Monday, January 07, 2013

Maths is a difficult study?


My general observation on our students has been Maths is the most difficult study area. How can i remove this syndrome in our students?

Warren Wolff • You cannot overcome all their reluctance. Frequently, this is because of poor preparation in the earlier years. For boys, it is easier than girls. Creata a scenario in which there is some problem that needs fixing, but you are not sure how. So, you go to the tool box, and meditate over which sequence of which tools should be "attempted". Sometimes, I relate about an engineer I knew who was making a healthy salary to solve several problems. One of them, he had been working on for over 3 years and still had no solution.
           

Jonathan Young-Scaggs • Teddy,
Could you be a bit more specific? What math course are you most challenged by in terms of delivering content in a accessible manner? Are the students "invested?" Math can be difficult because we have always told students that it is a difficult subject, but those of us teaching math ... I guess we never really had that tough a time. Why must our students hear anything negative about the material? We need to say just what we know. With some hard work and a little practice ... math is fun!! In fact it gets even more intriguing as we add complexity. Who would argue?? Calculus is easy ... its the algebra that makes the subject complex. Teach what needs to be done, and make space for the creative elements of student initiative.

Yeah, I know ... I have oversimplified the problem. But "the machine" must be moved forward by any means necessary. What are we to do? I wish our education system would place a much emphasis on math literacy that it places on reading and writing. Then we might not have to start sooo far behind. Catching up is where the real problem resides.

Your thoughts?
           

David Ball •

Maths is a challenge. And it is achievable. A good Maths lesson is a good lesson in any subject, not so different. Maths has the virtue of right, wrong and quirky. People refer to how easy Maths is without considering why people become phobic. Intimidation cows students. But rising to the challenge .. and possessing the arrogance to say to ones self "I don't understand what you mean but I will do this thing that others have" is helpful. I have dealt with students who were so low they couldn't follow simple instructions to solve routine algebra. Their knowledge is poor regarding mental arithmetic. Their problem is not to do with following an algorithm. There is a low percentage of students who by year 10 (15 years old) don't gain anything beneficial from further Math study. Most 'weak' students can benefit from remediation which targets basic 'skills' building their confidence. Online material is available for such remediation. I have run a battery of tests and assignments for building mid range students which has been successful from years 7 to 10.

Marie Kielty • We begin at the earliest years with well trained teachers. There is research published in 2008 in Developmental Psychology, "School Readiness and Later Achievement" by thirteen researchers at nine universities in three countries. (The principal researcher was Greg Duncan from Northwestern who is now at one of the universities in California.) The research showed that it is the math more than the reading that children know at the beginning of kindergarten that is the greater predictor of success by the middle grades. This was referenced in Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity, a publication of the National Research Council.

Teachers of young children typically do not have a good background in math and how to teach it. The National Reserach Council publication mentioned above has listed the trajectories, the sequence, for learning number. Because young children can count, parents and sometimes teachers assume that the children have the same understanding as we adults. They may or they may not.

This is especially true of understanding cardinality. In my work of training teachers I use several phrases in other languages. One of them is "isa dalawa tatlo apat". I ask the group what the phrase means to them, how does it makes them feel and then I ask them to share with those around them. It is eye-opening! The phrase has no meaning! The phrase is "1, 2,3 4" in Tagalog, one of the languages of the Philipines. If children do not understand cardinality, they need to begin counting at one when they add, instead of beginning with the last number and adding on. And the solution is counting on one's fingers into third, fourth, fifth grades and beyond.This has major implications in the children moving toward mental math.

When teachers typically teach addition to young children, they begin with the math terminology "plus", "minus", "equals" without teaching the concept of equality. Again this has inplications into second and third grade when children deal with problems such as, 8 +5 = _____ + 3. The blank means that is where the answer is put. So the problem becomes 8 +5 = 13 +3, in which students ignore the "+3".

Illinois has no endorsement in the teaching of mathematics below the middle grades!
The whole foundation for math has been laid by then!

Richard Catterall • One: Mathematics IS the most difficult area of study because it involves language at one further level of abstraction. There is nothing wrong with noticing that! and
Two: Mathematics can be made enjoyable and therefore students will want to learn even when it is difficult - in fact students thrive on challenge. There is so much richness in the study of mathematics that when the teacher is enthusiastic (and preferably knows plenty of mathematics) the students will want to know what there is to know.
Three: much of the learning in mathematics is sequential, so that if a student has a gap in knowledge, or understanding, or both, the gap needs to be filled before further learning past that point can take place - rather like trying to build a taller building on a weak foundation - and often students are very reluctant to do work that they perceive as "beneath their level". Explaining this and fixing the problem are probably the most difficult parts of teaching mathematics.
Four: New Zealand mathematics education has tried to address some of these challenges recently (in the last 7 or 8 years), and although there are still problems and difficulties the improvement in attitude of students towards mathematics has been significant. Get hold of a copy of the Numeracy Project booklets and encourage your local, regional and national governments to get something similar working there. And good luck! Remember that the probability of success is directly proportional to the amount of inspired work of dedicated teachers!

Richard Catterall • 

PS Something I have found useful in explaining to students why they need to practise exercises once they have grasped a new concept is the parallel between learning mathematics and learning to ride a bicycle (or learning anything actually). There are five distinct stages and many people become stuck at one of those stages and never get to the next: One: When you know you cannot - I cannot ride because I am too small/I cannot do mathematics because my parents tell me they couldn't, so I won't be able to either. Two: When you know you can - I have not tried the bicycle yet but I see my brother riding and he is a person just like me so I know I will be able to learn/I haven't tried this mathematics yet but all those other students learned so I can too. Three: When you try, and fail - I got on the bicycle then fell off and hurt my knees/I tried that problem but got it wrong. Four: When you try, and succeed - I got balance/I got a problem correct on my own. Because this is such a happy time, many people stop there and do not proceed to the fifth stage, which means that if a difficulty arises they fall off/fail again and sometimes drop straight back to stage one! Five: When you "overlearn" - I have done so much practice that now riding is automatic and I can concentrate on the traffic around me/I have done so much practice that now I can tackle new and different problems that are unfamiliar. :)



Seong Kim • Math can be as easy as 1, 2, 3, 4, 5, ... or easier, and it can be as hard as 2, 3, 1, 4, 5, ... or harder.

The same is true, too, for any other area in school discipline as language arts, science, history, etc. And the same is true, also, for any work that needs to be done.

If something is worth it, it is probably not easy, and is usually not free.
$50 can be a big money for many people, but can be a petty cash for many other people.
It is for sure though, no money is easy if it is earned.
And the same is true for math, too.
What matters in education is sincerity, and not efficiency.
           

Dixie Donovan • I teach the students (secondary level) that struggle in math. The number one issue is integers--dealing with the negative numbers. If they can't do that it doesn't matter what we try to teach them as it always falls back on dealing with the negatives. I've put together a research project on this that I will be implementing over the next two months. Hopefully I can find a way to help them understand this very basic but critical concept.

When I started teaching high school I didn't understand why integers was an area being taught. It's taken me eight years to realize that is the crux of the problem in understanding. If you don't believe that, look at your students and try to remove the various issues and get to the core issue. When you do, I bet it's the negative numbers.
                       

David Ball • Good luck Dixie but I think you will find it is more than that. Addressing individual needs is the imperative in classwork because if an individual won't be part of the lesson, you can have no end to dividing the class to achieve the lesson's end. As for dealing with negative integer operations, I find the visual aid of the creation of a table .. 3 +1 = 4, 3 + 2 = 5, 3 + 3 = 6 .. then reverse the table and continue .. 3 + 0 = 3, 3 - 1 = 2 .. etc by doing the table with the class you can illustrate the abstract rules. Some drill work to reinforce it. Make put together jigsaw puzzles out of student made problems .. get the students to make and do them. Paper exercises are still effective.
           

Seong Kim • As we all know, example is the best teacher.
So teaching negative numbers, too, we may want to begin with examples as the ones David showed above, or the ones below:

1 + (-1) = 0, 2 + (-2) = 0, 3 + (-3) = (-3) + 3 = 4 + (-4) = ... = 123 + (-123) = 0, etc.

-7 = (-1) + (-6) = (-2) + (-5) = (-3) + (-4) = (-4) + (-3) = (-5) + (-2) = (-6) + (-1) = 0 + (-7)

Then, for instance, move on to: 2 - 1 = 2 + (-1) = 1 + 1 + (-1) = 1 + 0 = 1.
And then, ask the students to come up with their examples, and then, ask them to say what they mean by a negative number. And if necessary, help them keep refining the definition they come up with until it is close enough or appropriate enough within the knowledge or understanding of the object they have now or at the moment.

Then, move on to the other operations in arithmetic, together with examples, of course.
Experiencing examples, and producing examples, along with correcting the mistakes and discussions on their findings, the students will approach the idea they need to get, and they will get it themselves.

And during the examples and discussions, we can add some spices to their cooking, and the spices are the theoretical facts, terminologies, notations, etc.
Adding those spices, we want to consider, of course, timing and the amount, and should be able to modify the spice in accordance with the students responses or feedback.
           

Rajinikanth Dhakshanamurthi • Build and maintain good relationships with students and Planning regular maths classes to be key skill to remove this syndrome. i believe this will work if only if the teacher is expert in the subject.
   

Jan Olivas • 

Making Math lessons and practice assignments as connected to real life as possible often pulls students back into the "teachable moment". When they see the relevance of what they are doing in the classroom to their lives, the light bulb goes off!






Teresa Katuska • I liked the bicycle analogy- it seems that our struggling students don't actually get to stage two. I think many students (at least in the United States) are influenced by parents and others who are readily willing (and even proud) to say "I can't do math" in a way that they would never say about another subject. I get very upset when I hear a colleague in the English department say this. My math students know that I read a lot, both for pleasure and to learn new things. We talk a good talk about learning "across the curriculum" and yet we tolerate a culture in which people brag about poor math skills. Math IS hard, but so is learning to read, playing the piano, hitting a baseball, or beating a video game. We need, at least from the early grades, to make sure that kids believe they can be "good at math".
           

David Ball • (in answer to a message) .. Yes, but there is a diminution in the analogy which is counter to reality. How the brain grows to pick up and acquire grammar is not perfectly understood. I have met better teachers than I who assure me there are some concepts some students don't get. There are a myriad of possible reasons for it and only some are addressed through basics mastery. A counter example is how the greatest math minds sometimes develop without mastering basics .. eg Ramanujan whose mistakes were as elegant as his proven theories. In some ways, slowing down learning to accommodate a few translates to slowing down everyone and promoting no one. I focus on the middle in big schools because there are always those willing to help the top and bottom. But also systemic resources are best allocated when the middle is the target.

Joseph Ventola • When I was a leave replacement and a substitute the way I had them enjoy and learn was make it realistic to them. Find out their likes and apply the lessons to that. Also, integrate the other subjects to math showing how important math is because it is in pretty much everything.

Kathryn Kozak • 

I teach mathematics at a community college. Most semesters I teach a developmental mathematics class. One thing I have noticed over the years is that students give up if they don't understand the problem right away. I, and my colleagues, believe this may stem from timed tests of math facts in Elementary School. I am now seeing this with my son. He says he is not good at math, because he cannot get his skills done fast. However, the other night he divided 120 in half correctly without even thinking about it. He is actually quite good at math. This may be why we see a gender difference in math, though I have no empirical evidence of this, or my other theory. But we do see students in college give up on a concept very shortly after they try it, if they don't get it right away. I would like to see a reduction in the emphasis in the speed of knowing the math facts, and more of an emphasis in the beauty, excitement, use, and fun of mathematics.


Cliff Cohen • Possibly if we paid math teachers more, more people with a real talent for mathematics would pursue a teaching mathematics. Math is undervalued and math teachers more so, therefore why teach math when there are so many careers that reward talented mathematicians.

Warren Wolff • Sadly, Kathryn, this all a result of the overall concept of a "throw away " and "instant gratification" society. Not sure significant improvement is on the horizon, BUT we just keep working at it.

Lucinda French • 

The use of calculators and computers has made this a very technologically dependent generation of young minds. Challenging to teachers as well as Grand-parents, is the pre-shcooler who is computer dependent. Laying the foundation for advanced thinking in a classroom for children using software technology that was only recently developed is the biggest challenge that I've encountered. Yet the principals taught in mathmatics dated back to ancient times and do not change. My experience has been to excite the student with the history of each topic and link the theory to the applied use. Kids need to see and feel the bicycle before even conceptualizing actually riding it!



Dixie Donovan • Kathryn--Interesting you should talk about the timed tests. My son is brilliant in math but his mind works faster than his hands. Orally he could do lots of math. When it came to the timed tests he was not completing them because he was being careful with his penmanship. Once I realized the test was the same each time, I taught him to memorize the 100 answers so he could concentrate on writing the answers and not doing the math. What did I learn? The timed tests were a load of hooey!

Don't get me wrong, learning by rote can be a good thing. I work with a girl with a very low IQ who can give me the answers to simple multiplication without the calculator. When she gets frustrated or unsure she then relies on the calculator. The point is, she's learned the basics and it had to me memorization and practice. But again, it's when I work with her orally. We don't allow for the students that work better that way. My son could do his math homework faster by dictating it to me.

Writing is good, but I believe knowledge is better. We seem to be willing to forego real learning in order to do things the way they've always been done. It means we as teachers have to be willing to decide what is important and change our methods. But even if we do and the students actually learn, there is a paper and pencil test waiting for them at the end to determine their fate.

Richard Catterall • Dixie - Yes paper and pencil tests are always there waiting; One thing I do with my senior students is to emphasize at the start of the year that my aim for them is for them to learn to think for themselves, and that results in examinations are a bonus, a usual by-product of being able to think mathematcally. After a year with me most of the students "buy in" to this aim, and are more relaxed about the final examination than many of their schoolmates. Many do well; but, more importantly, they mostly take away a confidence in themselves as mathematical problem solvers that serves them well later. I was fortunate to be able to teach my son in 2011 and he gave me feedback that these teaching methods have worked well for him.
One other thing I do, with every class, is to survey at least two (relatively) randomly chosen students (different each time) at the end of every lesson to ask: did you learn something? what? did you enjoy something? what? If I get less than three "yes" answers out of four I know to amend my teaching.

A Cron • Teddy, when can you get the parent to say "Math is easy for me" and I will help you, and they do. The problem has been ingrained into the psyche of your students...it will probably take a few generations of savvy teachers and parents to change the overall attitude...oops this sounds like a downer attitude, but there is no easy solution...

My mother was an educator and my father had a masters in economics, my parents never stated any thing about any subject being difficult for them...so I did not have this ingrained into me, my students do...I had a brilliant math student once, but since the mother and father had trouble with math they discouraged her abilities...this is a problem...
           

Richard Galbraith • Just a short comment, Cliff. Talented mathematicians don't always make talented mathematics teachers.
           

A Cron • I can verify that my first calculus teacher was an honor graduate with a master's from Rice University, very talented...could not connect with students and could not teach at all. Even math majors were failing...
           

David Ball • I love the antecedent belief that many express here that students are blank canvasses on which knowledge is placed. Research doesn't show that. Parents can be helpful. Teachers can be constructive. Students can be proactive. Some don't get it. Delivery can be improved. Instruction can be made more explicit. algorithms can be shortened to a few steps. There is a danger that good students get bored by pre-digested material. Getting back to the original post, my advice is to target the middle bands. Schools that effectively work with the middle tend to have happier, more productive top and bottom ends too .. and better public results.



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