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Mathematical Mindsets: Unleashing Students' Potential Through Creative Math,

Inspiring Messages and Innovative Teaching – Jo Boaler – Jossey Bass, Wiley.

ISBN: 978-0-470-89452-1

In “Mathematical Mindsets” Jo Boaler aims to communicate the importance of the

adoption of growth mindsets for mathematical achievement. The growth mindset

approach, a phrase coined by Carol Dweck in 2006, has recently become popular in

the education community, particularly in America. This approach suggests that

encouraging students to recognise that intelligence is malleable rather than fixed will

increase learning outcomes. Boaler specifically applies the growth mindset approach

to mathematics education and suggests that achievement will be raised if students’

perceptions of the subject are shifted from that of a fixed skill to something they can

explore and make sense of. Growth mindset approaches emphasise the importance

of teacher attitudes and the types of tasks that teachers set in the classroom. In

particular, shifting from closed-answer questioning to discovery-based learning is

encouraged. Within this book Boaler provides numerous activities that would

appropriately engage students; encouraging group discussions, problem solving and

enthusiasm in classrooms.

A strength of this book is that Boaler discusses good evidence-based practice such

as emphasising the exploration of students’ errors and the use of concrete

manipulatives. Careful error analyses allow educators to explore misunderstanding of

concepts and procedures and is a strategy that teachers have been encouraged to

adopt (Hansen, Drews, Dudgeon, Lawton & Surtees, 2014). In addition, there are a

plethora of studies that suggest that adaptive and flexible strategy use is a key

component of success in arithmetic (e.g. Geary & Brown, 1991; Vanbinst,

Ghesquiere & De Smedt, 2014). The methods that Boaler discusses in the book have

the potential to enable opportunities for students to apply these adaptive strategies in

a classroom scenario (p59). There is also a growing research literature on the

efficacy of concrete manipulatives to improve learning, especially in the early years

(Clements, 2000).

Boaler suggests that the growth mindsets literature has proven this type of

intervention to be effective: “Research has shown definitively the importance of

growth mindsets” (p34). This is a bold statement for an area of intervention and

research that is in its relative infancy. It is correct that some studies have shown

improvements in achievement following mindset based interventions, for example,

Paunesku et al. (2015) observed that growth mindset interventions increased

achievement in children who were at risk of dropping out of school. However,

educators should recognise that there have also been null results in studies of

growth mindset interventions on mathematical achievement (e.g. National College for

Teaching and Leadership, 2015; Rienzo, Wolfe & Wilkinson, 2015). Of course, a

plausible explanation for the null effects of growth mindset interventions is that the

control groups may already be experiencing a growth mindset messages from

existing high-quality teaching, but it is also possible that mindset interventions are

not as effective as Boaler suggests. It is also important to note that these

interventions may not have a positive effect on learning outcomes, but may positively

benefit children in terms of decreased mathematical anxiety or increased self-

efficacy.

A growth mindset approach may lead to increased confidence, decreased anxiety

about learning and thus lead to increased achievement in mathematics (Hembree et

al., 1990). However, the way in which Boaler presents evidence for her pedagogical

recommendations has to be questioned. There are numerous examples of an inappropriate use of

neuroscience to back up educational claims in this book. Of course within the

boundaries of a review it is impossible to address all of these statements, therefore I

have extracted some clear examples.

The word “neuromyths” describes misconceptions that are perpetuated due to

misunderstandings or misinterpretation of neuroscience research (OECD, 2002) and

there has been a recent concerted effort by the academic community to address

these. Neuromyths are frequently present in the book, for example (citing a

presentation by Carol Dweck, p.12), Boaler states that “Every time a student makes a

mistake in math, they grow a synapse” (p.11) and subsequently that “Mistakes are

not only opportunities for learning… but also times when our brains grow, even if we

don’t know we have made a mistake.” (p12). Boaler supports this suggestion with the

work of Moser et al. (2011) who measured event related potentials in 25

undergraduate students during a flanker task (a response inhibition task which has

no mathematical content) and also assessed their growth mindsets. The authors

established that increasing levels of growth mindsets were associated with

heightened neural activity in response to errors, concluding that high levels of growth

mindset was associated with closer attention to errors on response inhibition tasks.

This study tells us nothing about synaptic brain growth and has no implications for

mathematics teaching or learning (e.g. Neuroskeptic, 2016). However, Boaler

perpetuates this ‘mistakes grow synapses’ neuromyth throughout the book to support

the adoption of growth mindset methods. In the climate of emphasising evidence-

based educational practice it is important that decision makers, both in terms of

classroom practice and policy, can make judgments on the best scientific evidence,

thus neuromyths should be dispelled in order to facilitate this decision making

process.

As academics and practitioners from education, psychology and neuroscience

increasingly collaborate to inform intervention and practice, there is a need to

understand discipline-specific language. There are examples of misinterpretation of

terminology within the book, such as the use of the word “compression” (p37). Boaler

states:

“When you learn a new area of mathematics that you know nothing about, it takes

up a large space in your brain…. But the mathematics you have learned before and

know well, such as addition, takes up a small compact space in your brain…. Ideas

that are known well are compressed and filed away”

This statement implies a physical change in the state of the brain. Importantly, Boaler

suggests that compression cannot apply to rules and methods, but only to concepts.

This assertion appears to create a new neuromyth. First, a typical neuroscience

definition of compression would be “information coming from a large number of

neurons must be compressed into a small number of neurons” (Allen-Zhu,

Gelashvilli, Micali & Shavit, 2014, pg. 16872), but this is entirely different to physical

compression of space in the brain. Second, no evidence is provided that only

concepts, and not rules or procedures, can be efficiently stored in memory: in fact,

longstanding evidence suggests that this is not the case (Baroody, 1983; Campbell &

Therriault, 2013). More problematically, Boaler also cites research that cannot

support her statements. There is a pertinent example in which Boaler references a

study that only measured children’s behavioural responses to basic numerical tasks

but this study is used to make neuropsychological claims about the importance of

communication between the left and right brain hemispheres (p. 39), it is important

that implications are extrapolated from appropriate eveidence. This section again

perpetuates unhelpful neuromyths which have the appearance of scientific evidence,

but in fact make evidenced-based decision making in education increasingly difficult.

Boaler suggests that reducing timed assessment in education would increase

children’s growth mindsets and in turn improve mathematical learning; she thus

emphasises that education should not be focused on the fast processing of

information but on conceptual understanding. In addition, she discusses a purported

causal connection between drill practice and long-term mathematical anxiety, a claim

for which she provides no evidence, beyond a reference to “Boaler (2014c)” (p38).

After due investigation it appears that this reference is an online article which repeats

the same claim, this time referencing “Boaler (2014)”, an article which does not

appear in the reference list, or on Boaler’s website. Referencing works that are not

easily accessible, or perhaps unpublished, makes investigating claims and assessing

the quality of evidence very difficult. A more nuanced view of the connection

between fact knowledge and conceptual understanding surfaces from the

mathematical cognition literature. This literature emphasises the iterative

developmental process between arithmetic fact knowledge and conceptual

understanding (Rittle-Johnson, Siegler & Alibali, 2001). Of course mathematics

teachers should highlight “seeing, exploring and understanding mathematical

connections” (p70). However, to deny that speedy access to mathematical facts

cannot enable this type of processing seems unfounded (e.g., Fuchs et al., 2013).

The recognition that a balance between procedural knowledge and conceptual

understanding, along with other skills, is required for mathematical success is

paramount and has been suggested as a way in which to address declining

standards in mathematical achievement in North America (National Research

Council, 2001).

In a rather alarmist section in the book (pg. 144) Boaler extends her argument

regarding a causal link from assessment to anxiety to include the potential for

suicide. Rather than basing this idea on research evidence, this section refers to a

documentary “Race to Nowhere”, which explores the impact of academic pressures

(such as homework and testing) on the wellbeing of students. Boaler describes the

case of a student who received a poor grade on a maths test:

“… the grade she received did not communicate a message about an area of math

she needed to work on; instead, it gave her a message about who she was as a

person- she was now an F student. This idea was so crushing to her, she decided to

take her own life”.

Multiple factors contribute to the complex behaviour of suicide (National Confidential

Inquiry into Suicide and Homicide by People with Mental Illness, 2016) and its

reporting has come under intense scrutiny in terms of responsibility of

communicators (e.g. Samaritans, 2013). Clearly, it is important that such a claim

needs an appropriate evidence base, which is lacking in this section.

Finally, some of the presented study findings are made difficult to interpret as some

of the graphical information is poorly presented. For example, one experimental

study described in the book assessed the efficacy of a mathematical mindset

intervention (p51). The results, as plotted in Figure 4.5, pg. 51 appears to show the

performance of the experimental group declining compared to the control group,

which conflicts with Boaler’s accompanying text. Perhaps if the axis were labelled an

alternative interpretation would be revealed, but without the further explanation the

reader is left confused. This is not an isolated occurrence and readers could more

easily evaluate the information if graphs and figures could be more clearly presented

throughout the book. For example, Figures 7.3-7.5 are boxplots where the median

line has been omitted, rendering any interpretation impossible.

While “Mathematical Mindsets” includes a rich range of tasks and suggestions for

teachers that may be highly effective at creating a positive classroom culture, the

way the book presents its case is concerning. As scientists who study learning and

achievement it is vitally important that we accurately present scientific evidence and

theory to educators who are striving to enable their students to succeed. In view of

its promotion and perpetuation of neuromyths, I do not believe that “Mathematical

Mindsets” passes this test.

Dr Victoria Simms, Lecturer in Psychology, Ulster University, Northern Ireland.

References

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Compression with Neuroscience-Based Constraints. Proceedings of the National

Academy of Sciences of the USA, 111(47), 16872–16876.

Baroody, A. J. (1983). The development of procedural knowledge: An alternative

explanation for chronometric trends of mental arithmetic. Developmental Review, 3,

225–230.

Campbell, J., & Therriault, N. (2013). Retrieval-based forgetting of arithmetic facts but

not rules. Journal of Cognitive Psychology, 25(6), 717-724.

Clements, D. (2000). Concrete manipulatives, concrete ideas. Contemporary Issues

in Early Childhood, 1(1), 45-60.

Fuchs, L., Geary, D., Compton, D., Fuchs, D., Schatschneider, Hamlett, C.,

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