There's a difference between curriculum and pedagogy. Curriculum is all about what we teach. Pedagogy is about how we teach it.


There's also a difference between knowing how to do something and understanding what you're doing. In mathematics there are all kinds of "how-to", or computation skills, that kids learn and promptly forget right after the test; sometimes they forget before the test. The thing is though, it's difficult to forget something once you understand it.

  by  dkuropatwa 


A few weeks ago I was part of a panel on the Richard Cloutier Reports show on CJOB radio here in Winnipeg. There were four of us: myself, Paul Olson (President of the Manitoba Teacher's Society), Robert Craigen (Associate Professor of Mathematics, University of Manitoba) and Anna Stokke (Associate Professor of Mathematics, University of Winnipeg). Robert and Anna are one-half of the group behind the wisemath blog.


There are some things we agree on:

• All kids can and should learn basic computation skills (how to add, subtract, multiply and divide).
• It's important for kids to understand what they're doing, not just to be able to perform by rote.
• Manitoba's recent poor performance on the Pan-Canadian Assessment Programme test is not good news and we have some work to do in mathematics in Manitoba.
• We'd like to see Manitoba place at the top of future national and international tests of this sort.

  by  dkuropatwa 


Some things we disagree on. I believe:

• Learning with understanding should precede the learning of rote algorithms in mathematics.
• To say Manitoba has placed 10th out of 11 provinces and territories in the 2010 PCAP test is a gross oversimplification of the the data represented on page 24 of the report (pdf). (Those confidence intervals are important. A repeat of the same test would likely have Manitoba place somewhere between 6th and 11th place. This isn't good news, but it's a little more nuanced than "10 out of 11". People knowledgeable about mathematics should be helping the public understand these nuances and promote informed discussion.)

So the crux of our differences are two-fold:


(1) I believe Robert and Anna conflate curriculum and pedagogy and are reading the Manitoba Curriculum documents as pedagogical texts when they were never intended to be read that way. Curriculum tells us "what" to teach, not "how" to teach.


(2) Robert and Anna believe the teaching of algorithms should be student's entry point to learning the basic operations (+, -, x, ÷). I believe the algorithms should be closer to the end-game of learning the basic operations.


John Scammel blogged about his take on the views expressed on Robert and Anna's blog. John points out in the comments the clear distinction the wisemath blog draws between Mathematicians and Mathematics Educators and the populations we teach. In K-12 classrooms we teach all students. The student body in University is different. Students taking math at University want to be there. That's not true of many students in the K-12 sector; the challenges are quite different.


On further reflection, there's a third difference: public (and private) debate should be open and sidestep insult.


The wisemath site seems to reject any comments that debate the blogger's views.


What I've read in the comments on John's blog and on Anna's blog (The last sentence of the last paragraph was recently edited; it used to say all future mathematics education research has no merit as a result of the issues Anna took with the article she blogged about. I regard this edit as a positive evolution in her thinking.) seems to hold K-12 teachers in a disdainful light.


Here's the audio from the CJOB panel we sat on together. It was a 2 hour broadcast, without commercials it's about 58 min. I took out the commercials. We talked about much more than was broadcast in the moments we were "off air". That was also an interesting conversation; unfortunately we didn't capture it. Next time I'll bring along my mp3 recorder. ;-)



Download (53.2 MB)

There's a new curriculum coming. Gave a little talk about it and the text book that supports it. Here it is. You can download it if you like. All the links I shared in the session can be found by clicking around on the slides below.

MHR Math10 Workshop

View more documents from dkuropatwa.

UPDATE

Joe, one of the folks who attended the session on Friday, has shared the Graphing Calculator Lease Form he uses in his school. You can see it here. Feel free to copy and edit or or just print it as is.

Thanks Joe!

Image by dkuropatwa via Flickr

The other day I was talking to a teacher who works with EAL (English as an Additional Language) students. I was reminded of a summary of teaching tips I had assembled by scouring through a pile of research articles when my department was struggling with the issue of teaching a growing population of EAL students in our school. Unfortunately I've lost all the sources. This was originally designed to be shared with just my colleagues as a quick reference sheet.

It struck me that these tips are really good for all students.

By the way, that picture is an integral written using various languages and multilingual character sets. It makes perfect (mathematical) sense if you can decode all the numbers. Can you "read" it?

I hope other teachers find this helpful. Please add your own suggestions in the comments below.

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Strategies for Teaching EAL Students Mathematics

The strategies below have been collated from a variety of resources. While all may be specifically geared towards assisting the EAL learner in math, these strategies are likely to prove effective in supporting the learning of all students.

The Language of Mathematics and Teacher’s Use of Language

"Command of mathematical language plays an important role in the development of mathematical ability"

• Teach the language of the subject.
• Be aware of vocabulary that has different meaning when used in mathematical contexts.
  • e.g. positive, negative, table, irrational, etc.
• Mathematical operations signaled by several different words or phrases.
  • e.g. add, plus, sum, combine, increased by, etc.
• Provide additional "wait time" for student responses to questions.
• Be conscious of the vocabulary you use.
• Simplify sentence structures and repeat sentences verbatim before trying to rephrase.
• Rephrase idioms ("take a stab at it") or teach their meaning.
• Clearly mark transitions during classroom activities.

Explanations and expectations need to be articulated explicitly and completely. Don't simply expect EAL students to "pick up on" assumptions, unstated premises, or subtle nuances of meaning.

• Periodically check to ensure EAL students are understanding.

Create Language Supportive Classrooms

"Journal writing offers English Language Learners (ELLs) the opportunity to practice and develop their emerging mathematics discourse skills."

Some possible prompts for journal writing:

1. Construct a word problem about [this] picture that can be solved mathematically. Share your problem with a partner and solve it.
2. What is the most important idea you've learned in [algebra] this week and why?
3. Write a paragraph containing as many of these words as possible: ..........
4. List some things you must remember when answering this type of question or doing this type of problem.

Connect Mathematics to Students' Background and Experiences

• Young people learn best from their own and not other peoples’ experiences.
• Use students’ past experiences with mathematical terms to help give the terms meaning in a mathematical context.
• The introduction of a new term should be carefully orchestrated through repetition in context and through saying it aloud and spelling it.
• To learn mathematics successfully, many ESL students need a more multisensory approach to mathematics.
• Relate mathematics instruction to the “out of school” life of students.
• The implementation of “ethnomathematics” can help teachers relate mathematics to their students’ “out of school” lives.
• Use teaching methodologies that “contextualize” the subject matter.
• Be concerned about affective factors in the classroom.

Between the 10th and 15th year of teaching I discovered that what was needed for these children was not an emphasis on the academic but a meaningful interaction with mature adults. The relationship with a stable, mature adult is most important.

Vary Instructional Methods

"ELLs learn best when instructional methods and approaches match their individual abilities and learning styles."

• Use a variety of methods tailored to students' needs including direct instruction, guided discovery, cooperative learning, computer assisted learning, etc.
• Provide writing and other language development activities for EAL students.
• Use cooperative learning strategies.
• Encourage students to rephrase information or instructions orally.
• Use peer tutoring.
• Establish a "homework club".

Contextual Supports for Linguistic Development

• Write key words on the board and other non-verbal cues, wherever possible, to present key ideas.
• Provide written notes, summaries, instructions, and pre-reading.
• Where possible, use the students' native language to check comprehension and clarify problems.
• Communicate interest in students' linguistic development and set expectations.
• Respond to students' language errors.
• Establish a supportive environment for language learning.

In most subject areas, EAL students should be able to grasp essential concepts, if these are presented carefully, emphasized through repetition, and clearly distinguished from finer points that the students are less able to assimilate.

Assessing, Evaluating, and Reporting on Students' Progress

• Use a diversity of measures including portfolios, observations, anecdotal records, interviews, checklists, criterion referenced tests, etc.
• Design alternative assessment tasks including exhibits, dramatic renditions, interviews, writing samples, etc.
• Include questions for small group discussion and individual writing.
• Provide extra time on tests for EAL students to process the question in English, think about them in their first language, and respond in English.
• Simplify directions in English and/or paraphrase in students' native language.
• Permit students to use dictionaries or word lists.
• Avoid heavy reliance on multiple-choice and true/false tests with EAL students (these involve a lot of reading and often depend on comprehension of subtle shades of meaning)

Functioning all day in a second language is exhausting and demanding. Homework can take these students two to three times longer to complete.

Listen to Arthur:

2 min 59 sec

I've had the same comment surface in several, unrelated conversations I've had with colleagues lately. All math teachers. In each case we were discussing some aspect of the curriculum and at one point they invariably say: "Y'know, you just have to memorize that."

Really?

Driving my son and a friend of his home yesterday we were talking. They started talking about tests they have coming up in various classes. They listed those classes where "you just have to memorize that stuff." Again, math was one of those classes.

So really, what do we absolutely have to memorize in math? I do not teach memorization, although I too have told my students that they have to commit certain things to memory. In each case I emphasize they should not memorize individual facts, rather, they should identify patterns and recall the patterns. I teach mnemonics. (Is that splitting hairs?) For example, how many patterns can you find in these two columns of numbers?

09
18
27
36
45
54
63
72
81
90

So really, what do we HAVE to memorize in math?

When I returned to work after 10 weeks of parental leave I wanted to find a way to measure how well my AP Calculus students had learned and understood what a derivative was. So, the day I returned to class I challenged them to create a commercial that educates: What is a derivative?

Some of the parameters they have are:

• The video must educate people who know nothing about calculus.
• The video must be published on YouTube.
• Students may work in teams of up to 3 people.
• No images of students faces. Masks or other methods of obscuring their identities may be used.
• No algebra.
• Maximum video length: 30 seconds.
• Include as many different descriptions as possible in the time allowed.

The way we're doing this is I have published a video to YouTube describing the assignment and extended it as a challenge to the entire math community on YouTube. Students will "hand in" their commercials as video replies to mine.

I'm hoping other people in the math community on YouTube reply to the challenge. It will go a long way to adding another layer of authenticity to the assignment and create a more meaningful reason than "marks" to create high quality content: social credit.

Social credit is a powerful motivational force in the lives of teens; a force that hasn't fully been tapped in educational circles. It's behind the outstanding work some of my students do on their class blogs. I'm trying to be more deliberate about accessing that potential in new ways this year.

The video has just been published to the AP Calculus 2008 blog. The students commercials should be published by Friday morning.

If you have a moment, head over to their blog and leave them a comment telling them you're looking forward to their commercials. It'll help make real their sense of audience and perhaps encourage them to do their very best before they hit the publish button on their videos.

Calculus Commercial Seed Video

This is a mirror of a post from a new group blog called Striving Readers Kentucky. The post is by RolandOD.

It caught my eye because of the Developing Expert Voices project I'm beginning with my grade 12 Applied Math class. (More about this in my next post ... hopefully later today.)

I'm collecting samples of online presentations whose format my students can emulate or riff off of. None of them have any math in them ... this one does:

To quote Roland:

This group of undergraduate students from Northwestern obviously understands the content. If you want to share this video with your math teacher's I encourage you. Here is a link to the lyrics; just in case they want to explore the math behind the music.

And while I'm storing stuff here (in my outboard brain) for me not to forget let me just add these three vids ...

Another remix of Karl's Did You Know? presentation.

Jeff Utecht's Web 2.0 presentation ...

And this breathtaking video response to Jeff ...