Tag: The Reflective Educator (page 19 of 43)

I’m assisting our 4th grade teachers in finding resources for their upcoming unit on division, and I’m hoping to avoid either of these situations from arising (see the videos below). Actually, these videos could be great hooks to see if students can figure out what mistakes the people in them are making…

A key problem in both of these videos is that these (fictional) people are making key mistakes in their use of a standard algorithm which they clearly do not understand, nor do they seem able judge how accurate their solutions are. Now, my suspicion is that students will not make such egregious mistakes, but they may make more subtle mistakes based on the same principles.

The key components of any division instruction should help students by:

• emphasizing understanding what they are doing when they use a division algorithm,
• providing tools for students to estimate their answer,
• encouraging them to use their estimation skills to check the reasonableness of their answer,
• suggesting alternate techniques to double check the accuracy of their answer,
• giving time to see the connection between different division algorithms,
• embedding the use of division in a problem solving context.

If I were going to teach division, (after confirming that students understand multiplication very well) I might start with a problem about equal sharing for students to do that every student can access. Next I might gradually increase the size of the numbers in the problem until students start to see that their "naive" strategies may work, but are very inefficient. One obvious problem I can foresee happening here is that some of the students in many grade four classrooms already know a division algorithm, and may be reluctant to come up with their own strategies as a result.

Next, I would compare different strategies that students use in a whole group setting. We would spend some class time discussing the strategies and talk about the benefits and drawbacks of each. We might also explore some other alternate strategies not yet shared, perhaps using a gigsaw technique where each group explores a specific strategy, perhaps suggested by a video.

One interesting activity for students might be to watch the videos above and try and figure what mistake the people in the video are making when they calculate.

Here are some division strategies that your students may not have seen before:

• Double division
• Galley division
• Chunking
• Division "by parts" (see the third method for division listed on the page)

I’m also looking for projects and/or puzzles which require the use of division as a strategy for solving the problem/puzzle.

An aside: I am not convinced that these algorithms have much value for our students. More time spent understanding what exactly what division is would be more useful, I think, for our students in the long run, and then just give them a calculator. However, that is not my decision to make, it is up to the BC Ministry of Education to decide.

I read a recent article about the importance of early number talk with children and was pleased that this issue was being brought up. The article shares research on a few of the stark differences in how parents talk with their children about numbers. For example, parents tend to talk to their daughters about half as much about numbers as their sons. Parents also range in how much they use number words around their children from about a dozen times a week, to as much as 1800 times per week.

However, I felt that the list of suggestions the article had at the bottom was incomplete. The article’s author essentially makes suggestions which I feel will only help children develop an instrumental understanding of mathematics, as opposed to a more useful, interesting, relational understanding.

Here are some more things my wife and I do with our children from a very young age to help them develop a deeper understanding of numbers.

• We play games with our children that involve numbers. We roll dice, we play cards, we solve puzzles together, and we play hopscotch. Through these games, my children gain an understanding of the relationship between numbers and actions we take in the games themselves. We ask questions like "how many ways can you get a 10?" My son recently answered that question with this sequence of answers: 20 – 10, 30 – 20, 40 – 30, 50 – 40, etc…
   
• We talk about how we use numbers in our day to day life. We talk about fractions of food (that are physically present in front of us), and talk about the relationships between these different fractions. We cook, do our finances, and share as many uses of numbers as we can with our children.
   
• We look for patterns in numbers. We play with relationships between different numbers. We celebrate discoveries our children make. For example, my eldest son noticed that he only needed to remember the very last digit in a large number to figure out if a number was odd or even. Now, he delights in asking people to give him gigantic numbers like 30,938,309,830,983 and being able to tell right away if the number is odd or even. As a mathematician father, I try hard to balance between giving my sons space to come up with their own discoveries, and expanding where their explorations might go.
   
• I also try very hard to remember that when my son makes a mistake, with further experiences, it is likely that he will discover these mistakes later for himself. For example, my son was convinced for a long time that numbers went 90, 100, 110, 200. He did not understand place value well enough, but over time, and with further exposure to our use of numbers, this misconception of his has disappeared. He now has a very good understanding of place value up to 1000, although he still does not understand larger numbers (he will say things like 233 hundred thousand, 293 million, 389 billion, and 43), but I am confident that as he continues to be exposed to numbers in his day to day experience, he will understand these numbers better.
   
• We balance our discussion of patterns in numbers with other types of patterns. We look for patterns in art work, in sidewalks, in tiled floors, and wherever else they may form. We create patterns ourselves in our art work and notice where they came from. We doodle, we draw shapes, we watch clouds, and we look at maps, all of which help my son develop a sense of shape and space.
   
• We ask for evidence from our sons about why they think something is true, whether or not what they are saying is accurate or not. When they have discovered something, and provided solid evidence for why they think it is true, we celebrate it. We might ask questions like, "How did you discover that?" or "Wow, that’s neat. Does it work all the time?"
   
• We give our children plenty of creative time to explore the world through art work, Lego, blocks, reading, playing games, and other self-exploration activities.
   
• We see the development of our children’s numeracy as a process, rather than a race. I have no interest in accelerating my sons through the elementary school curriculum, instead I focus more on providing opportunities for enrichment.

The key to all of these activities is that we view numbers and quantities as ways of exploring, and we nurture our children’s sense of wonder about the world.

Andragogy is a theory of learning as learning applies to adults rather than children (pedagogy). According to Malcolm Knowles, there are 6 key components of adult education.

1. Adults need to know the reason for learning something (Need to Know)
2. Experience (including error) provides the basis for learning activities (Foundation).
3. Adults need to be responsible for their decisions on education; involvement in the planning and evaluation of their instruction (Self-concept).
4. Adults are most interested in learning subjects having immediate relevance to their work and/or personal lives (Readiness).
5. Adult learning is problem-centered rather than content-oriented (Orientation).
6. Adults respond better to internal versus external motivators (Motivation).

I fail to see how these six things are not also true for children.

The objects that adults produce as part of their learning should be different than the objects children produce. Adults don’t need to create posters (although this may still be a valuable learning experience depending on the context) at the same rate that children do during their learning.

Adults have some different external concerns (children, job, home, etc…) than do children that sometimes interfere with their ability to learn in a classroom setting, but these concerns are just different than the concerns of children, they are no less important to the learner.

Adults come to their learning with more experiences than children, and this may make any unlearning (if necessary) more challenging for them, but the fundamental process through which they learn should be significantly similar to the process children go through.

The primary difference I see between adult learning and children learning is how much power they are granted during their learning.

You may remember Paul Lockhart as the author of a Mathematician’s Lament. I’m currently reading his newest book, Measurement. I’m halfway through it and reading it every chance I get. Here’s my favourite quote from the book so far:

"All of the events — past, present, and future — of our whole ridiculous universe are writ on this one four-dimensional canvas, and we are but the tiniest brush strokes." Paul Lockhart

Of course, what Paul says is true. Here’s a great mathematical investigation: Given a football sized canvas, how large a brush stroke would all of the aggregate movement of the human species require, assuming the canvas represents the entire universe.

If you are a math teacher, or just want to understand what people find fascinating about mathematics, I recommend reading Paul’s book.

I have seen a number of people post reviews of their year in 2012, and it led me to review my own year, and here are some of the insights I had.

Most popular posts of 2012

1. Do iPads improve mathematics instruction? Maybe. (6938 reads)
2. Children are not railroad cars. (6775 reads)
3. We need social media etiquette. (5116 reads)
4. Profile of a phishing attempt. (4500 reads)
5. The difference between relational and instrumental understanding. (4355 reads)

Other stuff

1. According to my server statistics, I had over 1.7 million hits on my website in December, of which the vast majority were blocked attempts to post spam on this blog.
2. Google Analytics suggests that a more accurate number of unique visitors, excluding spammers presumably, is about 68,722 which is still significantly more than the previous year.
3. The most popular search terms which led to my blog were "David Wees", "Purpose of Social Media", "School paper use", "Relational understanding" and "Teachers are stupid." I’m flattered that 238 people looked for my blog by searching for my name, and I’m less flattered that 123 people found my blog by searching for "teachers are stupid." I decided recently to unpublish the post that was getting all of the hits for "teachers are stupid." I’m not sure I want to own that particular search term.
4. I started a dual role this year at my school, both working on our use of educational technology and supporting our teaching of mathematics. I love my job and am looking forward to the rest of the school year.
5. I attended 7 conferences during the year and presented 13 times (externally to my school).
6. I spent much more time with my family in 2012.

This was written by Ana Ver, the Learning Specialist for Science at my school. She tweets at @anainvancouver if you want to connect with her.

Winter break is coming! But three weeks away from school during the holidays doesn’t mean that education has to stop. Being an explorer, a scientist, an artist is not just a job– it is a behavior, a way of life, and is automatic when a master learner is in his or her element. Learning should be ingrained, even when there are no rubrics or marks or IB exams. When the motivation for exploring and learning comes from inside, you know that the real education has begun. How can you help your child become an explorer of the world? Here are some engaging activities to help kids be better observers, listeners, thinkers, scientists, artists, and explorers. Many more ideas can be found in Keri Smith’s book, “How to Be an Explorer of the World”.

What’s Their Story?

Sit in a public place and watch the world pass by. Make up stories about where people may be headed, what they’re doing, who they’ll be meeting. Could they be a world traveler or a time traveler? These observations and inferences, not to mention storytelling skills, will help kids become excited explorers!

The Hunt for the Perfect Poking Stick

Walk through a park or the woods or the beach, but keep your eyes trained on the ground. You’re looking for the perfect poking stick! It needs to be long, not too heavy, sturdy, and with a dull end so it won’t hurt anyone or anything. Use your poking stick to explore your surroundings—to turn over rocks and leaves, to poke at that dead jellyfish. Where will your poking stick lead you?

Iron Scientist

You’ve heard of Iron Chef—contestants are given a secret ingredient (fish eyeballs, Meyer lemons, birthday cake) that they have to incorporate into a dish. Why not rejig the formula as an engineering activity? Clear out the junk drawer, decide on a random secret ingredient, and challenge your neighbors. Build a bridge, a better home alarm, or a home for your pet rock. See what kind of creations emerge.

Play With Your Food!

Go to a grocer where you’ve never tasted most of the fruit or vegetables for sale. Document the sights and smells and sounds. Pick an interesting looking food. Ask the grocer his or her favourite way to prepare it. Buy it, bring it home, and have a taste test!

Found Faces

Take a walk around your neighbourhood with your dog and your camera. Look for objects, garbage, trees, clouds that look like faces! Take pictures of the faces and give them names. Alternatively, look for objects or designs that look like letters or numbers. Spell your name. Write out your favourite chemical equation.

Small Explorations

On rainy days: Find a puddle. Jump in it. Save the earthworms and slugs from the sidewalk.

Go for a bike ride. Bike as fast as you can up a hill. Turn around and coast back down the hill. Journal some metaphors for the feeling of the wind rushing by your face.

Find something precious to you. Bury it. Make a treasure map for your best friend to find it.

Turn all the photos in your house upside down. Hide around the corner and observe if anyone notices.

Create a stack of cards with mini-thought experiments on them, like, “If we went to Mars, what would we need?” or “What would happen to restaurants if you could miniaturize food?”. Keep them on the dining table. Discuss over dinner.

When adults model exploration and joy of discovery for their children, we help students become knowledgeable, inquiring, and open-minded reflectors, communicators, and risk-takers. We help continue the IB education outside of school. We help them become explorers of the world.

(Image credit: zeligfilm)

If you are using formative assessment to help guide your instruction and give feedback to your students, then you should be able to fairly easily predict how well your students will do on a classroom test. If you can, why are you giving your students the test?

Hopefully your answer isn’t "so they will have experience in a testing situation for later in life." To be clear, I still use classroom tests, but I am beginning to question the practice.

Can someone give me a better argument for classroom tests?