab) = 0

From this, we can show that ab and –ab have opposite signs and therefore that a positive times a negative is a negative. Using the fact multiplication is commutative, a negative times a positive is also negative.

Similarly, we can prove that a negative times a negative is a positive.

a + (−a) = 0
a × (−b) + (−a) × (−b) = 0 × (−b)
ab + (−a) × (−b) = 0

Since we know that −ab is negative, and the sum of these two terms is 0, therefore (−a) × (−b) is positive.

Prerequisite knowledge: The prerequisite knowledge for this proof is much less than the other one, but it does assume a fair bit of fluency with manipulation of algebraic structures.



Given that the goal of an argument that something is true is to leave the other person convinced of the truth of the argument, whenever anyone uses any justification, representation, or proof, it behooves one to check that one’s audience is left convinced.