I recently had an interesting conversation with a high school student and it gave me a lot of insight – not only into how she and her friends view their school but also how aware they are of the fact that they are responsible for shaping their future. This group of students are currently in grade 9 in a prominent public school in Gauteng with a rich history of academic, cultural and sports achievements with the ability to inspire unwavering loyalty in their students. She and her friends were starting to question aspects of their loyalty to the school. What interested me about this particular conversation was the fact that they were disillusioned with the way in which the school deals with “underachievers” in Mathematics. This came about after they had had a conversation with a former pupil at the school – now at university – who warned them to not allow the school to force them to change subjects from Mathematics to Mathematical Literacy in grade 10. The advice from the university student was that even if their marks were not quite in line with school expectations, they should never allow the school to limit their future study options in this way. Their (wise) consensus was that they should all keep their long term goals in mind, and not allow the school to influence any of them into choosing Mathematical Literacy if there were any chance that it may limit their study opportunities.
A more cynical view of this practice to “strongly encourage” students to choose the perceived “easier” Mathematical Literacy at the end of grade 9 , would be to assume that this school, and many others, are only concerned about their own matric pass rate and matric average with little regard for students’ best interests. There is another way to look at this, though, and it ties into the article by professor Jo Boaler (Stanford University), Unlocking Children’s Math Potential: 5 Research Results to Transform Math Learning
“There is a huge elephant standing in most math classrooms, it is the idea that only some students can do well in math. Students believe it, parents believe and teachers believe it. The myth that math is a gift that some students have and some do not, is one of the most damaging ideas that pervades education in the US and that stands in the way of students’ math achievement.”
The Science that challenges the “math is a gift” myth…
Because brains change, it is possible to learn skills not previously mastered, including Mathematics (Neuroplasticity)
Neuroplasticity is the ability of the brain to grow new neural pathways. It happens when we are exposed to new knowledge, have new experiences and master new skills. As these new neural pathways form, we are increasingly able to complete tasks (like mathematics problems) which we were unable to master before. As Boaler puts it, this means that students’ brains “can adapt and grow in response to any learning opportunity, and ideas that some students are not capable of learning high-level content should be rejected.” This is a very strong claim and it means that any student is capable of mastering school mathematics. Many teachers and parents will probably disagree based on anecdotal evidence and their own “experiences”, but if we take neuroplasticity seriously we have to accept its validity.
If this is the case, some may argue/ask why “real life” and “real classroom experiences” seem to confirm the “common sense” view, namely that some kids have a gift for Mathematics and some not. Perhaps some of the following may shed some light on this question.
Belief about ability determines success, irrespective of ability (including Mathematics)
The work of Carol Dweck on mindset is well-known by now. According to her research, we all have implicit theories of intelligence (TOI) which determine how we react to learning and challenges. For some, their TOI implies a fixed mindset and for others a growth mindset. Significantly, her research has shown that our success is more a result of our mindset and effort than of our genes. Simply put, if students believe they have a limited, fixed capability for Mathematics they will perform accordingly. However, the opposite will happen if they have a growth mindset where they believe that through practice and effort they are able to overcome obstacles. Again, the research on this is sound, but the problem is that if parents and teachers do not know or believe this they will use test results to judge students’ capabilities, ultimately leading to practices like sidelining students from Mathematics to Mathematical Literacy.
Mistakes and Struggles Lead to Success and do not indicate a lack of capability
This point actually ties into the previous one on growth mindset, but it is based on separate research that shows students with a growth mindset react differently to mistakes than those with a fixed mindset. Students, especially those with a growth mindset, learn and master content as they make mistakes and struggle. The authors remark: “We have therefore shown that growth-minded individuals are characterized by superior functionality of a very basic self-monitoring and control system.” This article shows clearly that “growth mindset” is not simply a “touchy-feely” concept, but that it has an actual impact on brain activity – in a measurable way. In order to get students to grow their capabilities, we need to provide them with open-ended problems where they do not “mainly get it correct.” It ties in closely with what Bjork (1994) calls desirable difficulties, and the fact that students seem to learn better and understand more deeply if they are presented with work that provides significant challenges.
What is interesting about this is that research has shown that it is often students who do well in typical (not necessarily open-ended) Mathematics problems that have a fixed mindset. These students are then often caught out later (at university) when problems are not quite as straight forward as those they had to complete at school. The challenge for Mathematics teachers and parents is to not get fooled into believing that students who only get their “math right” after multiple attempts are not quite as capable as those who get it right the first time. It is likely that it will turn out later that the “slower” learners actually grasped their Mathematics at a deeper level than their counterparts.
Speedy completion of Mathematics problems does not imply superior capability
The prevailing view amongst parents and teachers alike is that “quick = clever” and “slow = not so clever.” This is true for most subjects, but especially so for Mathematics, with many tests or quizzes designed to force students to provide their answers as quickly as possible. However, this is a completely mistaken view of reality, and perhaps one of the most well-known examples to the contrary is the 1950 Fields medal winner for Mathematics (the Fields medal for Mathematics is the equivalent of the Nobel prize), Laurent Schwartz, who famously remarked in his autobiography about the fact that he was always slow and is to this day (as quoted by Boaler):
“I was always deeply uncertain about my own intellectual capacity; I thought I was unintelligent. And it is true that I was, and still am, rather slow. I need time to seize things because I always need to understand them fully. Towards the end of the eleventh grade, I secretly thought of myself as stupid. I worried about this for a long time. I’m still just as slow. At the end of the eleventh grade, I took the measure of the situation and came to the conclusion that rapidity doesn’t have a precise relation to intelligence. What is important is to deeply understand things and their relations to each other. This is where intelligence lies. The fact of being quick or slow isn’t really relevant.”
Slowness is not an indication of a lack of intelligence or capability, but it could easily appear that way: We know from recent research that stress has an extremely negative impact on student performance. When students are put under too much stress – for example putting too much emphasis on timed tests – they will be unable to recall work that they actually know, or struggle to perform tasks that they are actually quite capable of completing successfully otherwise. The end result is a skewed view of many students’ capabilities, simply because they are not “fast enough.”
Parental and Classroom Language can make or break students
When cognitive researchers speak about the power of language, they do not refer to the obvious examples of derogatory language that are certainly harmful. Rather the focus here is on well-intentioned language that is nevertheless harmful because it conveys and perpetuates a mistaken view of intelligence – again mostly tied to fixed mindsets. For example, Dweck and colleagues (1998) have shown that praising students for being “smart” rather than for working hard had the exact opposite effect of what one might expect. When these students had the opportunity to choose between a difficult or easy task after the “smart” or “hard work” comments, more than 90% of the “smart-labelled” group chose the easy task, while the majority of the “hard-work” labelled group chose the difficult task.
In a 2012 study by Yeager and others, they divided high school students into two groups with both groups receiving critical feedback after a test, but the feedback from one group (without the teachers knowing the identity of the students) included the sentence “I am giving you this feedback because I believe in you.” A year later those students performed better in tests than those who did not have this sentence added to their feedback… Language matters and this is just as true for Mathematics as for any other subject. The point is, are we willing to accept the research and start acting accordingly?
But what about Common Sense?
For most of us, the idea that anyone can master mathematics goes completely against our common sense, understanding, and experience of the world. It just does not seem to tie in with what we “know” from experience to be true. However, the preceding discussion is a good example of science challenging our common sense understanding of the world, and if we are unwilling to accept the research, it simply shows to us our own difficulty in changing deeply ingrained beliefs. However, if we really have our students’ best interests at heart, we will take the research seriously and ask ourselves how we can change our teaching, our language, and our own mindsets in order to ensure that no student is forced to change to Mathematical Literacy when they are entirely capable of completing normal Mathematics, even if it is at the risk of the school forfeiting its 100% perfect matric record…
Dr. J (Lieb) Liebenberg is a Research Fellow at the Department of Informatics at the University of Pretoria. He has been involved in learning research and development since 2006 and has delivered academic papers on e-learning as well as published on the subject in peer-reviewed journals. Dr Liebenberg is the Chairman of the Optimi Academic Council.
His first mobile learning project was MobiMath which provided Grade 10–12 Mathematics learners with videos and assessments on mobile phones.
As Project Director for the University of Pretoria’s Health Information Systems, Data Capturing Training for the National Department of Health, he was also responsible for the introduction and use of mobiles for post-training support to more than 2500 learners throughout South Africa.
Since 2010, Dr. Liebenberg has been involved in the conceptualisation and development of the ITSI Solution which allows teachers and schools to optimise teaching and learning for the 21st-century. The solution is used by more than 220 schools and thousands of teachers and learners from both the private and public sectors.
He is passionate about connecting technology and the learning brain, making learning visible and removing fragmentation from the classroom in an effort to prepare students for the challenges of the 21st-century.
He is a member of the International Association for Mobile Learning and regularly participates in conferences internationally and locally.