Friday, March 29, 2013

Yessssssss, I can teach.

At times, my job is so amazing that it gives me goose bumps. One of those times is when I am able to witness that magical moment when a student teacher realizes she can teach. I wrote about this last year when one of my student teachers was so overcome by the moment that she stopped, smiled, and told the students how good they were being while I observed her. I think this happens to all successful student teachers; that moment when they finally know they can teach; but we're not often there to witness it. So the next best thing is to hear about it which I did this week from my colleague Prof. Valerie Bang-Jensen who is supervising Laken Ferreira in her student teaching experience. Valerie sent me, with Laken's permission, this wonderful excerpt from Laken's journal.

"As I was setting up in the morning I was surprised by how much I missed the class after not seeing them for 4 days. The students flooded in and had lots of stories to tell me about their snow day and the weekend. I hadn’t really realized how much my students valued routine and consistency. Our morning meeting and mother echo time was by far the best hour that I have ever taught. Everything went so smoothly. I counted down to morning meeting and made sure all students were a part of morning meeting by having the students who came in late do their nametags for the sticker box. During morning meeting I had a moment. Dani told me that Professor Whiteford spoke with her last semester about a moment that all student teachers have when they realize that they can teach and are a teacher. Instead of calling on students individually to share, I had everyone turn to an elbow partner to talk to about what they were looking forward to today. After about 45 seconds I brought everyone back by counting down from 5 and was amazed by how attentive everyone was. Within 3 seconds everyone was turned to me with his or her eyes on me and mouths shut. As I stared back at my 20 students present that day, I had the moment Dani described" (Laken Ferreira 2013).

Isn't that wonderful; you can just feel the joy of teaching; that sense of 'yes, I really can do this'. Well done, Laken.

Thursday, March 28, 2013

Being a Student - Becoming a Teacher

The transformative learning theory of Jack Mezirow has always seemed to me to be  the most compelling piece of adult learning theory upon which to base one's approach to teacher education. I must admit, though, that I have always seen its application as being more appropriate for my graduate students where, for many of them, the process of becoming a teacher is a significant change in their lives in the context of what they are currently doing.

Recently, however, I have begun to see where this theory applies to undergraduate students as they make their way through the four years of undergraduate education. I have always perceived, and treated, the undergraduate students I work with as adults so it seems logical to apply a theory of adult learning to their passage from being a student to becoming a teacher.

One of the primary tenets of tranformative learning is the idea that  it consists of "critical self-reflection and disorienting dilemmas to make cognitive adjustments to reframing one's world". There is a clear distinction between first year students who are exploring the teaching profession through introductory classes and those who are set to graduate having completed their student teaching experience; at least with those who have been successful. It's not just the fact that they are four years older but something far more complex and meaningful. At some point, for most around the middle to the end of their junior year, they no longer seem to be students attending classes. They have become pre-service teachers learning how to become teachers. Their levels of motivation and curiosity change; they begin to take ownership of their learning. The field experiences they have become sources of self reflection and disorientation from which they construct new outlooks on who they are and what they can do. They become "authors" of themselves as described in the introduction to transformative learning from the website linked above;

"Transformative Learning is a theory of deep learning that goes beyond just content knowledge acquisition, or learning equations, memorizing tax codes or learning historical facts and data.   It is a desirable process for adults to learn to think for themselves, through true emancipation from sometimes mindless or unquestioning acceptance of what we have to come to know through our life experience, especially those things that our culture, religions, and personalities may predispose us towards, without our active engagement and questioning of how we know what we know." 

"For us as adults to truly take ownership of our social roles, and our personal roles, being able to develop this self-authorship goes a long way towards helping our society and world to become a better place through our greater understanding and awareness of the world and issues beyond us, and can help us to improve our role in our lives and those of others."

Isn't that neat!

Tuesday, March 26, 2013

Why Wasn't I Taught That Way?

I love this picture because the sheer joy of learning is illustrated by both students; in a uniquely individual way by each student. Sometimes in my classes the sheer joy of learning or discovery is frequently met with the phrase "Why wasn't I taught this way?". This most frequently happens when we are exploring fractions or decimals as we did this last Monday night in my grad. math ed. class. Sometimes it's the simplest of ideas such as being able to count fractional parts like anything else or suddenly realizing how easy it is to find 1/2 of 2/5 if you just visualize 2/5 as two fifth pieces and take half of them.

The funny thing is is that students have been saying this in my math classes for 30 years. So why weren't they taught that way?  I have to assume that I am not the only math teacher educator advocating for a conceptual approach to teaching math? Every college math ed. text I receive for review stresses the importance of helping children understand what they are learning; every conference presentation I attend stresses the same thing as do the math coaches I work with. The bridges amth prgram used on many schools throught the country also focusses on this worthwhile goal. It's been this way for thirty years at least.

I find it hard to believe that elementary school teachers are just not putting into practice all the things they learn through pre-service and in-service activities. Perhaps it is the students who do not remember learning the material in such a way. Perhaps the experiences they have in high school overshadow the things they learned in elementary school? Perhaps they just don't remember learning it this way when they were in elementary school. Too many students have these aha moments to say it is just isolated students who did not have good math teachers.

The classic example form Monday night wa the use of base-ten-blocks to teach decimals, something I have been doing seemingly all my life, yet none of the students had experienced such a thing. It is indeed, a mystery of monumental proportions.


Monday, March 25, 2013

"Because It Is" says Mr Gove

There's a story I tell in my math ed classes about my daughter when she was in middle school. One day, after supper, she drew a 45 degree angle on a piece of paper and asked me "Dad, why is this 45 degrees here" pointing close to the origin of the angle, "and 45 degrees here as well ?" ponting to the angle rays some 6 inches from the origin. "Look how much further it is between the rays" she said incredulously. Never missing the chance for an instant math lesson we spent the next 15 minutes discussing how angles are measures of rotation about a point, using our arms to demonstrate, and no matter how far away from that point you go the number of degrees is always the same. She seemed wonderfully satisfied with this newfound understanding and so I ventured to ask what the teacher had said when she asked. "Because it is" she said; the teacher said because it is.

In our rush to cover 'everything' there is often little time to develop genuine understanding of the math we teach and, in this case, I think I'm being generous to my daughter's teacher. I don't think she understood the concept of rotation herself and so could not answer the question meaningfully. All too often though, especially in math, we want children to remember things, for later, without fully understanding them. This seems to be what Mr Gove, the current Education Secretary in the UK seems to want teachers in that country to do. "A mountain of data" his critics say, to be memorized by rote. Facts to be learned without any sense of what they mean or when they should be used. Things are so bleak that the teachers' union has passed a lack of confidence vote in the Education Secretary and HM Inspector of Schools chief.

Their goal is to have an education system that is "robust and rigorous", words that always seem to mean lots of fact retention and memorization especially in math. "Back to the basics" yet again; a basic lack of understanding. I call it the "Because It Is" curriculum.

Wednesday, March 20, 2013

Cell Phones, Mobiles and Maths

In my math ed. class last Tuesday I asked my students to use the calculator function on their cell phones. In fact, I asked them to do this several times; it was even written on the class handout as directions for completing a series of activities. The topic of the class was how to teach decimals, or decimal fractions as I like to call them, in the elementary school classroom. We went over a number of  activities using Base Ten Blocks in which we constructed different decimals using different parts of the base ten blocks to represent the one, or whole to which the decimals referred. We modeled 365, then 36.5, then 3.65, then .365 to show the relative sizes of each number. We then started to compute with decimals which is when we need the calculator function of the cell phones. Since every student had one it was much easier to use them than passing out calculators.

This made me start thinking about what type of resources are available for using cell phones in the classroom. So I started "Googling" and discovered some interesting trends. The first thing I discovered was cellphonometry that appears to have been very popular some 4 to 5 years ago judging by the dates on the blogs and websites describing this phenomenon.  Interestingly, there are still cautionary discussions about the use of  cell phones and mobiles in the classroom while other schools are conducting pilot studies. There are also tips on how to use cell phones as educational tools. If you expand your search to include the British term 'mobiles' you'll find even more resources such as this  40 Ways to use your Mobile.

In fact, with all the apps available now and the fact that many cell phones are mini-computers some college professors such as professor Shadrick Paris at Ohio University are using cell phones as part of their
teaching strategy. It's neat that professor Paris says "I'm all about cell phones. If a student is using it for the wrong reasons they're just missing out on class". Teaching and learning are indeed based on trust.

Monday, March 18, 2013

The Apostrophe's fate

I'm not a linguist nor am I a perfectionist or a nit-picker but I do believe in protecting the Queen's English, as we used to say. When I was a fourth grade teacher many years ago I remember spending many hours teaching the value and virtues of correct grammar so that one's meaning could be precisely communicated. One of the main topics of the fourth grade curriculum was the apostrophe and I remember going over carefully all the instances for using the apostrophe as well as those when one should not use it. According to this Googled resource there are an unfortunate 13 rules identified for using or not using the apostrophe. I wonder if they chose 13 on purpose?

The apostrophe has landed on my radar screen because of this interesting discussion going on in the county of Devon in England where they have decided, for now, to ban the use of the apostrophe on road signs. The Chairman of the local County Council thought apostrophes caused confusion! (Perhaps he was confused about when to use one). This was met with a swift rebuttal from Mary de Vere Taylor from Ashburton who said the thought of apostrophes being removed made her "shudder"."It's almost as though somebody with a giant eraser is literally trying to erase punctuation from our consciousness," she told BBC News. She said there was something "terribly British and terribly reassuring" about well-written and well-punctuated writing. "Some may say I should get a life and get out more, but if I got out more and saw place names with no apostrophes where there should be, I shudder to think how I'd react," she added. (From BBC News March 15). Clearly, all this shuddering is not a good thing!

The retention of grammatical rules in "public" places is an interesting issue in these times of "private" social media where the conventions of the English language are constantly under attack.  

Thursday, March 14, 2013

Tommy Sands' incredible Concert

His voice, his words, his tunes. It's hard to tell which is more compelling. Perhaps it's the way all three seem to go together so well; a sort of synergy of emotion and music; synesthesia for sure. His song "There Were Roses" has been my favorite song to sing with my band since it first appeared around 1985. He sung it last Tuesday and even though he has changed the names of the two main characters the song still has to be the best song for peace ever written. "An eye for an eye 'til everyone is blind".

The concert opened with the incredible Irish dancing of the McFadden Academy of Dance which concluded with the most remarkable a capella Irish dance, something I've never seen before. Tommy then took the stage with his son Fionan and entertained us with songs, stories and videos. One video in particular was of an old country bus driving along the Ryan Road that figures in many of his songs. There was also a video of him singing in Sarajevo with  Vedran Smailovic, the great cellist and  peace activist from Sarajevo. And here is probably the best version ever of Pete Seeger's, Where Have All The Flowers Gone.

But it was his stories of growing up on a small farm in Ireland and his experiences during "Thc Troubles" that made his message and his songs so inspiring.

Wednesday, March 13, 2013

Pi Day for Some but not All

 So it's Pi Day tomorrow in all the parts of the world where the date is written in the remarkably illogocal way of month/day/year. This is the day  when Pi is celebrated, usually in high schools, with all kinds of pies and various feats of memory such as those demonstrated by young  Pi, the hero in Life of Pi.

There's even, a whole website devoted to all kinds of things round, ratio, and pi. The Exploratorium in San Fransisco will be celebrating all sorts of different things on March 14, the 25th time it has celbrated PiDay. At you can find a plethora of different activities you can do with your students to celebrate Pi Day. You can have memorizing competitions to see who can memorize the most digits in the never ending Pi sequence of numbers; you can bake pies, or have students form a Pi chain of the digits in Pi. Sadly, there are no activities designed to help students learn that Pi is a ratio between the diameter and circumference of a circle. In other words, in every circle the circumference is just over 3 times (3.14 etc to be a little more precise) times longer than the diameter. (or 3.4159265358979323846264338327 950288419716939 937510 to be a little more precise, but not as precise as possible).

Amazingly almost all the student I have in my classes remember that the digitis of Pi go on for ever in a non- recurring pattern yet virtually not one of them seems to remember that it is a ratio. To try to remedy this I always bring in lots of different sized plates and have my students measure the circumferences and diameters and come up with the Pi ratio themselves. The U.S. Department of Education actually has this activity on the top of their list of things to do on Pi Day ( our Government really does do some things well).

And, of course, there are Pi Day jokes, here, and here, and there are even songs here.

So spare a thought for all those unfortunate people in the UK and other parts of the world where the date is recorded in the immensely logical format of day/month/year but who are, sadly,  not able to share in the joys and wonders of Pi Day.


Tuesday, March 12, 2013

Got a Problem?

We used to teach children to find key words in word problems that would tell them which operation to use to solve tehm. We also used to tell them to look at the numbers; they'll "tell" you which operation to use;  "altogther" means add and the numbers 30 and 5 probably means divide since there will be no remainder. We know now that such misguided advice does nothing to help children know which algorithmic operation to use, or, more importantly these days, which key to press on the cell phone calculator feature; add, sub, mult. or div. For example, simple one step addition/subtraction  problems of the type found in the elementary school can usually be classified  as joining, separating, part-part-whole or comprison. The key to solving such problems is to work out what is happening, find the question, select the operation to use to asnwer the question, then check and decide if it seems a reasonable answer. To help children work out what good problems look like we should share examples of  poor problems, or un-problems. In addition to being humorus such problems help us teach children what constitutes a solvable, meaningful problem.

Here are some examples of un-problems;

  1. If it rains for 3 hours on Monday how much will it rain for the next 4 days?
  2. If it takes ½ an hour for 3 friends to walk home how long will it take 5 friends? 
  3. If 2 students have 5 pet gerbils how many gerbils does each student have? 
  4. If 2 girls have 3 brothers how many brothers do 4 girls have? 
  5. What is the area of your hand if your thumb is 3 inches long and your middle finger is 4 inches long? 
  6. If the temperature is 62F today what will it be for the next 3 days? 
  7. If 2 squares have 8 sides how many sides do 3 triangles have? 
  8. If it’s 60F in Vermont and 72F in Maine what is the temperature  in New Hampshire? 
  9. If one train is traveling at 35 mph (clearly an AMTRAC train) and another similar train is going at 45mph what time will they pass each other? 
  10. If you eat 2 slices of your birthday cake today and ½ of what’s left tomorrow how much will you eat the next day? 
  11. If 140 students are going on a field trip and a school bus will hold 60 students how many school buses will you need?
  12. If 2 friends have 5 pets and one of them has 2 cats how many dogs does her friend have?


Friday, March 8, 2013

It's a Fractious world

Gosh, did we have fun with fractions in my math class yesterday. It was the second of two classes devoted to everyone's 'favorite' math topic.

On Tuesday we explored some of the fundamental concepts related to understanding fractions. We explored the idea of the whole in the sense that fractions at the elementary school level have very little value, or meaning,  if you don't know what the whole is to which the fraction refers. To test your understanding of this  concept decide whether you would rather have half the money I am holding in my right hand or a fourth of the money in my left hand. You would need to use your imagination to do this. The clear choice is that you cannot choose because you don't know how much money I have in each hand. A half is clearly more than a quarter but only if the wholes are the same  (or, to be really exact,  as long as the whole from which the quarter is taken is not more than twice the size of the whole from which the half is taken). We explored other fraction ideas such as the top number, numerator, is a counting or cardinal number, an adjective, while the bottom number, the denominator, is a nominal or naming number, a noun (teacher knowledge at the elementary school level). 

It is critical to help children grasp the fundamental ideas of fractions because they are so counter-intuitive compared with whole numbers. Yesterday we used our understanding of these fundamental concepts to look at some of the procedural knowledge of fractions such as performing the four operations.  We looked at finding common denominators using the fraction bars (in the picture) to change 1/3s and 1/4s into 1/12s so we could add them together. We also did this to compare; which is larger, 1/3 or 1/4 (assuming the wholes were the same,  of course)?  We then looked at multiplication of fractions by exploring simple ones like 1/2 x 2/5 (a half of two-fifths is one fifth) and 3/5 x 5/8 (three fifths of five eighths is three eighths).  Try this using the fraction bars. They work so well for this because you can hold up five 1/8 pieces and just take 3 of them to get three eighths.This is where the idea of the one or whole becomes so important because the one or whole of the 3/5 is the 5/8 and the one or whole of the 5/8 is the 1 (red piece in the pic). The question to be answered then is what is the one or whole of the answer, 3/8?

This example uses the equal groups concept of multiplication. We can also conceptualize this problem using squared paper. Draw a rectangle that is 5 x 8. Mark off 3/5 down the 5 side and 5/8 down the 8 side. The rectangle you have now created will be 3/8 of the large rectangle. Once  students begin to see the patterns they can generalize to develop the algorithmic rules for multiplying by fractions. Math is the science of pattern after all. It also helps keep my students' attention at 4:25 in the afternoon when I speak with my best American accent.

Try this division one. How much of a 1/2-cookie could you get from 3/8 of a cookie? Isn't that easier than "change the sign and invert the second fraction"? Well done Emily who solved it in less than a second.


Thursday, March 7, 2013

Today is a Prime Day

My daughter Marie emailed me earlier today to let me know that today is a prime day, 3/7/13. I replied thanking her and letting her know that she was indeed her father's daughter. She then wrote back; "Looking forward to Prime Monday next week! And then Prime Wednesday! and then Prime St. Patrick's Day!"

Isn't that wonderful. All those years we spent in the car together when she was a child learning to be aware of the math around us really is paying off as she negotiates her way through  her 30th year.

My mission in life as a math educator is to help young children become aware of the mathematical relationships around them; the mathematical patterns, numerical relationships, relative sizes and places of things; where there are circles, squares, perpendiculars and parallels, unknowns that can be found through problem solving and the aesthetics of it all. Prine numbers are a classic example of how unthinking, rote learned classroom math can destroy our sense of mathematical wonderment. Remember the wonderful definition and test for prime numbers that we all learned? "A prime number is a number that is divisible only by one and itself". A search of the Internet will reveal a whole variety of definitions each more complicated than the next.

To young children first coming into contact with the concept (about 3rd grade) we need something that illustrates the concept of the prime number visually, efficiently, meaningfully,  and clearly. This can be achieved by using the area or array concept of multiplication. Every multiplication fact makes either a rectangle, series of rectangles or a square: 4 makes a 1 x 4 or a 2 x 2: 9 makes a 1 x 9 or a 3 x 3 square (hence its name - square number): 7 makes only a 1 x 7 rectangle, 13 makes only a 1 x 13 rectangle and 23 makes only a 1 x 23 rectangle. So prime numbers make only one rectangle - unlike 24 which makes a 1 x 24, 2 x 12, 3 x 8, and 4 x 6 rectangles.