PDP 3232 Reflection 2

PDP 3232: February 4th 2008 - Reflect and Respond

Reflections, Connections and Questions re: readings

We received the article, “Reflections on Math Reforms in the U.S.: A Cross-National Perspective”, by Xiaoxia Newton and it raised some points that came up in class on February 4th.

In essence this paper notes how different mathematical training is performed in China than in the United States. I can infer that what she says also applies to Canada since I believe that there are many similarities between the two systems.

Ms. Newton notes that in China teachers of elementary mathematics are specialist teachers. In the U.S., however, mathematics is taught by generalists which in her opinion gives the “impression that elementary mathematics … (is)basic, superficial, commonly understood, and repetitive” (p.684).

I have to wonder how different mathematics instruction would be if all the teachers of this subject really enjoyed it, and were really motivated to teach it and were truly confident in their abilities. My impression is that this is a subject that many are more or less comfortable with. Moreover I think that while elementary mathematics my be basic (in the sense that it is “element” of thinking), it is anything but superficial.

The author also laments the lack of preparation time and the opportunity to meet with colleagues in the U.S. She states that in China a teacher only spends 40 % of their work day teaching. It is as if once a teacher graduates, they have all the tools that they need. In other words “they are done with learning and know everything they need in order to teach.” (p.684)

How much better would our teaching in math (or any subject) be if we had regular opportunities to meet with our collegues? How much better would mathematics instruction be if teachers had a chance to study new ideas and new approaches? While I can not answer this for sure, I suspect that having more specialized teachers would improve results.

Reflections, Connections and Questions re: discussions

During our last class we looked at a series of quotes about mathematical teaching especially by Jo Boaler. In the course of the discussion I noted something along the lines that mathematical instruction in elementary school is often the bridesmaid of core subjects. What I meant to say was the “perennial bridesmaid”. This is a reference to the idiom, “always the bridesmaid, never the bride”.

What I meant was the most elementary teachers are interested in reading. If you were to take a survey of their majors, I suspect English and Psychology are by far the most prevalent. I have met very few with a background in mathematics. I found this paper so interesting because it seemed to support my inference.

Next Steps…

This article suggests that mathematical instruction is done quite differently around the world. I would like to know more about how mathematics is taught, especially with regards to elementary mathematics instructions. How many jurisdictions use specialists at the JK to 6 levels for example? Does this have marked benefits in the outcomes of learning mathematics in those jurisdictions? If it does should we move in this direction in Ontario?

I suspect that such a reform would be extremely difficult because we would have to reorganize totally elementary schools and teachers would see their lives severely (if not adversely) altered.

Teaching Symmetry Through Art

Here is a video from a PBS station in Pennsylvania that demonstrates how a classroom teacher creates a simple symmetrical design and evaluates it with the class. He is integrating mathematics into art.

Although this video takes place in the United States the expectations are not unlike those in Ontario. For example, one expectation from the grade three strand of Geometry and Spatial sense states: " (the students must) complete and describe designs and pictures of images that have a vertical, horizontal, or diagonal line of symmetry (Sample problem: Draw the missing portion of the given butterfly on grid paper.)."

Here is the video. Happy viewing:

http://www.youtube.com/watch?v=GDzWNfZOriY

First Reflection for PDP 3232

PDP 3232: January 29th 2008 - Reflect and Respond

Reflections, Connections and Questions re: readings

The base reading for this reflection was “Four Practices That Math Classrooms Could Do Without” by Nick Fiori. I really liked this paper. It was short and easy to understand.

His first point was about giving students forty problems a night. When I was a classroom teacher and was first starting out, I did that. Why not? The parents loved it because they could understand what I was asking the kids to do. No one complained as it seemed like real mathematics was happening.

If my objective was to make faster human calculators, then I succeeded. Was it good pedagogy? I doubt it.

Point two (The third-person czars of math problems) made me recall an experience getting older. I can recall looking at one of my brother’s grade twelve math texts when I was still in elementary school. The book had little vignettes about famous mathematicians. I thought this was quite interesting. I had never thought before that mathematics was such a human activity. It did seem to be such a faceless pursuit.

Reflections, Connections and Questions re: discussions

In our discussions of this article, we spent time discussing what mathematicians actually do. If we were to actually ask children (and even parents) this question, they would have no idea. The article suggests that mathematicians spend most of their time looking for good problems to solve. (As Mr. Fiori said: “Teachers give problems/students give answers”. )

This is why change is so hard. Here is what I believe. When educators who follow the three part lesson plan talk of mathematics, they are thinking of problem solving. When much of the general public talks of mathematics, they are thinking of arithmetic. Perhaps if we banished the name “mathematics” and instead talked of “problem solving”, then there would be more clarity about what we were trying to accomplish.


Next Steps…

So what to do?

Step 1: Make curriculum as painless as possible. I liked the point from the discussion that stated that we should not send open ended problems home. The children will need our guidance to complete these types of questions. The parents won’t know what is expected and their frustration levels will rise. Instead we should send some home skill and drill questions. We don’t have time to practise these enough in the classroom and if the children get stuck, the parents will feel comfortable helping.

Step 2: Humanize mathematics. We can spend some time discussing the humans who created mathematics and why. What kinds of problems were they looking to solve. Mathematics is a creative activity on par with any art form. If we give it a human interest component, it will be more interesting to our students. I believe that this is hardly impossible task. I note that in popular culture, mathematics has been more frequently represented. In recent years, I can recall seeing at least two movies about mathematicians: Proof and Good Will Hunting.

It is interesting that mathematics is one of those subjects from which you can get either a B.A. or a B. Sc. degree in most universities.

Step 3: Celebrate good problems. I liked how one of the teachers in the class maintained a bulletin board with a difficult problem every week. For those children who solved the problem, they had their name listed on the board. There are mathematics competitions also where we celebrate those students who come up with answers. If we accept that finding good problems is part of mathematics, perhaps we need a way to recognize children who come up with good problems too.

Teaching Ideas Maths Site

Here is a great site from the UK. It has great ideas for all subject areas but I will concentrate on mathematics.

If you look in mathematics, there are suggestions for geometry, fractions, addition, subtraction, division and etc. You will find interactive PowerPoint presentations and so on. There are even pictures of other teachers' bulletin boards. It is worth your while to visit this site called Teaching Ideas.

Strategy Games

We have started a list of links for strategy games because we believe that game playing teaches problem solving. Children will play and learn mathematical processes without realizing it.

The games that we have included are simple. They require few materials. Classic ones, such as "Battleships", can be played with only pen and paper. This is a material readily available in any classroom.

My links about games usually go to Wikipedia. This is for two reasons. Wikipedia will explain the rules and will include links to online versions of the games. If a child does not have a partner, they can play online.

Homeschool Math Blog

Homeschool Math Blog is an extensive blog about mathematical topics by a dedicated blogger. If you view the archives on the right hand side, you will find a number of relevant topics for elementary school mathematics. Click here.

Vedic Mathematics

Vedic mathematics is a series of principles for rapid calculation. See my earlier link to the YouTube video for an example. For more on Vedic mathematics go click here:

Jim Loy's Mathematics Page

Here is a site with useful links about topics in Arithmetic. Many of them concern topics which are too advanced for elementary students. However, if you look under the section entitled: Arithmetic, Roman Numerals ... you will find a number of interesting topics that could be used for enrichment in junior grades.

Free Internet Math Resources

Here is a post from a blog of an American homeschooler who has a site called Let's Play Math. It includes links to numerous sites that provide free math resources. Every teacher likes free resources. Click here.

Math Games with Playing Cards

Math Games with Playing Cards Playing cards are readily and cheaply available. I believe that strategy games are an excellent way to teach mathematical processus, such as problem solving. The OAME has a great, free, link for games with playing cards. The link can be found here, and it is in PDF format.

YouTube Video of Multiplication

YouTube is a great spot to find videos on all sorts of topics. Mathematics is no exception to this. Here is one that demonstrates yet another algorithm for multiplication.

In Mathematics part one and two, my instructor encouraged me to learn and teach other ways of performing arithmetic calculations. Here is one that is doing the rounds of the Internet.

The video is entitled Multiplication Using Vedic Mathematics. I explained it to one of my coworkers who teachers grade three, and she taught it to her students. Her students were fascinated with the process and it was encouraging to see her students multiplying two digit numbers by two digit numbers in their spare time.

Does anyone have any alternative ways of calculating addition, subtraction, multiplication and division that they would like to share? If so, let me know, and I will post them.

Ancient Strategy Game: Nine Man's Morris

Here is a link for Nine Man's Morris. It is an ancient strategy game and it can be played with simple materials, such as paper and plastic counters. You could even make a board out of chalk and play it outside on a sunny day.

There are many variations. In general the more men in the title, the greater the difficulty. Click on the link here for the rules.

Report on the Expert Panel for Teaching Mathematics

I thought everyone might find this link useful. This is the report in PDF:

http://www.edu.gov.on.ca/eng/document/reports/numeracy/panel/numeracy.pdf