# 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 *un*important. 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.

Conclusion

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.

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This is all good, Bryan. I want to point out that Boaler completely circumvents the point of speed in timed drills when she says speed is not important in mathematics. That may well be true, as a mathematician I value rigour, insight and accuracy more than speed. However, those of us who have navigated to the heights understand the value of committing mechanics to the subconscious level — there is a bottleneck in short-term memory that prevents upward mobility in mathematics, and it starts kicking in around middle school and only gets worse if students are trying to process all levels of meaning consciously.

We must … MUST … stop thinking about the most basic level of meaning and perform mechanics automatically if we are to rise above that level. That is not to say that we stop understanding that level — it should all be there for retrieval. But we must stop THINKING about it and get on with the more sophisticated layers of meaning that enable us to soar in the layers of abstraction that make up mathematics beyond arithmetic. You illustrate this point well with your derivation of the quadratic formula. Observe how one does not think, at any stage in this derivation, about the meaning of addition, multiplication, squaring and square roots (well, perhaps the latter two, only fleetingly). We must be working at a level where we’re comfortable to forget about such things, knowing we COULD deconstruct every step to that level of meaning, but also knowing, through spaced repetition we have mastered material whose deductive basis is established to the point we needn’t continually revisit it.

Then … once having established the quadratic formula, we move on and avoid deriving it every time it is used. That’s the whole point of a formula — to allow us to circumvent the work of derivation over and over. The power of mathematics lies not in the habittual deconstruction of meaning to its lowest level, but in the confidence of the surety of those lower levels to the point where one needn’t do so. That is what Whitehead meant in his famous quotation where he said that mankind progresses not by thinking about things we’re doing but by the number of things we can do easily without thinking about them.

To me this truth came home some time during high school when I realised that, although I “understoood” the mathematics I was doing I was getting bogged down thinking about it, and I could not understand why it all seemed so hard when every step was easy. The thought came to me … “just trust the mathematics!”. I started taking that as a guiding principle … and it works! In every discipline there is a point where you have to start trusting that it will, in fact, hold itself up. Take your finger off the page as you read. Sing out that chorus without looking down at your music. Ditch the training wheels on your bike. And stop telling yourself which foot goes to the brake and which to the clutch (er … if you’re driving standard). TRUST the math.

Which brings me to my point: the purpose of speed in drillwork while learning mathematics is to help students achieve that “lift-off”, to stop counting on fingers, and to stop relying on rapid steps in one’s head for elementary arithmetic … it’s time to move on, and speed is the way to accomplish it. If young readers are indulged in reading letter-by-letter they may never read fluently. At some point they must jump to reading entire words, and then phrases at a glance. It is not a fault to fail to sound out letter-by-letter; on the contrary, it is the goal! And in mathematics the goal of speed in arithmetic is so that the individual steps become so smooth and automatic working memory is freed up for higher processes. Speed drills are not so that students will become “fast calculators”. They are to break the arithmetical habits analogous to keeping your finger on the page as you read. These needn’t be done in a high-pressure or high-stakes setting. In fact, the best teachers know how to make such drills fun, even exciting. And kids experience the thrill … of skill … that comes through drill …

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