Weekly Math Updates
March 2, 2006
You're getting 2 emails today, the update and then a stripped down comic that's going to make it easy to forward out to all your friends. You will still get the comics in this weekly update each week so don't think you need to subscribe to both lists. The new list will just be a comic linking viewers to the website. Please forward it to everyone in your address book and lets spread this comic all over the world and encourage people to sign up to get the comic. It's a great example of a picture saying a thousand words.
Here's this week's comic:
David Wright at BYU has circulated a petition among higher education math professors to help us push Utah toward adopting the California math standards. As soon as I can get a copy of the document I will link to it or post it on my site, but I wanted to point to a press article in the Deseret News speaking about it. I note with particular humor that one of the UVSC professors that refused to sign this petition said, "Research has shown that when students only memorize math facts, the learning is "fragile" because they don't understand the mathematical foundations of the facts." What does this guy think kids are memorizing? 3825 x 274??? Who memorizes everything? I'd love to see a curriculum that is actually focused on memorization. It doesn't exist, just like independent studies to support fuzzy math don't exist.
ASD Homework (link to assignment)
Attached to this email is a homework assignment that caused me to change my remarks to the school board Tuesday night. They were visiting my local school and I thought I'd throw them by speaking about something unrelated to math. Their annual report had a footer saying "educating all students to ensure the future of our democracy." Well, we're not a democracy and the founding fathers had a lot to say about that. We're a democratic republic. Anyway, one of my children got the attached document as a homework page and finished it before I even saw how ridiculous it was so I had to go and ask the district what was the purpose of presenting such a confusing document to the pupils. It's not even Investigations math. It's one of ASD's own supplementary pages!!!
If you look at it, there's 4 boxes and it asks the student to use 2 of the strategies on the page to solve 84 x 26. The first strategy shows 84 x 10, 84 x 20, and 84 x 5. When I add those up I get 84 x 35. It's beyond me how that strategy is supposed to teach anything but confusion. The second box is equally bad, giving the student more work to get to the answer than the first incorrect example. The third one is at least a valid set of problems that add up to the equation, but thankfully, my daughter took the 4th box to do the traditional algorithm, which is also the fastest and NOT EVEN PRESENTED in any of the other "strategy" boxes. Not only do we have to deal with Investigations math, but the district is supplementing it with their own materials made in Elbonia (if you're not familiar with that country, you need to read more Dilbert cartoons). It will be important to understand that country in a week or two.... :)
I wanted to share with all of you a couple of emails I received from a jolly chap on the other side of the pond (hope I got the lingo right Andy :)). This teacher in Britain found our site and was happy to share his insight into how Investigations style teaching ruined his country and they're just on the mend now. There's a couple of emails he sent, the second after reading one of the latest press articles posted on my site.
Andy Beety's Emails:
I came across your site by accident but am happy to share my experiences on the above with you. I live in Surrey (Great Britain) and have taught math at all ability levels in state schools at secondary level (11 - 18) in the UK for 25 years. I support you in your campaign to rid your schools of this style of "teaching" because in reality it's not "teaching" at all in the traditional sense. When I was involved in my early teaching career similar programmes of "investigations" or 'learning math by discovery' were the buzz words but I was always asking the same question... what are my students learning??? because, lets face it LEARNING is crucially important for a student. when students learn a mathematical skill it gives them confidence. I found that they learnt a lot less than I hoped for, they were very confused about what they thought they had learnt and crucially I felt I had let them down because I KNEW I could have taught them a lot more in a much shorter time period. An important point about Mathematics is its ability to delight and astound anyone of any age and there is no greater truism that the more Mathematics you learn the more you realize there is still to learn. There is no other subject in the school curriculum which allows students to gain satisfaction from feeling that they have done a Math problem and got the correct answer. This even gives me a great feeling at my age!!! "Investigations" removes this feeling of satisfaction from the student because they never feel that they have learnt anything and if they have they certainly can't tell the teacher what it is! Group work - whilst I have nothing against this per se, it should NEVER be the main focus of a teaching programme. It is also a complete nightmare for the teacher as they never know how much the students have learnt. One of the major drawbacks with group work was that inevitably the more able children were the ones who did the bulk of the work and the others just copied and thus learnt relatively little. It was then left to me to re-teach material so that I was sure in my own mind that ALL my students had the opportunity to learn certain key facts. Mathematics is a subject which inevitably requires certain skills to be TAUGHT by teachers and LEARNT by students. When students are able to build up a set of skills they then have the knowledge and the confidence to move on and learn more complicated Math. This is never more important than for younger pupils in what we call primary school (ages 6 to 11). It became so 'trendy' (fashionable) to not teach multiplication tables and basic operations with fractions and decimals that millions of young children were left with a big hole in their mathematical knowledge and it left them badly prepared for further work. We had to spend about 2 months teaching children to perform basic mathematics when they arrived in secondary school at age 11. School based assessment is also extremely difficult when students have been taught by an investigational approach. Do we set an examination involving just investigations?? if we do how do we mark it?? Take it from me it's extremely difficult. When it comes to national examinations we saw huge differences in the performance of traditionally taught students compared to those taught by investigations and this of course was the big worry. We were turning out a generation of mathematically illiterate students who were not equipped to face the world of work or Higher (University) education. Thankfully things are much better now!! We still have investigations and they count 10% towards the nationally assessed GCSE. We see them as a 'different' way to approach mathematics because let's face it we all learn from working together in a group and cooperating on a joint task. But we never forget that LEARNING is the most important thing so a list of 'learning outcomes' is given to teachers along with the investigation task and if the teacher feels that his or her class has not achieved these the class is given a refresher lesson to reinforce the learning outcomes. I found that the most difficult thing for educationalists (especially in Math) to admit was that they had made a mistake and were just plain wrong!! They want their dogmas and methods to be accepted by all no matter how many children's futures are affected. These people are not teachers and in many cases left teaching because they could not handle working in schools so they left to work in Education colleges passing on their ludicrous ideas to future teachers. This is a particular hobby-horse of mine and I could go on... however.. I would suggest that in your situation you should aim to empower someone to reach a compromise which allows both groups to save face. Introduce a programme which allows investigations to run alongside a more 'traditionally' class taught programme. The class taught programme should form the majority of the lessons with the investigations being seen as a different type of activity but with very clear learning outcomes. This will allow the 'investigationalists' to withdraw without losing face because I'm sure they know by now that they are in the wrong but are not prepared to admit it. I wish you every success in your task because I know that many children are missing out on what should be enjoyable math lessons.
I've just been reading some of the press reports and one of them left me utterly flabbergasted. It's in Deseret News 3/02/06. The district individual is talking about a lesson whereby pupils 'discover' how to multiply 22 by 44 and I quote " "The first thing you'd do (is) go through the whole thing about sets," he said. "You've got 22 sets or 44 sets with groups of 22." What the hell does that mean?? My English is pretty good but even I can't understand what he means. He goes on.. "Students divide into groups to discover methods that lead to the answer. Each group presents its method before the class. The teacher also presents the algorithmic way to solve the problem, which the district person called most efficient. Besides teaching efficiency, you're giving them flexibility" The first thing that strikes me is what a complete and utter waste of teacher and pupil time it is to have to go through that process just to learn (and I use that word assuming that they have learnt how to do long multiplication without a calculator) the process of long multiplication, and let's face it long multiplication is only a minor process which has to be learnt in order to progress to more difficult processes which involve it. ( If they spend this amount of time on long multiplication it's no wonder they are 2 years behind by 5th grade according to one article I read) The other point is that if the teacher then teaches the algorithm to pupils and says it is the most efficient way of doing long multiplication, what is the point of confusing pupils (because believe me that's what will happen using the 'fuzzy' method) and making them feel that they are losing confidence by introducing many different problems which because of their age they can't solve. My final point on the press article is the one about giving pupils 'flexibility' .... what a load of xxxx. He is assuming that pupils are able to sort through the maze of confusion caused by these and by accident decide by him or herself which is the most appropriate method to use. These sorts of ideas make me spit because as teachers we are there to lead pupils through the maze of confusion and tell them that this is, trust me I'm a teacher, the best method. If I were teaching this example of 44 x 22, I would ask the class as a whole for their ideas... some may offer 'it's 44 x 20 add 44 x 2' (we call this decomposition ) some may offer different ideas. I would then say.. OK how would you do 376 x 27 ???? This is where the teacher does his/her job and say that there are different ways of doing math and that's good but the most efficient way is this... using the standard method which works for all long multiplication problems. Pupils then practice the algorithm with say 20 questions, safe in the knowledge that Math is not a dictatorial subject and there are different ways of solving problems but we look for the most effective and often 'mathematically beautiful' way. These methods have been tried and tested over centuries and one of the ways we move on as individuals is by accepting that certain methods have been developed for future generations to use and not have to go back over old ground. One of the classic examples of this is in complex numbers and the concept that the square root of -1 exists. Now if you teach this for the first time the enquiring student will often say 'well how do you know it exists' ?? I replied 'well I can't prove it exists at this stage because you haven't done enough Mathematics to understand the proof, but I will prove to you that even though we are dealing with complex numbers all the processes of addition subtraction etc work and nowhere does any process not work when we are using complex numbers. This was enough to convince my students that much work had already been done on cn's and there was no need to reinvent the wheel. We then went on to discover that, despite little knowledge of complex numbers, they were hugely important in the field of engineering electronics, microprocessor design . I suppose my point is that mathematics is there to be discussed (not rediscovered) practiced and learnt in order to be used at a later date. I'm quite a pragmatist as far as math is concerned, I like to 'cut to the chase' and use math as a problem solving tool, because let's face it, mathematicians like Pythagoras devised his theorem for us to use, not to have to prove it time and time again before we use it. I could go on...it's a very frustrating time for you as parents I can tell that, but I'm sure you'll win through in the end. You can use any of my ramblings as you see fit - I've got nothing to lose!! We've been through all of this in the UK but unfortunately lost a lot of youngsters on the way. Please let me know how you are getting on.
Till Next Week,
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