I recently finished reading Mathematical Mindsets by Jo Boaler, a professor of mathematics education at Stanford University (although she incorrectly represents herself as a “mathematics professor” on p. 38) I had previously read only bits and pieces of Boaler’s work, until this book — which has been widely adopted by local school districts, and has become mandatory reading — showed up in my school mailbox over the summer.
I have to say that I find it somewhat troubling, the degree to which Boaler and her book/philosophies have been widely and almost universally accepted at face value around here. It’s not that I 100% disagree with Boaler, nor some of the activities and methodologies she claims are beneficial. Unlike the claims made by Boaler, it is not so black-and-white — there are some things of value in this book, and in some of the insights and potential learning activities that Boaler shares with us.
However, what I find troubling is:
(a) that there is an “all-or-nothing” approach (once again) to education, rather than recognizing that education is a delicate balancing act, and that any one-sided approach (such as the group-based-inquiry one espoused by Boaler) is not going to be highly effective, especially not when you consider that students and their learning styles are diverse;
(b) people are blindly accepting the claims and purported benefits at face value, without any sort of further research or critical analysis which reveals that there are actually numerous fallacies in the claims and conclusions Boaler makes, and that some of what she recommends is actually not only ineffective, but would actually be detrimental to students and their future prospects.
Indeed, it’s not so much that I disagree with Boaler’s statements; it’s that science and reality do not agree with many of Boaler’s claims. Problems include: misleading data manipulation; cherry-picked and insufficient research, much of which is actually antithetical to what the overall corpus of research has shown; and evident personal biases, in relation to personal/emotional anecdotes, as well as unsubstantiated opinions about what types/styles of math are “important” or “real.” As such, this book reads more like proselytizing instead of pedagogy; the conclusions made are more sales pitch than sound science.
Let me elaborate, fact-check style:
Many Problems & Falsehoods
Misleading Data Visualization
In numerous instances, Dr. Boaler uses figures known as “truncated bar graphs” in which the Y-axis has been truncated; ie. the scale does not start at zero. This is a textbook example of “misleading graphs” and, in fact, the curriculum I used last year specifically taught 6th grade students that such graphs are a misrepresentation of data generally intended not to enlighten but to obfuscate the facts. It must be noted that there are some occasions in which truncation is acceptable; this, however, is not one of them — these graphs would fall under the category of “how to lie with data visualization.” In fact, if these graphs were used in advertising, they would be labeled false advertising and would face consequences by the National Advertising Division of the Better Business Bureau. If misleading graphs are not allowed in advertising, then they certainly should not be allowed in the realm of research and academia, in books or reports that could have widespread consequences in education policy.
NOTE: I generally agree with the statements Boaler makes with these graphs, that “fixed mindset” is highly detrimental, and that conceptualization is a more important skill/process for math than memorization is… but I don’t approve of misrepresenting data in order to try to prove these points. Surprisingly, the data above (when represented truthfully) actually tells a different narrative: “Memorization is almost as effective as understanding Big Ideas & Connections.” What the data actually convinces me of is that memorization does have some value. Thus — as I stated earlier — a hybrid approach or partial adoption of conceptual activities like the ones in Mathematical Mindsets would be better than an all-out replacement of the “traditional” methods. Perhaps what needs to be memorized are not discreet “facts,” as Boaler claims math teachers try to enforce, but rather memorization of strategies, which is actually what most math curricula — even old-school ones — actually teach.)
Personal Anecdotes & confirmation bias
As passionate as I am about math, the English teacher in me continually cries “cite source?” as I read Mathematical Mindsets.
There is a surprising dearth of actual research being cited in this book (aside from the 22 references to Boaler’s own work.) Perhaps this is because many of the claims (aka personal beliefs/opinions) being made by Boaler simply aren’t supported by the research — in fact, many of them are antithetical to what the research actually shows! Perhaps that is why there is a high amount of unsupported opinions, assumptions, personal anecdotes, and “references” to things like blog posts (which are filled with even more anecdotes and opinions, rather than any sort of data or scientific research) in lieu of research studies.
There are at least 18 occurrences of Boaler sharing instances from her daughter and the ways her school/district were doing things “wrong.” Examples: “My own daughter was given very damaging messages by teachers…” (p. 17); “When my own daughter started times table memorization and testing at age five, she started to come home and cry about math. This is not the emotion we want students to associate with mathematics…” (p. 38)
These types of stories are a persuasive tactic called an “emotional appeal.” It is an effective technique to try to convince people to agree with you — for example, the reader may commiserate: “Wow! I, too, have a child who doesn’t like homework! So it must be bad!” What this really represents is a form of confirmation bias, and as such these anecdotes are not convincing in a scientific or research-based context.
False & Unsupported Claims
Claim: “Over the years, school mathematics has become more and more disconnected from the mathematics that mathematicians use and the mathematics of life. Students spend thousands of hours in classrooms learning sets of procedures and rules that they will never use, in their lives or in their work.” (p. 27)
Rating: UNSUPPORTED (CITE SOURCE?)
Before addressing specific the specific minutiae and incorrect claims made in Mathematical Mindsets, we must first examine Boaler’s personal bias about what math actually is. According to Boaler, schools do not teach “real mathematics” — yet never defines what “real mathematics” is. She states that “mathematics, real mathematics, is a subject full of uncertainty; it is about explorations, conjectures, and interpretations, not definitive answers.” (p. 21) Yet there are many, many professions in which we use math to solve problems and there are, in fact, “correct solutions” to those problems: carpenters need accurately-cut lengths of wood; bakers and chefs need precise proportion and scale of ingredients; mechanical engineers need precise gear ratios; etc.
“At its core, mathematics is about patterns. We can lay a mathematical lens upon the world, and when we do, we see patterns everywhere; and it is through our understanding of the patterns, developed through mathematical study, that new and powerful knowledge is created.” (p. 21)
True. The way I have explained it to people is this: “Language is the words we use to describe humankind; math is the language we use to describe the universe.”
However, we could also say “at its core, language is about symbols.” But, just because words are symbolic, doesn’t mean that’s the end goal… we don’t want to have students just identify symbols (or patterns)… we want them use them for some sort of practical purpose. The reductive definition of mathematics that Boaler uses defeats the entire purpose of why math was invented in the first place: math, like language, was invented to solve pragmatic problems. Math was never intended to be a form of navel-gazing or dreamy philosophy, as portrayed by Boaler; the earliest civilizations developed number systems, weights and measures, etc. to record transactions, make commerce more accurate and efficient, predict the weather and other agricultural phenomena, and myriad other practical purposes. In other words, I would argue that “real mathematics” has to do with solving problems in the real world. This is quite the opposite of what Boaler thinks is “real math.” What Boaler calls “real mathematics” is actually known as theoretical mathematics.
Thus, there is a serious disconnect between what is happening in schools and what Boaler thinks should happen in schools. Students learn sets of rules and procedures that they will use — when shopping, cooking, or in countless professions. Boaler talks a lot about “mathematicians,” but my goal as a teacher is not necessarily to create “mathematicians” like the names she mentions. My goal is to prepare students for the real world, including the applied math that goes into a whole variety of jobs, and required for everyday life.
According to the Bureau of Labor Statistics, there are only 3,500 mathematicians in the United States. Compare this to the number of occupations that require use of math skills — often including getting correct answers, and usually within a specific time limit: 24,500 actuaries; 30,000 statisticians; 72,400 aerospace engineers; 98,400 chemists; 316,000 electrical engineers; 328,600 computer programmers; and over 1 million other jobs that require constant calculations (which Boaler would argue could just as well be done with a calculator, but tell that to a baker with flour and dough all over his hands, or a carpenter already carrying various tools, and needing to perform cuts and measurements all day long.)
“(1) Joe can do a job in 6 hours and Charlie can do the same job in 5 hours. What part of the job can they finish by working together for 2 hours? (2) A restaurant charges $2.50 for 1/8 of a quiche. How much does a whole quiche cost? These questions all come from published textbooks and are typical of the questions students work with in math class. But they are all nonsensical. Everyone knows that people work at a different rate together than when alone, restaurants charge a different proportional price for food that is sold in bulk…” (p. 192) [No, not “everyone” knows these things, and they are not even necessarily true. Cite source?]
Yes, math is important, but “real mathematics” is not just about creativity and beauty and patterns and making new discoveries. It’s about solving problems.
Claim: “research has consistently found homework to either negatively affect or not affect achievement” (p. 107)
When it comes to talking about homework, Boaler is surprisingly regressive; for someone that preaches “innovation” and “reform” and “progress”, there is an overwhelming focus on “old-school” homework — worksheets, isolated lists of practice problems, etc. — without even acknowledging that newer forms of homework and tools that have been around for decades now (such as the personal computer) render those problems obsolete.
Once again, Boaler resorts to personal anecdotes / emotional appeals (“As a parent, I know that homework is the most common source of tears in our house, and the subject that is most stressful at home is math” p. 46), as well as citing a couple other sources… so let’s take a look at those scant few sources of “support” for Boaler’s clearly biased belief:
1) Boaler cites Alfie Kohn’s blog. For those of you who don’t know, Kohn has made a career out of being a sensationalist contrarian — an educational demagogue, if you will, who preys upon the things people “don’t like” (like testing and homework) and then constructs a false narrative to appease their confirmation bias. In other words, he tells people what they want to hear, without actually backing those claims up with facts and empirical evidence.
“Is it really possible to completely eliminate homework – or at least to assign it rarely, only when it’s truly needed – even in high school? We keep hearing from educators who say it’s not only possible but preferable to do so” the website boasts. And yet… despite all of the anecdotes, there is zero actual data or evidence on this blog page to say how it actually affected student learning, such as would be measured by assessments. (oh, but Kohn doesn’t believe in the use of any sort of measuring tools… he thinks high test scores are actually bad: “This assessment is borne out by research finding a statistical association between high scores on standardized tests and relatively shallow thinking.” and that “Measurable outcomes may be the least significant results of learning.” How convenient! How can you prove anything scientifically or make any sort of conclusions or data-driven decisions without measurement tools of some sort?)
Educators and policy-makers should run, not walk, from anything Alfie Kohn has to say. It is populist — yet almost entirely unscientific — garbage.
2) Boaler cites a white paper by “Challenge Success” (how convenient… yet another Stanford affiliation!), a group whose bias is evident up-front: “At Challenge Success, we believe that our society has become too focused on grades, test scores and performance.”
Ironically, though, despite the spin that they try to put on body of research, even their own white paper admits the same thing that I highlighted in my previous post, “The Million-Dollar Question: Is Homework Worth It?“:
“Harris Cooper (1989, 2001, & 2007) has reviewed hundreds of homework studies and is often thought to be the leading researcher on homework…there is almost no correlation for students in elementary school between the amount of time spent on homework and student achievement. In middle school, there is a moderate correlation, but, after 60-90 minutes spent on homework, this
association fades. The authors found a correlation in high school, but this also fades after two hours spent on homework.” (pp. 3-4)
This is quite different than Boaler’s assertion — what should actually be said is that there may be little to no academic effect in elementary school, but that large studies have shown moderate to significant correlation between homework and academic achievement in middle school and high school. Yes, there is such a thing as “too much” homework, but the evidence is clear: zero practice (homework) — which is what Boaler promotes — would be detrimental to students’ academic success. To state that “research has consistently found” that homework is negative or neutral is, quite frankly, wrong.
Here’s another interesting finding from Boaler’s own cited source (Challenge Success) which contradicts Boaler’s message that, if homework is given, it should not be graded: “In one study of low income ninth and tenth graders, the authors found that when students were given homework but had few consequences for not completing it, students showed an increase in disengagement from school. (Bempechat et al., 2011).” This research actually shows the negative results of Boaler’s stance of not grading homework (if it “has to be” given.)
My own experiences in the classroom have shown homework to be very valuable, when it was meaningful, relevant, given value, and enforced. My homeroom students who completed the assigned homework last year did well when tested; those who did not do homework suffered academically, as shown by the assessments. Note: Yes, I did grade the homework (but only at 10% weight of grade; this assigns the homework “value” as Bempechat et al. study suggests, but allows overall grades to be based on more relevant assessments.)
Claim: “teachers should abandon testing and grading” (p. 17)
Boaler’s reasoning behind doing away with both tests and grading basically boils down to one thing: they induce stress or anxiety. Boaler makes references to “stress” 16 times in this book, and “anxiety” 18 times. The underlying premise for most of her recommendations — that homework, tests, and grading should be eliminated — could be summarized as this: These things are stressful, and stress impairs learning and induces negative associations, so we should eliminate all of the sources of stress.
I think most people could agree that stress and anxiety are no fun. Raise your hand if you like stress. I sure don’t. But that doesn’t mean (a) that it is beneficial to entirely eliminate these scenarios; nor (b) that the situations described by Boaler have to be “stressful” or “anxiety-inducing”.
“When students are stressed, such as when they are taking math questions under time pressure, the working memory becomes blocked, and students cannot access math facts they know (Beilock, 2011).” (p. 38) This is true — it has been found that stress impairs cognitive functioning. The part that is unsupported, however, is the claim that timed tests are automatically stressful. (Whom does Boaler cite to support this claim? Herself.)
Likewise, Boaler claims “Grading reduces the achievement of students.” (p. 142), citing a few studies relating grading vs. formative feedback and the effect on motivation and performance. Most of these studies are from the 1980s, two of them being seminal works by Ruth Butler, as well as a more recent one by Pulfrey, Buchs, and Butera (2011.) That study replicated findings of Butler’s 1988 study, which essentially says this: “being graded can cause anxiety.” Yes, it has been shown that the mere awareness that you are being evaluated can cause stress or anxiety. And, as noted above, anxiety causes decreased cognitive performance, so it is not a surprise that — especially if they receive low marks — students become demotivated and that graded assessment would thus impair learning. (Boaler doesn’t mention that the findings actually showed that, if students scored high marks, it had a motivational and positive effect on further learning and perseverance.)
However, to propose complete elimination of an assessment or evaluation system is unrealistic, and — even if such a paradigm shift were to be made in public schools — students would find themselves ill-prepared to face the realities beyond the classroom. The fact of the matter is that students will be evaluated, rated, and ranked in life. It is inevitable, and I would argue that performance evaluations and ranking systems are actually necessary in society, to ensure that we have the “best person for the job” in many different situations. Whether it is at a job interview, obtaining a driver’s license, passing a bar exam, or a performance evaluation at work… they will be evaluated and, in most cases, rated (not just given feedback), even if it is as simple as “pass/fail” or “hired/not hired.” So, the premise that we can simply “do away with” grades or evaluation is an unrealistic one, and to do so in schools would be damaging, as it would not prepare students for the realities outside of the classroom.
It is ironic that Carol Dweck (author of Mindset: The New Psychology of Success) wrote the foreword to Boaler’s book, because Boaler completely exhibits a fixed mindset, not a growth mindset when she claims that we should simply avoid doing anything that could induce stress or anxiety. In a fixed mindset, you avoid challenging situations because they might lead to difficulty — and this is exactly what Boaler proposes we should do!
Tests have actually been found to be a powerful learning tool. At a “Learning and the Brain” conference in San Francisco a few years ago, I watched Dr. Robert Bjork present his findings on learning and memory gleaned through many years of research as a cognitive psychologist at UCLA. A few important elements from the cognitive science research refute Boaler’s claim that (a) tests should be eliminated, and (b) that anything stressful should be eliminated. First of all, studies have shown that testing can be a very powerful learning tool — that’s right, not just an assessment tool, but one that results in formation of long-term memory (ie. learning):
Dr. Bjork has also extensively researched “desirable difficulties” — these are interventions that cause difficulty, and thus can induce some level of stress and slow down learning… but they actually result in better long-term retention. [To be fair, Boaler agrees with some of these strategies, or similar ones, such as noting the problem that “most practice examples give the most simplified and disconnected version of the method to be practiced” (p. 42); ie. not presenting things with more open-ended complexity and diversity to more fully represent a concept.]
To try to simply eliminate any source of stress is actually quite detrimental to the student, because the fact of the matter is that they will, at some point, encounter those situations, whether it is tests or it is having to perform accurately in a time-limited situation. As much as we may not like the anxiety that it can induce, the reality is that such stressful situations are part of life, and it is better to prepare students to deal with them, than to simply pretend like they will never exist. Examples: standardized tests such as ACT and SAT are currently required for college entrance; after college, job interview processes often involve performance tests — especially for STEM careers. (In the many interviews I had for software engineering positions, I was tested in all of them. In order to get the job, I had to prove that I had the knowledge to perform a task… and, yes, there was a time limit! And, yes, there were “right and wrong” answers!)
In essence, Boaler says that, because timed tests induce fear/anxiety (and, thus, reduced cognitive function), we should eliminate them. This is actually harmful to a student’s future, because that future will involve being tested, assessed, or evaluated! Boaler’s approach is the exact opposite of what cognitive behavioral psychology tells us we should do to overcome phobias.
It is not “stressful” situations that are the problem; it is how we react to those situations. In other words, the only thing that defines a situation as “stressful” is whether or not it induces a stress response in us. What if we were able to reduce or eliminate that stress response, instead of trying to reduce or eliminate the stimulus/trigger?
One of the most recommended methods for overcoming phobia is called “exposure therapy” and it involves: “Gradual, repeated exposure to the source of your specific phobia and the related thoughts, feelings and sensations may help you learn to manage your anxiety.” (Mayo Clinic) In other words, when something is scary, the way to overcome that anxiety is to be repeatedly exposed to that fear-inducing stimulus, but in small, non-threatening ways to show that nothing bad actually happens.
I have found this does work to reduce test anxiety and the cognitive impairment that can result, as well. Instead of eliminating tests, I simply changed the way they are given, in a way that makes them less scary and, indeed, more of what Boaler called an “assessment for learning” (A4L), rather than just an “assessment of learning” (although, in reality, it serves as both!):
- I gave tests far more frequently, rather than less frequently. The more tests you give, the less of an overall impact each one has on your grade, so there is less fear that doing poorly on one could ruin your entire grade.
- More importantly, I gave the same assessment multiple times (but with different questions; concepts remained the same, but numbers and details changed) — ie. each test was both an assessment for learning (formative), as well as an assessment of learning (summative.) How did this work?
- At the culmination of learning a topic (including, yes, plenty of practice via “homework”, although many students were able to do that work at school due to morning and after-school opportunities we provided — there goes Boaler’s argument of “inequity”), I would assign the topic test.
- The topic test was called a “practice test” — the message was “Let’s see how much we have learned, and how much we still need to learn.” There was an incentive to try their best — if they liked the score they received, they could keep it (but, unless they received 100%, there was still room for improvement so they would get a chance to try again.)
- An item-analysis of errors from the test would allow us to examine the most common mistakes, misconceptions, or learning gaps, as a group in the classroom. We would analyze our weaknesses or confusions, do some more teaching and practicing, and then try again.
- A second test was given — sometimes this was yet another independent “practice” test to see how much we had grown after reviewing the things we had difficulty yet.
- When the final test was given, anxiety had been reduced, because (a) it was not a surprise what types of skills were going to be assessed; (b) their grades could not go down — they could only go up! They had already taken the test before, and if they did worse on this one… it wouldn’t count (so why not try and see if we can do better? It was a zero-risk proposition); (c) the message was not a high-stakes one. The message was “Our goal is not scoring 100%. Our goal is to learn. If you score any better than you did on the last test… you are learning! If you score worse? Then you are still learning, it just shows there is more learning to be done!”
That last piece — the messages that are used and the priorities that are communicated — are an essential part of building growth mindset (as shown by Carol Dweck, but similar messages are also outlined in Mathematical Mindsets), but what Boaler doesn’t seem to acknowledge is that it is possible to reduce stress and anxiety and build growth mindset while still keeping the valuable benefits of homework, testing, and grades.
Claim: “if (teachers) do continue to test and grade, they should give the same grade, or higher, for mistakes” (p. 17)
I understand the rationale behind this idea: the purpose is to build perseverance and motivation, by not making students fearful of mistakes, since mistakes and “failures” are a natural and necessary part of the learning process. While it is true that such a policy would definitely eliminate that fear of “being wrong,” Boaler’s recommendation would actually be harmful to students.
The tenets of behavioral psychology have been around for a long time, but they are completely ignored by Boaler in her recommendation. Operant conditioning, as shown by B.F. Skinner 80 years ago, has shown that, when people (or animals) are given a reward for a behavior, it provides positive reinforcement and encourages the behavior to continue or be repeated in the future.
In other words, if we reward making mistakes, that actually encourages students to make mistakes. In fact, if you give students a higher grade for mistakes than for correct answers, as Boaler ludicrously suggests, then this is actually a motivation to purposefully get the wrong answer.
The delusion here has to do with the value of mistakes. It’s not that mistakes are good — it’s that they are expected or acceptable, in certain situations… such as when learning. The only reason we want to celebrate mistakes is not because mistakes are inherently good, but because they show that a student put in effort and took a risk, when they could have otherwise simply quit or used avoidance tactics.
It is not the mistake, then, that is actually what we should be happy about — the mistake is just a byproduct of the actions we should be celebrating: ie. “giving it a try” and refusal to be afraid of failure.
So, Boaler has it wrong: it’s not the mistakes that we should be celebrating and rewarding (of course, nor should they be punished!) They should be acknowledged for what they are: not the desired goal, but a possible indicator that progress is being made toward the desired goal — that goal being a useful discovery or a successful solution to a problem.
What we should be rewarding is: effort; perseverance; modeling and communication. These are the actions that can lead to mistakes, so it’s good to communicate to students that making mistakes is normal and expected and okay as part of the process. But to reward mistakes sends the message that mistakes are the desired goal … and that is blatantly false. If you teach students this, they are going to have serious problems when they enter the workforce. NASA is not going to reward you for making mistakes when engineering a rocket; Apple and Google are not going to reward you for making mistakes when coding security features of their platforms.
So, it is best to keep things real and honest with students: mistakes are not “good”, but they are a natural indicator that good things are happening (effort & learning.) So, there are times when mistakes are “okay” (ie. when we are learning), and other times when they are not (ie. when we are actually performing.) In any case, mistakes are something to be accepted, but not rewarded.
Unconvincing Results from “Exemplary” Schools and Programs
Despite all of the anecdotes and stories presented in Mathematical Mindsets, what a lot of readers are looking for, really, is the answer to: “How can we fix this?” As in, how can we do better at teaching math to our students? Currently, the #1 metric (and perhaps the only “universal” one for the United States) to measure and compare learning from one school or a program to another is: standardized CCSS testing.
So, as I read Mathematical Mindsets, whenever Boaler would mention teachers, or schools, or districts who were doing great and innovative things, I had to admit that it all “sounded good” on paper and in theory, but the nagging question was always: “So… how well did it work? Did learning improve?” I wanted empirical evidence… data. Not just quotes from kids about how much they “liked it” (Which is a very common piece of support Boaler likes to use — cherry-picked personal narratives and qualitative anecdotes. Which is all well and good, but enjoyment level is irrelevant in a classroom if learning is not occurring…) I wanted to know: how well did they learn?
So, I tried to find unbiased, quantitative data the best that I could and… it turns out that test scores show the achievement levels are not so impressive as Dr. Boaler paints it to be.
Boaler begins Chapter 6: “I am passionate about equity” (p. 92) and asserts that “When mathematics is taught as a connected, inquiry-based subject, inequities disappear and achievement is increased overall.” (p. 103) Let’s see how true that is…
1) Life Academy
Boaler explains that she takes her undergraduate class each year “on a field trip to the incredible Life Academy, a public school in Oakland that is committed to disrupting patterns of inequity on a daily basis.” She states that “The accomplishments of Life Academy are many; the school has the highest college acceptance rate of any high school in Oakland, and the proportion of students who leave ‘college ready’ with California’s required classes is an impressive 87%, higher than at the suburban schools in wealthy areas close to Stanford.” (p. 99)
Okay… but… those look like cherry-picked stats… those are the types of numbers that can be manipulated, simply by giving students easy grades, or graduating them when they haven’t earned it. How much do they actually learn? Let’s see what GreatSchools.com has to say, based on actual performance:
Yet… here’s what the standardized tests show:
And how does equity rate at this school? Not good, either… the only “equitable” thing at this school is that they perform equally poorly:What does this highlight? It actually highlights a serious problem: schools can “fudge” success very simply through grade inflation. While this might help students get into college, it sets them up for failure once they arrive, and in the years beyond, because they have not been given an honest evaluation of what they know and what they still need to learn… it also creates an “uneven playing field,” giving an unfair advantage when colleges are comparing these students — whose grades appear to show that the students are proficient, when the test scores show that is not the case at all. You want equity? This is the opposite of equity.
Boaler’s previous seminal works have centered around: (a) “inequity” (based on race/ethnicity and socioeconomic status or, more often, gender); (b) traditional vs “reform” open inquiry-based instruction; (c) tracking vs de-tracked, heterogeneous groups.
While others on the internet have noted Boaler’s clear biases and unsupported conclusions in some of those studies, the one that stands out as being an influential work is her “Railside” study, which is referenced on various pages of Mathematical Mindsets, but mainly with regard to heterogeneous grouping on pages 119-120.
The data and conclusions have been called into question by others, including expert mathematicians — You can click here to read the report two university professors and a statisticians wrote when they dug deeper and found evidence that the conclusions Boaler drew were not accurate, based on the data, and that the methodology was essentially flawed. You can also click here to read Boaler’s rebuttal, calling the revealing of her data “harassment and persecution.” (I can only presume she would have the same response to me and the facts I am revealing in this blog post. However, I hope if she were to respond to the points I am making here, she would do so with logical and research-supported defenses, instead of resorting to the ad hominem diatribe she made in retaliation to Milgram et al.) She also has some of her facts wrong, stating “Milgram and Bishop’s ‘paper’ contravenes federal law that protects the human subjects of research as it identifies schools, teachers and students. Its identification of individual students breaches the Family Educational Rights and Privacy Act (FERPA).” I did not see any identification of individual students in their paper — they only identified cohorts. This is no different than the type of data reporting done on state standardized testing sites. In other words, it is legal. But the fact that she tried to use it as a defense says a lot.
The reason the (influential) results of the (questionable) Railside study are so concerning, is that several districts and schools have bought into the “inquiry-based” math curricula used, such as CPM (College Preparatory Math)… and the results have not been pretty. For one, Boaler claims (and provides lots of cherry-picked anecdotal quotes, of course) that enjoyment and motivation are increased… but when I searched for reviews and opinions about CPM online, the response is quite different than what Boaler suggests. Many parents, teachers, and students have complained about the program — about its avoidance of direct instruction, about its insistence on group work for all aspects, and about its (apparent) lack of efficacy. One group of parents was so concerned that they created a “Fairfield Math Advocates” group to combat the adoption of CPM, and has a whole lot of data to highlight problems with it, as well as a list of myriad other advocacy groups that have arisen to rail against it. (They also highlight how it caused fewer and fewer students — a disproportionate number of boys, especially — to complete the math program as years went on. So much for equity!) In fact, I tried to find positive reviews of CPM, and the only positive messages I could find were by Boaler and her colleagues, or people who had direct professional connections to CPM (but, if you don’t believe me, check it out for yourself)
The comments of these disheartened teachers, parents, and students also reveal a major flaw in the logic of “student-centered” / inquiry / constructivist philosophies: the premise is that people only really “learn” when they discover things for themselves. The problem is… people do not easily do this. It takes time. A lot of time.
Evidence indicates that modern Homo sapiens first appeared over 200,000 years ago. So, consider the following:
The concept of “zero” was not discovered until sometime between 300 B.C. and 458 A.D. Yes, left to our own devices to “figure things out”, it took humankind about 200,000 years to learn how to use zero as a symbol and a placeholder.
But, through the power of direct instruction, we can ensure that students know and use the number zero by the time they are 5 years old.
The above is an example that demonstrates what is true, ad infinitum, for everything we know about mathematics: Pythagorean theorem… algebra… etc. The fact is that all of these mathematical discoveries have some things in common:
- Conceptual understanding takes time (often a lot of time), and…
- They come about by first learning and understanding other, prerequisite concepts. Discoveries are sequential, they are not “Eureka!” moments (not even for Archimedes, who already had background knowledge and concepts that would be prerequisite to making new discoveries.)
Doesn’t it make sense, then, that we should accelerate discovery by first actually imparting the knowledge that it took humankind hundreds of thousands of years to discover? Rather than expecting students to “discover” 200,000 years worth of knowledge within a few short years of their lives, we can ensure that they learn those skills in a swift and efficient manner, and then they will be properly prepared to move forward into the realm of the unknown — the realm of theoretical mathematics.
A Few Good Things
Despite all of the shortcomings above, there are valuable nuggets of insight, new things to try, and ways to enrich/supplement existing mathematics programs. I think that is the crux of the message here: everything in moderation. So, what are some of these valuable tenets that can be used, at least partially, to improve learning?
Despite the lack of growth mindset when it comes to dealing with “stressful” realities like practice, assessment, and grades, this book does have a lot of good examples of tools and effective communications that can be use to bolster confidence and motivation and a “non-defeatist attitude” through growth mindset messages to students. It is worthwhile to take a look at the various verbal cues and responses, reflection questions, and more that do seem like they will bolster growth mindset in students (although you can also get these exact same ideas from Carol Dweck’s Mindset book.)
Pragmatic Suggestions for Group Work
There does not seem to be consensus on whether group work is superior to independent work (indeed, studies show that it is often beneficial, but detrimental for some students, and sometimes detrimental to the learning process in general), but at least Boaler admits that group work is fraught with challenges and obstacles to overcome. The problem lies when curricula like CPM (the much-touted program used in the Railside study) insist on group work for everything (even assessment), all the time. One beneficial/practical feature in Mathematical Mindsets could be the examples and recommendations for structuring and managing groups, including specific roles and methods for handling communication, reflection, and power/social dynamics (which are often the greatest hurdle in grouping.) Although there does not seem to be any actual empirical data regarding the efficacy of these group strategies, I am curious to give them a try…
Support FOR Common Core
Since the above skills — modeling, critical analysis, and communication — are now essential parts of the CCSS standards, it is not surprising that Boaler supports CCSS. She even admits that the standardized tests are better assessments of mathematical abilities than the older ones (it is surprising then, that she never seems to use CCSS standardized test data as a data point in the research or school examples…):
“One critical principle of good testing is that it should assess what is important. For many decades in the United States, tests have assessed what is easy to test instead of important and valuable mathematics. This has meant that mathematics teachers have had to focus their teaching on narrow procedural mathematics, not the broad, creative, and growth mathematics that is so important. The new common core assessments promise something different, with few multiple-choice questions and more assessments of problem solving, but they are being met with considerable opposition from parents.” (p. 141)
I have experienced this firsthand, and it is unfortunate. But, just like some people are too quick to jump on board something because it is new (like, ahem, those who uncritically accept the entirety of messages in Mathematical Mindsets), there are also those who think the status quo should never change, that it’s “good enough” despite the harsh truth that evidence is to the contrary. Hopefully, growth mindset training / interventions can extend beyond students and the classroom, to also be able to reach (and undo fixed mindsets) in parents and households everywhere.
The Importance of Modeling, Communication, & Critical analysis
“One of the most important contributions of the Common Core State Standards (CCSS), in my view, is their inclusion of mathematical practices—the actions that are important to mathematics, in which students need to engage as they learn mathematics knowledge. ‘Modeling with Mathematics’ is one of the 8 Mathematics Practices Standards…
The act of modeling can be thought of as the simplification of any real-world problem into a pure mathematical form that can help to solve the problem. Modeling happens all through mathematics, but students have not typically been aware that they are modeling or asked to think about the process.” (p. 194)
I agree with this, and — in my personal (anecdotal) experience — it is true that it is a major mind shift for students, who have not really been asked to discuss, analyze, model, or critically analyze the work they are doing. Thankfully, the CCSS curriculum is addressing that problem… but, in my experience, the quality of those experiences and skills in the current curricula varies widely. As such, it could be beneficial to supplement (not replace) it with some of the open-ended, puzzle/pattern-style activities Boaler highlights in Mathematical Mindsets.