Author: David Wees (page 29 of 97)

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?

Here is a question I was asked recently.

A student asks you why 0 to the 1^st 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:

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 = 2^x 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.

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?

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:

• Which shape has the largest area?
• Which rectangle is the biggest?
• Which shape is the most circular?

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

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.