Year: 2013 (page 4 of 15)

I created a new puzzle, based on something I played with when I was at the New York Hall of Science. The idea of the puzzle is that given three numbers, you are trying to balance both sides of the equation, with the constraint that different positions on each side of the equation are worth different amounts. At this stage, this is a prototype, so the numbers are chosen randomly, but the eventual plan is to build this balance puzzle into a series of challenges, each of which will probably have multiple solutions.

One hope is that a puzzle like this helps bridge the gap between arithmetic and algebra, by providing an activity which uses arithmetic operations but embeds algebraic reasoning. I also hope that this app helps reinforce that the equal sign (and the inequality symbols) are not indicators that one should do an operation, but rather that they represent a relationship between the two sides of the equation.

Check out the puzzle here: https://davidwees.com/balancepuzzle/

I’d love feedback before I move onto making this puzzle more robust.

(This is my son learning how to program in Turtle Art, at age 4)

As educators, we give work to students based on what we consider to be developmentally appropriate, and what we feel they have the capacity to learn, which then helps them develop exactly in the ways we expect. Isn’t this a bit of cyclical reasoning?

Shouldn’t we challenge our students to find out what the limits are to what they can do, rather than assigning them limits before we have sufficient evidence of what those limits might be? I’m not saying that the children we teach do not have limits, only that we do not know what those limits are without attempting to stretch them.

I don’t see what value, if any, my time spent managing my LinkedIn profile has given me. I have nearly 1000 connections on the site now, but I have only used it to contact people a few times, and people have only used it a few times to contact me. They could have contacted me in a bunch of other ways, through sites that I use much more frequently.

I feel like I’m throwing away my time working on my LinkedIn profile and getting nothing back for that time spent. I remember thinking that building connections there would be really useful if I decided to find a new job, but I have a new job, and getting that job had nothing at all to do with my LinkedIn profile.

This type of analysis, by the way, is related to mathematical game theory. I have a choice I can make: to continue using LinkedIn, or to discontinue my use. In my brief discussion above, I have resources I can allocate (time) and a hope for a benefit that I will get out of the activity (employment). Everyone else who uses LinkedIn has the same choice. If people stopped sending me connection requests or adding new skills to my profile, then deciding to end my use of LinkedIn would be easier, but I feel social pressure to continue my use of LinkedIn, and consequently, I apply social pressure to others to continue their use.

Thoughts?

I’m working on a set of possible questions one can ask their students (and teach their students to ask themselves) while they are problem solving in math. Note that these questions are related to the work of George Pólya from his book How to Solve It.

What would you add?

Questions to ask during problem solving

What are your assumptions?

• What happens if you change those assumptions?
• What assumptions have other people made?

Is there another way to solve it?

• Within your current assumptions?
• With different assumptions?

How is this problem related to other problems you have done?

• Can you solve a related problem?
• Can you simplify the problem, and then solve it?
• Can you find connections between this problem and other problems?

Can you explain the solution to someone else?

• Can they explain your solution to you?
• Can they explain your solution to someone else?
• Can you explain your solution without words?
• Can you explain your solution using only words (no symbols or drawings)?

What tools could you use to help you solve this problem?

• Are there any technological tools that might make the problem easier to visualize or manipulate?
• Are there any mathematical techniques that might be connected to this problem?

How can you justify your solution?

• How can you prove your answer is unique (if it is unique)?
• If your answer is not unique, how many different answers are there?
• How do you know your answer is reasonable?

Can you reflect on your problem solving process?

• How could you change this problem?
• Can you think of related problems?
• What is interesting about this problem?
• How could you generalize this problem?

(Image base created by Curran Kelleher)

In the course I am taking online, How to Learn Mathematics, one of the assignments is to create a poster that highlights the importance of mistake making during the process of learning mathematics. What I believe is that mathematics is both in the external world (and hence discovered) and also a construct of our minds (and hence created). When students learn mathematics, they explore the world external to themselves to develop an internal model of mathematics.

An aside: Can you help me figure out how to make my poster look better? I’d like to add some colour or something, but I’m not sure exactly where…

On Wednesday, the A2I team at New Visions participated in an excellent workshop from Harold Asturias on English language learners in a mathematics classroom. Harold made some points about the difficulties English language learners face in a mathematics class that really drove home this point to me; mathematics teachers are teachers of language too.

To illustrate my point, try this mathematics problem:

This is what solving a word problem is like for children who either can’t read at all, or who are illiterate in the classroom language. What about this problem instead?

Here at least the character set is familiar, and you can recognize some words of the problem too. Some of you with some knowledge of the history of mathematics may even recognize a name from the problem above and be able to find out what the text of this problem is. Note that for almost anyone trying this problem, the text of the problem is still pretty close to incomprehensible.

What about this version?

(Source: Wolfram Mathworld)

This is at least readable, but it is still a challenging piece of prose to read. Hopefully you can imagine some of the stages of understanding between the second and third versions of this problem, and how challenging this kind of problem is. (Here is another example, this time from a textbook written in Thai.)

You can have issues with language result from just regular classroom discussion. Speak the following sentence aloud:

"Please take half of the peas that I have, and give two of them to him, and leave four of them for her."

For someone who is just learning English, even though the level of difficulty of the individual words in the sentence above is low, they may still have problems decoding the meaning of just this one sentence. There are many, many other examples of "difficult to decode" sentences. Note that this problem is probably not just exclusive to English Language Learners! Many native speakers of English struggle with comprehension in both the cases I’ve described above.

One solution, that I have heard proposed by some, to this problem is to remove word problems from mathematics classes. I think this solution is short-sighted. As teachers of mathematics, we are not just teaching the content of our discipline, we are also teaching the language of our discipline, and word problems are a way to build a bridge between the everyday language students have, and the academic language that mathematicians use. By removing this bridge, we would shut our students out of mathematics. If we want students to learn the language of mathematics, we must build it up from the language students know.

Another solution is to build a different kind of word problem, one not so dependent on language. While I think there are benefits to this approach, we still want students to develop mathematical language. My suggestion is to use the Three Acts type problems, and write the text of the word problem together, after students have solved the problem. This way they get a chance to develop their mathematical reasoning skills, and then develop their vocabulary for that reasoning while working on the problem. Dan Meyer demonstrates a Three Acts problem with a room of mathematics teachers. Notice how he builds up the language of the problem from the language participants use to describe it. From here, it is not too far a leap to include more of the language mathematicians would use to solve this problem.

There are ways to scaffold the use of language in your classroom without reducing mathematical difficulty. You can use daily math talks to build your students’ ability to discuss mathematics. You can be deliberate in your use of language and carefully introduce vocabulary as needed to discuss mathematical ideas, and build a word wall of the vocabulary you have introduced. You can give your students opportunities to problem solve together so that they can support each other’s use of language. 

Here are some other resources you can use to help address this issue:

• Stanford’s Understanding Language page
• Virginia’s Department of Education 
• Discussion frameworks: Problem stem, Writing to explain, Think/pair/share, Turn & talk to produce something, Sentence frames or starters, Solo => partners => table => whole group

What other suggestions do you have to help all of your students learn the language associated with a math classroom?

Importance of questions

Questions, both those asked by teachers and those asked by students, are an essential part of education. Questions can act as goals for learning, and motivate student curiousity about what they are learning. All teachers can attest to the fact that students who feel motivated to learn will learn much more effectively and much more deeply than students who lack motivation.

Questions also frame ideas in ways that may offer a different perspective for teachers and students. Imagine yourself brushing your teeth. Now ask yourself the question: How could you explain brushing your teeth to someone who has never done it or seen it done before? Or you could ask yourself: What questions do you think an alien would have if they were to watch us brushing our teeth? Notice how these questions can take a mundane activity, and require us to examine the activity from a different perspective in order to try and come up with answers to the question.

If we consider questions to be an important part of education, then it is worthwhile examining what effective questions look like, and how they can be used inside a classroom environment.

Teacher questions

"[In multiple studies] researchers have found that in a single day teachers ask hundreds of questions of their students. In one study of third-grade reading groups, on average teachers asked a question every 43 seconds." On the other hand, students ask few questions. (Gambrell, Journal of Educational Research, volume 75, pp. 144-148)

Teachers ask many, many more questions than do students during the course of the school day, but the questions teachers ask are usually asking students to recall facts, rather than make connections or inferences. While it is not possible that every question a teacher asks will provoke higher order thinking, more questions that ask students to make connections and inferences will provoke student thinking. Asking questions that provoke thinking also allows teachers to model asking these types of questions so that students will get better at asking questions.

Dylan Wiliam (in his book Embedded Formative Assessment) suggests that questions teachers ask should either provoke student thinking, or help the teacher learn what the kids know, what they are able to do, and how they think. Teachers should find ways to give all of their students time and space to engage with the questions they ask, so that they are either provoking thinking in all of their students, or they are learning about what all of their students can do, rather than just a few of their students. Teachers should also be thoughtful in choosing what questions they ask, so that the questions are more likely to achieve their purpose.

Classrooms should also have big, broad questions that are asked, so that students can connect ideas from one class period to the next, and look for structure in what they are learning. These big, broad questions can be called Essential Questions, and Jay McTighe’s and Grant Wiggins’ book of the same name offers many resources for implementing and using them.

Teachers should also ask themselves questions, such as: What questions can I ask to provoke student thinking? Which instructional activity is best for learning this idea? What are my students learning? Why am I teaching this? What is the goal of education?

Student questions

According to Peter Liljedahl, students ask basically three types of questions:

• Stop thinking
• Proximity
• Start thinking

A "stop thinking" question is a question a student uses to end her or his thinking. Examples of "stop thinking" questions are "Is this the right answer?" or "How do I do this next step?"

A "proximity" question is a question students asks because you are near them. If you walk around a classroom and check in with students, you should notice that they ask you questions when you visit their space of the room. Some of these questions will be questions they have been waiting to ask you, but others of these questions will be ones that would never have been asked if you had not been in the student’s proximity.

A "start thinking" question is any question a student asks which is either a reflection of thinking they have done, or a question that they ask to expand upon their thinking.

Peter’s advice is to stop answering the first two types of questions, at least in ways which do not provoke student thinking. As for the third type of question, however you answer it should at most help the student explore the answer of the question themselves. One way that I have taken to addressing these types of questions is to use some of the questions George Pólya outlines as part of the problem solving process so that instead of just leaving students to find answers for themself, I at least give them some tools they can use to answer their own questions.

One way of highlighting "start thinking" questions is to keep a record of them, possibly publicly in your classroom. Imagine you had an area of one of the walls of your classroom where the deeper questions students asked during the year were displayed, some of which could be categorized as still open, and others which could be categorized as explored.

It is my experience that although by middle and high school students are not very good at asking questions, they improve in their ability to ask questions by having question posing modelled for them, and by being given space and time in which to ask their questions.

Resources

• Embedded Formative Assessment (Dylan Wiliam)
• Essential Questions (Jay McTighe and Grant Wiggins)
• How to solve it (George Polya)
• Classroom Questioning for Trainee Teachers (University of South Hampton)

• Dot patterns

  

  A dot pattern is a visual array of dots, flashed up long enough so that students can see it, but not long enough that they can count it directly. One way to use these patterns is to ask students to come up with how many dots they think they saw, and why they think they saw that many dots. One advantage of dot patterns is that the threshold for the ability for your students to participate is low, and so everyone can more easily see how many different representations are possible.

   

• Two dimensional patterns

  

  With two dimensional pictures, one objective for students could be to try and reconstruct the picture that is flashed in front of them. Students should try and describe the object using vocabulary they know, and that is as precise as possible to describe the shapes they saw.

   

• Three dimensional patterns

  

  You can also use three dimensional diagrams in much the same way that you use two dimensional diagrams for math talks. This particular diagram is created in Minecraft, but it is reasonably easy to create a similar diagram using isometric paper.

   

• Arithmetic problems

  

  Many arithmetic problems would be appropriate for math talks. As a teacher, you might want to select the arithmetic problems based on previous math talks your students have done, so that you have the potential for patterns toi emerge with a series of math talks around arithmetic. Conceptual understanding of a variety of different strategies for solving arithmetic problems will likely help your students develop much needed numeracy skills. Note that it is best to choose problems which can be done mentally by most of your students, so that they have a need to use non-standard algorithms to calculate the arithmetic.

   

• What’s my rule?

  

  Function machines are also possible, and depending on the function and your goals for the math talk, you can use a variety of different representations in order to display the relationship between the input and output of the function, including, for example, a table of values or a graph. As with all of the math talks, the objective is for students to talk about what evidence convinced them that they knew the function rule, and how they could check that their rule matches the given information.

Some other resources for math talks:

• Visual Patterns
• Estimation 180
• Math mistakes

(Image source: Wired Magazine)

If we teach as if everything we know is written in stone, our students will end up with the impression that knowledge is something static and unchanging, which is demonstrably false. A professor I had told me once, "Something is true when everyone stops complaining that it isn’t true." A corollary of this is that there are a lot of things which we assume to be true but which probably have some controversy associated with them.

Teach the controversy. Have rich conversations about why the controversy exists, and what each sides’ perspectives on the controversy is and what evidence supports their perspective, even for something as contentious as evolution. If we ignore the controversy, even when we don’t necessarily agree with some of the perspectives of it, we allow some of our students’ unspoken counter-arguments to remain hidden, and they will have little opportunity to make an informed and education decision for themselves.

Neil Postman gave a terrific talk on "Bullshit and the Art of Crap Detection" for which I recommend reading the transcript. He doesn’t have any solutions for the issue he brings up, but I believe that teaching controversy and asking students to present evidence that supports each side of the controversy is one of the best ways of giving thinking tools to students so that they can be armed with their very own crap detectors.

If I can’t provide evidence that something is true, it may be crap (corollary: the last sentence of the previous paragraph might be crap). What do you think?.