Sunday, November 22, 2015

On-Line Schools Disaster

I've always thought the idea of on-line schools or virtual education was one technological step too far. Now a new study  by the University of Washington, Stanford University and the Mathematica policy research group shows that students enrolled in on-line schools fair poorly when compared with those in traditional schools and classrooms. In the report by the BBC the problems are identified as many and significant and in some cases students have been found to lag as much as a year behind the counterparts in traditional schools. The report also identifies the "digital glitz" associated with the use of the internet and other technological forms of so-called learning.

On another on-line related but completely different topic, I discovered this past week that I have spent a third of the semester so far glued to Canvas. To be more precise, I have spent 160 hours loading my coursework for my three courses onto the Canvas Digital Learning Management System. This is in addition to the hours I spend each week in my classes face to face with my students. I wonder how much better off my students are because of this? I wonder how much better teachers they will become? I wonder what other criteria we should measure the success and value of  an on-line learning management system such as Canvas?

Just because we can, should we?  

Wednesday, November 18, 2015

Remembering; the Lost R in Education

So, in this world of instant access to virtually anything what should we require our students to remember? What sorts of things do we need to have at our fingertips? What things do we need instant access to that will enrich our lives? What is it that's just plain fun recalling instantly from memory. My son, Andrew, who has Down Syndrome, can recall almost everything that's happened in his life that he has enjoyed in vivid detail. He can remember what he was wearing, who he was with, what the weather was like and what snack he had when he went to the movies to watch a Harry Potter movie ten years ago.

But what about the factual information that makes up our adult lives? Is it important to distinguish a noun from a verb, to know who the Secretary of State is, to know the capital of Holland, or even where Holland is, to know how many ounces there are in a pound and how many pints in a gallon? Or is it important not to clutter our minds with minutia so that we have room for the important stuff like using computers and navigating the endless features of our cell phones?

I know which side I would come down on but I'm from an older generation that simply uses a cell phone to make phone calls and take the occasional picture. But my memory is full of the most incredible things apart from the number  of pints in a gallon, pounds in a ton and meters in a kilometer. I get an incredible sense of joy when I recognize a drumlin, an  erratic or a raised beach when I'm driving around the countryside. It's a genuine thrill when I can recall the name Copland in response to a crossword clue "composer Aaron". The fact that I can remember long-ago-learned facts when I can't remember what I did yesterday is somehow rewarding, reassuring and satisfying.

But what of today's younger generation who do  not seem to be committing things to long-term memory with the same gusto that we did in our youth. What will they have to recall in their old age? 

Making Meaning in Maths with Manipulatives

Ever since I was an undergraduate student learning how to be an elementary school teacher I have always believed passionately in the use of manipulatives to introduce children to new concepts, ideas and skills in mathematics.  It has always seemed the most logical way of starting children on their journey of mathematization, as Bob Wright of Math Recovery would say.

Well for the first time in my professional life someone asked me about the research basis for using manipulatives for teaching math. In the math ed. texts i use in my courses the use of manipulatives is advocated and illustrated with the introduction of every new idea. It's as if it's a no-brainer, something that is as natural as the sun rising each morning or snow falling at some point in a Vermont winter. So I took to Google this morning and tracked down an array of really interesting research-based articles in support of the use of manipulatives for teaching math at the elementary school level. It felt a bit like conducting research to see if more people used raincoats and umbrellas when it was raining than when the sun was shining but the results were startlingly interesting. Here they are;

Here's an NCSM article. NCSM is the national math leadership council and reminds us that we must also include the use of virtual manipulatives. This one is from the Journal of Instructional Pedagogies and gives an overview of the history of using manipulatives as well as the current research.

There are many publishing companies that also produce manipulative materials so it is probably only natural that they should also produce research to support the use of their products. This is a particularly good article from the folks at ETA.

And here's a student research paper on the topic from Marygrove College. Finally, here's a neat article from Sage Publishing that mentions Montessori education which is really where the use of manipulatives in teaching math all began.


Tuesday, November 17, 2015

Thou Shall Not Reduce Fractions

With the increased use of the word "reduce" in the sustainability and recycling movements there is even more reason now not to use the term in reference to fractions.

If we use the word "reduce" to change 4/8 to 1/2 we are giving students two clues to make them think that 1/2 is smaller than 4/8. The word "reduce" clearly means to make smaller while the two numbers in the 1/2 fraction are clearly smaller than the two numbers in the 4/8 fraction. We are, in fact, giving chidlren two clues to make them think 1/2 is smaller than 4/8. To students who have good fraction sense this is probably not a major issue but to the many students who struggle to learn fractions this metaphorical use of the word "reduce" can cause untold misconceptions and misunderstandings.

I have been recommending the use of the term rename or regroup to describe changing a fraction from one form to another. In my math and Diversity grad course last night one of the students suggested using the term "synonym". At first, this seemed quite inappropriate since synonyms do not have exactly the same meaning while 1/2 is exactly the same as 2/4 or 3/6. Then we started to think about about the use of the term and it began to seem more plausible The discussion went something like this.

While two  synonyms may  not exactly be the same it is really the context that requires the use of one over the other. For example, if you walked into a small grocery store and asked for a dozen big eggs you would get a funny look, perhaps, from the store keeper. The usual word to use when referring to eggs is large, a synonym of big. Now apply the same reasoning to fractions. 3/6 is a synonym for 4/8. Even though they are exactly the same size, given that they both refer to the same whole, each one is more appropriate in certain contexts. If you were adding 1/2 + 1/6 it would be more appropriate to rename the 1/2 as 3/6, or use 3/6 as a synonym for 1/2. On the other hand if you  were subtracting 1/8 from 1/2 it would be much more appropriate to use 4/8 as a synonym for the half in just the same way that it is more appropriate to use the word "large" when referring to eggs rather than the synonym "big". 

Monday, November 16, 2015

Math and Poverty

I learn so much from reading my graduate students'
reflections about the readings I assign each week. This is especially true when they relate their own experiences to what we are discussing in class. The thirteen students in my Math and Diversity grad class have such diverse backgrounds that they bring fresh and diverse perspectives to almost everything they read.

One such insight this week really made me stop and think about how we work with children from financially challenged families when it comes to learning math. The student shared an experience where families could sign up their children for a variety of different experiences that were being implemented beyond the normal hours of school. There were series of activities involving art, or science or math, in fact all the disciplines commonly found in the elementary school curriculum.

The observation of particular interest that the graduate student made is that  no children from financially challenged homes were signed up for the math experience. Each different experience had roughly the same number of students signed up but all the students signed up for the math experience came from financially stable or affluent families. The questions arising from this observation are interesting to say the least. Why do financially challenged families not see extra math experiences as beneficial for their children? Why would they rather sign them up for an art or a music experience?

 These are questions I will ask my students in class tonight and ponder for the next few weeks. What do you think?


Wednesday, November 11, 2015

Growth Mindset Maths is a Must

Well, I don"t care what Alfie Kohn says, and I do usually agree with him, but I think  Mindset Theory is the best thing I've added to my teaching repertoire since I discovered John Dewey's ideas of Inspired Vision and Executive Means back in 1969.

I've now introduced it in both my undergraduate and graduate math courses and the results have been great. This is especially true in my undergraduate math class where we've been exploring teaching everyone's seemingly least favorite maths topic, fractions. For some reason, my students nearly always seem to enter this topic with very little relational understanding of fraction concepts or fraction sense. It's as if they've slogged through endless hours of learning nothing but how to add, subtract, multiply, and maybe divide, using archaic, instrumental strategies such as "you can't add apples and oranges", or " invert and multiply"or "cross multiply" to name but a few.

They seem to have one revelation after another when they realize the power of the ONE or referent when when working with fractions. The idea that you can count like fractions the same as you can count anything else and the remarkable patterns fractions make like these two 1/2 2/3 3/4 4/5 5/6 6/7 7/8 8/9 9/10 and 2/1 3/2 4/3 5/4 6/5 7/6 8/7 9/8 10/9. Each forms a pattern approach ONE but never getting there.

Frequently, during class-time, we refer back to the Mindset class we had near the beginning of the course and they all remember Carol Dweck's maxim of "yet".  This idea seems to work well with the Learning Communities in the class where each member of the community bears a responsibility for making sure that every one in their group of 4 or 5 students is developing an understanding of the topic, fraction concepts and skills in this case,

The more I try to develop my Mindset language the more I see the students responding in a positive way. I feel like I am even more "on their side" so to speak than I thought I was before. My job is clearly to help them all succeed in developing the relational understanding of maths  required of being an elementary school teacher. 

How Maths Controls our Lives

Katy, one of the graduate students in my Math and Diversity class has a wealth of experience living, working and studying in a variety of different countries. This gives her a unique perspective on learning maths as she is well aware of how it differs in different countries no only in how it's taught but in the very nature of the maths itself.

Yesterday she shared, with great excitement, some of the remarkable things she was discovering about maths as a result of reading the book Here's Looking at Euclid by Alex Bellos. One story that I only half knew about was how some indigenous folks living in the Amazon rain forest have a very unique outlook on things mathematical.

Their number system for example consists of one and two, which are numerically form followed by threeish, fourish, and fivish, terms that I will need to read the book to fully grasp.

Katy also described how there are no standard units of measurement used to measure anything. For example, time is not divided up into units such as seconds, minutes and hours. There are no length or distance measurements such as feet or meters or miles or kilometers. This made me start wondering about how dependent our lives are on every conceivable unit of standardized measurement. There is almost nothing in our lives that cannot be measured with some standardized unit s is illustrated in this Dictionary of Units of Measurement.

I am sure that the indigenous people of the Amazon have units of measurement for some things that are part of their culture and that they all know but they clearly have no need of standard measures with which to communicate with the rest of the world.

So in my class yesterday I asked my undergraduate students to imagine living without standard units of measurement. I posed the question "Suppose you came to class when you felt like it and left when you felt like it or if there was another way of determining when class started and ended" . We didn't have a whole lot of time to get into a deep discussion and I hadn't really had time to think through where I wanted to go with it but the whole idea does make you start to explore how we live. How mathematically our lives are defined by how we used standardized measures.

Perhaps retirement is life without  the control of standardized measures?

Tuesday, November 3, 2015

Third Grade Math Problem

Once again the internet is abuzz with hysteria because of a third grade problem that was marked incorrect when, according to many, it was not.

The problem involved using  the repeated addition strategy to solve 5 x 3. When the student wrote 5 + 5 + 5 every one cried foul saying the student's answer was correct. The answer is clearly incorrect because it should be, as shown, 3+3+3+3+3+3.

Understanding the repeated addition concepts is one of the most important ideas in multiplication and division especially in problem solving. There are so many different skills wrapped up in this seemingly simple idea that we, as parents, owe it to our children to take the time to grapple with this issue and understand it so we can help our children and not dismiss  it out of hand becasue it is different fro the way we learned math.

In terms of a simple math procedure 5 x 3 is the same as 3 x 5 since they both = 15. This is called the commutative, or turn around, property of multiplication. But, if you apply this idea to real objects in real situations the two number sentences are not the same. One is three groups of 5 and the other is five groups of three. If you were working say with M&Ms 5 x 3 would be 5 groups of 3 M&Ms which looks quite different from 3 groups of 5 M&Ms. *****  *****  ***** or *** *** *** *** ***

Now, if you apply this to division, the reverse of multiplication, 15 M&Ms divided between 3 children is quite different from 15 M&Ms divided between 5 children. If we teach multiplication as repeated addition we can teach division as the reverse or repeated subtraction.

There are, of course, more concepts related to multiplcaiton and division but that's another story. So my advice to parents is don't knock something you might not understand just because it's different from what you learned 20 - 30 years ago. Take the time to learn it and you'll find that understanding math instead of just memorizing it is so much more rewarding and useful in the long run.