The transcripts included in the research paper "Mathematics in the Home" (PDF: http://www.researchgate.net/profile/Nicole_Else-Quest/public...) are really telling. Parents who are comfortable with math will tend to ask their kids questions and give little hints when they're stuck, whereas parents that are uncomfortable with math will tend to just blurt out the solution whenever things are taking too long, or encourage strict adherence to one particular heuristic for finding a solution. Over the long haul, the result is that mathematics will start to look like just one long arbitrary list of rules. Of course, that's no fun, and so kids of mathophobes become mathophobes themselves.
Plenty of help available, though, for people who do want to help their kids: "Help Your Kids with Math: A visual problem solver for kids and parents", "Math Power: How to Help Your Child Love Math, Even If You Don't", "What Can I Do to Help My Child with Math When I Don't Know Any Myself?"
This condemns children to repeat the cycle. They (completely rationally) decide that they're just 'not math people' because they can't follow lists of arbitrary seeming steps to do something that no one they have ever met ever actually does in daily life. And some of them grow up to be teachers themselves.
Sigh, your statement reminds me of one time when I was in 6th grade geometry and was asked to complete this one proof. The problem was intended to get us to apply a concept we had been taught verbatim, but I couldn't remember it well and so I completed the proof a different way by extrapolating other basic concepts we had gone over into a method we had not been taught. I remember feeling very proud of myself when I finished, as I had created something which in my 12 year old eyes was a "novel proof". I ended up being given 0 points on the problem because I did not complete my proof using cookie cutter methods we had been taught (even though my proof was correct and it was not specified that I use a particular concept in the instructions).
I remember complaining to my teacher about how this was unfair and she kept saying something akin to "well you didn't do the problem the right way". The incident pretty much killed any interest I had in math, which wasn't rekindled until I stumbled upon machine learning recently in my 20s.
There are a lot of other people out there who will unknowingly be very discouraging, to both children and adults, but their opinions have no bearing on our ability to learn. I remember being in music class in grade 1 and it was my absolute favorite class. Near the end of the year I was told that I wasn't very good at singing by my teacher. I never tried to sing again, but I learned later that I have good pitch and a decent voice. I am not really interested in singing now, but I might have learned a different set of skills if I didn't avoid it.
Is there a way that I could have been prepared so that being told by the person I looked up to for validation, that I was no good wouldn't make me stop trying to improve?
I don't know about that, but if there's a way to teach that, it would definitely be a good lesson. It would certainly be nice if teachers didn't do things like that, but even if you could guarantee they didn't, you won't stop the bullies or the critics or anyone other hater as you get older.
To the grandparent: I don't think getting upset is at all irrational.
Not helping matters is the fact that most of the reformers are themselves ignorant chuckleheads who identify the heuristics that skilled arithmeticians use and codify them as a NEW set of rules to be memorized...
Failure to learn the standard algorithms impedes progress. You are unlikely to do well in the next course if you are slow and can't handle numerous cases.
Learning alternate algorithms is great... once you have the standard algorithms down solid and are ready to prove equivalence.
Have a look at the sample chapter from David Tall's book...
That said, I've definitely seen some of that "let's just replace method X with method Y" – how harmful that is depends on whether it is just presented as a default or as The New Standard Algorithm, I guess.
Of course it isn't, because such lunacy wouldn't be supported by competent research. But the people who do education research, the administrators making curriculum decisions, and the concerned parents trolling about the dangers of "Common Core" on Facebook are distinct sets with little overlap.
I think physically intuitive math and abstract math should be explicitly identified and treated differently in curricula. Else, the result will be clumsy contrived attempts to find physical relevance to abstract concepts and weaker students falling behind. Dreary contrived problems must be replaced with fun puzzles. Puzzles share the attribute of abstractness-and-apparent-uselessness with portions of the math curriculum.
Somehow teachers and parents find it hard to accept that some topics in math curricula are purely in the abstract realm.
The other meta problem and perhaps a big cause of math anxiety is that the curriculum and system doesn't have much wiggle room to fail at understanding something. The most important quality for success in later life is perseverance when you're stuck or when you hit a wall. The education system teaches the exact opposite: "it is not ok to hit a wall or fail to understand something, you have to learn this right now, else you're done for!" There is very little flexibility to allow students to fail, persevere, and repeat. This is worsened by the needlessly frenetic pace of the academic year.
Edit: D'oh, got 3/4 + 1/7 wrong, it should be 25/28
My favourites are the Rob Eastaway / Mike Askew books:
(Amazon: Please have some really easy way to give me a clean url for sharing. I am literally sending potential customers to your website.)
For the .co.uk links, concatenate:
"amazon.co.uk/dp/",[10 character string in URL, aka ASIN]
So, your first link would be
I don't know how to draw a multiplication (and I think I'm not alone), but get some physical tokens that you can divide in 2 or 3 (like matches), and you can do all kinds of fraction multiplication and division with them.
I found out later that she hated math, and thought she was bad at it (she really wasn't, but that's another issue entirely...). However, she never let me see that, which was fantastic.
Math talent, like other intelligence, is inheritable. So until you've pried apart nature and nurture, the simplest assumption is that these kids are bad at math because they don't have much talent for it.
Google Turkheimer's Three Laws of Behavior Genetics for more on this often neglected field.
Did my recent ancestor who failed high school classes have math talent or not? And did my ancestors who didn't have a high school at all but could build a barn that still stands one hundred years later have math talent or not? How can we even talk about whether this thing is inheritable if we don't define it well and it manifests in such different ways in different environments?
The simplest assumption is that these kids are bad at math because math in our elementary and middle schools sucks! Look at the stats for immigrants to the US. If you came from impoverished places without educational opportunities, educational attainment rises over time. If you are a kid who comes from Hungary or Romania or Russia or South Korea, you'll be so much better at math than the US kids that it's not even funny. If your grandparents came from Hungary or Romania or South Korea and kicked ass at math, but your family has been in the US for a few generations, you'll suck as much as any other average American. If you're going to push heritability, you're going to have to face the conclusion that the water and air in America prompt genetic mutations that make people bad at math over a few generations. (I'm so glad I'm first-gen in part, but I fear for potential children.)
"The more the math-anxious parents tried to work with their children, the worse their children did in math, slipping more than a third of a grade level behind their peers."
The implication is that if the math-anxious parents did not work with their children as much, these children did better.
That's useful "know your limitations" information, but I don't see how it separates nature vs nurture effects.
Does this work the same way for parents who aren't comfortable talking or reasoning about (for example) history? Or literature? Or is math a special case that works differently from other subjects taught in schools?
There's a useful note in the article that maths-anxious parents didn't affect their children's reading ability. But it'd be fascinating to know whether (for example) poorly-literate parents have an effect on their children's reading ability, independent of other factors.
Really anything is - we've all seen it extend as far as prejudices. However, the degree of the influence probably varies wildly and I'd posit that "internal stress/stressors" (such as anxieties, fears) could have the largest overall influence.
Definitely. As a parent, you have to be careful not to let your own fears or anxiety influence your children. I'm sure it's a good evolutionary trait -- children who afraid of the same things as their parents probably survive longer in the wild. But you don't want pass along any irrational fears.
OK, assume they found a negative correlation between grades and the amount of help of the parents. I'm not buying this explanation. Sure, if your kid has worse grades then you help him/her more on the critical subject.
> we tested the interaction between parents’ math anxiety and the frequency of parents’ homework help while controlling for students’ grade, gender, beginning-of-year math achievement, and beginning-of-year math anxiety
So the study controls for baseline performance. It's definitely possible that some selection bias remains, but it looks like a reasonably well-executed study on a large sample (438 children) and I wouldn't discount the results out of hand.
My wife is more of a visual hands on learner and suffered through the same text books. In college she had a series of bad professors (like, cancelling class because the professor had a hangover bad) and this just cemented her view that she was bad at math.
All that long exposition is to say that people learn different ways and a "bad" method may result in a good outcome. Learning is a complex interplay of teacher, student, material and presentation that can be different for every person.
When I teach my kids I like to have them all in the same lesson even though they are different ages and doing different levels of work. The primary purpose of this is to just talk about math, make it normal to think and reason about it and show that it's not scary. Math concepts are often taught as a series of ever increasing obstacles to be jumped over. You jump over them until you reach a ten foot wall that you just can't summon the mental power to leap over. Then you are bad at math.
But math is more like a hike. you start out in the foothills where even your 5 year old can keep up with the rest of the family. Each footstep is a little closer to the peak. You are making progress and learning even if you only make it to the first mile marker. You get fitter and accumulate some equipment and technical skills for the mountaineering sections higher up as you go. It's challenging but fun and you have a sense of accomplishment no matter how high you are. As you get higher the vistas opened up to you show you the world in ways you could never see lower down and give you motivation to go forward towards higher peaks and better views.
Particularly for things like manipulating and simplifying systems of equations, the Saxon program was great, simply because it was built on repetition and practice - you would work through dozens of problems of a particular category over time. The other thing that was very nice about the Saxon program was that it would revisit older concepts - the problem sets would not just be on the current chapter concepts, you'd also have to work through material that had been covered over the previous 15-20 chapters. My mother is a special education teacher, and she's had success even with children that have serious memory deficits, because the repetition will pound the knowledge through into short, medium and long-term memory.
Moreover, most people have it exactly backwards... to the extent passing epigenetics to your children makes sense at all from an evolutionary perspective, it is passing adaptations, not weaknesses. If a parent has a hard time finding food, the stress of being hungry passes on to a child to make them smaller, i.e., less needing of food because they're not going to try to grow as much. Epigenetics is not "If you break a leg, your child will be born with a broken leg". That's just Lamarkianism, and Lamarkianism is still false.
While I don't think epigenetic math anxiety is particularly plausible (for, basically, the "math receptor" reason you state earlier), an adaptation in general may turn out to be a weakness in a particular case. If, say, it was possible to epigenetically pass on stress triggers (an "adaptation", because having a stronger, quicker fight-or-flight response to a particular environmental condition that is a frequent stress source might generally be an advantage) this might manifest as a weakness in some particular cases. Now, because its hard to believe that there are "math receptors", it seems unlikely that even if this was possible in some general sense, that it would be possible in some way where "math" was the specific sensitized trigger, but if it was...
Is this really a thing? Are there adults (except for those in the < 2nd std dev range, who are near-but-not-quite-at mental disability-level) who cannot calculate the amount of paint needed to paint a room?
For some people who are weak at math, it even becomes a badge of honor: "yeah, um, sure, we could calculate how we want to split the bill, but do we really want to be such nerds about it?"
Studies abound, too, showing that even grade school maths teachers often eff up when quizzed on pre-algebra.
A better example of following the rules without understanding would be percentages. Many people could tell you what buttons they push to answer "What is 37% of 240?" But many people would be lost if you asked them "What is 84 as a percentage of 270?"
Or "This product now costs £230. It has had a 20% discount applied. How much did it used to cost, before the discount?"
People I went to school could definitely work out the paint or percentage problems, but for a really good majority of them: only because they faced those exact questions in exams/tests.
A number of people can't do this.
One of the interesting hard mathematical results to come out of generalized learning theory, as used by machine learning, is that there is a fundamental relationship between the speed of feedback and the ability to learn. It is not merely a matter of "willpower" to learn from temporally distant feedback, it is fundamentally, irreducibly harder, and there's no way to "correct" it because it's not even wrong.
Homework grading (note the emphasis) as a learning mechanism has such a delay that its learning rate is effectively zero. And what feedback you get is often not even very much beyond WRONG!, which is not exactly rich with information itself.
Homework is only useful for the feedback and learning you get while doing it, and for the ability to assess the student because we live & die on assessments, give us a direct choice between assessment and learning and the system will generally prefer the former. So, I won't say "homework is useless" but we could save everybody a lot of time if we only checked that it was done and spot-graded it for assessment, rather than making the teacher laboriously, yet nearly entirely uselessly, grade stacks of homework.
If a kids approach to homework is to try everything and leave the answers he can't do for mum and dad, he'll be crippled in his learning. The point of homework isn't to get 100%, it's to learn.
For one thing (and this is probably most damning), classes are highly regulated by school administrators. Managers who try to hire obedient teachers. The few teachers who try to slip through are nevertheless horrifically constrainted.
Children want to learn; it's a deep human motivation that has to be beaten out of people. The adults I teach have to be scrubbed of so many educational scars and bad habits. If only I had time right now to go into depth of all the antipatterns, the symptoms.