How Much or How Many

One of the wonderful things about teaching is when you find a better way of explaining, presenting or demonstrating something that is difficult for students to grasp.

When teaching about the different types of simple math problems (as defined by Thomnas Carpenter) I use the phrase 'how many" in all of the problems except one in which I use "how much". The students always tell me, when they come to theat problem that I have made a mistake. When I ask why they think that they always say something like "well, it just doesn't make sense to say how much pennies of how much hours, or how much buckets". When I tell them that it  is not I who have made the mistake but them they always look somewhat confused.

To clarify, the problems are set out with no referents next to the numbers ush as "I I  have 6 ___________ and you give me 8 more ________________, how many _____________ do I have now? Their task is to fill in the referents so that the problem makes sense. So the odd problem similar to the one above ends with "how much ____________ do I have now.

The key to success with this particular problem is to change the referent from a discrete one such as pennies or hours or  candies to a continuous variable such as money, time or sand. A discrete variable refers to things that come in single units whereas a continuous variable can be divided in an infinite number of ways.

Yesterday in class I made two vertical  lists on the board ; one of continuous variables and another next to it of related discrete variables. 

Next time you visit your local supermarket check to see if the express lane says "10 items or less" or the more mathematically correct "10 items or fewer.

Taking the Calculating out of Maths

Five years ago Conrad Wolfram made a very compelling argument for taking the calculating out of math education in this TED talk. "Stop teaching calculating and start teaching math" is the slogan that was very attractive to those of us who thought that math was so much more than the drudgery of teaching the four operations of addition, subtraction, multiplication and division. For several years I agreed with him that we really needed to throw out the arithmetic component of elementary school math and focus on problem solving and other more appealing aspects of math, I also realized that this was a radical idea that would probably never fly politically, socially or educationally.

What it did do, however, was to challenge us to think about why we teach these time consuming procedures when children could simply use calculators or computers to complete simple calculations.

I think the answer can be found in Bob Wrights work with the Math Recovery materials in which he advocates for  mathematization as a process for getting children to think mathematically. The focus of Wright's work is clearly on the development of a child's mathematical thinking through the development of numeration and place value understanding. Part of this is process is the application of the algorithmic procedures as a means of applying and practicing this understanding. In other words, the purpose of teaching algorithmic procedures has changed from primarily applying them to problem solving  to applying and practicing conceptual understanding of numeracy, number relations and all the various aspects of what it means to think mathematically.

Wolfram MathWorld is a pretty cool place for everything maths.



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Cell phones and the NGSS Practices

For the past several weeks in my Teaching Elementary Science and Engineering course we've been looking at the three legs that form the three dimensional teaching characteristics of the NGSS program. I recently blogged about the activity we did to illustrate the crosscutting concept of cause and effect. The week before this we explored the disciplinary core ideas by planting 15 Bean Soup beans in hydroponic :gardens" (plastic cups) and the week before that we looked at the science and engineering practices by creating activities with bouncing balls.  The students were able to identify several of the practices that could have been developed through this activity but it was something else they did with the 23 different balls we had  that blew my mind.

As they bounced the balls next to a tape measure taped to the wall they found it difficult to say the exact height of the rebound without any real accuracy. So what did they do? They did what any self-respecting college junior would do in a science class, they pulled out their cell phones and took pictures of it. This would never have occurred to me.

But they didn't stop there. They realized that if they used the slo-mo video feature on their cell phones they could slow the motion down so that they could actually see the elasticity of some of the ball as they hit the ground as well as the exact height of the rebound. Isn't that amazing and wonderful?     

There are 12 Days in a Week!

I have a really wonderful group of students in my Math and Diversity class this semester. All 13 of them seem to revel in the aha moments that happen with great frequency as we explore the wonders of teaching math to students with diverse needs.

There are fiver areas of diversity identified in the course, each with their own particular set of characteristics that need to be considered. The five areas are teaching math to students who are English learners, students who have special needs, students who have a math disability, students in poverty and students who are high fliers. We've just completed the first area where we explored math and the WIDA standards, SIOP and wonderful things like comprehensive input. We even interviewed a group of international students to find out what the math they had learned was like and to explore some of the numerical aspects of their cultures.

Each class comprises an exploration of one of math such as numeracy and one aspect of one of the areas of diversity. Last week we explored place value and completed activities designed to get students to think more deeply about place value than they have ever done so before. We finished up by exploring different bases which is where there were so many aha moments. By the end of the session all the students were able to say why how there could be 12 days in a week and 202 weeks in a year. This is absolutely true if you are using base 5. 12 is "one-two" and not 12 so it is two 1s and one 5 which is 7 in base 10 . 202 is two 1s, no 5s and two  25s which is 52 in base 10. 

I Really Love Maths

 Tomorrow night I'll be giving the key-note presentation at the VMLC (Vermont Math Leadership Council)  annual meeting in Randolph, Vermont. The title of the presentation is I Really Love Maths. I plan to present a sequence of examples of things that have caused me to love the subject I teach to prospective and practicing teachers in three main areas; the theories that have influenced my thinking, the people who have influenced the way I teach, and the life experiences that have caused me to develop a lifelong passion for teaching maths.
  

Perhaps the two most influential theories that I was lucky enough to encounter early in my career were Skemp's idea of instrumental (fragile) versus relational (robust) understanding, and Shulman's conceptual and procedural knowledge. Being able to look at preK-6 maths through these incredible lenses has allowed me to clearly see what matters more and what matters less, as well as the flaws of some of the traditional instructional practices we have had to endure in years past, and sadly still do in some places today.

Two of the people that have most influenced my thinking are smiling at you right now. At least Sir Ken Robinson is. I looked long and hard but could not find a picture of John Dewey smiling. Sir Ken gave me permission to be creative while Mr. Dewey impressed upon me the value of experience in the educational process. To these names I would add Jo Boaler of Standford  whose work in teacher education in math has been completely illuminating in so far as it has shone the light squarely on the need for teachers to help  children  understand the math they are learning. I would also include Vi Hart who's videos on irreverence in math class are so inspiring. And finally I would add Carol Dweck who gave us Mindset theory with the unbelievable idea that everyone can learn math if they have a Growth Mindset.

And finally my experiences working with English Learners has taught me the humility that comes with standing back  and listening to the way people from other countries do math and think mathematically. The diversity  in the ways we count and communicate mathematically are one of the hidden riches of global thinking. I also think of all the students with disabilities I have worked with some of whom think very differently in terms of the maths in their lives. Some function, and very well too, with an almost exclusively  understanding of nominal number as opposed to cardinal number while others can seemingly compute in milliseconds..

I have been lucky indeed to embrace such diversity of thought, experience and practice during the past 50 or so years.   

Cause and Effect in the NGSS

We did a great activity in my Science Ed. class yesterday. In their Learning Communities (LC) (all my classes have learning communities of 4 or 5 students) the students had to find the causes for several effects related to the Earth. They had to explain the cause of day and night, the four seasons, the different time zones and the differences in temperature and hours of sunlight at different places and times on the Earth's surface.

To do this they had a globe, a balloon, markers, a flashlight and access to the internet. Their goal was to make sure that everyone in their LC had a working  understanding of  all these different phenomena by identifying the cause for each effect.  It was a great experience and gave rise to a lot of discussion, mind changing and a host of aha moments.

At the end of class I gave students a picture of my daughter standing astride the Prime Meridian in Greenwich, England. The picture was similar to the one above complete with the all important shadow.  The task for the students was to say which foot was in the eastern hemisphere and which was in the western hemisphere and to say anything else they could about the picture.

Non-Palindromic Numbers are Coolest

OK so this is not a palindromic number but it's way cooler than any palidromic number you can think of. 202 and 1,234,321 are palindromic numbers but they are exceedingly boring. If you reverse a palindromic number and subtract it from itself the difference is always 0.

But....... if you do the same with a non-palindromic number something amazing always happens. The sum of the digits in the difference always equals 9. In this example 3 + 6 + 6 + 3 = 18 and 1 + 8 = 9. This happens for absolutely any non-palindromic number. 7,829,572 and 2759287. The difference is 5070285 which gives a digit total of 27 which
                                                                    together equal 9.