“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.
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.
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.
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.
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.
Online learning can be enhanced by giving learners control of their interactions with media and prompting learner reflection.
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.
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.
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.
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.
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.
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.
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?
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
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.
Representationon a number line
Imagine we represent multiplication as jumps on a 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.
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.
So -3 × -3 is, therefore, a reflection of 3 × -3 across 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.
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 prooffrom 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
Axiom 2: (a + b) + c = a + (b + c)
Axiom 3: 0 + a = a
Axiom 4: There exists −a satisfying a + (−a) = 0
Axiom 5: 1 × a = a × 1 = a
Axiom 6: (a × b) × c = a × (b × c)
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).
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))
a = −(−a)
So now we know that if we introduce negative numbers a is equal to −(−𝑎).
0 = a + (−a)
0 = (0 + 1) × a + (−a)
(Axiom 3 and Axiom 5)
0 = 0 × a + 1 × a + (−a)
0 = 0 × a + (a + (−a))
(Axiom 5 and Axiom 2)
0 = 0 × a + 0
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.
−a = −a + 0 × a
(Corrolary 2 and Axiom 3)
−a = −a + (1 + (−1)) × a
−a = −a + 1 × a + (−1) × a
−a = (−a + a) + (−1) × a
(Axiom 5 and Axiom 2)
−a = 0 + (−1) × a
−a = 0 + (−1) × a
Now, finally, we can prove that (−a) × (−b) = ab.
(−a) × (−b) = (a × (−1)) × (−b)
(−a) × (−b) = a × ((−1) × (−b))
(−a) × (−b) = a × (−(−b))
(−a) × (−b) = a × b
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.
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.
“Sixteen,” I respond and then add, “How do you know that’s true?”
“Hrmmm. I know four times four is the same as four plus four plus four plus four and that’s the same as eight plus eight, which is sixteen.”
During my career, I’ve found that mathematical ideas include procedures, concepts, habits of mind, and declarative knowledge. Even in the brief exchange above, I claim that all four of these types of mathematical ideas are being used.
I define procedures as a sequence of steps intended to be used to solve specific mathematical problems, concepts are ideas that can be used with different procedures, habits of mind are general problem-solving strategies, and declarative knowledge is that which is known to be true without reference to other ideas.
Procedures: My son knows that if he wants to add 4 numbers, he first adds two numbers together, then the last two numbers together, and then these two results together.
Concepts: My son knows that one definition of multiplication is repeated addition and uses this idea to transform 4 times 4 into 4 plus 4 plus 4 plus 4.
Habits of mind: My son knows that if one is not sure how to solve a problem that one can often change it into a different problem that one can solve. In this case, my son decides to change the multiplication problem into an addition problem.
Declarative knowledge: My son knows that four plus four is eight and eight plus eight is 16 without reference to other ideas. It is often the case that things that are currently declarative knowledge are based on procedures and conceptual knowledge learned earlier.
This is why I find arguments about whether we should teach children procedures or teach them conceptually confusing — it’s not possible to do one or the other, students are always learning some mixture of all four types of mathematical knowledge.
“Knowledge is not tiny bits that we can count and represent by numbers, but a network of logically interconnected ideas, beliefs, and generalizations structured so it can be searched and used to work out and evaluate new ideas.” – Graham Nuthall, The Hidden Lives of Learners.