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Mathematical mindsets: unleashing students’ potential through creative math, inspiring messages and innovative teaching

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
Allen-Zhu, Z., Gelashvili, R., Micali, S., & Shavit, N. (2014). Johnson-Lindenstrauss
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.,
Seethaler, P., Wilson, J., Craddock, C., Bryant, J. Luther, K., & Changas, P. (2013).
Effects of First-grade Number Knowledge Tutoring with Contrasting Forms of
Practice. Developmental Psychology, 105, 58-77.
Fuchs, L., Geary, D., Compton, D., Fuchs, D., Schatschneider, C., Hamlett, C.,
Deselms, J., Seethaler, P., Wilson, J., Craddock, C., Bryant, J. Luther, K., &
Changas, P. (2013). Effects of first-grade number knowledge tutoring with contrasting
forms of practice. Journal of Educational Psychology, 105(1), 58-77.
Hansen, A., Drews, D., Dudgeon, J., Lawton, F., & Surtees, L. (2014). Children’s
Errors in Mathematics. Exeter, United Kingdom: SAGE Publishers.
Hembree, R. (1990). The nature, effects and relief of mathematics anxiety. Journal
for Research in Mathematics Education, 21(1), 33-46.
Moser, J., Schroder, H., Heeter, C., Moran T., & Lee, Y.-H. (2011). Mind your errors:
Evidence for a neural mechanism linking growth mind-set to adaptive posterror
adjustments. Psychological Science, 22(12), 1484-1489.
National College for Teaching and Leadership (2015). Closing the gap: Test and
learn. Downloaded from https://www.gov.uk/government/publications/closing-the-
gap-test-and-learn on 14th April 2016.
National Research Council. (2001). Adding it up: Helping children learn mathematics.
J. Kilpatrick, J. Swafford, and B. Findell (Eds.). Mathematics Learning Study
Committee, Center for Education, Division of Behavioral and Social Sciences and
Education. Washington, DC: National Academy Press.
Neuroskeptic (2016). Your brain on maths: Education neurononsense revisited.
Downloaded from http://blogs.discovermagazine.com/neuroskeptic/2016/03/26/brain-
on-maths-educational-neurononsense/#.VxSzM3pgjBw on 14th April 2016.
OECD. (2002). Understanding the Brain:Towards a New Learning Science. Paris:
OECD Publications.
Paunesku, D., Walton, G. M., Romero, C., Smith, E. N., Yeager, D. S., & Dweck, C.
S. (2015). Mind-Set Interventions Are a Scalable Treatment for Academic
Underachievement. Psychological Science. doi:10.1177/0956797615571017
Rienzo, C., Wolfe, H., & Wilkinson, D. (2015). Changing mindsets: Evaluation report
and executive summary. Downloaded from
https://v1.educationendowmentfoundation.org.uk/uploads/pdf/Changing_Mindsets.pd
f on 18th April 2016 on 9th April 2016.
Rittle-Johnson, B., Siegler, S., & Alibali, M. (2001). Developing Conceptual
Understanding and Procedural Skill in Mathematics: An iterative process. Journal of
Educational Psychology, 93, 346-362
Samaritans (2013). Media guidelines for reporting suicide. Downloaded from
http://www.samaritans.org/sites/default/files/kcfinder/branches/branch-
96/files/Samaritans%20Media%20Guidelines%20UK%202013%20ARTWORK
%20v2%20web.pdf on 2nd April 2016
Suicide by children and young people in England. National Confidential Inquiry into
Suicide and Homicide by People with Mental Illness (NCISH). Manchester: University
of Manchester, 2016.
Vanbinst, K., Ghesquière, P., & De Smedt, B. (2014). Arithmetic strategy
development and its domain-specific and domain-general cognitive correlates: a
longitudinal study in children with persistent mathematical learning difficulties.
Research in Developmental Disabilties, 35, 3001-3013.
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