The Reflective Educator

Education ∪ Math ∪ Technology

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Year: 2012 (page 1 of 14)

2012 in review

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.

 

 

How to Be an Explorer of the World

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.

 

Classroom tests

Classroom test
(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?

What does this mistake mean?

Here is a question I was asked recently.

A student asks you why 0 to the 1st power is 1. What do you do to help the student understand?

Any time a student comes to me with a question, I try and see if I can figure out what’s going on by paying careful attention to the student’s explanation of the thinking that led to come to their current conclusion. In other words, I make an attempt to assess what they understand and use this to diagnose what’s going on.

Depending on what issues I discover the student has, I might address this by addressing their pre-requisite misconceptions, create an activity for them to do, ask them questions to help them draw out a better understanding of the problem, or by involving a peer to have them help explain the concept.

In this specific case, if the student asks this question, chances are they are misremembering the "rule" that says that x0 = 1 for all x except x = 0 (at x = 0, the expression is undefined, which could possibly lead to a really interesting investigation for students). A student who misremembers this rule has probably learned it by memorizing it, and may lack understanding of where this rule comes from. It may also be that they do not really understand what exponents are. It may also be that the student does not yet understand division and multiplication thoroughly, as these are pre-requisites to really understanding exponents.

A simple investigation that may address this is as follows.

First we look at this pattern:

 Patterns in exponent rules

Next we ask the student, what do you notice about this pattern? What would happen if we changed all the threes to 7s or 2s? Would the result still be true? How could we generalize this pattern we have discovered? At this stage, the student might recognize their misconception and be able to move on. If the student is not proficient with exponents at this stage, they may need a more broad investigation into the other exponent rules. 

According to this summary of research on student understanding of exponents, they often feel that exponents lack connection to the real world (Senay, 2002), and do not understand the point of studying them. A student who struggles to to understand a concept involving exponents may be resistant to the concept simply because they cannot see the value in it. One way to address this is to choose specific examples where exponents make the problem or idea in the real world much more easy to understand. The Powers of Ten website may address this to some degree, as may the story of the farmer and the king. There are also many other resources available which may help students see the value in learning more about exponents.

Another option, if the student has a good understanding of graphs, is to try and have the student produce the graph of a function like y = 2x for x > 0. They can do this with a table of values using x = 1, x = 2, x = 3, x = 4, x = 5, etc…, and then attempt to fit a nice exponential curve to the existing points. Extrapolating the curve back to x = 0 should lead the student to see that the result is not likely to be 0, and if they are lucky, they may get close to the actual value of 1. This may make another approach, like the experimental one given above, seem more realistic.

Another option is to look at decreasing values of exponents.

Halving powers of 2

Again, the purpose here is to get the student to look at the pattern and try and draw some conclusions. What value of the question mark would make this pattern work nicely? This will help students see that the rules we are using are intended to create consistency within our use of exponents, and preserve some of these patterns.

While I am working with the student, I try to use good questions. If the student asks questions, I try to respond in such a way so as to make them continue thinking. My objective is to provide students with tools they can use to figure out the solution to mathematical problems themselves so that we can work them toward being independent from their teachers.

Can you recommend some other ways I could help this hypothetical student understand exponents better?

Activeprompt examples

Sample photo with 4 different shapes on it.

This morning I generated three new Activeprompts related to area and understanding the relationship between dimensions and the area of an object. I’m hoping to use these as examples of questions teachers can use to prompt good questions from their students and generate discussion around mathematics.

Examples:

I sent out the prompts via Twitter, and one person responded with a great question.

 

 Of course, this is exactly the kind of question I’m hoping students come up with.

Inquiry into positive and negative integer rules

Our department had a meeting recently where we discussed the need for more investigative approaches in our teaching. We all use investigative approaches at least part of the time, but some of us disagreed about whether it was possible to approach every topic with an investigation. 

One of the specific topics that came up was "the rules for multiplying positive and negative integers". Here are some ideas we came up with:

  1. Give students calculators and have them try out different calculations with different signs. They are likely to quickly discover the "rules" for multiplying and dividing positive and negative numbers, but I am not sure if they will understand why the rules work. Still, it’s a step in the right direction toward student discovery.
     
  2. Have students generate patterns when they are multiplying positive and negative numbers like so:
    3 x 4 = 12
    3 x 3 = 9
    3 x 2 = 6
    3 x 1 = 3
    3 x 0 = 0

    Prompt the students to see what patterns they notice about this list of multiplications. Ask them to extend the patterns another couple of rows. They will have hopefully noticed that the second number in the multiplication is decreasing by one, and that the answer is decreasing by three each time. They will need to have a good understanding of multiplication, subtraction, and negative numbers to be able to be successful in this investigation. This will be an excellent opportunity to formatively assess students on their understanding of negative numbers to see if they can extend this pattern to 3 x -1 = -3.

    Have students repeat this process for another set of similar multiplications but perhaps with the pattern flipped around slightly. For example:

    2 x 5 = 10
    1 x 5 = 5
    0 x 5 = 0

    You can then set up other similar patterns which should result in the students developing similar rules for other forms of multiplication.

    5 x -3 = -15
    4 x -3 = -12
    3 x -3 = -9
    2 x -3 = -6
    1 x -3 = -3
    0 x -3 = 0

    There are some advantages of this approach. First, the students will see that the rules for multiplying positive and negative integers come somewhere; they are in essence necessary to preserve the internal inconsistency of multiplication with these patterns. Second, the students will necessarily get some practice multiplying some of the numbers together with a purpose.

     

I’d like to try and figure out a visual investigation which doesn’t seem completely contrived. I could imagine some sort of animation involving positive and negative areas which could be useful, but it would require significant preparation ahead of time to ensure that students have a solid sense of multiplication as area of rectangle model before using it.

 

Philosophy of Education

People learn through a process much like scientists do, discovering the world through observation. They either consciously or unconsciously hypothesize about how the world should work, collect data, compare the data they have collected to see if it fits in their theory, and then revise their theory if they feel enough evidence has been found. In this way, people construct an understanding of the world around them using what they know as a framework for understanding. Like a scientist, each piece of knowledge a learner is connected through a personally developed taxonomy, and it is through these connections that knowledge is stored, retrieved, and built upon.

Each piece of knowledge people gain has to be fit into their personal schema. At first, people will adjust their hypothesis to make facts fit which seem inconsistent, but eventually if enough contradictory data is collected, people are forced to revise their ideas. This is part of the reason why students have so much difficulty learning topics for which they do not have any background; they are constantly required to create and revisit their hypothesis, and to build theories about the information they are receiving. Learners often struggle to transfer information from one domain to another; their personal schema may not include a connection between the two domains, and so they may be forced to seek new hypothesis about the new domain.

It is crucial during this process that the learner feels comfortable to make mistakes. Instead of feeling pressure to have exactly the right answer, learners must be willing to work through the entire process of learning, and discover their mistakes for themselves. Although it is possible that an individual learner will have an incorrect theory which fits all the facts as they are collected, it is much more likely that conflicts exist between their theory and the data. We also need to be cautious of what types of student questions we answer as a teacher; we should stop answering questions that stop students from thinking for it is through thinking that students will be able to resolve these conflicts and improve their model.

The role of a teacher in this process is to provide an environment where learners are likely to be able to explore ideas, and be presented with feedback on their understanding of these ideas on a regular basis. The teacher must also act as a learner in this process and be open about how they are learning so as to model this for all of the other learners in their learning space. The teacher, with their students, shapes the learning space so as to expose students to new ideas, and to explore the existing worlds the students inhabit.

As I am myself a learner, I attempt to live according to this framework as well which means that I actively try to make meaning of what I observe, study what interests me, and explicitly refine my models of how the world works.

Activeprompt

Riley Lark recently shared Activeprompt, which is a way for one person to create an image prompt that can be used for a variety of different purposes. Riley released the code for Activeprompt as open-source, but in the programming language Ruby, which I do not know. I’ve created my own version of his project in PHP. I am also releasing my code (for non-commercial purposes only).

Here is a video explaining the project.

 
How to use this program:

  1. You upload a picture and write a prompt to go along with that picture.
  2. You send the link to the poll to whomever you would like to respond (like your class).
  3. You open up the other link listed after you create an image to view the responses to your prompt as they are posted.
  4. You can view a gallery of different prompts here (requires login: please let me know if you see anything inappropriate here).

I have plans to add a log in (so there is some level of security on what is posted), the ability to book mark prompts, and the ability to clone a prompt (allowing you to reset the results for a new class).

What other features would you like to see? How could you imagine using this with your students?

Landfill Harmonic

Find a little more than 3 minutes, and watch the following trailer for the Landfill Harmonic movie.

My favourite quote from this trailer:

People realize that we shouldn’t throw out trash carelessly. Well, we shouldn’t throw away people either.

I wonder what the world would look like if we all lived by Neil Degrasse Tyson‘s creed:

“The problem, often not discovered until late in life, is that when you look for things in life like love, meaning, motivation, it implies they are sitting behind a tree or under a rock. The most successful people in life recognize, that in life they create their own love, they manufacture their own meaning, they generate their own motivation. For me, I am driven by two main philosophies, know more today about the world than I knew yesterday. And lessen the suffering of others. You’d be surprised how far that gets you." [Emphasis mine] – Neil Degrasse Tyson

Bean counting and place value

One area of mathematics which I strongly suspect many students have problem understanding is place value. It is an important abstraction for students to understand, and without understanding it, it is unlikely that students will progress very far in arithmetic (and then will likely struggle in algebra later).

Here is an activity my friend David Miles told me about years ago which I would very much like to see in action some time.

A big bag of beans

Give the students a very large amount of beans (or something similarly small and dry) to count. For younger kids, give them a smaller amount, and for older kids, give them a larger amount.

Start by asking them to estimate how many beans are in the bag. Perhaps ask them to give you a number which is probably more than the number of beans, and a number which is definitely less. It doesn’t really matter how good this estimate is, the idea is that by asking students to give an estimate, and then letting them compare their estimates later with their more accurate answers, that students may improve in estimating.

Next, ask students to work in groups to count the beans. Give them LOTS of time. Give them some very small cups they can use to help them with their counting which should ideally hold about 10 beans maximum. If you need to use larger cups, ask students to restrict themselves to only putting 10 beans in at a time. While kids are counting, if they aren’t keeping track somehow of their numbers, count loudly to distract them, forcing them to keep track of their results. Don’t give them any paper or pencil, just the cups.

The idea is, the cups are too small to hold many beans each, and the students don’t have enough cups to hold all of the beans. What they will end up having to do is to choose one cup to represent ones, when this one fills up they will have to create another cup to put a bean in to represent 10 beans in the first cup, and when this cup fills up, they will have to create another cup to represent 10 beans in the 10-bean cup (or 1 bean represents 100 beans) and this leads to what place value is, at least for numbers greater than 1.