The Reflective Educator

Education ∪ Math ∪ Technology

Pandemic Math

While many people think of 2020 as “that year from hell that just kept on going”, I also think of it as the year that mathematical literacy became an obvious necessity. In many different stories this year, mathematics featured prominently as a way to understand the world.

As everyone knows, in the first months of this year, the pandemic started. Not surprisingly, a lot of coverage in 2020 focused on the number of cases, the spread of the Coronavirus. What might be missing for that coverage is the mathematical literacy required by the people consuming it.

Here are a few areas where mathematical literacy is needed in order to deeply understand the arguments being made by epidemiologists and policy makers in relation to the pandemic.

 

Exponential Growth

Number of new cases over time

Understanding these graphs requires mathematical sophistication. For example, the graph above shows the number of new cases over time. From my experience working with kids, I know that many of them perceive flat areas on graphs as areas where no change is occurring. However, this graph is showing new cases, which is related to the rate of change of the number of people infected, not the total number of cases. Many people interpret this graph as meaning that the spread of the virus was levelling off in April, but that part of this graph is still showing roughly 30,000 new cases a day!

The general public was also introduced to a new term for them, R0, also called the basic reproduction rate. R0 = 1 means that each person who gets ill, on average, infects one other person while they are able to infect others. R0 = 2 means that each person infects 2 other people on average. Any R0 > 1 corresponds to exponential growth of the infections. Communicating to people that R0 = 1.1 is much, much better than R0 = 1.5 has been challenging. The table shows the relationship between each generation of infection and various R0 values. Notice how much of a difference a small change in the R0 values makes.

R0 = 1.1R0 = 1.5R0 = 2
1000 infections1000 infections1000 infections
1100 infections1500 infections2000 infections
1210 infections2250 infections4000 infections
1331 infections3375 infections8000 infections
A table showing how R0 impacts total infections

 

Ratios and Proportions

A Facebook friend recently argued that the Coronavirus vaccination must not be working because more vaccinated people had died in the past month than unvaccinated people, 52 to 48. First, let me just say that any people dying from this disease is heartbreaking. That being said, this argument fails to apply some basics of ratios and proportions.

In the area of the world my friend lives, roughly 90% of people eligible for vaccination are vaccinated. Suppose this corresponds to 80% of the people who can potentially contract the disease and that my friend lives in a town with 100,000 people. 52 of 90,000 people is a much smaller percentage (≈ 0.06%) than 48 of 10,000 (≈0.5%). In fact, if one considers the relative rates of death, being vaccinates increases one’s odds of survival by almost ten times, relative to being unvaccinated.

 

Probability

In a brilliant and interactive essay meant to argue for the power of wearing masks, Aatish Bhatia uses probabilistic arguments to show that one mask is better than no masks, and that each person wearing a mask is much better than one person wearing a mask. This argument unfortunately relies on people understanding probabilistic reasoning.

Two commonly fallacious arguments rooted in misunderstandings of probability are the use of anecdotal evidence to argue against probabilities (“My friend was wearing a mask, and they still got Coronavirus”) and assuming that low probability means zero probability (“Those scientists said that the vaccinations would protect us from the Coronavirus”).

The first argument forgets that when one finds an example of an event occurring, there are many examples one may overlook of the event not occurring. The second argument assumes that the goal of vaccinations may be to prevent infection, serious illness, or death, when the goal may simple be to reduce the probability of these events occurring.

 

Statistics

One way we know the impact of the Coronavirus in terms of mortality is indirectly through a statistic called Excess mortality. Essentially, our society and those around the world to varying degrees, keep track of the typical rates of death from various ailments. Excess mortality is the difference between the typical rates of death from all known causes and the existing rates of death during a crisis like the Coronavirus pandemic. When the reported rates of death exceed the expected rates of death by a statistically large enough margin, we can attribute the excess deaths to whatever change in environmental factors currently exists, such as the pandemic.

Understanding this argument requires one both to understand the prior year deaths are highly consistent making future years’ deaths predictable and that the deaths from the two current years completed during the pandemic differ significantly from this average. These ideas might be intuitive for some versed in reading statistical plots but are not at all obvious to someone with a poor background knowledge of statistics.

 

All of these areas of mathematics are already part of a typical school curriculum. People may have learned them in high school, but have subsequently forgotten them. However, it is more likely they never had an opportunity to learn them, as they are often part of the optional math courses towards the end of high school. This to me reinforces the argument that a greater level of mathematical literacy is necessary more than ever for people to be fully informed citizens.

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.
Source: https://xkcd.com/263/

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?