* Stereotype threat.
* High school students thinking "I was good at math in 3rd grade, so I need to be good at math now."/"I was bad at math in 3rd grade, so I will be bad at math now." while not taking into account the differences in the subject and themselves over that time.
* Students never being told that math is actually difficult and requires perseverance, encountering a struggle for the first time in math, and taking it as a personal failing rather than the expected order of things.
By far, the most common reason I see it is how they were treated by math teachers. One student had a teacher who made students go up to the board and write their incorrect solutions on the board, and they had to stay there until they "figured it out". Many have been told they were stupid, or told they would just never be able to learn. That experience is way more common than many teachers and administrators want to admit.
There's also the pacing issue. Many students get lost in a math class at some point in their early learning career, and then there's no real way to get caught back up. This compounds with teachers who don't really understand math, who are just teaching algorithms and rules and don't know what to do with students who are lost.
I've had good success just by meeting each student where they're at, and letting each student progress from where they are. I always look for what students do understand, and build on that understanding. It typically takes students one or two classes being treated well to start really recovering from that earlier anxiety.
Following pupils pace is a failing of mass education (in the UK) in general I feel. At primary school I was disengaged from maths because it was so facile.
To some extent I've seen that with my kids too -- for example entering nursery they do counting to 3 (!). But my youngster already counts beyond 20.
We sought to use Flexi-schooling to mitigate this effect but unfortunately it's down to individual schools and our headteacher isn't interested (I think that contradicts the Education Act, in spirit at least).
Very true. When we started doing algebra I simply didn't get it until my older sister sat with me. After a few weeks I went from total failure to saying "algebra is easy". Without my sister I probably would never have caught up.
This happened to my sister (now an AE with several graduate degrees). It set her back years, even though she could do the math she believed she couldn't. This caused her to perform poorly on exams. Sometime in high school she started performing better, but it was in college when she finally realized that she was good at it. That particular teacher was fired before the year was up, but that doesn't help the kids she hurt.
I think this bullet point overlooks other factors contributing to the idea that the math anxiety is a personal failing.
For example, kids doing poorly in math observe other kids that grasp the concepts quickly and don't fall behind. Some get A's without struggling.
The article/study of this thread is about primary & secondary school and at that young age, the math of Algebra, Trigonometry, etc was not difficult for me. It was tedious (e.g. endless manual matrix multiplication exercises) -- but it was never difficult. Some kids really do struggle with basic topics like fractions, percentages, reciprocals, etc.
The first time math was "difficult" was Calculus I. (What's this delta-epsilon proof of a limit all about and why do I care?)
Yes, one has to practice math but kids are not stupid in that they see some of their peers master it easily. The kids understandably conclude that their brain isn't optimized to easily understand math. Explaining to those slower kids that "math is difficult therefore you must practice more" actually increases their math anxiety.
* The kids who appear not to struggle may be struggling at home the night before, or working on math a lot more.
* Math will become difficult for anyone who pursues it far enough.
One can tell the struggling child that others "may have struggled the night before" but that explanation will not remove the math anxiety because it will not convincingly explain all the performance differences the child sees.
For example, in the USA, a primary/secondary teacher will lecture on a new math concept on the whiteboard and there will often be ~15 minutes left over to start the homework assignment. The kids "good at math" get started right away without any visible struggling -- and may even finish the homework before the bell is rung. The slower kids become self-aware that they need the teacher to help them more often. There is no "previous night" to hide a struggle. The slower kids see immediately that some of their peers really do pick up on new math concepts easily without practice. It's their comparisons of their ability to their peers that triggers some of that math anxiety.
I remember a test in high school where the bonus question was a delta epsilon proof.
It was the only question I got full marks on.
On all the other 'easy' ones where you had to memorize the derivatives of fundamental functions (sin, cos tan...) I made many mistakes.
The teacher wrote: "You understood the most difficult part, as usual..."
I got poor marks on the test, but it's one of my proudest memories as a math student.
For the non-math subjects, some people do say "I'm bad at spelling", or "I'm bad at grammar". At a higher level of language skills, some people say "I'm bad at writing speeches or wordsmithing an email/Powerpoint to explain things".
Personally, I've always been weak at grammar. Intellectually, I know the difference between "whom" and "who" but I inevitably forget the extra "m" in the appropriate spots. It's not a matter of insufficient practice either. I've been practicing English for decades since I was in diapers! On the other hand, I never forget the minus sign for the derivative of "cosine(x)" being "-sin(x)".
It's like we're forcing children to learn how to use an astrolabe and then blaming them for finding it complex and inapplicable to their daily lives. Google maps provides the answers they want with a better user experience. I would have astrolabe anxiety, I don't have Google maps anxiety.
Unfortunately there is a pseudo-religious connotation to mathematics. It is "pure", "beautiful", etc. Teachers rarely encourage students to change, improve, disrupt, or invent math.
Mathematical notation and pedagogy is continuously evolving and to think that we're at the global optimum right now both ignores the negative reviews users are constantly providing and ignores the history of math itself.
I think that may only be the case at lower levels of math education. If I had to guess, I would say that notational rigidity is enforced by teachers who are themselves not confident in math. It's easier to behave like a cargo cult than it is to attain true understanding.
The higher you get in math (undergrad, postgrad, research math) the less people care about notation. The only thing that matters is that your arguments are clear to a person suitably versed in your field. If you need to make use of some special notation for a one-off purpose, then you simply explain it beforehand.
All of his code used variables like "x", "y", etc.
His code was a steaming pile of shit to review or decipher.
Pedagogy could be improved significantly.
There are people who do.
There was a common theme to some of the problems these students had: if you didn't quickly grasp a concept in the specific way that particular teacher taught it, the class would soon move on to the next topic anyways. And sometimes teachers would not teach one particular thing very well, but it was something the rest of the class's instruction depended on. This left people adrift for the rest of the curriculum. If they took the class over again, they could get stuck at the same point because the class moved at the same pace, often with the same instructor. That's both incredibly frustrating and a huge waste of time and money!
I had no clue what those concepts were for at least a year, but I managed because I always knew I could take both sine and cosine and say "well I need the bigger number in this scenario ∂or the smaller one in that one, so it must be (cos or sin)"
When I saw a picture of a circle and an angle that basically said "Cosine is how far you are horizontally and sine is how far you are vertically" it was so damned simple I hated that I hadn't learned sooner. How did I miss it? Was I sick that day?
For the teaching methods, I'll defer to A Mathematician's Lament by Lockhart. Kids are taught to memorize and often by teachers (esp in grade school) who don't understand the math. When I got my teaching credentials in secondary mathematics, one of the points driven home was that a large factor that led many grade school teachers to avoid secondary (high school) was fear of the math.
Case: my daughter's 7th grade teacher taught the rules of exponents. Part of it, he said anything to the zeroth power is 1 and anything to the oneth power is itself and nobody knows why, it is just one of those things. Fast forward a year or two, and kids are lucky to remember a factiod right. A simple pattern matching exercise would have helped kids rediscover the rule if needed.
Interesting My experience with my (now 11 year old) child was that the focus was heavily on trying to get to an intuitive understanding of how things work with zero memorization.
On the flip side, if you don't know e.g. your times tables by heart, you are in for a shitton of trouble when it comes to doing the higher level stuff. The best way to achieve the arithmetical competence it takes to successfully do math at secondary levels and above is drilling. Educational history is littered with the wreckage of alternative attempts: New Math, some of the approaches derided as "Common Core", etc.
> Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching
Evidence for the superiority of guided instruction is explained in the context of our knowledge of human cognitive architecture, expert–novice differences, and cognitive load. Although un- guided or minimally guided instructional approaches are very popular and intuitively appeal- ing, the point is made that these approaches ignore both the structures that constitute human cognitive architecture and evidence from empirical studies over the past half-century that con- sistently indicate that minimally guided instruction is less effective and less efficient than in- structional approaches that place a strong emphasis on guidance of the student learning pro- cess. The advantage of guidance begins to recede only when learners have sufficiently high prior knowledge to provide “internal” guidance. Recent developments in instructional research and instructional design models that support guidance during instruction are briefly described.
You can develop skills to compensate (e.g., starting with small primes and working up instead) but it's still an extra step in the process.
This does not necessarily affect a student's overall understanding of polynomial functions, but can hang over their head in the actual work with them.
Mathematics from there on continued to mostly be the memorizy-algorithmy kind, but with enough of the intuitiony-thinky kind sprinkled in and no clear guide to which was which or even that the teachers were asking us to do something fundamentally different that anyone who hit a big speed-bump at factoring didn't have a hope of keeping up.
[0, EDIT] Maybe 7th or 8th? It's been too long, IDK.
From where I sit the mountains between me and deep mathematical understanding sure look like they're mostly memorization and drilling until various identities and mechanical operations are second-nature. Some of those peaks are surely "ah ha" moments or creative thinking or whatever but... not most of it. Maybe on the far side, but I can't even see that far. And damn is that far side ever removed from what 99% of people need math for. Not the way a literature class might be, but the way a Latin or Attic Greek language class might be. It's that far off and round-about and well into diminishing-returns territory.
School districts in the US answer this in a couple different ways. Many define the number of credits you need to earn, and many define the level you need to reach. But that's compounded by all the issues in math education that occur before high school. How do you hold everyone to the same bar if some are coming in with elementary skills and some are coming in already beyond high school? Many systems don't serve either group well, and the effects can be devastating to people in both groups. How many of us on HN had high-level math skills and had to endure years of boring required math classes?
I had an interesting exchange with another math teacher once. My school requires everyone to move forward from where they're at, and to reach a skill level appropriate to your post-high school goals. Many students end up just becoming well-grounded in Algebra 1, which is great. Another school in my district requires students to get through Algebra 2. We were discussing this discrepancy at a district meeting, and I was being looked down upon for "only" bringing some students to Algebra 1 level skills. I looked this teacher in the eye and asked, "How does a D- in your Algebra 2 class compare to a B- in my Algebra 1 class?" (They have to reach B level work to earn credit, which is why we are okay with them staying in Algebra 1 work if they need to.)
His response? "Well, at least they've seen Algebra 2 level work before they leave high school." The arrogance and uselessness of that statement echoed through the room that day. I would love to see this conversation play out more around the country.
His book covers what he used to think and do in his earlier years as a (successful) teacher, what he has learned since then from academic studies and other teachers, and how applying these principles and techniques has improved his students' learning.
"How children fail" is another excellent book on how not to teach maths.
 - https://en.wikipedia.org/wiki/Radian#Conversions
I noticed a particularly odd example of it with my mother, who "hates maths" but has worked with finances all her life. If you ask her what 10% of 200,000 is, she'll say she doesn't know. If you ask her what sort of return she'd expect on a $200k investment she'll say, no shit, "you'd want 10% per year, so... $20,000.".
It's not that she's incapable of doing it. Her mind just shuts down when faced with the question.
Underlying this is the fact that teaching is a discipline by itself. Understanding the material is maybe halfway to quality teaching.
I'm 31 years old now with my BA in Computer Information Systems and I still don't know all of my multiplication problems from 1's to 12's, and have difficulty solving basic algebra problems.
I've always stated that I hated math, and that schools are always forcing students to learn certain types of math just to pass/graduate. Instead I think the schools should focus on math that the students can immediately start applying to their daily lives and jobs; no child working at a fast-food chain is going to even try to apply calculus to their job.
My issue is that not everyone learns or interprets the language the same. Some like myself take longer to process a problem to form a solution, but in school you're on a timed schedule so people like me struggle.
Also, some teachers only teach the way they learned and understood the language and don't really care or think that other students may learn from different ways/perspectives.
I know I can learn any type of math, I just need it explained and demonstrated to me in simple english that even a 5th grader could understand and not with college words that I need a dictionary to comprehend.
I secretly deep down want to be great at math and to learn and master, algebra, trigonometry, geometry, calculus, etc. so that I feel competent in the computer industry to solve basic and more advanced problems. But instead, I am reclusive and feel dumb and that I will never be great at any type of math discipline.
But people could also be experiencing this for any subject. I don't see why this is limited to math.
I think my math anxiety started in high school as a result of teachers moving through material too quickly in a learning style that didn't match up with how I learn. Once you get a little bit behind it can be really hard to find your way back in.
Millers were commonly demonized and scalegoated for neither doing honest work with the rest of the peasants and instead maintaining "strange" machinery that did the work for them and they worked with knowledge but neither as a part of clergy nor nobility. Worse yet knowledge outside of known and respected sources.
Merchants, owing to astoundingly bad economic theories were considered to have to be engaging in fraud because the idea of the value of a fur coat being higher in the mountains than the desert never occured to those not dealing with the bleeding obvious of the profession. They were also widely considered "non-productive" - without irony given nobility.
Impetus theory is another sign of the disdain for the practical - how long upperclass and "learned" men would use a theory of projectiles in triangular motions that any artilleryman or archer would burst out laughing at from how obviously wrong it is.
One common factor in even more modern times is ignorance and its self-perpetuation. People fear what they don't understand and avoid it leading to not understanding and more fear.
It probably isn't just from this feudal history and the causal link may be tenuous but the tendencies have been around for a long time.
"The UK is facing a maths crisis: according to a 2014 report from National Numeracy, four out of five adults have low functional mathematics skills compared to fewer than half of UK adults having low functional literacy levels."
And yet the pass rate at GCSE and previously GCE O Level has been fairly stable at roughly two thirds passing give or take a couple of percent for 60 years...
Downloading report now.
=== Summary of Key Findings ===
o We conducted a literature review into the long-established relationship between maths anxiety and performance (those with higher maths anxiety tend to have poorer maths performance). We conclude that this is likely because anxiety interferes with performance and poorer performance increases anxiety, acting as a vicious circle.
o In our large sample of British children, we investigated the relationship between maths anxiety and developmental dyscalculia. We found that whilst more dyscalculics than typical children met criteria for maths anxiety, the majority of those with maths anxiety had normal performanc.
o In a separate group of Italian children, we participated in research looking at developmental change, gender differences and specificity of maths anxiety. We found that unlike general anxiety, maths anxiety increases with age. The relationship between maths anxiety and performance becomes more specific with age – in younger, but not older, children, this relationship disappears after accounting for general anxiety. See Maths anxiety: Gender differences, developmental change and anxiety specificity for more details.
o We have identified, in our large British sample, anxiety subgroups. These may increase in complexity with age. In our secondary school students, we found that those with anxiety specific to academia (high maths and test anxiety) had poorer performance than those with higher, but less specific, anxiety. We conclude that this may reflect a dual path in anxiety development and maintenance.
o In our smaller subsample of British students, with whom we conducted further testing, we looked at the relationship between various cognitive variables and maths performance. It seems that a myriad of factors are associated with maths performance, but that basic numerical processing is not (unpublished data).
o In another Italian sample, we investigated specific memory subtypes and their relationship with maths anxiety and dyscalculia. Whereas maths anxiety appears to be associated with a deficit in verbal working memory and perhaps also visuospatial working memory, dyscalculia is associated with deficits in visuospatial memory; both short-term and working memory are affected.
o Our qualitative research has shown that children of 9-10 years are able to discuss their experiences and origins of mathematics anxiety. Mathematically anxious children seemed to describe negative events with less contextualisation. They were also more likely to discuss physical sensations in their maths classes and clearly articulated some of the negative consequences of maths anxiety.
=== Conclusions ===
Each of the completed projects within our study further reveals the complex, multifaceted nature of mathematics anxiety. It is likely that mathematics anxiety is not a simple construct with only one cause – rather, it can emerge as a result of multiple predisposing factors including gender, cognitive abilities and general predisposition towards anxiety, rumination or panicking under pressure. This helps to explain why mathematics anxiety is robustly correlated to a small degree with many constructs (e.g. test anxiety, general anxiety and mathematics ability). We have clearly shown that emotional and cognitive mathematics problems dissociate and therefore require different intervention strategies. Our qualitative analysis of structured interviews suggests that children as young as 9 are experts in their own experiences in mathematics and this can be harnessed to further understand the thought processes underlying maths anxiety. This brings us closer to design effective prevention and remediation programs for mathematics anxiety.
=== Recommendations ===
o The 9-item modified Abbreviated Mathematics Anxiety (mAMAS) scale developed by this project proved to be a reliable tool for investigating math anxiety in school context.
o Teachers need to be conscious that individuals' maths anxiety likely affects their mathematics performance.
o Teachers and parents need to be conscious of the fact that their own mathematics anxiety might influence student mathematics anxiety and that gendered stereotypes about mathematics suitability and ability might drive to some degree the gender gap in maths performance.
o Hence, for parents and teachers, tackling their own anxieties and belief systems in mathematics might be the first step to helping their children or students.
o With our research showing that maths anxiety is present from a young age and goes through significant developmental change, we suggest focusing further research on how maths anxiety can be best remediated before any strong link with performance begins to emerge.
o The qualitative part of our research shows that children are able to verbalise the suffering that mathematics anxiety causes them. Our qualitative research also points to several potential causes of maths anxiety that could be focused upon by further research.
o Teacher training should clearly highlight the role of both cognitive and affective factors behind maths learning in schools.
o Policy makers should be conscious that emotional blocks can have substantial impact on learning potential.
o Emotional and cognitive problems require completely different interventions.
I like the part about "for parents and teachers, tackling their own anxieties and belief systems in mathematics might be the first step to helping their children or students," which I think is super important.