Tag: The Reflective Educator (page 20 of 43)

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.

I’m gathering resources for a teacher for her unit on fractions. I thought I’d compile what I have found here:

• Refraction

  

  This game allows students to develop some intuition around fractions in a game environment. The Puzzle school is also developing their own version of this puzzle, which will have the added capability of teachers adding their own puzzles.
   

• Cuisenaire rods

  

  These rods come in different colours corresponding to specific lengths, and can be used to create activities which will help students understand fractions better. For example, try and find how many green rods it takes to create a red rod. The only problem is the rods will all end up being fractions out of 10.
   

• Burnt recipes

   

  Cathy gives an excellent explanation of how she uses recipes to create a problem where students have to use equivalent fractions to solve the problem.
   

• Horse races

  
  This game is intended to teach equivalent fractions, and give a visual representation of different sizes of common fractions. The objective of the game is to move all of your horses from one side to the other, taking turns with an opponent attempting to do the same.
   

• Use real life examples

  
  (Image credit: kobiz7)

  When I give my son a piece of an apple or cookie, or many other similar situations, I often use the language that includes fractions such as "half an apple" or "a quarter of an apple." This helps my son develop a model for what a fraction is based on things in his life with which he is familiar. I recommend this approach for teachers as well but make sure that these examples come across as natural sounding, rather than forced or contrived.
   

• Fractions in Music

  
  (Image credit: m kasahara)

  Music is full of fractions! If you teach your students to read sheet music, or at least listen to music carefully to here the difference in the lengths of the notes, then they will be learning another representation of fractions. I could imagine an entire unit on fractions based on music. As well, musical notes made with simple instruments often involve fractions, so students could build a music instrument (using jars of water for example) and tune their instrument, which will hopefully help them notice the simple ratios that result.
   

• Visual Fractions

  This website has a bunch of simulations that students can use to work through visual problems related to fractions. You could also, as a teacher, look at their simulations and turn them into activities students do in class.
   

• Cynthia Lanius has put together some resources here that teachers can use in order to develop lessons on fractions for their students. Some of the puzzles she has includes are fantastic.
   

What are some other resources or ideas you have to help students understand conceptually what fractions are? We have plenty of examples of practice students can do once they understand, so I’m hoping to find resources which help students understand.

Updated:

Since I created this slideshow back in 2013, I’ve learned a lot about formative assessment. In particular, I’ve learned that formative assessment (also known as responsive teaching) is not an action or a task but a process. For a good read about formative assessment, I recommend Dylan Wiliam’s book Embedded Formative Assessment.

I had a discussion with the director of my school about formative assessment, and we talked about some misconceptions around its use. I decided to create a presentation with some examples of formative assessment.

Here is the link to the slides if you want to modify them (just make a copy of the slides).