Newtown: what have we learned?

As the one-year anniversary of the Newtown tragedy approaches one cannot help but think that we have done little to improve the quality of school life in a way that will help avoid the replication  of such terrible events.
Since that fateful day almost 200 children have been killed by guns and yet nothing significant has been done to curb the sale and possession of fire arms.

There has been a raft of misguided strategies such as those identified here  where people and organizations have grasped at straws in order to try to protect those who are most vulnerable. Some of the suggestions have seemed pathetically comical such as throwing erasers and cans of soup at gunmen or having bullet-proof white boards available. At one point I even thought I might have to include instructions on how to use a gun in my math education courses when arming teachers was being suggested.

In this article, the piece I find most hopeful is the section on what steps should be taken to help prevent such tragedies. Interestingly, these suggested steps are based on extensive research and give hope that there might be a solution since it appears sadly, that our culture is too dependent upon guns to give them up. According to the article schools should develop a strong emotional climate based on listening, trust and caring; reduce bullying and remove the code of silence. In other words, we need to create school environments that are sensitive to students who are having difficulty; an environment in which it's alright to seek or offer help to an individual.

This sounds like a wonderful set of goals and yet do we really value these attributes in our places of learning. Recent trends in Education would suggest we don't. What seems to be more important is creating climates of competition in and between schools where many schools are identified as failures based on a narrow set of test scores. "Race to the Top" even implies that life in schools is a competition where only a certain number pf participants can succeed. In the UK there are even "League Tables" where schools are compared based on student test scores. Not surprisingly there is a clear correlation between those schools that do well and those where the students' parents have higher incomes.

A competitive climate of success and failure will not encourage an environment of trust, caring and sensitivity researchers suggest is the key to making our schools safer places for students and teachers. We owe it too our children, our students to be better than this.   

ELL and Math resources

Here's a really neat Powerpoint  resource I came across recently that looks at teaching math to English learners from a different perspective; that of the TESL person as opposed that of me, the math person. 

There is an absolute wealth of information in this presentation especially the linguistic identification of the different types of words used in math.

And here is another really good account from the Virginia Department of Education that also gives some historical background to the theories involved in teaching math to English learners.

Finally here's a book that I probably should have read some time ago but have only just discovered.
It is the CALLA handbook. The book describes the development and implementation of the Cognitive Academic Language Learning Approach to teaching English learners.

K-5 Math Teaching Resources

The Common Core has done one thing if it has done nothing else and that is to stimulate growth in the educational section o the public sector.  A Google search of "Common Core math" yields "36,200,000 hits in 23 seconds". Many, many of these are publishing company's attempts to cash in on the need for new materials to meet the demands of the soon-to-be required Common Core math (at least in those states that have signed up).

Over the past year or so I have been bombarded with websites offering all kinds of things usually promising to make my job easier ( I wish they would promise to make it better) Every-so-often I poke around on my own looking for something I can use i my courses and recommend to colleagues. Here's one that I find quite amazing because it is so simple and  straight forward and the math is absolutely wonderful. The work-cards and images for arrays are terrific and are so much better than the rectangles one so often sees that I find confuse arrays with area.

How many eggs are there? A student who understands the concept of the array as multiplication will say 4 x 6 while the student who doesn't will count them one ob one.

Here is the K-5 Math Teaching Resources; it really is worth a visit. I wish I knew who the authors were though.   

Recognition of Math and English Learners

It was such an honor to receive the 2013 Balomenos award as the math educator for New England  at the recent ATMNE conference in Killington. (That's Mary Calder, VCTM president presenting the award). It was a recognition that teaching math to students who are English Learners requires something different, that we need to stop and think a little more carefully about what we are doing. We are now beginning to recognise that there are significant differences between the math students learn in other countries and the math they learn when they arrive in US classrooms.

Differences can exist in so many different ways from the simple algorithmic procedures we use to the complexity or simplicity, as is so often the case, of the counting or number  systems being used. There are also so may other ways that the quantitative aspects of a culture can differ. The strangeness of our measuring system must drive English learners used to the metric system absolutely crazy. Imagine trying to estimate distance in kilometers, or weight in kilos or length in centimeters?. What is your hand span in centimeters? Now measure it to see how close you are. Try the same thing in inches and I bet you are more accurate.

The numbers inherent in unique cultures are also very different. Our lives are defined by numbers and quantitative relationships whether we like it or not. House numbers, SSNs, clothes sizes (we are different sizes in different countries), numbers of States, shires counties or Provinces, stripes or patterns on flags and so on. Every country has a mathematical profile that affects the individuals who live there.

Then there are differences in the math education. The recent PISA test results show several Asian countries at the top with the US and UK fairly well down the charts. I always find such comparisons fairly futile and frustrating since there are so many difference that are not considered. In Shanghai, for example, which came top, there are reports of students working under incredible pressure, studying 15 hours a day and all weekend, There are also reports that teachers get a lot of time during the day for professional development and I am sure there are many other reasons why the differences in scores exist.

So when we are teaching math to English learners we should never dismiss their math difficulties as the result of their limited English. There are so many other reasons why they might be having difficulty. It could also be that they are not having difficulties in math at all; just difficulties communicating what they know but we should never make this assumption. We owe it to our students who are English learner to find out about their math, the math  they know and understand.   

English Learners and Counting

As I continue to interview students who are English learners it is so interesting to observe different things which when put together create a more solid understanding of the issues EL students face when learning maths in US schools.

The difficulty English learners have in hearing the difference in the pronunciation between the teen numbers (fifteen) and the decade numbers (fifty) is well documented. The 'n' phoneme at the end of a word is rare and quite difficult to say in some languages and therefor is very difficult for some people used to speaking those languages to hear.

I remember discovering this several years ago working with a group of Somali students who kept saying "twenty" after they had counted by tens to ninety. Yesterday I was interviewing a third grader form Burma and he almost made the same error catching himself at the last moment. The student has a wonderful cheerful disposition and immediately said "teen is 1 and ty is 0" as if it was something he had been taught to help him differentiate between the two. I must follow up to see if this is in fact true.

The other thing I am noticing is that many of the English learners I am working with find it very difficult to skip count by 2s an 5s. Even when they have an excellent grasp of place value up to reading, writing and modeling 4-digit numbers they still have considerable difficulty skip counting.
This apparent lack of a sense of pattern in number could be quite an issue. This is something I will begin to watch for more closely.

Edshelf; A Great Resource

The world of iPAD Apps is truly bewildering but here's a great resource with what appear to be genuine reviews and collections sorted by topic. It's called Edshelf and is free but you have to sign up. It also looks as if it is free from the fine-print that can get you into trouble if you leave a negative comment of a product. Here's the math manipulatives page
which gives you a sense of how usefuol people are finding each of the different Apps.

There's information about sources of funding for purchasing iPADs such as this Apple information and  and here is the US Department of Education funding site. There are also many YouTube videos featuring iPAD math Apps.

Math Triangle of Meaning

I've been on sabbatical this semester researching ways to help students who are English Learners learn math in US classrooms. Part of the project has involved interviewing both adults and children who have settled in the US having grown  up in other countries, just like I did. This week I have interviewed a couple of third graders who were born in Burma and came to the US via Thailand.

Both students spoke English well so communication was not an issue. In the first interview the student did something quite remarkable in that it was so exactly in line with a particular theory I use in my math education courses. It is always so neat when this happens because it reconfirms the value of a theory-based education. When we can make an observation and relate it directly to a particular theory, or piece of a theory, as teachers we are in such a better position to know where to go next in the instructional process.

The particular observation happened as I was interviewing the student on his understanding of basic numeracy; more specifically being able to read, write and understand 3 and 4 digit numbers. The triangle of meaning is a piece of theory that identifies the  three component of a piece of math understanding; the idea, the word and the symbol. For example 5 is "five" and *****. Put one of these at each  corner of a triangle and there are 6 relationships (e.g hold up five fingers and ask how many, hold up 5 fingers and ask student to show the correct numeral). Students may  frequently have some of the relationships but not all 6.

Yesterday the student kept reading 4 digit number incorrectly. For 4,582 he would say "forty-five thousand and eighty two". So I asked him to write numbers such as 4,275. He did this correctly every time. I then asked him to read back to me the numbers he had correctly written and he read them incorrectly using tens of thousands as in the example above.

He could model the numbers using base ten blocks and he could write them when given them orally  but he could not read them even when he had written them. The important lesson here is to remember that just because a student can write a number doesn't mean to say she/he can read it, or knows what it means.   
  

A Ray of Hope for Education

In the depression and gloom of the current movement of  the private sector take over of public education there's a ray of hope that has to be developed from its quiet, gentle nucleus into a roaring inferno that will  extinguish the darkness that is enveloping our schools.

That ray of hope, a beacon of light, is embodied in the writings of Diane Ravitch and others with  visions of what education should be. In this particular piece she asks why we have to treat schools like sports teams with leagues containing winners and losers. Race to the Top has done much to engender this analogy by implying the education is a competition in which everyone is racing to get to the top.

The interesting thing about that analog, for that's surely  what it must be, is that it implies so many different ways in which some people are better than others. If it is a race it implies there are winners and losers; that not all people can be at the top. Even if everyone could reach the top it wouldn't be the top anymore because 'top' is a relative position implying that those on top are on top of those below those below. If everyone was on top they couldn't be on top because there would be no-one for them to be on top of.  This makes the whole idea of Race to the Top rediculous.

As Ravitch says ' We must think and act differently. If we do, we will not only have better schools, but a better society, where people help one another instead of finding a way to beat out their competitors".

Now the big question is how can we get the private sector to think in that same collaborative way where individual people come before dollars and the profit margin? 
 

Dark Days for Education!

One of the good things about the impending Common Core is that it is designed, in theory,  to just  describe a set of standards expected of students at different grade levels in ELA and maths. Unfortunately, that is also one of the really awful things about it. In essence it is open to just about any interpretation including what is happening in New York State where modules. are being prescribed for implementation at each elementary school grade level. If this eloquent presentation is an accurate description of what is happening all is lost and these are dark days indeed for elementary education in New York State.

I cannot imagine what it must be like, to teach these modules where content is dispensed in carefully timed segments regardless of  student understanding and participation. One has to wonder who is in chanrge in Oneonta. Then, of course, there is the corporate control over the curriculum such as that exerted by Pearson and others. SBAC and PARCC, the two approved assessment companies, will make a fortune out of the mandatory assessments associated with the Common Core.   The private sector has probably moved faster than any other aspect of our culture to "make a quick buck" from the implementation of the CCSs. It's not just in public schools where the private sector is taking over. Future teachers are going to be required to complete an electronic portfolio scored by ETS in order to get their teaching licenses. Control over the quality of teachers in the future will pass from the State to the private sector.

Ultimately, the private sector will have direct control of the education of "young consumers". Perhaps schools themselves will be taken over by the private sector. Students from kindergarten on will wear school uniforms with company logos on them; classrooms will be equipped with all the company's latest products, teachers will say "Math class is brought to you today by Walmart". Teachers will be required to drive Fords in schools owned by Ford and wear clothes purchased at Kohls at schools owned by that company. 

 

Jo Boaler Almost Has it Right

I really like what Jo Boaler has to say about maths education. I was one of the 40,000 who followed her Stanford course this summer on teaching maths. Her research over the years has shed much light on many different aspects of mathematics from the best way to teach to how to get people to open their minds about what maths and maths education are all about.

Sometimes, however, I fear she is guilty of making  over-generalizations in a way similar to those who believe that math should be memorized and learned the way it was 50 years ago.

In a recent article in The Atlantic she  avers that  "Speed doesn't matter, and there's no such thing as a "math person."". While I would agree that the two ideas of a) the need for speed, and b)  only people good at maths can do it, are two misconceptions that plague the real and joyful  study of maths there is difficulty in stating the situation categorically as Boaler does.

Clearly, the argument she gives against the need for speed in problem solving and so on is exactly right but I believe there are times when speed is a good thing such as in recalling facts. Having quick access to certain pieces of information is empowering and makes life in general easier and better. Generally, waiting for your mom to cut your food or for your dad to tie you shoe laces is OK when you are 3 or 4  but not when you are 9 or 10.

Again, to say there is no such thing as a "math person" seems to overstate what we really believe. There are clearly some people who will always be better at maths than others in just the same way some people will excel in sports or art or as musicians. I guess I still have to  agree with Howard Gardner that some people have a propensity for being able to think in certain ways while others may not. What I think is the heart of Boaler's assertion is Carol Dweck's idea of fixed versus growth mindset. In other words, so many, many people end up believing they are not good at math because they couldn't solve problems quickly, couldn't remember facts and were made to feel foolish by uncaring teachers resulting in a lifelong belief  that they were not good at math.  No-one should ever say again "I'm no good at math".

When we teach young children maths we should be sensitive to the way we respond to their efforts and accomplishments in maths class. We should encourage them to take time to think through what they are doing and to use what they know and understand.  We should encourage them to do the best they can but recognize a situation where, for one reason or another, a student might need extra help in the form of a different example, method or strategy to understand a particular idea or develop a particular skill, Everyone should have the opportunity to become the best "maths person" they can be. 

  

Math Apps for iPAD2s

Finding good APPs for the iPAD2s in your classrom can be a time consuming business, and expensive too if you pay for them. One way to reduce the amount of time and download them with a greater degree of certainty is to use a reveiw center like this one at Ed Shelf. It has a whole bunch of features that add to it's value such as being able to make "collections" of your favorite Apps which you can then post for other to share.

This strategy significantly increases the sense of trust one can have that the "likes" and recommendations are not being posted by those who might profit from increased sales. The site is also recommended by the IT folks at St. Mike's and one that they all use.

More and more schools are investing in iPAD2carts so that more students have access to this neat learning tool. There's a new Mac Lab

Math and English Learners

This past weekend I gave a presentation on teaching math to English Learners at the NNETESOL conference at the University of Southern Maine campus in Gorham, Maine. Titled My Math Counts Too, the presentation focused on selected issues related to teaching math to students who are English language learners or English Learners (ELS) as they are now referred to. I have made several presentations at this conference over the years and they are always well attended. I have the sense that there is a great need for more research/information/support and so on for teaching math to students who are English learners. Quite often the math ELs have learned is quite different from that in the US classroom which can present quite a challenge. For example, the new CCMSs require that all students be taught the standard algorithms. Many EL student will, however, have spent many years learning algorithms
that are quite different from those defined as 'standard' in US classrooms!   

I always begin these presentations by addressing some of the assumptions we tend to make when teaching math to English learners. For example, we tend to think that math is the same the world over but there are so many differences between the math of different cultures that we really cannot make this assumption. The procedures and strategies students are taught as well as the mathematics of the culture can all be significantly different. It can also be assumed that a student's difficulty in math can be attributed to a lack of English competency but if the student has never had the opportunity to learn math no amount of instruction in English is going to make any difference to the student's math skills and understanding.

Next fall I will be offering a course titled Math and Diversity which will include a section on teaching math to English learners. Offered through the Graduate Education Program at St. Michael's College the course will focus on this and three other strands; teaching math to students with special needs, students in poverty and students with math disabilities.    

Early Chidlhood math

Here's a really useful and interesting resource a friend in Early Childhood Education recently sent me. It's research-based and has a wealth of ideas and suggestions for teaching math to preschoolers and kindergartners. The really neat thing, however, is the suggestion that we need to make math an integral part of children's lives in the sense that they see things through a mathematical lens from a very early age.

I have a feeling this is similar to the idea of mathematization developed by Bob Wright in the Math Recovery materials. As we know, manipulative materials are essential lfor helping children develop all sorts of mathematical and quantitative relationships but it's the student's ability to use these skills and apply these concepts abstractly that enables them to think mathematically.

The paper also suggests that children gain experiences in a variety of math topics such as measurement and geometry in addition to fundamental ideas of number. I wonder what the effects of the increased use of iPADs and other tech-based learning tools will have on the development of student's mathematical ideas?

Math TED talks

Tracy Watterson reminded me today of the wonderful Arthur Benjamin TED talk about the magic of Fibonacci numbers.So I did a quick search to find all the TED talks about math. and started watching them (one of the joys of being on sabbatical). So far I've watched The Math and Magic of Origami which shows how Origamists eventually turned to math to develop rules and new origami projects. This was quite amazing in terms of the mathematical laws and rules that govern origami. I've also watched the Benjamin one of the Fibonacci numbers which really brings them to life beyond their usual natural occurrence.

I usually find myself agreeing with the speakers especially those like Dan Meyers who insists that we really do need to do something to make math more captivating for students of all ages. But, occasionally I do find disagreement with the ideas being presented. One such TED talk is the one given by Conrad Wolfram entitled Teaching Kids real Math with Computers. His fundamental idea is that we have to "stop teaching calculating and start teaching math". His fundamental error is that he defines "basics" of maths as calculating using paper. This is an error that is so frequently made by people who would have us teach the way we taught fifty years ago. He completely ignores the idea of numeracy and the fundamental understanding of place value and base ten which comprise the real "basics".

After you watch the Wolfram TEDtalk read this article by Karen Fuson about standard algorithms in the CCMS. She makes a good case for "calculating". The real question then becomes:  "What is the standard algorithm?" especially if you are and English Language Learner.

If you have a favorite math TED write about it on this blog.

PS The Ken Robinson TED talks on Creativity in Education are also really good.

Are Standard Algorithms for All Students?

   53 
x 25
  Every so often I get asked to interview students who are enigmas in the classroom; students who are difficult for teachers to understand; students who, perhaps, do things slightly differently. When this happens I use one of the several Interactive math Interviews I have developed over the past ten years. designed to engage the student in a conversation with a purpose, the interviews provide a framework or jumping-off point from which the student's thinking can be thoroughly explored. 

As an aside, I recently came across this study at Florida State University  in which the researchers found out, after using up a 2.9 million dollar grant, that;

         "When early elementary math teachers ask students to explain their problem-solving strategies
         and then tailor instruction to address specific gaps in their understanding, students learn
         significantly more than those taught using a more traditional approach. This was the conclusion
         of a yearlong study of nearly 5,000 kindergarten and first-grade students conducted by
         researchers at Florida State University".

To me, this seems like studying whether it's raining or not but watching people week after week to see if there's relationship between  umbrella use and inclement weather. WHY WOULD TEACHERS NOT ASK STUDENTS TO EXPLAIN THEIR PROBLEM SOLVING STRATEGIES? Clearly, I'm not the only one amazed by this somewhat banal finding.

But to return to the remarkable 3rd grader. When asked to solve the problem above he said; "Hmmmm, four 25s are 100 so 40 are 1000". All in his head but was getting a bit confused so I suggested he write it down which he did. He then said "so, twelve 25s are 300 which leaves just one 25 which mans the answer is 1325". I was pretty impressed.

I probably should have given him one with "un-nice" numbers to see what he would do but I ran out of time. My hunch is that he would have worked out a similar way of doing it using his incredible understanding of number. The real question is; Should he have to learn how to do the Standard Algorithm since he will most likely be tested on it in the upcoming SBACS? 

What do you think?

 

Eleven/Twelve/Thirteen

The time has just passed 14:15:16 on 11/12/13. These three numbers are probably three of the most interesting numbers we have in our numerical series. Eleven  has a really interesting history and basically means one left over after ten in several different ancient languages. One can see the lev - left connection. Similarly twelve also has a similar connection with two and lev being sort of smashed together to make twelve. Then thirteen is the first number to adopt the teen suffix although for some obscure reason it is not threeteen which would make things easier for many to pronounce. Like fiveteen it probably got contracted to make things a little easier to say.

This is all good and well, or well and good, unless you happen to be a person who grew up with a wonderfully logical numbering system like most people who live in Asia where things tend to go ten-one, ten-two, ten-three and so on. How much better it would have been had we had the foresight to change things around a bit to make it easier for our young people learning to count. Oneteen, twoteen, threeteen, fourteen etc would have been so much easier.

Two of these numbers have also been embellished with all sorts of wonderful extras. Twelve even has its own name; dozen, and thirteen is famous for its triskaidekaphobia label. Thirteen is also a baker's dozen, a phrase which has a checkered and often gruesome set of possible origins. 

And remember this date means nothing in the UK where it is written 12 - 11 - 13. Their 11-12-13 will be the upcoming December 11!!!!!!!.

Common Core Math Standards Ignore the Role of Pattern

    

I still cannot decide if I think the new Common Core Math Standards (CCMSs) are the best thing since sliced bread, just abother phase in the development of American Education or an imminent disaster. I've just spent three days at the ATMNE (Association of Teachers of Mathematics of New England) conference listening to a variety of speakers and presenters sharing their ideas and strategies for implementing the standards in classrooms througout New England. What concerns me most is that all the people I respect in the field of maths education all seem to have different views about the CCMSs. Most are passionately against or passionately for them; there seem to be few indifferent opinions. 

The focus on the math practice standards is probably a good change although the recognition of the importance of finding patterns in mathematics seems to be completely missing. Yes, finding structure is included as a practice standard but this is not the same.

For example in one of the presentations on misconceptions in fractions the presenter used an example of comparing the fractions 5/6 and 6/7 I think it was. Children, apparently, gave all soprts of answers and predominantly said they were the same because there was only 1 difference between the numerator and denominator in each fraction. Most teachers suggested converting to decimals or finding the common denominators.

But, if we taught children all about the inherent patterns in math they would be able to compare them easily. 

Look at this pattern; 1/2, 2/3, 3/4, 4/5, 5/6, 6/7, 7/8, 8/9, 9/10. Numerators and denominators increase by 1 and each fraction is getting progressively closer to 1. In other words the bit left out of 1 not included in the fraction is getting smaller; 1/10 is smaller than 1/2, We can use models to show this.

Look at this pattern; 1/2, 2/4, 3/6, 4/8, 5/10, 6/12. All are other names for a half. In each one the numerator is half the denominator; the numeraotr increase by 1 and the denominators increase by 2.

Now look at this pattern; 1/3, 2/6, 3/9, 4/12, 5/15, 6/18. All are other names for 1/3 and in each the numerator is a third of the denominator with numerators increasing y 1 and denominators increasing by 3.

Just for fun, look at the half pattern again and find a fraction between 1/2 and 2/4. How about one and a half thirds? 11/2 is between 1 and 2, and 3 is between 2 and 4. It's actually 3/6.  Now try the fraction between 2/4 and 3/6.   

Maths Should Be Fun?

For some reason we seem to be going through a phase in our culture where math has to be fun. Judging by the incredible number of websites that have sprung up with mindless activities that you can play on-line and be subjected to a barrage of ads, or buy at great cost the business world has discovered there's money to be had with people's current dissatisfaction with public education. One such company IXL seems to appear everywhere and has probably the most useless activities of any to be found on-line.

Of course math should not be the incredible tedium it once was and still is in some places but neither should it be fun. It should be CAPTIVATING, INTERESTING, MEANINGFUL, RELEVANT and above all else, IT SHOULD MAKE SENSE. If everything in life was fun we would not know what fun was. Fun is something we have to relax and enjoy ourselves when we are not working. Learning math is work, it is something that, hopefully, is all of the things in CAPS above but it cannot be fun; not all the time. There are some things like muliplication facts that just have to be learned.

There are times of course when what we learn in math can lead to fun, When we are playing games that involve math knowledge and understanding such as cards, board games and many of the tech-based games that children play. Have you ever thought about how much math there is involved in playing Wii or the DS games? (Wii Play Together). There are also some great Apps for the Ipad2 that help children learn math facts and concepts in a motivating way.

OK, so there are some fun games for helping children remember their math facts but these should only be used after children have developed strategies for remembering their facts. Here's my current favorite on-line math activity. It's called Arcademics. Have fun, er, I mean, be captivated and try to
remember your facts.

No Cursive; no hand calculations?

I recently read this neat Huffington Post piece about the demise of cursive handwriting. Well, demise maybe premature as the conclusion in the article was that cursive writing now belongs as part of the elementary school art curriculum.

So, if we can cast aside the century-old idea of creative handwriting as a mainstay of the elementary school curriculum then it's time we cast out the four algorithms, hand calculations, used to perform addition, subtraction, multiplication and division. I hasten to add that I am referring to the vertical form of algorithm where you put one number above the other with a line below and follow a procedure. We must, of course, still teach the concepts of addition, subtraction, multiplication and division in all their forms but it's time we used technology to do the tedious part, so to speak. Learning the algorithmic procedure does nothing to enhance students' mathematical skills. But, there it is in the Common Core Math Standards (CCMS) due to be implemented in 2014. 
 
Conrad Wolfram suggests we should be using computers to take the drudgery out of hand calculating in maths by allowing students to use calculators and computers to perform these tedious hand calculations. 

I was beginning to go along with this idea until I read this piece by Karen Fuson and Sybilla Beckman about the inclusion  of the "standard algorithm" in the CCMSs. Their argument is persuasive and cites the practice students gain in developing their understanding of place value and base ten when they are involved in the completion of hand calculations. It is part of the process of mathematization that students go through when they are learning elementary school maths. The use of manipulative materials and technology are important in developing ideas but it is what happens in the student's mind that is the most important aspect of learning math; the ability to abstract and manipulate ideas mentally.

Standardized Testing out of Control

As I was watching the Harry Potter movie, The Order of the Phoenix, with my son Andrew this afternoon it occurred to me that a passage from this movie is the perfect allegory for what continues to happen to education in the US.

Ever since the implementation of NCLB we have become slaves to the insidious blight upon our education system that comprises our constant need to test students from their earliest school experiences. Why do we do this? Why do we place such  incredible value on a momentary glimpse of what a student knows, usually through some spurious paper and pencil medium.

The Harry Potter movie made me think of this because when Professor Umbridge replaces Dumbledor as Head of Hogwarts she implements strict testing procedures for all the students. Learning goes out of the window and is replaced by teaching for the test in just the same way that "high stakes" testing occurs in all schools now in the US. Schools are defined as failing schools based on test scores on tests  which measure the narrowest, and often the least worthwhile, aspects of learning.

I am not the only one feeling this way. An AP item in the local press today describes how more parents are opting their kids out of standardized testing. 

There are so many reasons why standardized testing is so innocuous for just about everyone directly involved. It tends to only measure those things which are easy to "measure" such as recall and recognition. There are few tests that can effectively measure understanding and sense making.   The worst thing, however, is when it is used to measure teacher performance. This opens up the whole education system to all kinds of deviousness such as the Atlanta debacle of several months ago.

Of course we have to assess students to find out what they know and understand but there are so many better ways of doing this that through "easy to administer and score" tests. A I read the paper this morning it occurred to me that for my entire professional life, almost 45 years, society has been generally displeased with teachers and the field of education. Why is that? No other aspect of human endeavor has had to put up with such constant confrontation or complaint about lack of performance; not doctors, nor lawyers, nor automakers, nor plumbers, nor electricians, nor librarians, nor bakers nor candlestick makers.

What does our culture really want from the education system?      
  

Thinking Fractions

If you think about what you are doing when working with fractions everything becomes crystal clear. The worst thing to do is to visualize the algorithm you learned in elementary school; something like 1/2 + 2/3 or 1/4 x 4/5 or, heaven forbid, 1/2 divided by 1/4. If we do this then we are confined to thinking instrumentally procedurally about something that is comprised of only procedural knowledge. We fall back on senseless rules like "find the common denominator" or "cross multiply" or "change the sign and flip the second fraction". learning tricks like this will probably get you a better score on a traditional math test (like those used to compile the TIMMS report)  but will do little to help you understand what fractions are all about.

So for 1/2 + 2/3 think about different ways you could make 1/2. You could say it's the same as 2/4, 3/6, 4/8 and so on. Now do the same with 2/3. It could be 4/6. Aha, no need to go any further if you know that you can add fractions that have the same denominator. So 3 sixths plus 4 sixths is 7 sixths. We can count sixths just like anything else. Now you can see how you can get sixths by multiplying 2 by 3 if you need to do more difficult ones. But at least you now understand why.

1/4 x 4/5 is more challenging because the x (multiplication symbol) and the whole concept of multiplication of fractions can be different from using it with whole numbers. Visualize 4/5 using the model in the picture above. Now take 1/4 of that 4/5 and you end up with 1/5. The key to understanding this is seeing how the size of the 1 to which each fraction refers changes. The 1 of the 4/5 is the red 1 above whereas the 1 being referred to by the 1/4 is the 4/5. Really we're asking what is 1/4 of 4/5? Once you start to see the pattern, the relationship between the numerator and denominator then things get easier. Try 1/3 x 3/5 or 1/2 x 2/7 or 1/4 x 4/9. If there is not relationship between the numerator and denominator you can do it another way. For example 1/2 x 3/8 is 3/16. This can be done conceptually by halving the size of the fractional pieces. 1/16 is half of 1/8 so 2/16 is 1/2 of 3/8.

1/2 divided by 1/4 is really asking how many 1/4s are there in a 1/2. This is easily conceptualized by thinking about a football game; how many 1/4s are there in the first 1/2? Clearly there are 2.

3/8 divided by 1/4 is a little more difficult because the referent of the fractions changes. Again, think how many 1/4s are there in 3/8 by visualizing the fractional pieces. Compare 1/4 (a yellow piece above) with 3/8 (three dark blue pieces and you'll see there are 1 1/2 quarters in 3/8. The referent for the 1 1/2 is the 1/4.

from this conceptual understanding it's easier to "mathematize", to use Bob Wright's term,  what is going on by working out how the procedural knowledge of operating with fractions works. 

Simple Algorithms are a Waste of Time?

As I start my sabbatical semester  I seem to be           generating more questions than answers as I explore ways in which children  learned math before immigrating to these shores. My goal is to interview as many recently arrived people as I can from Nepal, Vietnam, Bosnia and Somalia, the four countries  most widely represented in the Burlington, Vermont, school district. 

The more I learn about the maths in different countries the more amazed I am that we have assumed for so long that math is the same the world over. But that's another story.

The question I have been wrestling with for quite some time now is why we still teach the four basic algorithms as illustrated above. The procedures involved in teaching all four algorithms to the level expected by sixth grade still consumes a large part of the elementary school curriculum even though almost everyone these days has a personal electronic device that will do the calculation in a nanosecond. The saddest part of all is that so many students who are able to complete the procedures have little idea about when to use them and even worse, many forget how to do it because, in real life, they never have to do it. When was the last time you worked out with a pencil 398 divided by 72? 

Above is a traditional addition algorithm although for some inexplicable reason the smaller number is on top. (remember how you were told to put the largest one on top as you would need to do this in subtraction?) The example above also has the wonderfully anachronistic "carry" as a command of what to do with the ten ones which magically become just 1 (interestingly ten 1s become 1 ten which changes the word 'ten from a noun to an adjective - the linguistic essence of regrouping).

So think about those two numbers 15 and 29. In what relationship could they be?

How about two groups of people, one of 15 and one of 29 being joined together into one group. Maybe combining two college classes because a professor is unwell.

How about a large gathering of people, such as a school staff, in which you know there are 29 women and 15 men and you want to know the total number of people. There are no separate groups; this is what we call part-part-whole.

How about two bags of candies? One bag contains 29 candies and the other bag contains 15 more than the first bag? This time we start with two groups and one of the numbers is not a group at all but the difference. Or one bag contains 15 candies and the other contains 29 more?

How about wanting to know  how many  free tickets you started with when you have given 15 away and now only have 29. You can even use addition to find the answers to problems that solve separation.

We could so easily set up the relationship like this; 15 + 29 = ? This would help children develop the real sense of a numerical equation along with the most important thing in algebra; the idea that the = sign means "is the same as".

If only we could spend  our time developing children's  number sense and their ability to recognize  mathematical relationships then use a calculator for the drudgery part instead of the endless hours we waste with mindless, boring and tedious  algorithms.  

Math in Singapore is Elitist

I always find it interesting how much faith people tend to put in test scores. School districts will spend hundreds of thousands of dollars, even millions of dollars, on math programs that are shown to produce great test score results. Such a set of scores is the TIMMS (Trends in International Math and Science Study)  report which shows the top five nations as Singapore, Korea, Hong Kong, Chinese Taipei, and Japan. The US comes half way down the "second tier" countries in mathematics.

If everything in each of the each of the 64 participating countries was the same then one could logically compare the results of the test scores and make comparative judgments. Sadly, everything is not the same, in fact, almost nothing is the same. For example, the numeracy system in most Asian countries is much easier to learn and more logically derived than it is in most western cultures (e.g. the word for eleven in most Asian languages is simply ten and one). Children also spend much more time in school in Asian countries than they do in the US; a lot more of GDP is spent on Education in many Asian countries (20% in Singapore) than in the US and formal education has a different cultural value and identity in each country.

It's not like soccer teams playing in the World Cup where the game is exactly the same in every country. The playing field is the same, the number of players is the same and the rules are the same. It's the same with any international sport. Education is different, it is not a sport. 

Interestingly, text book publishers in  the US  have made the most out of this divisive test score reporting by marketing a program in the US called Singapore Math. They cite the incredible successes of the Singapore education system as if everything were equal.  I wonder how teachers and students would perform in Singapore if they were held to the same humanitarian ethics as we value in the US. Singapore's education system is described as elitist and a meritocracy. Children are "streamed" in fourth grade which means they are put into ability groups based on test results. This means that if you don't make it by age 11 you probably never will. We used to call this the 11+ in the UK until it was abolished in1976. Basically, the claims made by the publishers of Singapore Math are based on the education of only part of the population. Teachers in Singapore do not have to spend time with differentiated instruction, RTI and so on.

Worse still, in Singapore children with special needs have no rights to a public education.In fact, they are not even required to attend school.  There are Special Education Schools  that parents may choose to send their children to.  The last words on this website are "The mission of SPED schools is to provide the best possible education and training to children with special needs so as to enable them to function optimally and integrate well into society".

 How can they possibly do this when they have been segregated for their entire education?

How To Learn Math

For the past week I've been auditing the on-line course by Jo Boaler of Stanford University. The course, How to Learn Math raises so many wonderful  issues about how we need to develop a more user friendly and conceptually-based  way of teaching math; how math needs to be seen as a creative activity and how we need to get away from the idea  that math is a closed set of procedures to be memorized. Now while these are things I have been advocating for for almost my entire professional life there is something in the course that is having a radical influence  on the way I see my role as a teacher of teachers of elementary school math.

This is the idea of fixed versus growth mindset. developed by Dr. Carol Dweck as described in her book Mindset.   This is clearly an  incredibly important issue for helping children become better at math especially those students who, because of our current system of  testing, have pretty much given up on ever achieving anything mathematically. To be told you are a mathematical failure at age 6 and have no hope of changing that is the world of the fixed mindset indeed.

But there's another application of this incredible idea which is to apply it to parents in terms of their attitudes toward math education. Over the years I have engaged in may discussions, arguments and  even confrontations, some even in public forums, with parents who truly believe that the math they learned by rote in school 30 years ago "was good enough for them so it should be good enough for their children". Sadly, the proponent of the fixed mindset are frequently successful businessmen who point to their success as the reason for maintaining the"no pain, no gain" approach to learning mathematics.

If we are to bring about the evolution of math education to a truly conceptual approach we have to do much to bring about a cultural shift in thinking about mathematics education, a task made more difficult by the disaster of the "new math' activities of the early 1970s in the last century. Perhaps the only way to succeed is to show that the methods advocated by Jo Boaler and the rest of us really do improve scores on tests.

Presentism and Time

 As I listened to the interview with Douglas Rushkoff, author of Present Shock, on NPR this morning I was reminded of just why it is so important to continue to teach children about time using an analog clock and not  a digital one. It's probably OK to use both later in life but we ignore the value of teaching  about time using  the analog clock or watch  at out peril; or at least the peril of our children.

                                                          It is the analog clock face that gives children a sense of time. The circle made by the clock face is an analogy

for an hour, or, more indirectly, a day or a night. The fact that a circle has no end point helps children develop the idea that hours pass continuously from one to the next and that time is constant. The digital face also gives us the language of time in a spatial sense with a "quarter past",  "half past" and quarter 'til" or "a quarter to" (as they say in the UK). In addition, the analog clock face gives us a comparative sense of time in terms of elapsed time or how much time is left. Usually, when we check the time, we do so to see how much time has passed or how much time is left; we seldom look at the clock just to see what time it is.

Without any reference to the analog clock face the digital clock face gives us but a fleeting moment in time. It tells us what time it is at the moment we look at the clock face. There is no before or after, no sense of space. It provides us with no clues for making comparisons between two times; we cannot instantly decide how much time there is left. The digital clock face is the essence of "presentism" or, as one speaker on the radio show said, "immediatism".

I wonder what effects  "presentism" is having on children's ability to learn how to tell time?

Vermont Math Scores

While the recent disclosure of the lack of growth in math scores in Vermont is depressing by far the most depressing situation is the fact that we are still using these  NCLB measures to evaluate student performance and, ultimately, schools and teachers  The way the AYP, annual yearly progress, targets are set up is almost like saying the average height of children must increase by a half inch each year. When we know what children are developmentally capable of learning and understanding at each stage of their lives why do people think we can constantly increase that natural development without changing something else.

There are, no doubt, places where the quality of teaching mathematics could be improved but to set "standards" so high that only half the population achieves them suggests that either the standards have to be reviewed or the method of evaluation has to be changed.

What is of much greater importance right now in math education, the "something else" I refer to above,  is to get students to enjoy, like and be captivated by mathematics. Part of the reasons why more students do not do well in mathematics is because they find it so deathly dull and irrelevant. The tests focus on memorization and recall and do not always test those things in math that are most important and meaningful in the lives of young children..

Our culture in general also has a terrible time with math. Many people are not afraid to say "I'm  no good at math" but no-one would say publicly that they cannot read. A recent news item about T-shirts being sold at The Children's Place store just adds to the way math is portrayed as being un-cool and not something to be enjoyed and valued by young people.

For the past few posts I have been suggesting that there are ways of making way captivating while at the same time maintaining the rigor and precision called for in the Common Core Standards. We have to do something before it is too late.

Math is NOT for Checkbook Balancing

I've lost track of the number of times I've been told that math is for balancing your checkbook, not that I was counting, of course. It is probably testament to the dire methods of teaching math that our fore fathers and mothers suffered through that math is now consigned to this anachronistic task. Anachronistic because most people today have instant access to their bank accounts through cell phones and so on so the need to "balance" the bank account using basic arithmetic no longer exists.

Saying that we need math to balance a bank account is like saying that we need to be able to read and write so that we can write checks drawn upon that bank account. If t his were true there would be no creative writing activities, no poetry., no language exercises in school that helped children play with language and learn all the wonders of alliteration, onomatopoeia and so on.

Hmmmm, that would make the English language arts just like math at the elementary school level wouldn't it!

Math is the wondrous study of all things quantitative and relational in this world and a good many other things too. The study of math by young children should should include exciting topics such as the numerical relationships in the Fibonacci sequence, the way multiplication facts are related to addition facts, the patterns made by fractions are the symmetry in the place value system we all use.

We need to captivate young children's attention in math just like we use exciting stories to fire their imaginations and motivate them to read on the English language arts.

A New Math Identity

The identity property of addition is 0 while the identity property of multiplication is 1. There are many other ways of defining identity mathematically but a new line of research in math education is revealing a completely different math identity.  I am currently reading a very interesting book in which the authors describe ways in which students identify themselves with math. The book, The Impact of Identity in K - 8 Mathematics by Aguirre, Mayfield-Ingram and Martin, brings a long-needed focus to the way students, and teachers, see themselves in the context of mathematics.

The authors raise interesting questions about how students see themselves in the context of math in relation to how they see themselves in other activities. Frequently students will have very positive identities in all sorts of things related to sport, art, music, language and so on but when it comes to math their self esteem plummets. This is  particularly true of students with diverse needs such as those who have special needs, come from disadvantaged homes, or are English language learners.

Since we live in a time when fewer and fewer students are choosing to enroll in  math courses in high school or college or, indeed,  follow math-based careers we have to do something to change these negative identities to positive ones at an early age so that students find math interesting rather than boring and irrelevant.

In recent posts I have been sharing the way I think math needs to be made more user friendly by relating it to art, and real life while at the same time maintaining the same standards of academic rigor as those required in the fields of the English language arts. In other words, we need to find ways that captivate students interests so that they see math as interesting and relevant as well as stimulating and challenging. 
  

Maths is the Science of Pattern

So here's the latest in the penny fractals I've been making on the wall outside my office. This one shows the development of the Koch curve through four iterations. If the thin line of pennies at the top is 1 (one) or 3/3 then the next one down is 4/3 (count the "sides" = 4 over the horizontal space =3. Each successive one is 4/3 of the previous one so the next one is 16/9 and the next one is 64/27.

There are so many interesting relationships between numerical patterns. Look at this neat relationship between the Sierpinski triangle and Pascall's triangle. If you color all the even numbers one coler and all the odd numbers another color you get a Sierpinski triangle. There's also a relationship between Fibonacci numbers and Pascall's triangle. If you add the diagonal numbers in Pascall's triangle you get the Fibonacci sequence.

Wouldn't it be wonderful if young children could learn these magical mathematical relationships in elementary school rather than learning math as a disjointed unrelated group of rules and facts. It would be much easier for them to remember their addition, multiplication and subtraction facts if they saw the relationships between them.

NCTQ advocates return to blackboards!

Having read the National Center for Teacher Quality (NCTQ)  report on teacher preparation throughout the country I was eager to see what they considered to be model programs. The report was so damming of such a large percentage of the 2000 or so teacher education programs reviewed that  the few programs they  held up as model programs must be remarkably good.

So, being a math educator I went straight to the Louisiana State University video to see what NCTQ thought was a model approach to teaching elementary school mathematics. The commentary by the professors, students, teachers and administrators had been clearly carefully scritped to include what we all know is good about math education as well as the importance of having close ties with mathematicians in the university math department. The video clip began with a college professor filling a blackboard with equations very much like he could have done in the 1950s or 60s.

This remarkable anachronism, the use of chalk and a chalkboard,raises serious questions about the LSU program. I haven't used chalk or a blackboard for at least 10 years since we discovered that chalk dust and computers do not live well together.There was a SMARTboard being used in a public school classroom later in the video but it looks as though LSU is still using blackboards in their university classrooms. This would imply that prospective students cannot use their laptops in the college classroom, that there is no instruction in the use of a SMARTboard and that students are not taught how to teach math through the use of technology such as iPads and interactive on-line activities.

One of the other videos at one of the other colleges also showed the extensive use of a blackboard.
I find this quite remarkable.

Mathematical Perspectives

One of the neat things about maths that makes it so exciting is the way you can look at the same thing from a different perspective. You can see 4 as 2+2. But, and this is what is so neat, you can see 4 from an infinite number of different perspectives. You can see it as 3 + 1 or 19 - 15 or as the square root of 16 or as the square of 2. You can see it as 4,326, 296 - 4,326,292 or as a square of 2 inch sides. It's also a Lucas number and a quartet such as the Beatles. It's a golf foursome or two couples out for dinner. Tetra and quad also mean 4 as do quatre and vier. Here's even more stuff about 4 .You can see it as 11 in base three or as 100 in base four. Isn't that neat!

I see math everywhere because I think that way. I get excited when I see an array of numbers that are similar or are sequenced. I like it when patterns appear out of nowhere like the 7272727 in my last post. Perhaps I have a well formed logical-mathemical intelligence ala  Howard Gardener's multiple intelligence theory . I am by no means a mathematician and I would struggle to complete some upper level  H.S. math classes but I see the fundamental constructs of mathematics with great clarity and in great depth.

I believe that these basic constructs such as the idea of part-part-whole are what we should be teaching young children so that they can see the world through a mathematical lens. For far too long we have clung to what the Victorians identified as "basic math" defined by the 4 operations of addition, subtraction, multiplication and division. These are no more basic to mathematics than declarative sentences and conjunctive verbs are basic to learning to read.  

Numercial coincidences are cool.

I love it when a numerical coincidence occurs. It's neat when you see several numbers the same or a sequence of numbers crop up in a series of unrelated things. It's like a quantitative harmonic convergence or QHC.

I had a neat QHC this morning when I was writing the check to pay my electricity bill. My eyes were thankfully distracted from the large dollar sum incurred through the extensive use of AC during the past month by the check number, 7272.

As I recorded the number in my check book register I wrote the check number followed by the date which turned out to be 7272 7/27 or 7272727. Not only is it a wonderful sequence but it is also a palindrome. Isn't that way cool.

The mathematical equation above is also a remarkable QHC. It's even more amazing because, as far as I know, this is the only time this happens. It's even more amazing still because the total of each side of the equation is 365, the number of days in a year.

12/12/12 was probably the greatest QHC that most of us will experience in our lifetimes. My birth-date turned out to be a neat QHC after i emigrated to the US. In the UK it was 19-12-46 which was depressingly boring and unspectacular. Now, it is 12 -19 - 46 which is really neat because sometimes I say "twelve nineteen forty six" as if it were 12-1946 and people think I have left out the day.  

Here's another neat QHC. Have you ever tried multiplying a number by 9 and adding the digits in the product as in 2 x 9? It equals 18 which, if you add the 1 and 8 you get 9. Now multiply 18 by 9 and you get  162. Again, add the digits and you get 8. Just keep multiplying by 9 and the digits will always add up to 9.

 

Intrinsically Interesting CCMS

At last, I've found an organization that believes mathematics education has to be something other than rigorous, precise, challenging and boring. Hats off to the Wisconsin Department of Public Instruction who believes that math should be not only "intrinsically interesting" but should also involve "collaboration, discourse and reflection" and be "meaningful and engaging". The image from their website even shows students smiling, engaged and clearly collaborating as they, presumably, explore meaningful math activities on their computers.

I came upon this refreshing web-site by "Googling" Common Core Math Standards Resources, a search that yielded 2,640,000 relevant hits in 29 seconds. When I "Binged" the same phrase I got a staggering 49,900,000 hits. Many of the resources are authored by States (e.g. North Carolina) , Foundations (e.g.The Noyce Foundation)  or professional organizations (e.g. NCTM )  but by far the greatest number are from  commercial organizations (e.g.Office Max)  hoping to cash in on the re-education of the nation's teaching force.

Having clearly only scratched the surface of this vast ocean of resources it is becoming clear that some of the authors really do understand the intent of the writers of the CCMS while others don't. Becoming a connoisseur of such resources is probably as important as constructing a meaningful understanding of what the CCMS are all about.

There are some excellent resources at Ve2,  The Vermont Educational Exchange. The Phoenix Rising article, in particular, by  Hung-Hsi Wu  is a really good read.



 

Fractals from Pennies

 Here's a great activity for helping students see the beauty of math. These are fractals made of pennies (cents) stuck to the wall outside my office. There's a Sierpinski Triangle, a Koch Curve and a Koch Snowflake. They are very easy to make and use the iteration of a simple pattern based on three pennies. There are many mathematical relationships in terms of both a numerical and spatial sense that you can  develop all the way from kindergarten to high school algebra.

Here's my student Lydia Koch's wonderful Math eNotebook assignment on pennies which started the penny fractal idea.  Isn't it an amazing coincidence that her last name is the same as two of the fractals!

There are so many things ot do with fractals when learning math. Here's the Cool Math fractal page , and here's the NCTM fractal generator.

Of course, the sky is the limit when it comes to fractals so here are a few more really cool fractal links.

The Fractal Foundation,
Vi Hart's Fractal fractions.
Fractals in Nature