Missing Messages from Jo Boaler’s Maths Video
“When students were given problems to solve, and they did not know methods to solve them, but they were given opportunity to explore the problems, they became curious, and their brains were primed to learn new methods, so that when teachers taught the methods, students paid greater attention to them and were more motivated to learn them.” -Jo Boaler
In Jo Boaler’s most recent mathematics announcement on YouTube, she gives her audience four important messages to think about. However, it is what is missing from these four messages that is more important for us to reflect upon.
Message One: “Everyone can learn maths to high levels”
Honestly, I really appreciate how they begin this section of the video. I agree that it can be detrimental to hold the belief that some people are “maths people” and some are not. Many of these harmful misconceptions are cultural – why is it in Western culture that we are proud of the fact that we have never used algebra in our post-high school lives? I do believe that removing these hidden messages from socio-cultural contexts, where possible, is very important. This said, this is not the focus of this section of the video, so let’s see what is mentioned regarding learning mathematics to higher levels.
Looking at the “BRAIN EVIDENCE!” leads us to a chat about neural pathways. While I fully agree with Greg Ashman’s critique of the language of this portion of the video, the discussion, I believe, is to motivate a concept of learning that most of the current theories (cognitivism, constructivism, cognitive load) agree on: humans recall past experiences or draw upon pre-existing knowledge to help them “make sense” of current stimuli and connect it to past knowledge. For example, the firing of a synapse may be us drawing on past experiences/knowledge, and creation of a new neural pathway might be indicative of new knowledge amalgamating with old knowledge.
Of interest here is Boaler’s discussion of the three week program in which participants worked for ten minutes on a subject. This study found that this amount of time was enough to cause permanent neural pathways to form. This is not terribly surprising, as some of the best ways to learn new material is repeated and spaced out practice (see How We Learn by Benedict Carey). Boaler’s student then states “If a three-week program can do that, what do you think a year of maths class can do to your brain?” I assume that if repeated and spaced out practice would be utilized more effectively in mathematics classes nationwide, a lot could be accomplished.
Unfortunately, Boaler seems to be at odds with memorization (see here and here) claiming that timed tests and memorization lead to maths anxiety. While it is true that high-stakes testing can lead to adverse effects, Boaler’s message is often misconstrued as “all testing is bad“, when this is not the case at all: low-stakes testing with feedback is known to be one of the best tools to enhance fluency and long-term storage of maths facts. The blog here can lead you to some of the relevant research on this subject.
Missing from Message One: Yes, you can learn mathematics to higher levels, but one cannot build a house without a strong foundation. Repeated and spaced out practice with mathematical concepts is vital to learning the fundamentals needed to access higher mathematics. Yes, it is true that your brain creates neural pathways when you learn new materials, and when you practice these facts over and over, the pathways become stronger. Testing is an excellent way to ensure fluency with maths facts – just keep it low-stakes.
Message Two: “Believe in yourself”
The next message we are to consider is that believing in ourselves as learners is important. The group goes on to briefly discuss Carol Dweck’s work with Growth and Fixed Mindsets. I want to be clear that I think having a belief in one’s own ability is a great thing; however, this message doesn’t contain the full truth. More often than naught, we fail to take out of Dweck’s message that learning comes with much effort and practice. If we fail to put in the desired work to better our knowledge of a subject, then the statement “I believe in myself.” bears no weight.
On a related note, let’s consider the argument of intrinsic motivation. Recently, Greg Ashman posted this blog reminding us that it is actually achievement in mathematics that predicts intrinsic motivation, and not the other way around. He cites a longitudinal study that focused on Grade 1-4 students that found higher achievement in mathematics led to higher motivation in mathematics. Similarly, Dörnyei noted in Teaching and Researching Motivation that motivation seems to be at its highest when students are competent with the subject material. So how do novice learners become competent with subject material in mathematics? By spending a lot of time developing content knowledge. So perhaps Jo Boaler has it backwards – maybe it is strong mathematics content knowledge that leads to a belief in our ability to do mathematics.
Missing from Message Two: Believe in yourself, yes. But belief in yourself can only take you so far. Eventually you will need to develop content knowledge in mathematics, and developing this knowledge takes time, hard work and effort. The more content knowledge you have in mathematics, the more likely it is that your achievement in mathematics will increase. Higher mathematics achievement may then lead to higher intrinsic motivation, self-efficacy and confidence – leading you to have more belief in your abilities.
Message Three: “Struggle and mistakes are really important”
Again, putting the odd language aside, the message being delivered here is misleading: “When people made mistakes their brains grew more than when they got work right.” While I agree that neural pathways are being created whether we answer incorrectly or correctly, I certainly want my students to be forming ‘good’ neural pathways that will help them search out an acceptable answer or procedure. If a student consistently answers incorrectly, and is not corrected, we know this ‘bad’ neural pathway will become stronger. These misconceptions can be very difficult to change later in our mathematics career, as new information now begins to consolidate on this misconception.
Perhaps Boaler was trying to motivate the idea of productive failure – situations in which a learner is set up to struggle with a problem just outside of their current abilities with only little guidance from the teacher. In this case, I think it is worthwhile to discuss the work by Manu Kapur. In this study Kapur followed 75 grade seven mathematics students in Singapore. Some students were a part of a traditional lecture and practice class, while another group was a part of an ill-structured problem-solving class with no structure or support. The later group, at the end of the unit, had a consolidation class where the teacher led a discussion on important concepts. Of interest is that the students in the productive failure group outperformed those in traditional lecture style on a test containing mostly well-structured response questions. A possible downside is that those students in the productive failure group also reported lower confidence in their solutions.
To round out this section of the video, the group discusses Michael Jordan and the failures he made as a basketball star (oddly enough they don’t mention the obvious failure of his 1993-94 MLB career). To me, this analogy doesn’t work given the context of the video. If an engineer stated “Twenty-six times I was trusted to build a bridge that was stable, but didn’t. I have failed over and over.” I don’t think this engineer would have much success in their field. In mathematics we do often make mistakes; however, as we progress further in our craft, we strive to minimize the errors with the facts/procedures that we are most fluent with. The use of Michael Jordan here simply does not add to Bolaer’s claim that “mistakes are important” and distracts the viewer from engaging in something of substance.
Missing from Message Three: To say that an individual’s brain will grow more making a mistake than with a correct answer is misleading. It is true that neural pathways will form in either case, and it is important to try to make ‘good’ neural connections. The more repeated practice a student has with a misconception, the harder it will be to remove this misconception later. We have also seen that perhaps struggling with a mathematics problem is important under certain conditions. Kapur shows us that there is such thing as productive failure – that is students may be able to consolidate information easier directly after struggling with specific problems. However, we note that these students also were not confident with their solutions. Thus, we conclude that the teacher utilizing productive failure as a teaching strategy must be well-trained, as we have already seen low confidence can lead to low achievement and motivation in mathematics.
Message Four: “Speed is not important”
There are many messages in this section that are misinformed. Let’s begin with “speed is not important”. I disagree with this claim to a certain extent. I do not expect my students to be The Flash on their tests of recall. However, if I am expecting them to perform the standard algorithm for multiplication, I do expect their basic fact recall to be decently quick. How fast is decently quick? I think that depends on the student. What I am looking for is that their working memory is not getting bogged down computing maths facts while thinking about the standard algorithm question. If this is happening then my student is too slow and needs to revisit computations of maths facts before coming back to the standard algorithm.
So what justification does Boaler give for speed not being important? Well, one of her students mentions that “some of the best mathematical thinkers of the world are really slow”. While I do not deny that mathematicians think deeply about problems, let’s also realize here that our top mathematical thinkers are often thinking about complex multi-dimensional problems. I doubt very much they take several minutes/hours/days/weeks to compute a multiplication fact. But this is beside the point – Boaler is using the argument that ‘mathematicians are slow’ as evidence that ‘speed is not important’. The use of one specialized cohort to generalize to all cohorts is false (even if we set aside our argument above that the types of problems being tackled are different).
Other phrases that are used in this section of the video are “maths isn’t just about calculations – they may be the least interesting part” and “maths is not about memorization and is not about calculations.” Boaler then goes on to discuss that it is more important to see deep connections, think creatively and see mathematics visually. While I agree that it is important to see connections in mathematics, I disagree that calculations are uninteresting. Often in mathematics, calculations lead us to general procedures or proofs. For example, the calculations used to get to the quadratic formula are given below. Notice that these calculations show us the procedure of completing the square (can you find it?)! If we had dismissed the calculations believing that they were uninteresting, we may not have seen this deeper connection – something that Boaler advocates for.I find her second statement that “maths is not about memorization and is not about calculations” very misleading as well. Memorization is a very important strategy to use when learning new facts and should not be undersold. We have all used memorization as a strategy to learn before – how else would we have conceptualized cardinality of numbers? Dr. Ansari from Western Ontario reminds us of this – he has shown us that our ability to retrieve maths facts is a predictor of achievement on the PSAT test. Those students who did not have maths facts memorized (used a quantitative-based strategy) did not succeed as well on the test as those who had facts memorized.
Missing from Message Four: First, speed is not important, but it is also not unimportant. Fluency with basic facts is important when working on more challenging problems. If we are constantly slowing down to think about basic facts, we begin to lose sight of our original problem. Second, it is true that mathematics is more than calculations; however, calculations often lead us to the bigger connections in mathematics, so they should not be disregarded. Finally, memorization is an important strategy to learning new information. Blanket statements such as “memorization is not important” downplay the importance of this teaching tool for teachers.
Overall, we can see that the messages shared in this video are quite misleading or incomplete. At the end of this video they promote Boaler’s site, Youcubed because it has “real maths” on it. I am skeptical that real mathematics resides on this website (you can see my review of one of the activities that was devoid of mathematical language here) – however, I promise to keep an open mind as I review more of her activities in the future.