Tuesday, January 15, 2013

Maths difficult?


My general observation on our students has been Maths is the most difficult study area. How can i remove this syndrome in our students?
13 days ago
     
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.
12 days ago
           
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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?
12 days ago
           
           
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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.
12 days ago
           
           
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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!
11 days ago
           
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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!
11 days ago
           
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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. :)
11 days ago
           
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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.
11 days ago
           
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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.
11 days ago
           
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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.
11 days ago
           
           
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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.
11 days ago
           
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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.
10 days ago
           
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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!
10 days ago
           
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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".
10 days ago
           
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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.
10 days ago
           
           
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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.
9 days ago
           
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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.
9 days ago
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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.
9 days ago
           
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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.
9 days ago
           
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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!
9 days ago
           
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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.
9 days ago
           
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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.
8 days ago
           
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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...
8 days ago
           

           
Richard Galbraith • Just a short comment, Cliff. Talented mathematicians don't always make talented mathematics teachers.
8 days ago
           
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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...
8 days ago
           
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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.
8 days ago
           
           
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Seong Kim • By the way, good math teachers are talented mathematicians, too, aren't they? 
It's simply because good math teachers know very well how math works as well as how to teach.
8 days ago
           
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A Cron • But not all talented mathematicians can teach, this does not say a talented math teacher is not a talented mathematician...most of my math instructors were teachers talented in teaching mathematics...
7 days ago
           
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Jonathan Young-Scaggs • @A Cron -- Right talented teaching of the subject of mathematics does not require one to be a "mathematician." But a talented math teacher must pass the passion of math on to their students. When that is achieved then success as gauged by student achievement will silence the critics. Mr Kim (above) understand that the challenge is the inner workings of math. Most of us have seen YouTube videos or late night commercials about tricks to improve children's mathematics knowledge. These are just tricks and memorization techniques. Nothing against memorization, but who needs to know how to multiply 4-digit numbers in their head?
7 days ago
           
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Warren Wolff • If you can learn the tricks as a student, it actually develops confidence and reveals that math CAN be fun. We engineers make decent math teachers too. I think my HS students would agree. Sadly, we math teachers are waaaay misunderstood by many administrators who are the very ones who hate math.
7 days ago
           
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Seong Kim • If you know very well how math works and how to teach, together with the passion of math, you can make a good math teacher, no matter what field you may be in. 
What's more important is though, the fact that the students learn from you, so you want to be a good example yourself. Show and Do yourself first, what the students need to do to see how math works and what they can do with it. 
And in teaching, what matters is sincerity, and not efficiency.
7 days ago
           
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Ilirjan Cane • The fact that mathematics is the most challenging subject in school is not a syndrome; it is a reality. There are, of course, individual variations with regard to time and effort a student spend on studying it, but for each subject such variations exist, and other factors, such as motivation, genetic predisposition, and learning environment also influence these variations. So my question is: Do you really think that math is not the most difficult subject for students? I am afraid that you will have a hard time convincing them about it.
7 days ago
           
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Richard Galbraith • Too general a statement. There are many students for whom math is not the most difficult subject, and, for example, English is.
7 days ago
           
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Cliff Cohen • Actually, the problem is that many parents tell their children that mathematics is hard, worse, some teachers do as well, and worse of all, many students believe them. We who know better need to tell students how exciting mathematics is, and the exciting things that can be done with just a little mathematics. Mathematics part of our lives. Most people can shoot a gun or rifle and hit a target with a little skill and practice; not even thinking about the math involved, but with just a little math, you can hit a target with a cannon or launch a missile hit an unseen target.
7 days ago
           
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Richard Sippel • I agree with Cliff, my parents tell their kids they were not good at math and instill in them that they will not be good at math either. Before they ever step in my room they have decided they will not do well. It is difficult but doable task to get some confidence built up in my students.
7 days ago
           
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Deborah Crichlow • I think the answer to this question is complicated. There is no one answer to this question since there are so many reasons to ask it. For example, is the math hard because students have poor math (including problem-solving,visual literacy, and critical thinking skills), is it hard (for us and them) because they are not motivated, is it hard because the pace is too fast, is it hard because of our style of teaching for the students we have is not effective, or is the math hard because there is no support for students who need remediation? There is no magic bullet. Teachers need to gain experience, continuing education in their subject, and mentors to help them be the best that they can be. If mentors are not possible, then they should work with each other in professional learning communities to discuss their problems, which might be school-wide, rather than just in your classroom. 

I don't need to say that the student of today is very different than those of ten years ago. We are in the computer age and with that comes changes in our social culture. As teachers, we must adjust to these changes to get students to buy into what we are offering them. They must see education as relevant to their lives.
7 days ago
           
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Richard Catterall • Ilirjan - Yes. Richard G. - No; all other things being equal Mathematics is one abstraction more difficult than a person's first language (though parts of it may be easier than parts of subjects as taught in schools), Incidentally it is the same part of the brain that processes language as the part that processes mathematical abstractions, so anyone who can read, write and speak their own language CAN learn Mathematics, it is just hard work. That does mean that the person who can do Mathematics can do anything else too, which is why researchers are finding mathematical ability a good predictor of future success in learning. Cliff - Yes; Mathematics is fun, exciting and incidentally useful. I try to emphasise the first two as parents, employers and the whole wide world put plenty of energy into telling students how important Mathematics is for jobs (compare the reasons people learn Art!). Richard S - Yes; teaching Mathematics is mostly about building confidence - it's a con game! We take the students through all those five stages of learning and get them to give themselves so much evidence that they can do Mathematics that they never drop back to stage one again, and then they can do the rest for the rest of their lives. That's an exciting business to work in. Deborah - Yes, it is complicated, so we need to use every available strategy and piece of research to assist us. A wise old teacher once said to me that, knowing that you can lead a horse to water but not make it drink, our job is to give it salt! We strive to continuously learn ways to be salt for our students so that they drink deep of the mysteries of Mathematics. 
One tactic I have been using lately (a couple of years) is to invite "intelligent questions" about Mathematics from my students, and accept all questions as "intelligent" and find ways to answer them that lead to more mathematical learning. This does sometimes mean that I introduce advanced ideas to junior classes; but I usually only give them a taste then return to the spiral of learning they need to advance on to gain a complete understanding. Thank you folks for all these answers to Teddy's question. They have helped inspire me to keep learning more and teaching more Mathematics.
7 days ago
           
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Pamela Kranz • Great thread. Sadly, I've found in elementary school that teachers are more comfortable teaching reading and writing than math. I taught 6th, still in elementary, and spent a lot of time showing kids that they understood more than they thought they did. "I don't get it" was met with, "OK, but what do you understand about the problem-we have to have a starting point so I can help you". It took awhile to build their math confidence, but we always made growth. Oh, and not a fan of timed tests.
6 days ago
           
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Paul Sinjani • I struggled with math in my school days, in fact, I failed. I tarted learning maths when I went college. This was so because I was always thinking math was tough and everyone said so. In college I met a lecturer who had a different attitude and approach to math. She made fun when teaching it and she assured me that I was a smart student. That boosted my interest for math and I have been learning it. In fact I have taught grade 8 and 9 math and all my student passed. 

We need to make the teaching of math very practical, applicable and relevant to life. Math is all around us and is with us. We need to help students realize this. Get students engaged.
6 days ago
           
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Seong Kim • If you are a math teacher, and your students like you, your students will probably do math pretty well or very well, and more importantly, at least, will not grow objection against math or study of math. 
You don't have to look beautiful or attractive to get liked by the students. 
And you don't have to please your students, either. You are a friend of theirs. 
All you have to do is to be sincere and considerate, and always listen to them. 
Listen to them, and see what they need, and then, do it. 
Ask questions so that they can reveal what they need, and then, do it. 
So don't just see it or say it. Just do it. 
If it doesn't work, revise it, and do it again.
6 days ago
           
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Brian przybylski • I have read some tintilating comments on this thread. comes down to: Be excited about teaching it and get the students on your side. I sing and dance(poorly) and generally act the fool in class, but my students show up, and some times they get the picture.
5 days ago
           
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Richard Galbraith • Brian, I agree, up to a point. Some of us are not effective singing or dancing. I also did weird things to get their attention and it worked most of the time. However, all students do not react favorably to that sort of thing. You must find what works for you (most of the time) and "hope for the best."
5 days ago
           
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Louise Berman • This is from an article I read and thought to publish it in the school newsletter to parents. 
. "The frequency of number talk in the children’s homes had a big impact on how well the youngsters understood basic mathematical concepts such as the cardinal number principle, which holds that the last number reached when counting a set of objects determines the size of the set (“One, two, 
three—three apples in the bowl!”). A subsequent study by Levine found that the kind of number talk that most strongly predicted later knowledge of numbers involved counting or labeling sets of objects that are right there in front of parent and child–especially large sets, containing between four and ten objects. 
Though it may not come naturally at first, parents can develop the habit of talking about numbers as often as they talk about letters and words. Some simple ways to work numbers into the conversation: 
• Note numbers on signs when you’re walking or driving with children: speed limits and exit numbers, building addresses, sale prices in store windows. 
• Ask children to count how many toys they’re playing with, how many books they’ve pulled out to read, or how many pieces of food are on their plate. 
• Use numbers when you refer to time, dates, and temperatures: how many hours and minutes until bedtime, how many weeks and days until a holiday, the high and low the weatherman predicts for that day. 
• With older children, math can become a part of talking about sports, science, history, video games, or whatever else they’re interested in. 
With practice, parents and children alike will find that math makes a very satisfying second language"
5 days ago
           
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Marie Kielty • Excellent ideas, Louise!
5 days ago
           

           
Tamara McAnelly • Why do you want to? Something has to be difficult... challenging. Math is a discipline. Nothing worth anything comes without struggle or trial. I can not tell you how many times I have cried over the frustration of math... but I LOVE IT!! It's part of the game.
4 days ago
           
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Warren Wolff • Judicious application of disequilibration!
4 days ago
           
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Seong Kim • If students can build strong foundation of algebra, they can reduce the syndrome. 
Doing algebra, they need to know not only numbers, expressions, etc., but how they work and how they mingle with the arithmetic operations. 
Algebra matters, together with geometry. 
What actually connects problems to solutions is algebra.
3 days ago
           
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David Ball • Seong, that sounds reasonable. In the US, following Sputnick, the Bourbakist math movement emphasised set theory for similar reasons. It became a fad and was largely felt to have failed as abstracting concepts doesn't help most learners. It was followed by concrete materials to introduce concepts. Now computers allow us to pursue individualised mastery. My preference for mass instruction is "introduce a concept. Practice it. Drill it. Use the time to set its place among the scheme of things and allow homework/improvement" There will always be a performance band from good to bad. The idea being to pitch to the middle. Let school process handle the rest. The idea that we can teach all equally and get equal outcomes is not observed in research.
3 days ago
           
           
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Louise Berman • I did not reference my quote above so here it is now: 
Why It’s Important to Talk Math With Kids 
March 2, 2012 | 8:27 AM | By Annie Murphy Paul

http://blogs.kqed.org/mindshift/2012/03/why-its-important-to-talk-math-with-kids/

I am not sure about copyright on this stuff and don't want to break any laws or offend people.
3 days ago
           
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Cecilia Villabona • I am fascinated by this thread started by Teddy with "......Math is the most difficult study area..." to which we have many responses from " ....for boys, it is easier than girls" (did you mean Math, Warren?) with ideas like to be entertaining, be liked, be a mathematician, make sure students know Algebra, use the right sequence,introduce Math early, and more to remove the syndrome that math is hard. 
No mention made to curriculum and methodology, it seems. Or to ability, except by Richard who says that anyone who can learn a language can learn mathematics. 
Then Paul shared with us that only when a college lecturer " made fun when teaching and assure me that I was smart" was he able to learn it. 
Here is what I believe: Mathematics learning required using our ability to think abstractly, and in general about 5% of us are naturally abstract thinkers. Some of us in that category become math teachers. If when we do we approach our students with the viewpoint that they should get it as easily as we do, we might become misunderstood by administrators and parents who would like to see more students get it that just 5%, understandably so. Our challenge is to reach the 95% who in most cases speak a language or two but loose confidence in themselves and end up believing they are not smart. 
In this moment all of the suggestions given come in handy, but often the curriculum is our enemy, taught to students in the order in which mathematics was discovered, for no other reason that it has been so, with the pressure to put our students through the high states test mill when at times they are not even abstractly ready. No wonder they end up thinking they are not smart! 
To the risk of making this too long, now on methodology. When elementary teachers are math anxious, they are not capable of building confidence in their students, when we as teachers assume that it is hard to learn math and only some can do it that is what happens. 
When we make students feel insecure and unsafe because they do not have the right answer, they go to a safe place to not think and not say anything for fear of being wrong, when we choose easy tasks for them since we believe that is all they can handle, we prevent them from doing the hard work, when we give them the answer too soon we encourage them to give up, oh I could go on forever....
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Richard Galbraith • Cecelia, You almost did go on forever!!!(just kidding) I think we all would like a simple answer but, as you have described, there is no one answer and it certainly is not simple. Those of us who have taught for a lot of years have seen curriculum, method, and just about everything else cycle several times.
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David Ball • Well put Cecilia. I only wish more math teachers were abstract thinkers. I'll settle for the ones who use abstract thoughts .. which is probably most .. but which .. (I'll stop that line) ..I think the issue of anxiety is more closely related to the failure of authorities to specify what is required .. they fail because they want more and accept less .. which is confusing. Then add math phobia. Thing is, we want a lot because some can handle it. I love the "Joy of Mathematics" and its sequel. But they aren't books you simply hand to children. They need to be read and understood and applied when the time is right. That takes someone special. I think what we are seeing is more a lack of leadership confusing junior initiate instructors.
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Richard Catterall • So, David, (and Richard, and Cecilia, ...) here is the challenge: take some leadership in the USA and begin programmes like the Numeracy Project we have used here. You have the opportunity to do some important work for your country. It will cost you the time and energy and effort involved in lobbying local and state and national administrators, and you will very likely (two sigma probability I'd say) get no financial reward nor any recognition for doing it; but the result will be a numerate population in the next generation - a noble goal. What do you say?
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David Ball • Love it Richard. I live and work (?) in Sydney Australia. Cecilia is doing exactly what you suggest with a wonderful program crossing Finland and the US ( http://app.pathstomath.com ). I have developed a program which lifts the middle band performance of a suburban multiethnic HS in Sydney. I have a stumbling block which I could use international help with related to death of school child Hamidur Rahman (I'm not at fault). I am unemployed yet working full time .. but I can't help myself .. I am a teacher, born and bred ..
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Cecilia Villabona • Richard my life mission in my entire teaching career has been to teach my 95 % non-abstract thinkers and to help them experience Aha! moments as they learn mathematics. Have I changed the status-quo in either Colombia or the USA? Not really but I have worked with over 10,000 learners and have experience a lot of rewards (non-monetary) which are very valuable to me. 
Thanks for the endorsement David, I appreciate it. Yes, we are combining expertise from my teaching years with Finnish materials to create what we consider a necessary change agent.
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Wallner Alina • First course as a student, our dean told us that we should get prepare for one of the most difficult Faculty and won't be 4 easy years. And he was right, in the first year we start in a number of 220 students and we graduated unless than 60%.
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Cliff Cohen • Responding to Wallner Alina: When I start college as a Math major, I was informed that I was in the top 2% of entering freshman majors. Thirty-five of us so designated began an "by permission of the professor" two year program of mathematics courses, and at the end of two years only 17 of us were left. I am not sure what happened to the other 18, but personally I found it challenging to not be the one of the top two or three students in my math class.
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Teddy Kafwanka • I am really being inspired with a lot of ideas on how to influence children positively on this study area. It is good to learn beautiful ideas on this link. You have been giving me excellent strategies and applications. My approach in my presentation has changed. My expectation for the year 2013 are better placed with much confidence.
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Merden Bryant • I used to teach in a public science high school for intelligent but underprivileged students. Students who get to enroll in that school have to pass a qualifying examination in English, Math, and Science. 
Despite being on the honor roll upon elementary graduation, many of my students before did not have proficiency in performing operations on fractions, integers, decimals, exponents. The most surprising thing was they shun problem solving even if the problems are procedural, algorithmic, and simple translational problems. Not only that, I also found their speed in performing math exercises alarming. Moreover, they believe that it is okay to perform low in Maths because according to them, Math is the most difficult subject anyway, like what they'd been told from their previous teachers and from the people in their circle. 
To address their weaknesses, I dedicated a month of my time budget for the school year in giving remedial instruction. I promised them in the beginning that with their cooperation, I would be able to help them realize that contrary to what they believe, math is the easiest subject. However, they had to go through a great overhaul of everything about mathematics. Firstly, I discussed place values then mental addition and subtraction by separately dealing with thousands, hundreds, tens, and ones. They never thought that addition can be easy and fun. Slowly, their speed improves because our practice exercises were like this: For 10 items of mental arithmetic, I would state number 1 exercise and check right away. It's like a quiz bee. I'd say right away, "Exchange with your seatmate!". They would shout and tremble. During the checking I would model the mental computation necessary to compute the answer. However, even if some could not answer no. 1, I'd say, " If you get no. 2, I'd give you a bonus to cover up your mistake in no. 1. I kept doing that for every number. As a result, their speed improved. Some quick and accurate thinkers would get double the item score. However, I kept reminding them this: Would scores in tests matter after finishing school? I underscored that it's learning that matters and what matters most is that everyone is learning and all of them would become mathematically literate which would be good for the country so all of them would be able to help the country prosper. I made it a point that the vision is for them to become great math thinkers to be able to contribute to society in the future. I made it clear that they should have a uniform score in each quiz. Their grades would only differ in Oral Participation and in the Quarter Exams. Basically, all exercises have to be answered by group, by pair or individually. A student can choose. For those who choose by group, they had to tutor each member so that a validation question/s would be answered by one or two of them because I had to call randomly member/s to represent the group. 

2. Before I discussed fractions, I had to make sure that students master these concepts: even number, odd number, composite number, prime number, Least Common Multiple, and Greatest Common Factor. I exposed them with common sense problems related to LCM and GCF. I emphasized prime numbers because prime numbers are the keys for tackling fractions, fractional exponents or radicals, and logarithms. I introduced the continuous/progressive division as an algorithm after presenting them a paper-folding activity where Least Common Denominator can be derived concretely resulting to conceptual understanding. Because of these, they realize that mathematics is an interesting subject. 

There are many things a teacher can do to make abstract concepts in math concrete. However, it takes a teacher with a heart--full of patience and understanding. 
I wish I could relate everything I used to do when I was a teacher. But I don't think I have the time in the world to tell it. This group rocks!
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Clifton Ellis • don't give them home work. Make them do the work in class under your supervision. Make them study in groups for each test (complete a study guide). 

Peace,
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Warren Wolff • Typical socialist solution. An idle mind is bad. Load 'em down in class and give them a wad more to do at home. Turned out a bunch of outstanding products, despite their complaining. Some refused to cooperate and flunked. Many have come back and said, "You had the right idea, Mr. Wolff. I have since grown up and made something out of myself. You taught me more than you will ever know." Guess I have a good report card!
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David Ball • I have been to dysfunctional schools and outstanding schools. You can't do homework at dysfunctional schools, although homework isn't the problem. At outstanding schools the homework wasn't dull drill work. It may well be the case homework is impossible at some schools .. but that suggests they are failing in their mission. There is a pathway out of such a bad situation .. and that pathway includes homework at the end. Looking to the future where learning is more individualised homework will still exist. Te simple reason for it is that to raise a student to the end of year 10 requires about 1000 hours of math instruction (not consecutive!) and that is not sufficient to cover the material and have the student internalise the material. Home time should ideally allow that time to expand to 2000 hours. Which illustrates why dysfunctional schools where homework is not done fare so badly in public performance.
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