Monday, January 26, 2015

The Math Practice Standards

I've read a lot recently about the math practice standards as described in the Common Core State Standards and have come to the conclusion that if the CCSSM content standards are the 'what' of teaching math the the CCSSM practice standards are the 'how' of teaching math. In other words, teacher must know the math practice standards so well that they become an integral part of the teacher's interaction with the student.

Any behavioral changes we need to implement in our interactions with students takes time but I can feel myself slowly changing when I interact with my students during my math  education classes. My focus is also changing when I observe my student teachers teach math. This morning, for example, I observed a student teaching third graders all about multiplication facts She did a great job drawing the students' attention to the structure made by the facts on the multiplication table, a 10 x 10 square. She also had the students identify the regularity of counting by 4s or 5s. And, if this wasn't enough, she also gave the students some problems to model mathematically to demonstrate their understanding of the multiplication facts.

The key to the implementation of the math practice standards has to be in how we interact with students mathematically; what we ask them to do and how we ask them to do it. The activities themselves are not going to do this. We need to consistently and conscientiously ask, require, tell, suggest, model, demonstrate, and use any other appropriate verb that will, over time, help students integrate the math practice standards into their lives.

They really are dispositions by which to live our mathematical lives. 


Thursday, January 22, 2015

Understanding and the CCSSM

Every semester, at this time, I introduce my students to the idea of understanding maths. It's not an easy task especially for those graduate students who may not have experienced learning elementary school maths for many years. Even with some of my undergrads the process of having to understand something they have known by rote, "off by heart",  for some times seems like a pointless task.  But, we have to understand the maths we are planning to teach so we go through the topic carefully to make sure everyone is on board.

In every field of study or human endeavor there are giants; people whose ideas, or deeds, have stood the test of time and are as fresh and relevant today as they were back then. One  such person in math education is Richard Skemp.

In 1976, Richard Skemp, a British educator published a definitive article in the journal, Mathematics Teaching, in which he described two types of understanding of mathematics; instrumental understanding; "rules without reason", "rote learning", "pure memorization",  and relational understanding; or understanding the what, why, how, when, connected with,  and so on; the type of understanding to be found in the CCSSM. Having found that my students have a hard time remembering the terms I now call them fragile understanding and robust understanding which always seems somewhat onomatopoeic.   

The wonderful thing about robust understanding is that it not only makes problem solving so much easier, it makes the retention and recall of all those illuminating but tedious  math facts children have to learn infinitely more efficient and effective. If, for example you can relate the multiplication facts to the addition facts they suddenly have a new structure that gives them sense and aids in meaningful recall. 6 x 5 for example is 5 x 5 plus another 5. Then if you can count by multiples of  5 the mental structures you are creating makes forgetting almost impossible.

We all learn that Pi is  3.14 and just about everyone knows that it goes on forever without establishing a repeating pattern. Every high school hallway is adorned with it on pi-day on March 14. (July 22 in the UK).  But ask anyone what it means and very few can tell you that it's a ratio between the diameter and circumference of a circle. Every circle is just over 3 times further around the outside than across the center.

This is what the CCSSM are designed to do.    





Wednesday, January 21, 2015

Mathematize and the CCSSM

Some time ago when I was learning about the Math Recovery program and reading the wonderful books co-authored by Bob Wright I came across the verb 'mathematize' and its noun counterpart "mathematization". I always thought it was a wonderful way of describing what math education at the elementary school is all about.

In his book, Developing Number Knowledge,  Wright defines the term (p15) this way;
                  Mathematization means bringing a more mathematical approach
                  to some activity. For example, when a student pushes some
                  counters aside and solves an addition  task without them,
                  we say they are mathematizing, since it is mathematically 
                  important to reason about relations independent of concrete
                  materials.

Others define it as "reduction to mathematical form" (Merriam Webster), "to treat or regard mathematically" (The Free Dictionary) and "explaining mathematically" the Collins dictionary.

The really, really interesting thing about all these definitions is the idea of reduction or movement from real life, concrete situations such as that described by Wright, to the symbolic form of symbols and algorithms typically used in math. This is completely opposite to the way math has traditionally been taught and  how it is sadly still taught in poorly taught math classes.

A classic example occurred in my math class yesterday when I asked students what 1/2 ÷ 1/4 meant. No one knew. My hunch is that if you randomly asked 100 people on the street only a handful would be able to tell you that this meant how many quarters are in a half. What really makes this intriguing is that most people would tell you to change the sign, flip the second fraction, multiply and get the answer 2. 

In other words people have not gone through the process of  mathematization when they have learned this procedure. A real world, concrete idea has not been "reduced to a mathematical form". They learned the mathematical procedure without any sense of what it meant or connection to any concept or relationship. There was no derivation, if you like, from a concrete experience or idea  to a symbolic,  mathematical relationship. This happens all the time in math.

Students are taught a square number is the result of "a number times itself" instead of a number that makes a square.

They are taught a prime number is "a number divisible only by 1 and itself" instead of a number that can only make one rectangle (e.g 1 x 7 or 1 x 13).

Students are taught the symbolic mathematics first and not the idea so they cannot be mathematized. Instead, we should be mathematizing them from concrete experience  to symbolic representation. This is what the Common Core State Standards for Mathematics is trying to achieve.
 

Saturday, January 17, 2015

Confidence; the Forgotten CCSS Math Practice Standard

One of the interesting things about the Common Core State Standards for mathematics is that nowhere does the word 'confidence' appear. Yet it is, perhaps, the one thing that will help young children, and many adults too, develop their understanding and skills in the field of maths. All eight math practice standards, in some way, imply the development of confidence.

I'm always very aware of this concept at the beginning of the semester when I'm getting to know my new students. There are also activities I do in which I ask the students how confident they are in their responses. For example, we use a math manipulative called Unifix cubes which are colored plastic blocks that fit together. As they explore the materials I ask students to describe them so that they can get to know how to use them. I ask them how many different colors there are After a minute or two someone will say "ten" and I'll say "how confident are you". This seems to stop them in their tracks as if I am challenging their counting skills. They start looking through the box of cubes again to see if there are any more and after a few minutes they'll say they are sure there are ten.I then ask "on a scale of 1-10 how sure are you? The usual response is 9 or 9.5.
I then ask, How sure are you the sun will rise tomorrow morning?  "Aha 10" they say. So why not a 10 with the number of different colored Unifix cubes?  For some reason math seems to promote this lack of complete confidence.

I think confidence in maths comes from a variety of sources but primarily I think it comes from how well supported are the things we know and understand. If we can see something as part of a pattern, or part of a system, or part of a family, or part of a schema, or part of a series, or part of a predictable structure,  or related to something else we know or understand then our confidence ill be increased.
If we know or understand something in isolation, without these connections, then we have far less confidence.

So, taken together, the Common Core State Standards for Math Practices add up to the development  of a student's confidence on the content standards they are learning.       


Thursday, January 15, 2015

Numbers, numbers, numbers.

This is my four hundred and first entry in my Mostly Math Blog making my last one my 400th. By blogging standards this is probably quite insignificant but for me it is quite the milestone. As a math geek fascinated by numbers and the relationships between them there is always something special about a nice round number; something to be celebrated as one does one's 21st birthday or one's 50th wedding anniversary.  Numbers, by themselves, are the markers of our life stories, the moments of significant achievement, the tale of the tape measure so to speak. They bring specificity to life as well as precise comparisons. The also bring a sense of accomplishment or failure, success and challenge. Imagine a life without numbers where we lived in a land of 'ish" in which we only used words like some, many, few, lots to quantify our lives. I am told there are some indigenous peoples in the Amazon jungle who have three counting words; 'one', 'two', and 'many'.

This is all fine and good but when it comes to problem solving in he elementary school the numbers really are just the details. It is what is going on with all the words that surround the numbers that is the most important thing to be thinking about. Does it really matter that much if one train is going 40mph, and not 45mph, and the other 60mph, and not 70mph, when we have to solve the problem involving when they will crash into each other, or how far apart they will be in half an hour. The numbers are, of course, important for calculating the specific answer but they don't tell us what we have to do. It's all the words around the numbers that we have to decipher in  order to work out how to solve the problem.

If Mary has 16 Hotwheel cars and Michael has four more Hotwheel cars than Mary has, how many Hotwheel cars does Michael have? To solve this problem the numbers themselves are pretty useless. The most important thing is what the words around the numbers are telling us is happening. The understanding of the situation we can construct from deciphering the words is what tells us what we have to do in order to solve the problem. Recognition of the numbers themselves only provides us with the accuracy for our solution. Our understanding of the linguistic structure will tell us if this is a joining, separating, comparing or part-part-whole problem. 

Tuesday, January 13, 2015

First Week of Classes at St. Mike's

Another week, another semester and another year. The first week of a new semester is always a time filled with mixed emotions. The excitement of meeting new students, as well as graduated undergraduates returning as graduate students, is mixed with the nervousness of of the unknown. Even after forty years of doing this the first half hour of a new class is a nerve wracking experience that I hope never goes away. I always tell my students that every new class of students they meet whether they be kindergartners or graduate students should be met this way.

One of the things I always try to identify at the beginning of a new semester are those students for whom maths is a chore; those students who dislike or even hate math; those students who misguidedly say they are no good at math. Recent research, especially in the area of Carol Dweck's Mindset Theory   suggests that the way we feel about ourselves in relation to maths directly affects our ability to learn maths. Very often, this negative self concept is something engendered by the actions of an uncaring or thoughtless teacher or a bad experience many years ago. Very often it is not a person's fault that they think they are no good at maths and there is absolutely no reason in the world that any adult can continue to not understand maths. 

One of the things I always try to do in the first class is to give students examples of how exciting and interesting it is to teach maths to young children One of the stories I always tell is of a kindergartner I once met several years ago who was convinced that 6 subtract 6 was five. Nothing we could say would convince him otherwise, so finally we asked him to show us with his fingers. He put up five fingers on his left had, pointed to each one with his right index finger and counted them out loud; "1,2,3,4,5". He then put up his thumb on his right hand and said "6", touching it with his nose. He then said, while removing his right thumb,  "OK, now I take away 6 and I've got 5 left".

According to the cognitive level of development of his numeracy skills he was absolutely right of course. He was in that halfway place of moving from the nominal or naming use of number to the cardinal or counting use of number; something we all go through in our early years. He was fine with 5 because that's an easy one, a handful, but was still working on the other single digit numbers such as 6.  

Learning about how children lean maths is the best thing in the world. I have also decided to use "maths" instead of "math" all the time. "Mathematics" is a plural; so too should be the abbreviation. 

Saturday, January 10, 2015

Progresso Needs to Learn Science,

Planning for and implementing the Next Generation Science and Standards (NGSS)  during the next eighteen months is going to be a monumental task. Unlike the Common Core standards only 26 States have agreed to sign on to implement the NGSS; Vermont is one of those States and the Vermont Agency of Education has already started holding regular meetings for those interested in getting a head start on implementing the NGSS.

Fortunately, the NGSS include a significant section on the development of children's engineering and technology skills, as they apply to science and engineering. The LEGO robotics sytems such as WEDO, Mindstorms and NXT, all of which we use in our courses at St. Mike's, will form a large part of this part of the NGSS. There's even a fledgeling robotics club at the college run by a group of students.

But the science part of the NGSS is probably the most important aspect and the piece that should be given most attention. During the next 18 months schools should be investing in professional development activities to help teachers implement the NGSS in 2016. They will need to collect materials and activity resources for each grade level as well as find time within the already busy school day to implement meaningful science activities.

So why the picture of a can of Progresso soup? Every time I see the Progresso soup commercial on TV I cringe. Anyone who has ever conducted the string-soup can telephone activity in an elementary school science classroom knows that it doesn't work unless the string is kept very taut and doesn't touch anything else. The commercial shows the string from the soup cans dangling loosely rendering it completely useless. If ever there was a need to improve science education in our culture this is a wonderful illustration of it. How come none of the hundreds of people involved in the production of this commercial didn't realize this error?