Evenly Odd or Oddly Even

If there's one thing that drives me crazy in maths it's the sayings and phrases we've come up with over the years to supposedly help students learn maths more efficiently. Two, in particular drive me nuts.

"You can't take 5 from 2" is a particularly useless thing to say when you are doing the subtraction algorithm such as 62 - 25. Children typically learn this when they are in second grade but then have to unlearn it when they get to 5th or 6 th grade and encounter negative numbers. It should be immediately apparent 5 is more than 2 so you regroup 62 into 50 + 12 so you can subtract 5 from 12 and 2 tens from 5 tend to get 37. Actually it's easier to use negative numbers. Just say 2 minus 5 is -3; 60 - 20 is 40, then 'subtract' the -3 and you get 37.

The other one that really gets to me is "an even number is a number that is divisible by 2".  This pearl of wisdom often crops up in crossword puzzles and should include "to give another whole number" to be accurate, Every number is divisible by 2 in some way, except 0. We can, of course say that even numbers end in 0,2,4,6,or 8 while odd numbers end in 1,3,5,7 or 9. Conceptually, even numbers comprise  pairs of things in them while odd numbers always have one thing by itself.

One way to think about this is to see the school bus on the left as if it was made of cubes. One at the front for the hood and then pairs stacked on top of each other for the rest of the bus. No matter how long the bus is, or how  many pairs you add, the number of cubes will always be odd.

Now look at the second school bus without the hood, or the odd block at the front. No matter how long the bus is, or how many pairs of blocks you add the number of blocks will always be even.

Another interesting thing about odd and even numbers is the way they impact our culture. For example, in Japan odd numbers are preferred while in the the US we prefer even numbers.

Odd and even numbers even elicit emotions as this Guardian report illustrates. Odds and evens even have gender assignments as this Kellogg study reports

It's Never Too Early to Count

It's never too early for young children to interact with the world of maths. Children learn the fundamentals of what we call quantitative literacy in the same way they learn to speak and read at a young age: through experience and practice with someone who knows what they are doing.

As I mentioned a few posts back my daughter Marie and her husband Erik are doing an amazing job helping their son Lachlan learn the intricacies of counting and other aspects of maths They don't force it on him, make him complete math activities, or even call it maths. They just make him aware of the quantitative and geometrical aspects of his life  as he interacts with the world around him. Currently, he is coming to terms with the oral number name sequence up to twenty. He nearly has it except some of the teen numbers are a bit random.

Quite remarkably, at two and a half, he also is beginning to develop a sense of cardinality. This is when you put quantities to the number names, Right now when hes says, "one, two, three, four" etc he is just saying a sequence of words, a little like reciting the alphabet. He has cardinality with two; he can identify two objects that are the same. This is an important idea because you cannot count rationally unless you know what you are counting. He can identify two fingers or two tractors or two people. The fingers, tractors and people are the referents of the counting words, the things to which "two" refers. In early rational counting the identification of the referent is important because we can develop the idea of counting as the process of  "one more"; three is two and one more altogether. The word "three" now refers to the objects which were two and one before they were joined together to make three.

This is not as easy as it sounds because there is also the ordinal and nominal use of number. The ordinal use of number, first, second, third  really doesn't come into play at this point in the learning to count process. But the nominal use of number, using numbers to name things, does. In the picture above the numeral 1 appears above a single tomato, as do 2,3 and 4. It's easy for a young child to name each tomato as 1,2,3 or 4. This would be a good activity for teaching the numerals once they had been learned orally. But to teach  rational counting, or the cardinal use of number, you would need a picture with one tomato next to the 1, two tomatoes next to the 2 and so on so that the numeral becomes a number associated with that many.

Making maths a part of everyday life for young children is easy if you know what you are doing. Bedtime Math is a wonderful resource I have mentioned before.    

The 3Rs - Ridiculous Regulated Rubbish

Just about everyone thinks of Reuse, Reduce and Recycle when "the 3 Rs" are mentioned these days. Everyone, that is, except the British Government who still, inexplicably,  refer to primary education as "the three Rs" even though only one of them now begins with R (reading  writing and maths).

In addition to introducing the school "league tables", as if education was a sport, the Conservative Government has now increased the level of difficulty of the standardized tests they give to primary school (elementary school)  children which means, somewhat obviously, that the scores of students are going to be lower than they were with the old test last year. Just last week I blogged about how tests only measure other tests, an assertion that seems well supported by the British Government's latest advice to parents to "ignore the latest test results".

Unlike Canada, where each minister is an expert in her or his field, the British (and US) governments appear to revel in the idea of putting people in charge of things about which they know nothing. Nick Gibbs, the current Minister of State for Schools at the DfE in the UK was an accountant before he became a politician. It seems remarkable that schools and children are compared using tests that measure a narrowly defined set of skills and knowledge at a particular moment in time using a specific medium and think that anything useful can be obtained.

To then say that only 53% of children "meet the standard" is absolute Ridiculous, Regulated Rubbish.

Counting Grandchildren

One of the wonderful combinations in one's life, as I am discovering, is to be retired and have a grandchild. It took me quite a while to make the adjustment to being retired but no time at all adjusting to having a wonderful grandchild.

Since I was a professor of maths education for most of my working life my brain is pretty much dominated by the idea of number so when I retired all sorts of numerical thoughts began to spring into my mind. Most of them were things that had never occurred to me before or things that I may have taken for granted. For example, I no longer meet upwards  of 80 or so new people, students,  in my life each year. There were usually around this many students in the courses I taught each year. And then when I supervised student teachers in public school classrooms I would frequently meet another 100 or so  new children each year. Only now that I am retired have I become acutely aware of just how much I learned from all these new people I met each year; the interactions, the term papers I read, the classroom events and experiences all conspired to enrich my life each year, I miss these interactions so much and frequently wonder if I did take them for granted. I don't think I did but it all looks so different now.

So I now have time to watch and interact with my two and a half year old grandson as he comes to terms with the wonderful world of maths. He has already learned to say "Sierpinski triangle" and can pick one out in a whole bunch of different triangles. There is nothing quite like hearing him say those two words and pointing at an example of one, Numerically, he is going through the process of learning the number names and, dare I say it, has already the beginnings of a sense of cardinality at least with two and maybe three objects. His mom, my daughter Marie, took my maths ed. grad course several years ago and so is really in tune with the growth and development of a child's counting skills. She demonstrates so wonderfully the two most importance things in teaching maths. First the importance of observing the student, her son, and second, just how much a full understanding of the most basic mathematical ideas is to the teaching process. She doesn't push maths on him at all but just makes him mathematically aware of the world in which he lives.  

Even something as seemingly simple as helping a student count requires a deep understanding of the ordinal, nominal, and cardinal use of number. Lachlan, my grandson, is currently learning the sequence of the number names. He can number name more or less up to twelve but hasn't yet quite got the teen sequence. I say number naming because he really is not counting yet in the true sense of the word  apart from, perhaps, "twoness" and "threeness". When he number names he is just learning the order in which the number names occur. He has, yet, no sense of "fiveness", for example. More next time on the nitty gritty of learning to count.  

Tests Only Measure Other Tests!

The PISA test results were released this week and once again the US is slipping in math test scores when compared with test results from 60 or so other countries. Of the 35 industrialized countries included the US ranks 31st.

When I was a fourth grade teacher many, many years ago in England we changed from one reading test to another one year without changing anything else; the reading program and the way I taught children to read remained exactly the same. The most incredible thing happened. All my students' reading abilities improved by more than a year. In other words, my fourth graders looked like they were suddenly reading as well as fifth graders. What a remarkably effective teacher I must have been that year. Not really since I did nothing differently. It was because the reading test was clearly easier or standardized differently from the previous one. This was my first experience of tests measuring other tests.  

In fact this very same phenomenon is mentioned at the end of the Hechinger report of the PISA results in these words:

"However, one test released last week —  the 2015 Trends in International Mathematics and Science Study (TIMSS) — showed a surprising gain for the U.S. in 8th grade math.  Historically, PISA and TIMSS tests have shown contradictory results for Eastern European countries and Russia, as they perform much better on TIMSS than on the PISA test. Scholars will need to explain the divergence for the U.S. this past year".

The PISA and TIMMS tests are clearly different and measure different things in maths. It is so easy to point fingers when you see bare test results such as these reported in the media but there are so many other things to consider. 

Take the maths, for example. Maths is not the same the world over. There are vast differences in the math itself before you even start to teach it. In most Asian countries, for example, the numbering system is so much easier for young children to learn than it is in the US. The teen numbers follow  the pattern ten-one, ten-two, ten-three and so on rather than the difficult eleven, twelve, thirteen, etc system we use. Look at which countries are at the top of the PISA table! In some countries such as Singapore children are given vast amounts of homework to do each night and not all children have the opportunity to attend public schools. There are differences in the sizes and cultural diversity of the countries being compared which can all affect score averages on a simple test. 

Like pretty much everything in our lives we can probably do better in maths education but we need to use caution when comparing test scores obtained from students in vastly differing cultures. 

Maths Is Easier When You Understand It.

I sometimes wonder just what we have to do in maths education to demonstrate that things are more easily remembered, used and developed if the learner understands what she/he is learning. This interesting piece of recently published research supports, for at least the millionth time,  that when students learn conceptual knowledge, along with procedural knowledge of maths, it is much, much, much, more effective. The neat thing about this piece of research is that it is for high school maths which has tended to lag behind the research on this topic at the elementary school level.

Probably the best way to distinguish between the two types of knowledge is the idea of pi. If I were to ask you what pi is you would probably say "3.14 but that's all I can remember". If that's all you know about pi then you have a small piece of procedural knowledge that you can plug into equations to find the area or circumference of a circle. Now, imagine that you have some conceptual knowledge to go with this. For example, knowing that pi is a ratio between the diameter and circumference of a circle would be immensely helpful to learning all sorts of more advanced maths. It would also be helpful for doing things in one's daily life. The fact that pi is a ratio means it is just over three times further around a circle than it is across the middle of that circle. If you think of pi as a fraction, 22/7, then the circumference could be 22 inches, or feet, or miles, and the diameter would be 7 inches, feet, or miles.

If I asked you to count by fives you would probably say, "5, 10, 15, 20, 25" etc assuming that I meant you to start at 0. So try counting by fives again starting at 3. The first few will be tough but you'll soon see the repeating pattern of 3s and 8s. Seeing the pattern is a piece of conceptual knowledge because you are applying your conception of counting by 5s and 10s rather than just parroting the numbers.

We have long known that conceptual knowledge of maths is as crucial to learning maths as procedural knowledge. Jo Boaler, the amazing Stanford professor and founder of YouCubed  was one of the first to demonstrate the importance of conceptual knowledge in a wonderful piece of research when she was at Liverpool University in the UK. She discovered that elementary aged students who were taught conceptual knowledge along with procedural knowledge did much better in high school maths that those who were only taught procedural knowledge at the elementary school, One of the interesting findings of her study, if I remember correctly, was that the conceptual/procedural knowledge taught students didn't. do as well on the end of elementary school maths tests as the procedural knowledge only taught students. The reason: the tests only tested the students' procedural knowledge of maths; a problem we still grapple with today.