Friday, April 27, 2012

The Reasons for the Seasons

This coming Monday I have about an hour in my Teaching Math/Science class to teach the Earth Science part of the K - 6 science curriculum; clearly not enough time to "cover" or explore everything K - 6 students need to know and understand. So, like each area of the science curriculum for which I have about an hour, we explore a sample topic. (This is pretty much standard procedure for most teacher education programs). I always try to select something from which I think the students would benefit most in terms of the development of both their own understanding and the work they will be doing with K-6 students. 

For Earth Science I do an activity with them that helps them develop a better understanding of why it is generally hotter near the Equator and colder near the poles. An activity appropriate for 5/6th grade students. We also explore the reasons for the seasons as well as day and night; all in ths pace of about an hour. The diagram above is typical of what you see in text books and can give rise to all kinds of misconceptions so we do something similar but make it into an activity.

The students sit in a circle and pass a globe around while I shine a flashlight on it from the center of the room. By carefully establishing some activity rules like; the globe is always tilted toward the same wall as it orbits the "Sun", and, the rays from the Sun must be kept horizontal, and,the Earth orbits the Sun and rotates on its axis counter clockwise, we can pretty much develop an understanding of the concepts of insolation, the seasons, and day and night.

We finish off the activity by exploring websites like this one, 24Time  that gives the time anywhere in the world as well as showing how darkness is coverning the Earth. You can see if it is tomorrow or today somewhere, whether it is day or night, and how long the days are at any location. Websites like these really allow us to help students develop a much more robust  understanding of what are pretty difficult concepts. 

Hopefully, my students will never teach their students that it is hotter at the equator because it is closer to the Sun; one of the most common misconceptions in the whole universe of science.. 

Thursday, April 26, 2012

Zero, Nought, or Nothing

Noughts and crosses is a popular game usually played with a pencil and paper especially during a long and boring lecture by two overly competitive students. I still call it this even though it is inexplicably called tic tac toe in the US (If you Google the origin of the name you'll find it has to do with the sound Roman pencil's made when they hot the ground - still inexplicable).

The word 'nought' is the British equivalent of zero and has given rise to some interesting phenomena such as the 'early noughties', the first ten years of the current millenium as in '04 and '08. Shakespeare loved 'nought', of course, as in " That it yields nought but shame and bitterness" from Henry V.

Today,  the concept of zero is something we must focus on developong with young children as they learn to count and understand the intricacies of our base ten system. The idea of the concept of zero was first introduced as an "empty set" through the disastrous New Math of the 1960s. Math educators are still frequently erroneously tarred with the same brush when any mathematical ideas are suggested that differ from traditional mathematics; but that's another story.    

When we teach children how to number name from 0 - 100 we must start with zero so that they recognize it as a number with a place on the number line. Later when children start to develop their sense of cardinality, the ability to recognize and name a quantity, they can start with 0 before they have any objects to count. The importance of developing this idea only really becomes apparent when they encounter the base ten system.

Traditionally, children have been taught to think of 0 as a "place hloder'. The problem with this idea is that it doesn't help in the development of the child's conceptual understanding of place value. For example, in the number 306, the zero stands for no tens. The idea of the "place holder" simply means that it keeps the 3 and 6 apart which does nothing to develop an understanding of what each digit in a number means. It's another  example of the useless metaphors that are sprinkled throughout traditional math.  

Monday, April 23, 2012

Farthings and Cents

Every so often a blog topic just lands on my desk. Today, Jonathan Silverman, my art education colleague extraordinaire, presented me with a 1939 farthing. It is identical to the one in the picture and brings back the most incredible childhood memories. We used to be able to by two "chews" for a farthing. A "Chew" was a candy I am certain was invented by an enterprising dentist.

Farthings were in use on the UK up until 1960 when they became no longer legal tender. A farthing was a quarter of a penny or 1/960th of a pound. This would be about 1/600th of a US cent at today's rate of exchange, roughly speaking. Eleven pence and three farthings was a popular price for an item when a shilling (12 pennies) sounded too much; just like $9.99 in today's currency. The farthing was also used as a description of a high wheeler bicycle which, in the UK, is still called a Penny Farthing because of the relative sizes of the wheels.

Two farthings made a halfpenny, or ha'penny, which was removed from legal tender in the UK in 1969. Both the farthing and ha'penny had been around for 700 or so years before they met their demise. I can well remember the discussions we would have about how silly and annoying such pittances of coins were especially with the imminent demise of the Canadian cent. I wonder how much longer the US cent has to go? I wonder how many there are in "penny jars".

You could almost study the entire history of Britain by exploring changes in the simple farthing     

Saturday, April 21, 2012

Probability and Confidence

Every time I teach I learn something new and never has this been truer than with my graduate class this semester. Last Thursday we were exploring issue of probability. We started out developing the concept linguistically trying to identify those things in our lives that were likely, probably or certain. It always amazes me how so many people don't like to say something is certain; even the sun rising tomorrow morning.

We talked about the difference between theoretical and experiential probability and how the latter is so much more interesting. We tossed coins and used an automatic on-line coin tosser that tosses a coin 1000 times in twenty seconds. I can't think of a better way of showing the law of large numbers.

We concluded class with a really interesting activity in which you place five cubes in a bag; the cubes can either be, say, red or blue. The task is then to work out how many of each by pulling one out, recording the color, putting it back in, shaking the bag and pulling another. This continues, not for a predetermined amount of time, but until the person pulling the cubes feels confident enough to infer the number of each color. I've done the activity many times before but have never had  such an interesting discussion about it. We decided that not only did it demonstrate the relationship between probability and confidence, something that is taught in statistics classes, it also demonstrated something about one's personality. Some students must have pulled 30 or 40 cubes before they wanted to say how many of each color there were while others wanted to say after just 10 or 15. Isn't that neat? It's almost like a personality test. Some wanted certainty while  others settled for a risk.

The sad thing is that we will not perhaps be able to do this in the future until 6th grade as probability is not included in the elementary school Common Core math curriculum. This is sad in a way because without the topic of probability math takes on an air of certainty. Perhaps this is not a bad thing for young children and something had to go?

On the other hand, when you stop to think about it, the concept of probability is involved in just about every aspect of human life. Every time we wonder if it will rain, or whether our favorite team will win, or whether we will get an A on a paper. In fact any time there is uncertainty.

Wednesday, April 18, 2012

A Custom Built Text

I did something this week that I've never done before; I designed my own custom textbook for my Teaching Elementary School Math and Science course. I have used a wonderful textbook by John Van de Walle in the course for many years but I wasn't using all of it. The book costs an amazing $158 and so didn't want my students to spend all that money on a book  I only partially used. The part I didn't use was pedagogy or how-to of teaching math. For this part of the course I use the Bridges program which is used extensively in school throughout Vermont and the US.

What I did want to use was the pedagogical content knowledge or 'what to teach' section of the book as I feel very strongly that my students need to understand the math they are teaching even if it is something as basic as subitizing.

So after several emails with the very helpful Pearson rep I now have a custom made book that includes only the pieces I need and with a really neat Matisse painting on the cover. You can even see some numbers in the picture if you look carefully; math and art are so interrelated. The book will only cost my students $123 and will be completely up to date with Common Core info, unlike the used copies of previous editions that are available on-line. They can also get it conveniently at the college book store.

Subitizing, by the way is one of the "basics" of math. It is the ability to look at a number of objects and quantify them without having to count them. Many animals can do this, with crows being quite spectacularly good at it. Most people can subitize up to 5 to 7 objects randomly placed although it takes significantly longer to subitize quantities after 3. We arrange dots on dice and dominoes into patterns so that they are easier to recognize (or, perhaps, subitize). Double 6 dominoes are familiar but try looking at a set of double 12s; the 7 and 8 and 11 arrangements are quite a challenge.

Monday, April 16, 2012

Science or Engineering?

Last week I posted a blog entry about Dr. Ioannis Miauilis, the Director of the Boston Museum of Science and Engineering. Like Dr. Miauilis I have always believed that the study of the natural world is not enough to fulfill the "science" component of the elementary school curriculum. To ignore design technology (engineering) would be like teaching children to read without talking about literature, or teaching children to compute without teaching them how to problem solve.

To help my students differentiate between science and engineering
(design technology) we talk about the origin of the questions we ask, the sources of the inquiry. In science the questions arise out of the natural world; what is magnetism? how does it work? what are the five senses? what is the water cycle? In design technology, or engineering, the questions arise out of how we use our scientific knowledge to solve human problems. How do we use magnetism to enhance our lives? How do we use our knowledge of the water cycle to grow better crops.

In the science part of class today my students explored water through the concept of drops of water. In the engineering or design technology part of the class they used a copy of the local newspaper to support a small washer as high as they could above the tables.

An application that brings the two together in a unified way is to explore the motion of a pendulum in terms of what causes it to swing faster or slower and then design a pendulum clock that swings once every second.

The design technology part of the science of magnetism can be exemplified with  Mag Lev trains or cow magnets. (My students would not believe there was such a thing, even though I was holding one, until I showed them one on a website).

Sunday, April 15, 2012

Starting the Journey to Becoming a Teacher

Yesterday, I took part in the Spring accepted students day at St. Michael's college. As an Education professor my job was to answer questions from students and their parents about our elementary school teacher education program.

I met some wonderful families, the sons and daughters full of excitement and anticipation, the mums and dads nervous and apprehensive; I remember having those same feelings so well ten years ago when my own daughter was searching for a college. For a couple of hours I answered questions about State teacher licensure reciprocity, about what our program was like in terms of field experiences, whether courses could be transferred from one college to another and how our program compared with others. Two prospective students stood out for me.

The first wanted to be a music teacher and wondered if we had a program leading to music licensure. I told her that we didn't and that not many colleges, especially in Vermont, offered that somewhat restricted licensing program. I suggested that she might want to think about getting her elementary education licensure  through a double major in Education and Music. She clearly loved music and told me enthusiastically about all the instruments she played. I could just see her as a second grade teacher in 6 years time turning every student in her class, and probably the school, onto music. I have nothing against dedicated music teachers, of course, but when the classroom teacher is a musician it makes such a difference.

The second student seemed to want to be an elementary school teacher but his parents were terribly concerned about their son's writing skills. I told them how St. Michael's had the Writing Center to help students improve their writing but they seemed more interested in wanting to know how much their son would have to write; would the required papers be any more than five pages long?  I tried to let them know gently that elementary teachers must be able to write since it is one of the main things they do. I suggested that his passion for becoming an elementary school teacher would really have to drive him to improve his writing skills if he wanted to succeed.

I wonder if  both students will choose to attend St. Michael's. If they do, I wonder which one will choose to begin the journey of becoming an elementary school teacher. Perhaps they both will!

Friday, April 13, 2012

What Are Schools For?

I was listening to the local NPR radio  station on the way to work this morning and the recent teacher strike in central Vermont was being discussed. The interviewer asked if parents were relieved that the strike was over suggesting that it must have been difficult for parents finding someone to watch their children for six days. The interviewee agreed saying it had put quite a strain on local families.

The exchange made me wonder, not for the first time, why we have schools. What is the purpose of the school, of education?  Imagine being from another planet and listening to this exchange on NPR; what would you think? Perhaps you would think that schools are places where parents keep their children during the day so they can go to work. You might conclude that, as communities, they employ people to look after them and keep them gainfully occupied during the day. They employ people to take them to and from school and other people to feed them at lunch time.

All joking aside, the purpose of schooling has changed significantly over the years as times have changed, to use a line from Bob Dylan. Schools change in response to all kinds of local and global forces. The one I remember most was the changes in the science curriculum in the western world when the Russians launched Sputnik in 1957.  Changes have occurred in what we expect students to learn, how we expect them to act and what we expect them to be able to do as a result of their education.

It used to be fairly easy to predict the future and have a sense of what students would need to know and be able to do but with the current rate of change and uncertainty about the future things are becoming more difficult. Gone are the days when a job out of college or high school was for life. One wonders if the Common Core, due to hit schools in 2014, might be out of date before it even arrives. I often wonder if those creating the Common Core have examined what comprises the curriculum carefully enough given the ever increasing rate of change we are experiencing.

In my own field of math education, arithmetic still takes up a significant part of the K - 6 math curriculum in the Common Core. Do we really still need to spend so much time teaching children how to subtract 38 from 92 with a pencil or pen? Shouldn't we be acknowledging that basic numeracy and problem solving skills are so much more important things to learn,  and that cellphones, computers and calculators can now easily perform such menial calculations. Wouldn't it be better to spend the time helping students become quantitatively literate and be able to work out which one of the four operations is required to solve a particular problem?   

Perhaps it's time that thinking and understanding replaced knowing as the primary goals of education?    

Thursday, April 12, 2012

The Value of a Place

I can remember being told, as a student in primary school, to put a comma every three numbers when writing large numbers. I was told it would separate the hundreds, thousands, millions etc. I remember dutifully doing this with absolutely no sense of why or feeling any nearer to being able to read very large numbers.

Today, we still use the commas when writing large numbers but
unlike the instruction of the last century, we now teach why we put a comma every three numbers. It certainly does separate things but, more importantly, it shows the repetition of the ones, tens and hundreds every three numbers. First we have ones, tens and hundreds of ones, then we have ones, tens and hundreds of thousands and then we have ones, tens, and  hundreds of millions, and so on. If we learn this repetition of the place value referents it makes reading and understanding large numbers much easier.

It also helps students develop a far more realistic idea of what zero, or nought, is.  I remember learning 0 as a "place holder", a bit like a pot holder, perhaps. What did that really mean? The 0 holds a place when there is no number there, perhaps. What nonsense!  The 0 means there are none of that particular referent in this number as in 23,405, where the 0 means there are 0 tens. In this number, 305, 877  the 0 means there are no ten-thousands; there are 3 hundred-thousands and 5 thousands but 0 ten-thousands.

'Nought', by the way is the British word for zero and became quite popular during the first decade  of the current  century with years like '03 and '06 which was referred to as the 'noughties'.

Tuesday, April 10, 2012

Three Friday 13ths This year

As you probably already know there are three Friday 13ths this year and they are exactly 13 weeks apart. The second is this coming Friday. The last time this happened was in 1984, the year made famous by George Orwell's novel of that name. I remember reading it in the mid '60s and wondering where I would be in 1984. Little did I know I would be in my third year in Vermont. The next time there will be 3 Friday 13ths in one year will be 2040. What will we all be doing then? Here are some interesting facts about 13. The Costa Concordia started sinking on the last Friday 13!

13th is an ordinal number, like first, second, third, fourth, fifth etc. We don't often talk about a thirteenth of something in a fractional sense but we do use third, fourth, fifth etc as fraction words. Yet another quirk of our strange English language. Fraction words are the same as ordinal words after you get beyond second/half. I remember working with a 2nd grade teacher once to teach fractions. We taught the students that a third as a fraction was one of three equal parts among other things. When the children lined up for recess one of the students looked up at me and said, "I'm third in line but I'm one of 16 equal parts!", referring to the 16 students in her class.  We discussed the two meanings of the word 'third' but I was never sure she understood.

Numbers occur in our lives in lots of ways; cardinals (counting numbers), ordinals (sequence numbers)  and nominals (naming  numbers). Room numbers, house numbers, telephone numbers and just about any number used to identify something is a nominal number. If you can replace the number with a name then it is a nominal. There are also classifications such as shoe sizes, the Beaufort wind scale, the Richter scale and Mohs rock hardness scale as well as rates such as prices, and measures such as temperatures. Each numeral (the written grapheme) has a different meaning when used as a different type of number.

Next time you fill your car up with gas, or petrol, look at the pump to see how many different types of numbers there are. It will take your mind off the price of the gas or petrol.  

Monday, April 9, 2012

Engineering for Kindergartners

This morning I gave a presentation at the STEM conference in Stowe, Vermont sponsored by the Vermont Science Teachers Association and the Vermont Council of Teachers of Mathematics. My presentation on teaching math to children with Down Syndrome went well but what really caught my attention was the keynote address by Dr. Ioannis Miauilis, President and Director of the Boston Museum of Science. He had the group of science and math teachers riveted from the moment he said that engineering should be a part of the K - 12 curriculum and it should begin in kindergarten.

The most interesting  part of what he advocates for is the idea that we spend so much of our time in science education classes focusing on the natural world and so little time on the man-made world. We spend so much time teaching things that are so remote from students' lives, such as lessons about volcanoes and coral reefs to students in Vermont, and so little time on things that are part of children's lives such as how drinking a milk shake through a straw really works.

When I teach science education to my students I always try to make the distinction between science and design technology, which I think, is close to Dr. Miauilis' concept of engineering. In science the questions arise from the natural world whereas in design technology the questions arise from how we use science to deal with the natural world. In science we can study the properties of magnetism but in design technology we can use our knowledge of magnetism to solve problems. Even though there are standards for design technology in the Vermont Grade Level expectations I fear they are seldom addressed.

Perhaps it's time to implement, as Dr. Miauilis did when he was a professor at Tufts University, courses such as those on the science of cooking or the science of fishing. In other words, perhaps science needs to be more relevant to students' lives in a way that will attract more students into the formal study of the sciences and engineering. Such courses would have the same academic rigor as the pure science courses but would be much more relevant to students' lives and interests.

By the way, Dr. Miauilis is far more entertaining, engaging and humorous as a keynote speaker than he is in the YouTube video.

Friday, April 6, 2012

$3.99.9 a Gallon

I've always wanted to ask for exactly one gallon of gas to see what would happen. How much change would I get? The only place in the US, in fact in the world, where the smallest legal tender denomination can be split into tenths is with gas or petrol prices. I suppose it is because the prices are very visible on large signs and gas stations tend to often be next to each other and, as we all know, a penny difference really matters!!!! I wonder what will happen in Canada now that they're eliminating the cent as legal tender?

The .9 is, of course, pretty meaningless since every oil company does it and it's usually so small that it's virtually impossible to see from a distance. Anyway, it's always there so you always ignore it. This is a wonderful example of just how quantified our lives are and how certain numbers are significant and other are not.

Perhaps, tomorrow, the price will have risen to 3.99.99!

The gas prices in the picture above are actually petrol/diesel prices in the UK. They are in pence and are for 1 liter of petrol. A liter is equal to about .26 US gallons so there are around 4 liters in a gallon. The rate of exchange between dollars and pounds sterling is around 1.6 dollars per pound. So the cost of petrol in gas equivalent gallons is roughly 4 x 1.48 x 1.6 which works out to around  $9.47 a gallon.

Somehow, $9.46.9 doesn't seem to make a whole lot of difference.

What does make a difference is that 70% of cars in the UK are diesel and get about 70 mpg and places are much closer together so you don't have to drive so far.  

Wednesday, April 4, 2012

Do the Math - Do The Art

One of the new Open Education sites that is devoted to  education is the Kahn Academy. There are hundreds of short videos available to those wishing to brush up on some aspect of their math or many other subject areas. Generally, the ideas in the math videos seem to be explained well but it is certainly a contrast with the Vi Hart math videos which, I must admit, I much prefer.

One of the phrases that has always sort of annoyed me related to math is "do the math" which is used any time someone wants you to compute or calculate in some way. It's as if that's all math is.

I was watching one of the Kahn Academy math videos, the one titled "The Beauty of Algebra", when I could barely believe my ears. After a brief explanation of how to use algebra to find a percentage of a price of something he says, quite remarkably, "now do the math" as he calculates the exact reduction in price. So what has he been doing up to this  point and what will he be doing after this point? Clearly he has not been doing math for the first several minutes of the video because he now decides to do the math.

Now I must "do the writing" to continue this entry so that you can "do the reading". We must also not forget to "do the driving" to get home after work, "do the cooking" when we get home and prepare the evening meal, and "do the sleeping" so that we are well rested for tomorrow.

And, of course, don't forget to "do the art" when you feel the need to doodle in  a long meeting.

Monday, April 2, 2012

Amazing Graduate Math Ed Class

I have the most amazing graduate math education class this semester. The students just will not let anything go that they don't understand. I have always believed that we should not teach anything we don't understand so this group of students is really keeping me on my toes.

Two weeks ago we were exploring fractions. We first looked at fraction concepts such as  the idea that fractions only have value when we know the size of the whole or referent to which they refer. (Would you rather have half the money in my right hand or a quarter of the money in my left hand?). We then went on to look at how to develop the fraction algorithms for the four operations. Addition and subtraction were fairly straight forward but, of course, multiplication and division are a completely different kettle of fish.

We also played around with fraction word problems which, again, is OK for addition and subtraction, but quite confusing for the other two operations. The other problem with fraction problems is that it's hard to get away from food; "if you ate 3/8 of your pizza.......?". And, of course, pizzas are only ever divided into eighths!

Then someone said "I was always told that "of" means multiply as in what is 1/2 of 1/3". Yes, it always has, I thought, but why? What is the conceptual connection between the word 'of ' and the operation of multiply? I asked several mathematicians and could only come up with nothing more than my original thought which was that it is just a linguistic connection as in "automobile means car" and "house means home".

My daughter, Marie, even suggested using the word "times" as in "a half of a quarter is a quarter, half a time" (4 groups of 3 is 3 four times). Then I started thinking about the area concept of multiplication as in the image above. in a 1 x 1 square an area which is 1/3 (blue side) by 3/4 (red side)  provides a conceptual connection between of and multiplication. Voila................................. maybe.