Fighting Confirmation Bias Is Like Fighting Gravity (so let’s stop fighting it)

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When you learn about confirmation bias two things are usually explicitly stated. Confirmation bias is inescapable and that we should do everything we can to escape it. This is like saying it’s impossible to escape gravity but you should do everything you can to try to escape it. 

Is it possible to hack our bias to confirm our beliefs in such a way that we don’t  need to feel like we’re constantly fighting gravity? 

To answer that question we need to understand confirmation bias more fundamentally. 

As I scroll through my social media feeds I feel confirmation bias. More specifically, the posts that confirm my beliefs feel good. They bring me joy. I’m more likely to share them. Those that don’t confirm my beliefs make me feel uneasy, frustrated, or annoyed. I’m more likely to be skeptical of those posts and try to find holes in the logic, which will relieve my discomfort. It will make me feel like I needn’t change my beliefs to embrace this new information. I might even share my skepticism with my followers. 

Confirmation bias operates constantly in background as we go about our lives. If we believe women are bad drivers, we pay attention to examples of women driving poorly and put less weight on men driving poorly. If we want the Tigers to win a baseball game, we will notice when the umpires favor the other team and hurt our team.  If we think “the left” is a cancer on society, we find examples to confirm this and ignore counter examples. Likewise for “the right.” Confirmation bias is always there, telling us where we should direct our attention.

Confirmation bias rides on the rails of the beliefs we have in our mind. In fact, in some sense it’s not so much a bias as it is our brain’s primitive defense mechanism searching for beliefs counter to our own and “protecting” us from them. We perceive ideas that don’t map onto our current beliefs as threatening – hence the negative affect. The negative emotion signals our brain to heighten awareness and seek safety. We find “safety” (hits of dopamine), by poking holes in the ideas attempting to infiltrate our current beliefs. That brings us comfort, but decreases the likelihood we integrate new information that may be true. 

If we are biased to confirm our current beliefs because we instinctively view new ideas as threatening, it’s hypothetically possible to adopt and preserve beliefs that embrace new ideas. That is, we can believe that new ideas enhance our current beliefs. 

Here are a few specific beliefs we could embrace. 

Nobody’s cornered the whole truth on anything

We might start by downloading a belief in our brains that no one person, or group, has a monopoly on truth. This includes you. If we believe this then when we encounter a belief that we disagree with, we will seek out the parts of the belief that enhance our understanding. We’ll seek to integrate both viewpoints, yielding a richer understanding of the given concept. When we find the valid points within the opposing viewpoint, it will confirm our belief (which will literally feel good). 

Essentially, if conformation bias drives us to find information that confirms our belief, then we can set it about the business of finding information that disconfirms our currents beliefs so as to enhance our total understanding what is true. We believe that diverse viewpoints are necessary to a complete understanding, so we seek out those viewpoints, and when we find the truth contained in them, we get the dopamine hit and the reinforcement.  

Our ego isn’t interested in what’s true, only that we’re safe

I’ve read “Don’t Label Me” by Irshad Manji twice and I’m working to become a Moral Courage Mentor. (Moral Courage is Irshad’s approach to psychologically healthy diversity and inclusion training.) In her book and in the course I learned much about the ego, or “egobrain,” as Irshad calls it. This isn’t the “woo woo” ego from Freud – it describes our innate, primitive, threat-detecting system. It’s what’s driving confirmation bias as I’ve described it above. 

Let me pick a totally hypothetical and unrealistic but illustrative example of the ego brain at work (that definitely didn’t happen…). The other day I had a… difference of opinion with my wife. We were getting things ready for a garage sale, which we had discussed would be good to spread over two days, Thursday and Friday. On Wednesday afternoon it became clear to her that we were not going to be ready to have the sale on Thursday. This was not clear to me. As I was cooking dinner she brought up several things we still needed to do to prepare, to which I thought (and probably said) “I guess we’ll stay up late and do them.” She mentioned how she was getting pretty anxious about publicizing the sale without those things done, to which I thought (and probably said), “Well it will be fine, we can get it done.” Finally, she said, “Do we really need to have this sale tomorrow?” 

At this moment I felt my ego say “That’s what we planned and lets just stay up late and do it. Why do we have to change things last minute?!?” However, I’m getting better at noticing when my ego is talking and when my calm, rational mind is talking. This was definitely ego. I thought a bit longer before ejecting the first reaction that came to my mind and realized that it was not necessary to have the sale the next day. My ego wanted me to cling to my old beliefs, my wife presented an idea that was counter to them, and I took the extra second to put my ego to rest, consider the problem more carefully, and ultimately concede that she made a good point. My belief about the importance of sticking to the plan was incorrect and defending it was irrational. 

As I hope you can see, the ego is generating much of our confirmation bias. In fact, we could call our “bias to confirm” our tendency to “protect our current beliefs.” Herein lies another belief we can adopt that would be healthy to confirm: we must routinely speak truth to the power of our ego if we are to update our beliefs about the world. 

Every person is a “plural” 

We have biases towards other people based on the labels they either ascribe to themselves or that we ascribe to them. These labels help us build a caricature in our mind of that person. We reduce them to labels, extrapolate all of their other characteristics from the labels, and then judge them. We quickly categorize a person as someone worth listening to or worth ignoring. 

While this labeling, categorizing, and judging makes navigating our lives easier, it’s unfortunately a house of cards that only fuels confirmation bias. We see a person with a MAGA hat on and we believe we know nearly everything about that person. When he does something that confirms our mental model of a MAGA-hat-wearing person, our belief is confirmed. If he does something that runs counter to it, we either don’t notice or write it off as an anomaly, not to be taken seriously. Pick your favorite tribe to hate on – the same thought process applies. 

However, if we decide to look, we will find that below these labels every person conceals a richer personhood, revealing that they are dynamic and multifaceted. Manji calls a person that consistently bucks their labels a “plural,” and reminds us that if we look (and listen) hard enough we’ll find that every person is a plural. 

If we believe that each person is a plural, then confirming that belief means we pass on snap-judgements and assume there’s more to them than the caricature we’ve built in our mind. 

If we believe that each person is a plural, then we’ll seek the complexity of each person. When we find it, we’ll confirm our bias, thereby reinforcing the assumption that each person is a plural. 

“Wait, you can’t just choose what to believe!”

Sure you can. We do it all the time. Sometimes we don’t realize we’re doing it, but we do. Many times it feels like reasoning leads to concluding that a belief is true, but just as often, if not more, we want to believe something is true and seek out the justification later. 

I think Apple makes better phones than Android makes and better computers than PCs. I’ve got, I think, good reasons to believe this but at no point in my life did I take a year and do an objective analysis on the features of each brand of technology. I had a couple good experiences with Apple products in high school and I’ve been happily feeding that belief ever since. 

More seriously, if we dig into our beliefs deeply enough I think we all get down to a priori assumptions that either consciously or subconsciously adopt. (Books have been written on that topic and I don’t have the space to explore that here. But if you think my argument falls apart because that claim is false, please let me know in the comments!) And, since humans are dynamic, we sometimes change those foundational beliefs. For example, I might go much of my life and assume that people are generally good people. I might then have an experience where I see the dark side of humanity and conclude that people are, in fact, generally bad. 

Is either true in a fundamental sense? 

How would we begin to answer such a question even remotely objectively? 

We can say that adopting either of those beliefs will impact the course of an individual’s life in meaningful ways, right down to daily interactions with other people. I think we can also conclude that a critical mass of individuals adopting either belief will have society-wide ramifications. Finally, in some sense one can choose to adopt either belief – and suffer the consequences. 

Now, not all beliefs are are equally true or pragmatic or will result individual or group-level flourishing. Not all are equally Good. Another book-length exploration would be required to explore what we mean by a Good belief, but for the narrow purpose of this essay let’s assume that it means developing a form of confirmation bias that doesn’t require us to constantly fight ourselves. 

I’m arguing that, given the malleability of our beliefs, we should adopt the the following: 

  1. None of us knows all of the truth which means the other people we interact with must know something important that we don’t – we should listen accordingly.
  2. Our ego often blinds us from the truth in an effort to maintain our current beliefs. This means we need to constantly be mindful of when it’s at work and keep it in check. 
  3. Every person is a plural. This means that we will be slow to put people in boxes and we’ll seek out the characteristics that demonstrate the individuality in their character. 

I’ve adopted these beliefs and I can tell you that my mind is in a healthier place. Remembering that people are plurals keeps me looking for the nuance in their personality. It motivates me to keep looking beyond the caricature I’ve built in my mind. Keeping my ego in check helps me avoid arguments for the sake of being right, as I explained in the story about my wife. I’m not perfect, but more often I find myself taking a breath to respond thoughtfully as opposed to reacting quickly. Remembering that I don’t know everything about anything motivates me to engage with those I disagree with to figure out what I’m missing. Finally, I find myself gravitating towards people who seem to believe the same things. 

In short, in feeding my confirmation bias I gain a richer understanding of nearly everything. 

Jo Boaler, Tracking, Education Research, and Honesty

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A few years ago I read Jo Boaler’s book, “Mathematical Mindsets” and I thought it contained some good ideas. There were a few things that I thought were not realistic or would be difficult to scale, but overall I found the book useful. Our department read it together, and I remember a colleague pointing out that Boaler often cited her own research. That revelation made me more skeptical of her work, especially when she provided citations.

Was she only seeking out data that confirmed what she already believed?

Despite this I would have considered myself a “fan” of her work. In subsequent years I used some of her resources from Youcubed and subscribed to her email newsletter. The content was a mixed-bag of resources and opinions, but she was clear in her book and in her work that “tracking,” the education model in which students are separated into courses based on their ability, needed to be eliminated. It was unfair, meant weaker math students weren’t exposed to “rich” mathematics, perpetuated inequities, and the research showed de-tracking was better for all students. If we wanted to fix mathematics education in the United States, according to Boaler, we would need to de-track mathematics, helping teachers to incorporate “low floor, high ceiling” tasks that are accessible to all students at a particular grade level.

My intuition about de-tracking is skepticism for reasons I might explore in another post. For now I want to focus on what’s happening in California surrounding their proposed mathematics curriculum, the de-tracking program San Francisco Unified School District implemented in 2014, and how advocates, Jo Boaler chief among them, are using misleading (or missing) data to push for policy that has little to no evidence to support its adoption.

This NY Times article gives context regarding the curriculum and the debate around it. From the article:

The California guidelines, which are not binding, could overhaul the way many school districts approach math instruction. The draft rejected the idea of naturally gifted children, recommended against shifting certain students into accelerated courses in middle school and tried to promote high-level math courses that could serve as alternatives to calculus, like data science or statistics.

The draft also suggested that math should not be colorblind and that teachers could use lessons to explore social justice — for example, by looking out for gender stereotypes in word problems, or applying math concepts to topics like immigration or inequality.

Evaluating the Success of SFUSD’s Framework

The evidence that advocates are using to promote this curriculum is largely from San Francisco Unified School District’s de-tracking program which they implemented in 2014. (“It also promoted something called de-tracking, which keeps students together longer instead of separating high achievers into advanced classes before high school. The San Francisco Unified School District already does something similar.”) They claim that they reached all of their goals with the program and show progress along many metrics . But a group of data scientists, teachers, lawyers, parents, and students put together a report outlining how SFUSD is either actively hiding data (as some California Public Records Act requests, California’s version of FOIA, have been ignored), intentionally misleading the public, or is simply incompetent.

The group, Families for San Francisco, put together a report (cited in the NY Times) explaining the problems with SFUSD’s nationwide campaign espousing the success of its new framework. They share evidence that de-tracking was an overwhelming positive decision for the students in SFUSD. Here are a few revelations from the report worth noting. They begin by going through each of the three goals outlined by SFUSD, the claims made by SFUSD about progress on those goals, and their analysis.

The first goal was to “Reduce the number of students forced to retake Algebra 1, Geometry, or Algebra 2 by 50% from numbers recorded for 6/2013.” SFUSD claims a “dramatic increase in student comprehension” and a drop in Algebra 1 repeaters from 40% to 7% in a press release from 2017. Here is the analysis of this claim from Families for San Francisco.

Facts: The grade distribution we received from SFUSD showed no improvement at all in Algebra 1 grades. The repeat rate did come down, but only because in 2015 SFUSD eliminated the requirement to pass the Algebra 1 California Standards Test (CST) exit exam as a condition of progressing. The effect of this change was later partially acknowledged by the Math department in the speaker’s notes in one of their presentation slides in 2020: “The drop from 40% of students repeating Algebra 1 to 8% of students repeating Algebra 1, we saw as a one-time major drop due to both the change in course sequence and the change in placement policy.” Finally, in conducting our review of SFUSD’s claims, we were unable to obtain any such “longitudinal data” they refer to nor could we replicate the repeat rate numbers quoted by SFUSD using data obtained via a CPRA request. We have deep concerns that SFUSD is claiming credit for student achievement that is either untrue or unsubstantiated by the data or both.

The second goal was to “Increase the number of students who take and pass 4th year math courses (post- Algebra 2 courses) with a C or better by 10% by 6/2018.” SFUSD claims that “456 more students, or 10.4% more students are taking courses beyond Algebra 2 in 2018-2019 than were in 2017-2018.” Unfortunately, this claim is misleading. Here is the analysis from Families for San Francisco.

Facts: Enrollment in advanced math classes at SFUSD has gone down, not up, and SFUSD has produced no data about pass rates.Advanced math is commonly understood to mean courses beyond Algebra 2, including Precalculus, Statistics, and Calculus; however, SFUSD’s claim that its enrollment in “Advanced Math” enrollment has increased depends entirely on counting students enrolled in its “compression course” — a third-year course combining Algebra 2 with Precalculus. The problem with this framing is that the University of California (UC) rejected SFUSD’s classification of its compression class as an advanced math course due to its failure to meet UC standards for Precalculus content. Once we exclude the enrollment data for the compression course, the enrollment number for advanced math shows a net decrease (emphasis mine) from 2017-2018 (the final cohort prior to the implementation of the new math course sequence).

The third goal was to “Increase AP Math enrollment & pass rate for Latino & African American students by 20% by 6/2018.” SFUSD claimed that “(a) ‘AP Math enrollment has also increased over a two-year period from 2016-17 to 2018-19’; (b) that ‘AP Statistics enrollment has increased 48.4%’; and (c) that Latinx AP Math enrollment increased 27% over the same period.” Families for San Francisco’s investigation found these claims inconclusive because they were unable to get all the data necessary to verify them. Here’s what they write.

Facts:Whether SFUSD met its original goal to increase Latinx and African American AP Math enrollment by 20% from June 2014 to June 2018 is unknown because in spite of our requests, SFUSD has not produced complete data for this period. For the two-year period from 2016–2017 to 2018–2019, African Americans are not listed among “subgroups who met or exceeded the 10% growth target” and SFUSD has not disclosed any performance outcomes. The five-year data for school years 2016-2017 through 2020-2021 shows that enrollment by African American students has fluctuated from year to year while enrollment by Latinx students has been more or less on the rise. And because SFUSD does not release data on the pass rate for AP Math exams, its success rate is unknowable.

Not only are the pass rates unknown, the enrollment data available shows that the claim of increased AP math enrollment is misleading.

Meanwhile, the claim of increased AP Math enrollment overall is misleading. The number of SFUSD students overall taking AP Calculus is down. The number taking AP Statistics is up but it is concentrated at three specific school sites (Lowell, Ruth Asawa SOTA and Balboa). The other schools showed no significant increase.

It seems clear that SFSUD did not meet their goals and is intentionally spreading misinformation with data “showing” they’ve reached their goals. This is frustrating enough as it demonstrates a clear effort by supporters to confirm their own bias and manipulate the data to mislead the public. But Families for San Francisco also points out that new inequities were introduced since the overhaul of their math program (which, keep in mind, is a model for the new California math sequence). For example, by the end of tenth grade “Algebra 2 enrollments of Black and brown students have declined because most students cannot afford the costly work-arounds afforded by their white and Asian counterparts.” Read their report for a more detailed explanation of why, but essentially it comes down to parents that can afford a workaround will do so, and those that can’t likely won’t.

None of this has stopped advocates from pushing the narrative that de-tracking, at least in the approach that SFUSD took, is good for students. Jo Boaler, a researcher in math education, should be able to take a dispassionate look at the evidence and conclude that the experiment failed. But she seems unable or unwilling to do this.

Here’s a tweet from Boaler in 2018 pointing to SFUSD data.

Here’s another one.

For a more detailed argument from Boaler and other math education experts, here’s a piece entitled “Opinion: How one city got math right,” which concludes with the statement, “We congratulate San Francisco Unified on its wisdom in building math sequences that serve all students increasingly well.”

That piece is from 2018, but she doesn’t seem to have revised her opinion, at least not publicly. I’m subscribed to her newsletter and in one she sent out in August entitled “New Evidence Supports De-Tracking” she links to a recent paper by her and David Foster. Throughout this paper she cites her own work to justify claims. It doesn’t appear to be peer-reviewed and is not published in a journal but hosted on her Youcubed website. The research in that paper looks promising, but given everything I’ve laid out above it’s difficult for the average educator to tell if this is real evidence or if the goal posts have been moved. Would a 23-page report by an interested organization yield all the same problems as discovered by Families for San Francisco in their extended report analyzing the data from SFUSD?

I don’t know the answer to that. I do know, however, that this case study represents a problem endemic in education research. I’ve yet to go to a serious professional development session in which research couldn’t be found to support whatever intervention the organizer was promoting. When educators seek out research on different topics in education it is very difficult to find a consensus. Given that, as of 2014, less than 1% of education articles were replication studies and that “replications were significantly less likely to be successful when there was no overlap in authorship between the original and replicating articles,” educators, in my experience, are understandably skeptical of education research. Boaler and other researchers that agree with her on this give us a reason to maintain that skepticism.


No one is immune to confirmation bias. The longer you’ve maintained a viewpoint, I suspect, the harder it is to let go of that viewpoint. However, one of the antidotes to confirmation bias is surrounding yourself with honest people who hold a diversity of viewpoints. Surrounding yourself with people who agree with you, or who disagree but won’t point out the errors in your thinking, means you will remain in error. If you are a person with the ear of a great number of educators, that means those educators who listen to you uncritically will also remain in error. If you are rewriting a curriculum for a state whose standards have an outsized impact on standards adopted throughout the country, then your errors will have an outsized impact on math education in general.

Jonathon Haidt explains how easy it is for humans, and social scientists, to fall into the trap of motivated reasoning in this piece, “Why Universities Must Choose One Telos: Truth or Social Justice.”

A consistent finding about human reasoning: If we WANT to believe X, we ask ourselves: Can-I-Believe-It?”But when we DON’T want to believe a proposition, we ask: Must-I-Believe-It?”This holds for scholars too, with these results:

Scholarship undertaken to support a political agenda almost always “succeeds.”

A scholar rarely believes she was biased

Motivated scholarship often propagates pleasing falsehoods that cannot be removed from circulation, even after they are debunked.

Damage is contained if we can count on “institutionalized disconfirmation” – the certainty that other scholars, who do not share our motives, will do us the favor of trying to disconfirm our claims.

I’m not claiming that everything Boaler says is incorrect and I’m sure her intentions are good. As I mentioned above, she is indicative of a much wider problem. I’m saying that someone who either knowingly manipulates data or can’t see the error in her analysis of the data shouldn’t go on perpetuating those ideas without criticism from educators who care about math education. In most situations educators don’t have the time or expertise to truly evaluate the claims made by education researchers. In this case Families for San Francisco did the legwork and revealed that the emperor has no clothes.

I encourage math educators that read this to share their report widely, counter claims and proposed changes that are based on SFUSD data, and promote an environment in which leaders in education are concerned about truth and pursue it through viewpoint diversity. Since Boaler is a leader in progressive math education, it’s up to people who support her work to point out how her analysis is flawed. We must call on her to stop pushing for policies that don’t help, and may actually harm, students in the name of falsely vindicating her ideas. It may be easy to dismiss critique’s from the “other side” as they will always have critiques. It’s much harder to dismiss a careful critique from within one’s tribe.

If this ordeal shows us anything it’s that we must be careful who we valorize – and that we must keep our eyes open and be ready to criticize their ideas when appropriate.

Antiracism in your school: 9 ways to keep the conversation rational and unifying

A few years ago, I noticed that the words diversity, equity, and inclusion were steadily gaining in popularity, especially in K-12 education. At first, I couldn’t see a problem with the concepts. But as I dug deeper I discovered that much of the movement behind these words, although advanced by people with the best of intentions, was contradictory, illogical, and somewhat unethical. Take the defining of every action as either “racist” or “anti-racist, for example. It’s easy to find examples that seem to be neither, but more importantly this framing necessarily divides the staff of a school and narrows the set of ideas discussed, rather than diversifying it. A search for structural barriers to the learning of minority subgroups of students should be taken up by every district, but I’m deeply skeptical that the captivating ideas of the current moment give us the tools to identify and remove those barriers.

Although diversity, equity, and inclusion initiatives have been issues of focus in higher education for a while, the suite of ideas, which I term “the social justice suite of ideas”, such as antiracism and white fragility, skyrocketed in popularity in K-12 schooling shortly after the killing of George Floyd by the police. As the ideas began to gain traction in K-12 schools and school districts—beyond just living on Twitter—I decided to create guidance for K-12 administrators and school leaders, like has been done for university leadership and administrators, to ensure they can avoid the nearly inevitable pitfalls that come along with the social justice suite of ideas. 

I want to be clear about something at the outset—if there are structures in schools that create a disparity in outcomes between different groups of students then we should make an honest effort to understand them and to have a clear-headed discussion about altering that structure. But I reject the assertion that disparity is only caused by racism and the conclusion that the social justice suit of ideas is the only remedy for the disparity. 

Notably, there is no lite version of diversity, equity, and inclusion. It’s not as if you can do training for a year and be done with it. Because, according to the social justice suite of ideas, all inequities are structurally caused, the remedy is diversity, equity, and inclusion training, and if inequities persist, then you need to do more of the “work”. But the social justice suite of ideas asserts that inequities will always remain—advocates of these ideas proclaim that the work is never done, which points to one contradiction in these ideas. As Robin DiAngelo, a prominent figure advancing the social justice suite of ideas stated, “I will never be completely free of racism or finished with my learning”.  

There are ways for administrators and school leaders to ensure that this movement brings about positive change. First, understanding the language of this movement is important because many of the words and phrases used seem inherently good. (Who wouldn’t want to be “anti-racist”?) New Discourses has an encyclopedia of the terms used in the social justice suite of ideas. Then, when teachers, staff, and community members propose implementing the suite of social justice ideas in your school, I suggest following these nine steps:

  1. Define all the terms at the outset so everyone is clear as to what is being discussed, and then agree upon the terms. 
  2. Demand specificity in regards to the problem being addressed. If the charge is racism, continue to demand specificity. Do not accept the charge that we are all racist to some degree and the gap can be fixed by all of us interrogating our own racism. 
  3. Demand to see data that demonstrates there’s a problem. Make sure that the data presented is data that everyone agrees is meaningful. For example, someone might claim standardized tests are racist. A reasonable question might be, “if they’re racist then why should we care about disparities in their results?”
  4. Look for proxies that might get at the root of the problem or help more students. For example, if racial or ethnic disparities exist, are there other factors that correlate with the racial disparities, like poverty or childhood trauma? 
  5. When activities or professional development are suggested, demand to see research supporting the efficacy of the activity or training. Be careful here—there is a lot of pseudoscience masquerading as research. Make sure the research actually supports the professional development and is relevant to the data used to demonstrate the problem. Simply because the author has a Ph.D. does not mean the preceding text is high-quality research. 
  6. Do not let the advocates define you. The social justice suite of ideas has created a binary— you are either a racist or you are anti-racist; there is no place to stand in between. This relies on a redefinition of the word that makes the bar for “racism” so low that anyone can and will trip over it. If you resist  suggestions from the social justice suite of ideas in the ways that I mentioned above, you risk being called racist. Do not cede this linguistic territory. Begin compiling a list of all the ways you’ve reached out and supported minority students and communities. This almost certainly won’t be good enough―if there’s any kind of achievement gap across any metric then there must be racism causing it, according to the social justice suite of ideas. But it can be used to demonstrate to the broader community (within and outside of the school) that you are in fact working on supporting students from different backgrounds. 
  7. As a follow-up to number six, you might want to consider beating the advocates to the punch. What disparities might the group be aware of or will find? Point them out to the group and propose research-based ideas for how they might be remedied or how you are attempting to address them currently. 
  8. Be very careful with appeasement. Resist the thought, “well if I give them this then they should be happy”. Staunch advocates for these ideas won’t be happy until all subgroups are achieving at the same levels. As I mentioned above, the solution, in the minds of advocates, is to advance the social justice suite of ideas, even if those same ideas have so far failed to fix the problem. They will contend, there must be more racism amongst the staff and the staff must not be working hard enough to eradicate it—the staff must not be doing enough antiracist work.
  9. Draw lines. As you learn more about what advocates want, make sure that you draw lines that you won’t cross. A reasonable line might be that you won’t approve mandatory implicit bias training. As one of the early researchers on implicit bias stated, “mandatory (implicit bias) training has the potential for backlash”. The Implicit Association Test, on which the training’s justification rests, has plenty of staunch critics. They cite the low test-retest reliability, weak evidence that implicit bias leads to actual discrimination, and questionable methodology in early meta-analyses. When drawing lines, be sure to engage with a diverse set of perspectives. 

If structures in your district hold back certain groups of students then you should work to remedy or eliminate those structures. But you should do this in a rigorous, scientific, and careful way. Book studies of “How to be an Antiracist” or “White Fragility”, diversity training, or implicit bias training will not provide solutions to the problems. 

While listening to Mike Strambler, professor of psychiatry at the Yale School of Medicine, in a roundtable discussion put on by Heterodox Academy’s HxK-12Education Community, he made a point that resonated with me. While discussing the problems with the social justice suite of ideas as they relate to K-12 education, he pointed out that no one in education wants there to be massive disparities between different groups of students, and that rich discussion can come from trying to eliminate structures that lead to those disparities. But the way that happens most effectively, he suggests, is by doing the following: defining the goals at the outset, identifying the metrics that will be used to evaluate those goals, and settling on methods that arise from bringing in a diverse set of viewpoints into the discussion. An effective solution is more likely with this approach (and division amongst staff less likely), than one grounded exclusively in the social justice suite of ideas. (You can see the beginning of that discussion here).

Further Reading:

You’ve been mandated to do ineffective training. Now what?

Diversity, Equity, and Inclusion in K-12 Professional Development: The Mission Versus the Reality

Responding Constructively to Mandated Diversity Trainings

We’re Teaching Slope Fields at the Wrong Time

In our textbook slope fields come during the differential equations unit, which for the last 8 years made sense to me. But every year there were groans from students and comments about how “pointless” they are.

Well, here’s why students think they’re pointless.

They already know antiderivatives. So they can take many differential equations and find the family of functions whose derivative is given. Many of the basic slope field problems can be somewhat easily antiderived, especially once students know about separation of variables.

“But, wait. Not all differential equations can be antiderived!”

This is true. And I point this out to students. They don’t really seem to care when they have to plot 25 line segments on a sheet of paper. For one problem. Especially once I show them how easily a computer can plot slope fields. I also think it’s in part because they don’t connect to very much. We just do the section and move on.

So I’ve been thinking about this for a few weeks, on and off, trying to figure out how to motivate the lesson. Here’s what I’ve come up with.

When I teach a derivative rule (the power rule for example), the next day I teach the antiderivative. It’s fairly easy for students to follow and then remember, because they just learned the derivative the previous day. Antiderivatives are just the process of undoing a derivative. This is as opposed to how my textbook does it, which is save all antiderivatives for the integration chapter. Anyway, instead of doing that, I think I’m going to show slope fields in between. In other words, I’ll teach a derivative, then use a related differential equation to get students to think about what a “slope equation” means and plot it, and THEN teach the antiderivative rule.

It would go something like this:

  1. Teach the power rule.
  2. The next day give students something like dy/dx=x – 1 and ask them to think of it as a “slope” equation. Inputs are whatever you want, outputs are slopes. Ask, “What will this look like?” Help students work out a sketch of the slope field.
  3. Then teach the antiderivative rule.


There will certainly be some students that figure out the antiderivative rule prior to or during work on the slope field(s). But at least this provides more of a motivated use for a slope field:

“What does the function whose derivative is ______________ look like? Well, we have this slope equation and it gives us the slopes of tangent lines at whatever point we want. Let’s use that to build a graph that gives us an idea of what that function (or family of functions) looks like.”

You could use this throughout the units on derivatives, including things like implicit differentiation. For instance, “We know how to derive x^2+y^2=9. But what if you’re given something like dy/dx=2x/y? What would that family of curves look like?” And then you can use a slope field.

It’s possible that this isn’t a great idea. Maybe it’s better to just leave slope fields as their own thing, drag the students through them, and move on. But I do think it would help students understand that a differential equation is a “slope equation”, and that it’s useful. I also think it might help students better understand what a “solution” to a differential equation is and how we might visualize it. Last I can imagine students would come to the “differential equations” unit already fairly comfortable with a lot of the concepts in it.

Any thoughts on this? Drop them in the comments!

Update: The first activity inspired by this post is located here.

Another update: The second activity, involving trig functions, is here.

“Why do we need this?”

My students in precalculus class have asked a few different times over the last couple of weeks, “When will I need this?”. I encourage them to ask this question of their teachers, because the teacher should be able to answer it. But when it comes up in the middle of a lesson I don’t really have time to answer it fully, so here are my thoughts, in a thought out format.

  • The blunt answer is that this is an elective and no one is making you take this. And if someone is making you take it (your parents, for example) then I can’t help you. I wouldn’t be satisfied with this answer if I was you, but it is true.
  • We don’t know and you don’t know what you’ll end up doing, so it’s better to err on the side of a broad knowledge base in high school.
  • These are things you need to understand if you’re going to take more math. Why might you take more math?
    • You might want to study math
    • You might want to go into a math related field
    • You might want to go into a science related field where calculus, at a minimum, will likely be required
    • You might want to go into pre-med or pre-vet (or pre-law?), where you’ll likely have to pass a calculus class
    • That’s what this class is designed for. That’s it’s stated purpose. It’s to get you ready for calculus, regardless of the reason you might end up taking calculus.
  • I admit, math is also a gatekeeper. It’s a way to check your level of competence. I’m not making a statement about whether that is right or wrong, but it is true. See a great exploration of this idea in Bad Drawings here.
  • But more broadly, math is giving you a language to describe and understand that world. Sinusoidal behavior (which is what we’re studying now) is everywhere. Anything that works on circular motion or has periodic behavior can be described by sine or cosine waves.
  • It also helps you exercise your rational thinking and pattern finding muscles. We don’t always do a great job of helping you work the pattern finding muscles, but the reasoning muscles are exercised every day. When you solve a problem you are reasoning. You put together a chain of logical statements, either explicitly or implicitly, to arrive at a conclusion. And those statements aren’t the same for each person, which is a great thing. There’s often more than one logical way to arrive at the same conclusion. But this mindset, of taking on assumptions and arriving at logical conclusions based on those assumptions is a vital component of thinking critically. It’s how we persuade other people. It’s how we make progress. It’s how we call BS on people when they make claims or statements that don’t make sense.
  • Mathematics, and the different parts of mathematics, can be viewed as games of sorts. You are given some rules (axioms, theorems, etc.) and then you see what is true given those axioms. You see how far you can push the rules and how far you can get in the game. But they’re often better than an arbitrary game. Many times they have implications in reality (although this isn’t a necessary condition for interesting mathematics and is not necessarily where mathematicians spend all their time). And sometimes those implications aren’t seen at the time, but turn out to be useful later on in science. A mathematical truth is a truth about reality itself.

The fact that we attach grades to mathematics does minimize some of this. It’s not a fun game to play if there are real consequences (in terms of grades) for not understanding the rules quickly enough. And I get that. But I reject the notion that we should throw the whole project out the window because what you’re learning on a Tuesday in precalculus might not help you navigate your grocery shopping trip.

Is a classroom really a laboratory?

It’s hard for me to track back and know whether I heard this somewhere or whether it just popped into my head, but I imagine the former. It usually comes in the form of a thought I think I should tweet.

“Your classroom is your laboratory.”

But then there’s a secondary thought that immediately follows.

“Maybe, but not for everybody.”

Many teachers don’t view their classroom as a place to experiment. I think there are a couple reasons for this. First, the accountability movement. Putting more pressure on teachers by putting more weight on how their students do on standardized tests and other accountability rules means teachers are less likely to take risks. They’re more likely to implement the district approved, research based lesson plan, over and over. This is because, if something doesn’t go well, then they are “covered”. In other words they failed but they failed trying something that the people “above” and “around” them said should work. This is fundamentally different than trying something that you developed on your own.

It’s playing it safe.

Intimately tied to the previous reason, the second is a fear of risk taking. There is a fear that if you take a risk and fail then you’re somehow a worse teacher.

Nope, you’re a better teacher because you learned something valuable about educating young people.

The students might know you made a mistake (gasp). Your administrator might know you made a mistake (GASP). But this is what education is about. You are a professional, you should be treated as such, and you should be able to take calculated risks in your classroom without fear of what might happen if things don’t go as planned.

In fact, the more we design our lessons to go as planned, the more we fail to adjust for the learning. It means that I’m dictating the pace, flow, and motion of the class to the point that I’m (likely) not being responsive to students needs. My time spent with my students is rigid and doesn’t flex based on their understanding of the concepts. Everyone has to decide where they’re going to land on the “rigid to flexible” scale but always erring to the rigid means less responding to student’s needs.

And I want to be clear. Some of you may be thinking, “look, you can’t just try a bunch of stuff on a whim.”

To which I say, of course not! Scientists don’t do this either. I’m not suggesting that teachers shouldn’t use research to inform their practice. I’m just saying that there are effective learning tasks that haven’t been developed yet and you might be just the person to develop one (or two, or ten, or a hundred…). Helping a student to understand a concept or idea is a complex task for a lot of reasons. This means that every lesson plan needs your mojo. It needs your creativity. It needs your flexibility. It needs to be a mash up of research, your knowledge, your creativity, your students’ knowledge, your gut, and a plethora of other factors.

pexels-photoIn that mashup it’s not unlikely that some aspect is going to flop. We need to be okay with that and learn from it, because there will also be days when you hit a home run. And there are few greater feelings in this profession than designing a creative lesson that just kills it. If you stay in the box all the time it’s difficult to capture that feeling.
Teaching is a creative endeavor. It has to be. But if we don’t work in an environment where risk-taking is encouraged and valued then it’s difficult to grow, and get better.

A bit of Coercion is Okay

Let me start by establishing that I recognize the problems with grades. I understand that they encourage learning to be competitive, often don’t provide a great window into students’ understanding of specific concepts, and cause kids to only worry about the number and not the learning. So if you’re in that camp, I’m Right there with you. I get it. However, I’ve still got a foot in the other camp. Let me explain.

In an ideal world students would be driven to learn by their own curiosity and passion for learning. This ideal is unrealistic. It certainly happens in bubbles and in spurts, but generally speaking, much more learning in school is being done because we require students to do it. This is not to say that we shouldn’t constantly be trying to tap into their curiosity and to intrinsically motivate them. I’m just saying few students are intrinsically motivated to learn polynomial long division or stoichiometry, for example. So, as Tony Robins pointed out in his TED talk “Why We Do What We Do”, motivation is a balance between the intrinsic and the extrinsic. If we go too far in either direction we’ll likely fail to get the learning outcomes we desire.

Following this line of reasoning I sought to solve the problem of low engagement with the videos that I have in my flipped classroom. I badly wanted my students to seek out the information in the videos because they realized they were an important part of understanding the math. However, I could tell many students either weren’t watching the videos, weren’t watching them with engagement, or were making an honest effort but still weren’t getting the most out of them. I’ve toyed with adding some sort of quizzing feature in my videos before, and the last couple of years have used the WSQ model, but I hadn’t done it wholesale. I decided that with the start of the new trimester I would have each of my students sign up for EDpuzzle and I’d just simply make them watch the videos and answer questions along the way (by giving them a small grade for doing it). Also, they would be due by certain dates. I told this to my students on the first day and waited for the complaining as the trimester rolled on.

But they loved it.

After a couple of days and a couple of videos the students, in both hours, commented on how they liked it better than the previous way. They -liked- having the questions in the video (they said it helped them stay focused). They appreciated the structure that the due dates gave them. I had students that never watched or took notes on videos before, asking me questions while they were watching them. Even though I am now taking a grade for the videos and the quizzes within them, my students are happier because of the change in format and I’m happier because students are getting more from the instruction.

The moral of this story is, if you’re struggling to get students to do something via intrinsic motivation, a little extrinsic motivation probably isn’t going to hurt. Especially if the result is more learning and engagement.

-Note: I thought about the fact that in life there is rarely an “EDpuzzle program” and a teacher making sure you participate in everything you’re asked to participate in or that will benefit you. However, I think the learning and engagement that I’m getting outweighs the life lesson that I’m losing. I’m sure there are differing opinions on this and I’d love to hear your thoughts in the comments.

Relation to Key Topics in Ed Tech (Deeper Learning through Technology)

In this post I’d like to lay out a couple ways that my grant engages with current topics in educational technology. My grant addresses a major topic and that is 1-1 device initiatives. Schools and districts all over the state are starting to give devices to each of their students to use in class. One of the problems with these initiatives is that they are often top down and the result is poor implementation. Even though devices can be used to increase deep understanding, they are frequently implemented ineffectively. My grant proposal provides several ways in which students will leverage the technology in effective ways.

Although I wasn’t explicit about this in my outline or write up, one other issue in ed tech that I’ll be addressing if I get the devices is the balance between high tech and low tech tools. I have had success with students using collaborative whiteboards for example and tablets cannot replace those. When considering which technology to implement it’s important to not focus only on high tech tools.

Last, the entire overarching goal of this technology is to get students to think deeper about mathematics. To me this is what we should be trying to leverage technology for in all disciplines. Technology has great potential to be a catalyst for critical thinking if it’s implemented carefully. A side effect of effectively implementing these technologies is that ideally they give students ideas for how they can support their learning in the future. We want students that are digitally literate and that can leverage technology to become effective lifelong learners.

Proposal Evaluation (Deeper Learning through Technology)

In this post I’d like to briefly elaborate on how I’ll evaluate the successfulness of my implementation of tablets for every student. (Read the outline of my plan here and the details here.) There are a few simple ways to measure the effectiveness of the devices. The first, and simplest, will be looking at student’s grades on a year to year basis. For instance, were the averages of my calculus classes worse or better in the year that I implemented tablets compared to previous years? I can also look at how specific students did in my class compared to their previous math classes. But beyond that, to really drive at my original goal (helping students consistently reach a deep level of understanding in mathematics), I’ll have to look beyond their grades.

The first method I’ll use will be their blogs. I’ve had students blog for three years, with some of the prompts being reused. I will be able to look back and compare blog posts on specific prompts. Students’ writing is one of the best places to understand how deeply they understand a concept. I also will gather students’ perceptions of how they felt the tablets helped them understand math. They will have several years of experience in learning math more traditionally to compare my class to. In addition to surveying students during my class, I will follow up with students after they leave high school. One of the problems I identified was that many students struggle in mathematics beyond high school. I will survey students about their success in college level math and ask them how they felt my integration of technology contributed or detracted from that success (or lack of success).

I think it’s also valuable to videotape my classroom on a somewhat frequent basis to both evaluate my teaching and my students’ learning. The first thing I’ll be looking for is growth in a number of areas. How are my students improving in the quality of their discourse around mathematics in my class? How frequently are students using graphing utilities on the tablets without being prompted to use them by me? How is my instruction evolving based on the increased formative data I’m getting? The process of videotaping will require reflection on the tablets’ effectiveness and I think this is best done through writing. I will blog on the success and failures of their implementation and adjust accordingly my teaching accordingly.

I think the advantage to implementing tablets is that they are such powerful, dynamic tools. This is advantageous because if I have an idea that doesn’t go well (maybe having students make video lessons for example) there are countless more ways to use them, so I can dump that activity and use them in different ways. With the combination of hard data (grades) and more subjective data (like blogs and surveys) I will be able to decide if the technology is effective or not and make adjustments accordingly.  

The Total PACKage (Deeper Learning Through Technology)

In this post I’d like to specifically lay out how pedagogy, technology, and content will be balanced in the context of my current classroom while utilizing this technology (tablets for every student). You can read the basic outline of my plan here, so I’ll get right into the details. My ultimate goal is for students to leave my classroom as deeper mathematical thinkers and I didn’t take the decision to write a grant for tablets lightly. I considered what is required for students to understand concepts deeply and settled on a few necessities: writing, exploration and play, and feedback. There are countless ways to achieve these in the context of a math classroom, the problem lies in accomplishing them on a regular basis. Tablets provide a means by which we can accomplish these aspects of learning as frequently as possible (or as frequently as pedagogically sound).


When I first settled on these three aspects of learning as a focus for this grant I began searching for what other math teachers do to increase these things in their classroom. The first thing I noticed was the emphasis on providing lot’s of collaborative space (which essentially meant whiteboards) as well as having a student centered environment. I currently have my classroom organized into pods and each pod has a collaborative whiteboard on it. In addition I have large whiteboards on three of my walls. I currently have a fairly student centered, collaboration focused classroom. In addition, I teach higher level math where much of the content cannot be accessed by simple “hands-on” manipulatives that are common in lower mathematics. This is to say that I considered “low-tech” options but have either covered them (collaborative space) or don’t need them (hands-on manipulatives).

I then looked to “high-tech” options. I decided that most of the technology I looked at would be most effective in the hands of my students. For instance, if I only had one tablet in my classroom I might be able to show a demonstration with it, but students wouldn’t be able to explore the mathematics on their own. Once I decided on devices, I needed to choose between tablets or laptops. I decided to go with tablets for two main reasons. First, they are better for creating things that require a camera (video projects, interesting image projects, etc.). Second, they are much easier to draw or write on.This comes up frequently in mathematics and is especially handy with some formative assessment tools (Nearpod for example). This often much more intuitive for students than using an equation editor. As outlined below, the positive changes these devices can bring about are many.


I currently have my students maintain blogs to increase writing in my class. On days in which there is a blog post assignment I must reserve a computer lab. This usually is not a problem but it requires breaking up whatever we are doing in class to go to the lab. Students usually don’t finish the writing at the same time so students that finish early are frequently left with not much to do. I want writing to be more of a “regular” thing in class and less of a field trip. I think it would send the message that writing is simply an important part of the learning process. Having a class in which each student has a device means that the flow of class doesn’t need to be interrupted for writing and students can do the assignments as soon as they reach that point in the lesson.

Tablets also provide students with powerful tools for exploring higher level mathematics. Software like Desmos, Geogebra, and Wolfram Alpha demonstrations allow students to easily play with math and make difficult concepts more accessible. Pedagogically speaking it makes it easier for a teacher to allow students time to engage with concepts prior to formal instruction. This gives students a chance to actively engage their prior knowledge with the new ideas (Bransford, 1999). They can begin to construct understanding of concepts and then the teacher can follow up with a more formal lesson to help the student make the final connections. The more frequently this happens, the more the student owns the learning.

In addition to exploration and play, historically many higher math concepts (slope fields, solids of revolutions, complex graphs, etc.) were only accessible through sketching graphs (by hand) or through graphing calculators. Both have severe limitations in their capabilities compared to the aforementioned utilities. (For instance, see the activities on composition and operations on functions located here. This is a concept that’s very difficult to see purpose in, but in creatively using Desmos it becomes one of the more engaging topics in the unit.) With these new tools students can access content that historically would’ve seemed mysterious. Granted, I could take students to the computer lab whenever I wanted them to use these tools, but the situation where these tools are more beneficial than drawing or graphing calculators comes up so frequently in higher math that it’s not reasonable to do so. Not only does this technology make these tools consistently available to students, but it also can help make my pedagogical decisions more in line with the constructivist philosophy of learning.


In order to decide how to teach a concept (or especially re-teach a concept) it’s helpful to have an understanding of your students’ understanding of a given concept (or group of concepts). To do this teachers are frequently formatively assessing their students. The more frequently we formative assess the better idea we have what students know. One problem with it is that much of the time it’s based on our “feel” for their understanding. For instance, we might use questioning frequently to get an idea of their preconceptions or misconceptions. The problems with this are that it doesn’t give us specific data about our entire class, we often overestimate our students’ understanding of prerequisite skills, and it’s difficult to narrow general understandings down to specific concepts. There are countless tools designed for tablets that give very specific formative assessments to students and provide valuable insights into their learning in real time. These assessments can happen frequently and drive instruction almost on a minute to minute basis is every student has a device.

When trying to design tasks in which students construct knowledge we must do our best to gain insight into how far they’re coming in their understanding. The better our understanding of their understanding, the more effective each class period is for students. In addition, this gives the students an understanding of their weaknesses and in preparation for their summative assessments they will know which concepts to focus on. Lastly, this allows students to develop a good study habit. Instead of just reading about concepts students should use self quizzing to reveal misunderstandings and force themselves to try to recall information or concepts. This forces them to honestly evaluate which ideas are stored in long term memory and which ones they merely thought were stored there.

In the context of my current classroom and teaching assignment, tablets can make the three main areas of teaching (technology, pedagogy, and content) blend together well. We can dig into content more easily and deeper than before. We can create more situations in which students play with mathematics as an entry point to lessons. We can assess in low stakes situations more frequently and effectively. Students can write on a regular basis. Tablets, implemented effectively, give my students countless more opportunities to understand mathematical ideas more deeply.


Bransford, J. (1999). How people learn brain, mind, experience, and school. Washington, D.C.: National Academy Press.