Tag: The Reflective Educator (page 6 of 43)

As I watched my son over the past few days learn about shape, I am struck by not only how much we need to learn to make sense of the world, but also by how even simple things cannot be taken granted as known by children.

My youngest son, who is about 20 months old now, is learning how to take a small piece of yellow plastic that forms a shape (an oval, a circle, a hexagon, and 4 other shapes) and push it through a hole that is only slightly larger than the shape he is pushing through it. The task requires him to pay close attention to both the shape he has in his hand, the orientation of that shape, and the hole he is trying to force it through.

At first he pretty much chooses shapes at random, and tries to jam into the hole. Eventually he finds a match and gets a shape through the hole, or he gives up. He rarely turns the shape in his hand, or the container the shape goes into.

Eventually he learns that he has to often rotate the shape in his hand, and so he picks up a random shape and tries to shove it in a random hole, and when it doesn’t work the first time, he says, “No…no…no,” and rotates the shape a bit to try again.

Soon he has made a few matches, and is always able to easily find the right hole for the circular shape and the oval shape, and soon after the plus sign shape. For all of the other shapes, he continues to try to randomly match shapes to holes. He looks at the shape to identify it as one of the ones he knows, but he does not seem to connect the general idea that the shape he sees in the piece should match the shape he sees in the hole.

The next day, I am surprised to discover that he can match almost all of the shapes, and frequently looks at the shape to see which one he has, and then which hole he should put the shape in. He often rotates the shape several times when he is sure he has the right hole. When he is not sure, he tends to give up quickly and go back to a shape with which he is very familiar.

In a few days, he goes from basically using a random matching strategy to carefully looking at the shape in order to be able to match it. At this stage the only shapes that stump him regularly are the regular pentagon, and the regular hexagon. 

I notice however that his ability to do other puzzles does not seem noticeably improved. It’s like his learning is restricted to this one very narrow context, and within this narrow context, he has either just learned to match each of the individual patterns or possibly he recognizes a small generalization; for this puzzle the shape should match the hole. As the months progress, I will continue to watch how his understanding of shape grows and develops.

It is fascinating to me to see that shape is a learned concept and that even what seem like simple generalizations are learned. It makes me wonder what concepts my students may not have fully developed, even by the time they arrived to me in high school.

I’m currently working on creating a sample sequence of lessons for teachers to use for a geometry unit. At this stage, students will have been exposed to (but will not necessarily have learned all of) geometric transformations, constructions, and some review on geometric vocabulary.

My objective is to create a sequence of lessons which include:

• embedded formative assessment,
• opportunities for students to discuss student thinking,
• opportunities for students to make sense of the idea of proof,
• opportunities for students to prove geometric proofs,
• opportunities for students to engage in mathematical inquiry,
• engaging tasks that help get kids excited about mathematical arguments.

The last time I taught geometry, I think I maybe managed to to hit 1 or 2 points from this list, so I have given myself a tall challenge. I need help.

One of my university professors once told me, “Something is proven true when everyone stops arguing that it isn’t” so with that in mind, here is a vague idea that I am considering and need help fleshing out:

Basically, students would spend their time during this unit making interesting geometric observations and then attempting to prove to each other that these arguments are true. Part of their time would be exploring geometric objects, possibly through constructions and possibly through looking at (interactive?) diagrams. During this time, when they see something that they think might be true, they create a proof (in everyday language that they understand) that it is true, and then they present their argument, either in small groups or to the whole class.

Here is an example from my imagination. Suppose students have access to this online geometric construction tool (or lots of paper, a compass, and a straight edge) and when playing around, one of them creates this construction.

Student 1: “Oh wow, that’s pretty cool. I made a regular hexagon.”
Student 2: “It sure looks like a hexagon, but remember what Mr. Wees said? It might be just really close to a regular hexagon. How do we know for sure it is one?”
Student 1: “Hrmm. A regular hexagon has all the sides the same length, and this looks like all the sides are the same length, so it must be a hexagon.”
Student 3: “How do you know for sure the sides are all the same length?”
Student 4: “Yah, maybe they are like, one or two pixels off or something.”
Student 2: “How did you draw this? Can you show me how?”
Student 1 shows her group members how she came up with the construction. “See, it’s a regular hexagon.”
Student 3: “I notice that when you made each of the sides you used a circle.”
Student 4: “Yeah, I noticed that too.”
Student 2: “And all of the circles you used were the same size.”
Student 4: “Well, not all of them. There are bigger and smaller circles.”
Student 2: “Okay well all of the smaller circles are the same size.”
Student 1: “How do you know that?”
Student 2: “They all have the same radius, see? These two circles have the same radius, and these two, and these two. They are all connected so they all have the same radius.”
Student 4: “Ooooh, I have an idea…”
Student 1: “Me too! Those radiuses that he pointed out are the same as the lengths of the sides of the hexagon so if they are all the same, then all of the sides of the hexagon are the same size. Done!”
Student 3: “Is it possible to make a hexagon where all the sides are the same length, but the hexagon is still not a regular hexagon?”

 

Obviously this is an idealized situation, and maybe a bit unrealistic but this is where I would like students to end up. What kinds of classroom conditions would lead to students being able to do this? How would the class be structured? What kinds of supports would a teacher have to give to help support students?

Why do we teach math?

(Image source)

It could be because the mathematical procedures that are taught in schools will be useful to students later, but I am pretty sure this is false. Almost everyone forgets those procedures as they get older because most people in our society use virtually none of the procedures they learned in school in their day-to-day life. Obviously there are engineers, mathematicians, and scientists who use the mathematics they have learned, possibly on a daily basis, but I think if you dig deeper into the work they do, many of these people use tools to help to do their work (like Mathematica, for example), look up the finer details of mathematical procedures that they do not use often, or who use only a very specialized portion of their mathematical knowledge regularly.

It could be that we want to expose students to different ways of thinking about the world. In this case we would be less concerned with the exact set of mathematical procedures they have learned, and more concerned with learning mathematics as a way of thinking and knowing. I see little evidence that this is an explicit goal of mathematics instruction given that; the students are assessed only on the procedures, teachers are assessed on their students understandings of those procedures, and that the set of mathematical procedures we want students to know is so prescribed such that it is virtually identical around the world.

It could be that we would like students to learn transferable problem solving skills. In this case, we want to teach mathematics in such a way as to promote the likelihood that students will be able to transfer what they learn to other areas. Cross-disciplinary study would be the norm, rather than the exception. It turns out that “teaching skills that transfer” is not as simple as one thinks. In fact, my understanding is that most of the times when people learn skills in one context, they do not end up transfering those skills to other contexts. Instruction that aims for transferable skills has to provide opportunities for students to make connections between different areas, reflect on what they have learned, and develop metacognitive strategies so that students think about their thinking. What evidence is there that these types of activities are a regular part of math classes?

It could be that we would like students to see the beauty and elegance of mathematics. One way to do this could be through exploring mathematical art. Another might be to look at some famous examples of truly elegant uses of mathematics. We could also ask students to talk about mathematics in the abstract and come to a shared understanding of what elegance and beauty in mathematics mean. As far as I know, none of these activities is a common one in math classes. It is depressing to me that this way of thinking which has so much beauty in it is shared in such a way that almost no one in our society ever gets to experience beautiful mathematics.

If one or more of the reasons I suggested above is something you think is a good reason to teach mathematics, how are you ensuring that you meet this goal with what happens for students in your classroom? 

What other reasons are there to teach mathematics?

I recently read the Common Core standards for Math for grade 6, which is where the concept of variable appears to be introduced. The standard in question reads:

Write, read, and evaluate expressions in which letters stand for numbers.

I tweeted this out, and one person responded with this observation:

@davidwees in d/dx[f(x)]=mx+b …d, f, dx, f, x, m, and b all have completely different meanings, and only m and b “stand for numbers”

— John Chase (@mrchasemath) October 27, 2013

Clearly this definition of variable is limited, and possibly misleading.

How do you define variable with your students? How do you introduce the topic?

JavaScript is one of the primary web programming languages. It is the language used in Google Apps Scripting, and in HTML5 interactive documents. Chances are pretty good that most of the interactive features that you see in websites were coded in JavaScript.

Much of the power of JavaScript comes from its power to manipulate and modify HTML, which is the markup language that websites are written in. Therefore, if you want to learn JavaScript, you will sooner or later need to learn HTML (and CSS) as well.

You will want these things to get started:

1. At least one web browser. This will be where you run your code. I recommend using Firefox with the extension Firebug installed. Firebug is incredibly useful for finding bugs in your code, and for debugging. Google Chrome is also useful for some simple debugging as well. Note: JavaScript does not always run the same in every browser, so one reason to have multiple browsers is to test that the code you write works in all of them.
2. A good text editor. I recommend Programmer’s Notepad for a PC, and TextWrangler for Mac, both of which are free.
3. A language reference. I personally like Mozilla’s reference here, 
4. Some understanding of how web technologies work. This reference guide, again by Mozilla, is useful.
5. A folder on your computer dedicated to your projects. It is a good idea to be organized from the beginning.
6. Blocks of time in your schedule to spend working on programming. Programming is something that will take concentration and focus, and is hard to learn in 10 minute intervals.
7. I also recommend buying a reference/tutorial book. I found JavaScript for Dummies helpful as a starting place, for example.
8. A JavaScript library that professionals use (I recommend jQuery). This will allow you to create more interesting projects, while at the same time helping you develop an understanding of JavaScript ideas.
9. An understanding of the design cycle is useful.

Once you have the things above, my recommendation is try to duplicate someone else’s project. This will help you ensure that you have your development process set up correctly on your computer. Your JavaScript files are loaded by the HTML page you create, and so end up linked in it. Here is a sample template you can use for your JavaScript development. There are a lot of ideas that will want to unpack if you use this template.

My workflow for programming in JavaScript is to have the text files I am editing open in one program, and the output open in a browser. When I make changes to my code, and want to see the output, I simply save the changes, and then refresh the page that has my JavaScript in it.

In a future post, I will share a screen-cast of starting my own JavaScript project, and talk through the creation of this project.

When I was 8, my father gave me my first computer. There were two things I could do on this computer; program or write. I found programming to be much more interesting! So with a little bit of support from my father, and a copy of the Wang BASIC reference manual, I began the long process of teaching myself how to program.

Here is basically what you need to learn how to program:

1. A goal. Decide on what you want to make. You may find that you can’t make it, but there will probably be some small part of it you can get started on.
2. You need resources to help with syntax and structure. This could be a good book, or a website, a class, or any combination of the above. I tried programming classes a couple of times and found them pretty boring, but this likely does not apply to every programming class. There are lots of interesting books on learning how to program in a variety of languages.
3. The patience to struggle, especially in the beginning. When you are first getting started, programming is like non-stop problem solving. It is challenging! 
4. Time. Learning to program, and to do it well, is not something you pick up in an afternoon.

If this is something you are interested in, you can join a new section of the Edtech community on Google+. This is basically intended to be a peer support group, along with (hopefully) some mentors to help people get started, and to help people when they get stuck. While I cannot promise that joining this group will guarantee that you will learn how to program, I think it will help. Just be willing to lurk, and if needed, ask questions in the “Learn to Program” discussion forum of the group.

P.S. If you know anything about programming and are willing to help out, please join the group and introduce yourself in the section called “Learn to Program.” At the very least, feel free to offer suggestions for starting places for teachers.

My wife and I decided a couple of weeks ago to withdraw our son from our local community school and homeschool him. We realized that the constraints on the school, and the choices made at the school were going to prevent him from getting the exercise, play, and intellectual stimulation he needs to remain healthy in body and mind.

We were required to send a letter to the superintendent of the school district as the first step of our legal requirements in NY state in order to homeschool our son. Here is a copy of that letter (I have edited out the portions that identified which school and which teacher he had).

Dear [Name Withheld],

We are sending this letter of intent as required of Section 100.10 of the Regulations of the New York State Commissioner of Education. We intend to homeschool our son, who is in grade two, for the remainder of the 2013-2014 school year beginning immediately. We would like explain to you why we are making this decision.

We moved here recently from Vancouver, British Columbia, where our son attended a public Montessori school. In this school, our son learned self-regulation, and as such had tremendous control over what he learned, and when he learned it. The school day started with Tai Chi, and our son had a morning recess, a lunch recess, and physical education every day. It was never difficult to get him to school, as he loved going to school.

This is not the situation this year. We are finding it much more difficult to motivate our son to wake up and get ready for school. Instead of telling us how much fun he had and how much he learned at school, we have to struggle to pull out of him what happened each day at school.

This is not because of our son’s current teacher. She is a caring, hard-working, and thoughtful person. We have no complaint with her character or her ability to teach.

We are concerned about the prescribed teaching method our teacher has been asked to use, specifically the excessive test preparation. We could easily offer this type of education ourselves after a trip to the local education supply shop. What then is the benefit of sending him to school? What the children need is to interact with each other and have at least some time to make their own discoveries which requires independent time and exploratory activities within the classroom.

Our son’s school does not offer a morning recess. At lunch time, unless they are participating in one of the supervised events occasionally run by staff, the children at my son’s school cannot run around. My son only has physical education once a week during which he has so far learned how to walk around a gym. Fortunately my son does have one dance class and one art class each week, but overall, he does not receive sufficient physical activity through school to keep him healthy and to help him focus.

We believe that children need these essential elements in order to become healthy adults; creativity, play, intellectual stimulation, exercise, and opportunities to collaborate with and learn from their peers. None of these elements is present in at my son’s current school in a sufficient degree. We can see that the school is fighting a losing battle to maintain some physical activity and art, and from the research we have done, these are often lacking in many of the public schools in New York City.

My son recently told us that his current school is the “No fun school” but that he is “learning to adjust to it”. Our fear is that in an effort to make school more academically rigorous, many of the things that make school worth attending are being removed.

The definition of rigorous, according to the Merriam-Webster dictionary, is:

• very strict and demanding

• done carefully and with a lot of attention to detail

• difficult to endure because of extreme conditions

With the exception of the attention to detail, we do not believe that these things should be applied to school, particularly not school for children ages 5 to 11. Our son’s current school is teaching children that learning is a chore to be done, and not something to enjoy and to love.

We understand that relatively recent legislation in New York State, where teachers and principals are judged based largely on the test scores of their students, is to blame for this situation. This legislation seems like a gigantic experiment that lacks sufficient evidence to justify such a punitive policy and we see no reason to experiment with our son’s education in this way.

Instead of allowing our son’s school to drain the love of learning from our son, we are removing him from your school’s care.

Sincerely,

David Wees
Vasilia Wees

Today I observed a teacher using this tool built by Jennifer Silver to engage her students in mathematical reasoning. It was a powerful reminder to me of the intersection between effective uses of technology to provoke thinking in students, and the pedagogy used to support that student reasoning.

First, the teacher brought up the interactive diagram up on her Smartboard, and then she asked a student to come up to change the slider values. She repeatedly asked students to say what they noticed each time the slider was changed. She took the time to have multiple students clarify what they said, to have their peers restate and respond to each other’s reasoning, and to have students take the time to make mathematical observations. She engaged students in collaborative mathematical thinking for 30 minutes. At the end of the class, at least 10 students came up en masse to play with the interactive diagram themselves and continued to ask her questions and make observations. She had to promise them she would email them the link to the diagram so that they could continue to play with it themselves.

The point here is that the technology made the conversation easier. Instead of creating 20 different examples of graphs and seeing what happens as each variable is changed, students were able to visualize the changes, both in the graph representation, and in the formula representation. When asked if they noticed anything after the “Point on the line” slider was changed, one student said they noticed the Intercept-slope form of the equation did not change. Another student responded to him with “that form of the line doesn’t depend on which points you use.”

It was fantastic.

What exactly does our assessment measure? I watched my 7 year old son complete an online assessment of his fluency with addition facts last week, and I noticed a few things the assessment measured unintentionally, at least to some degree.

• It measured his ability to decode the symbols presented (eg. 2 + 10 = __).
• It measured his ability to find the keys needed to answer the question on the keyboard.
• It measured his ability to combine the pairs of numbers together to come up with an answer.

The tool used appeared to make an incorrect assumption – that it was measuring fluency with an addition fact exclusively. In fact the feedback it offered to me the parent said exactly that.

However, there were some important things not measured by this online tool.

• It did not measure his ability to explain his reasoning to others.
• It did not ask him to show multiple solutions for finding his answer.
• It did not present a meaningful context, and measure my son’s ability to apply his understanding to that context.
• It did not check to see if my son had gained any transferable understanding.
• It did not allow my son to talk to peers about his solution.

When we look at any assessment tool, we should ask ourselves, what is possible to measure with this tool, what may be unintentionally measured with this tool, and most importantly, what is not measured with this tool.

Virtually all digital assessment tools I have looked at are very good at the low-hanging fruit of automated responses to trivial questions, and almost none of them help answer more important questions about student understanding.