Is there a Learning Museum?

Once again, Twitter has inspired me. This morning, I saw the following tweet from the National Math + Science Initiative:

Here's the money quote from the article: 

It also reinforces the emerging concept of “free choice” learning, which holds that people get most of their knowledge about science from someplace other than school or formal education.

What caught my attention was the phrase "free choice" learning. The idea that people take responsibility for their own learning is also found in Cambourne's work. This got me thinking about ways to change schools so that they more closely resemble museums where people learn, which led to this tweet:

The perspective that schools currently serve as "fact factories" comes from Sir Ken Robinson and Seth Godin.

I began to wonder what would be in a Learning Museum (not to be confused with Joe Bower's Museum of Education) and decided that it would have rooms dedicated to these topics:

• What does learning look like?
• How is a learner different than a student?
• When and where has learning occurred?
• Why is learning important?
• What if I want to be a learner?

A google search found a Museum of Learning but no Learning Museum. Therefore, I decided to open a virtual Learning Museum. As curator, I will gather learning artifacts, but I cannot do it without your help. If you have any exhibits related to learning that you might like to share, please include links to them in the comments. I will try to gather them into the different rooms so people can stop by and improve their ability to learn. Again, from the article:

“The holy grail of science museums is not to provide someone all the knowledge they need, but to inspire them, to become a launching point,” said John Falk, an OSU professor of science education and national leader in the free-choice learning movement. 

The Learning Museum is now open. I hope it inspires you. Please visit often.

What are the chances of a “Pig Out”? Part II

Last post, I shared with you the first half of an article I wrote with Dr. Mary Richardson (here is another article I wrote with Mary addressing fair games). Learners have just predicted the chances of rolling a "Pig Out" in the popular Pass the Pigs game. They are now ready to explore the experimental probability associated with this question.

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The Activity (continued)

After learners share their subjective estimates, they are ready to test them by developing an experiment, as described on the Activity Worksheet.  In the activity, learners create possible plans for determining an estimate of the chances of a “Pig Out”.  Once plans have been suggested, the class discusses the plans, agrees on a plan to utilize, and carries out the plan.  One possibility would be to have each pair of learner toss the pig dice several times in order to see how often a “Pig Out” occurs. For example, within each pair, the following tasks are assigned:  one member tosses the pigs; and the other member records whether or not the pigs landed in the “Pig Out” position.  After one learner completes ten trials, the students switch tasks and ten more trials are conducted.  This provides partners with a total of twenty trials that can be used in determining an experimental probability.

Once the learners have conducted the experiment, we ask them to refine their original guesses.  When asked how they could be more certain of the probabilities, most respond that a longer experiment (more tosses) would result in more certainty.  However, they recognize that it is unrealistic for each of us to toss the pigs a very large number of times.  Instead, we decide to use the results from the entire class to get a better estimate for the probability.  We draw a number line on the whiteboard.  Each learner writes his/her own result from their tosses on a sticky note and posts the note appropriately on the line in order to form a dot plot.  After we have discussed the data for the whole class, everyone makes a new estimate for the probability.

Learners are asked to share their new probability estimates.  Most learners select the class median or class mode as their new estimate.  We estimate that a “Pig Out” is rolled one time in every five rolls.  Of course, differences in desktops, and the very nature of the experiment may produce varying results.

Conclusion

The important concept in this activity is that one cannot simply count outcomes for every object and assign a theoretical probability to any given outcome.  By exploring objects that cannot be assigned theoretical probabilities, learners seem to have a greater understanding of when a theoretical probability can be assigned.  Furthermore, learners recognize the need to generate data in order to obtain experimental probabilities

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The exploration did not end there for us. We found several websites dedicated to Pass the Pigs. They contained online versions of the game and data collected in secondary classrooms. We used the data to update our experimental probabilities and look at new questions.

Sadly, the websites we found are no longer, but Wikipedia has data on rolling a single pig 11,954 times. Maybe you could use these to collect your own data to answer your own questions. For example, in order to win the game, you want to know when to pass the pigs. Therefore, the next question we asked was, "How long until we "Pig Out"? We came up with two ways to answer this question but I bet there are more.

What are the chances of a “Pig Out”? Part I

Several years ago, I wrote the following article with my colleague, Dr. Mary Richardson, for MCTM's Mathematics in Michigan (Volume 43, Number 1). Given that the issue is no longer available, I thought it would be appropriate to share it here. Because of the article's length it has been edited and will be split between two posts.

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Introduction

Middle grades learners should be provided ample opportunities to explore probability and statistics topics.  In particular, learners should be encouraged to collect, organize, and describe data, summarize the data, make predictions and conclusions based on the data, and test these predictions and conclusions (NCTM 2000).  [Update: this connects to CCSS 7.SP.C.6.] In the following activity, learners use the Pass the Pigs dice game to design a simulation experiment in order to estimate a probability. 

The Game

Pass the Pigs involves two tiny rubber pigs that players throw like dice (rules). Each pig can land in six different ways resulting in various point values.  A turn consists of a player taking as many rolls as he or she dares until: (1) deciding to stop and recording the total score for that turn, (2) a “Pig Out” is rolled and the player records a score of zero for that turn, or (3) an “Oinker” is thrown and all points accumulated in the game thus far by the player are lost.

The Activity

In this activity, learners explore the experimental probability of obtaining a “Pig Out” in the Pass the Pigs dice game.  The activity works best if learners have experience calculating measures of center, including mean, median, and mode.  Learners work in groups of two.  Each pair has one Pass the Pigs dice game and a flat table or desktop on which to work.  Each learner has a sticky note and a copy of the Activity Worksheet.

To begin the lesson, each pair of learners examines their pig dice.  We discuss the possible outcomes if the pig dice are simultaneously tossed.  Each learner is then asked to estimate the probability that a “Pig Out” will occur when playing Pass the Pigs.  Learners should write down and share their estimates and the reasoning behind them.  This leads into a discussion about different types of probability.  In addition to the two types of probability traditionally targeted in middle school (theoretical and experimental), this activity provides the opportunity to introduce learners to subjective probability.  Learners recognize that this is a situation for which there is no easy way to determine a theoretical probability and no data on hand for them to use to calculate an experimental probability.  Therefore, they must assign probabilities based upon what they think will happen (subjective probability).

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To be continued... In the meantime, please estimate what you think is the probability that a "Pig Out" will occur and share it in the comments. Thank you.

"What if they come at you with a pointed stick?"

Brain Camborne has spent his career researching learning in natural settings in an effort to "find an educationally relevant theory of learning." This research led to Camborne identifying eight conditions present to support learning. The figure below comes from his book, The Whole Story, and shows how he envisions the conditions working together in a language arts classroom.

Engagement is central to these conditions; "the necessity for engagement underlies the whole plan, guiding and influencing all of the teacher's actions and interactions with the learners." Camborne found that three factors impacted learners' level of engagement (see figure to the right). 

In watching this Monty Python sketch, I was reminded of these factors and how they are too often ignored during instruction. [Sometimes it takes the extreme to become aware of the obvious.]

Why did only four cadets show up for the self-defense class? What might they expect from past lessons? Did they not show up because they felt they did not have the resources (guns, 16-ton weights, or tigers) needed to be successful? Were they uninterested in learning how to defend against assailants wielding fruit (instead of pointed sticks)? Or did they feel unsafe (and rightly so)?

More importantly, how do we unsure that our lessons address potential, purpose, and protection?

Where do you stand on teaching mathematics?

Today my Teaching and Learning Middle Grades Mathematics course begins their unit on teaching mathematics. This follows units on doing mathematics and learning mathematics. We begin this unit by taking a simile survey that asks respondents to complete the following:

Ideally, a mathematics teacher is like a(n):

a. Coach

b. Doctor

c. Entertainer

d. Gardener

e. News Broadcaster

f. Orchestra Conductor

• Choose the simile that you believe best describes a mathematics teacher and explain your choice.
• Choose the simile that does the worst job of describing a mathematics teacher and explain your choice.
• Is there another simile that does a better job than these of describing a mathematics teacher? If there is, then what is it and what makes it better than these?

This survey follows similar simile surveys done for doing and learning mathematics. The doing mathematics unit had integers as one of the topics, so I asked my learners to rate the "doing" similes from -5 (the worst) to 5 (the best). We created a life-sized number line as a way to share and compare our choices. 

Rational number was the mathematical focus in the learning unit. For this survey, learners assigned their worst "learning" simile a 1 and then decided how many times better each other simile was than the worst one. The life-sized number line went from 0 to 1 this time. Learners computed the following ratio for each simile: simile rating to best simile rating. (i.e. The best simile was a one.) Again, we shared and compared our choices by standing on the number line.

The teaching mathematics unit concentrates on geometry. Today, I will ask them to stand in one of four corners representing strongly agree, agree, disagree, or strongly disagree for each simile. It is certainly a simpler process than the others but it often results in some interesting discussions.

Ideally, a mathematics teacher is like an entertainer. Where do you stand?

Is mathematics about product or process?

Several years ago, a group of mathematics educators at GVSU decided to address an issue prevalent in our course, Mathematics for Elementary Teachers I. Many preservice elementary teachers in this course viewed mathematics from a product perspective. They considered only the answer as important. In response, we attempted to be more explicit regarding the mathematical processes by revising the course using the NCTM Process Standards as a framework. Below is one of the units that we developed around the theme of polygons.

In an effort to model promising teaching practices while maintaining the focus on the Process Standards, we introduced learning centers addressing Communication, Connections, Problem Solving, Representations, and Reasoning & Proof. The following centers came from a unit with Pentominoes as the theme. The preservice teachers rotated between these centers over the course of a two-hour class at the end of the unit.

Most of the ideas for the centers came from articles in Teaching Children Mathematics - NCTM's journal supporting improvement of pre-K-6 mathematics education. The content foci for the Pentominoes Unit were Measurement and Data Analysis and Probability. Some of the centers were closely related to the activities done during the unit, such as the Connection Center. Others, like the Reasoning & Proof Center, were new contexts that applied ideas found in previous lessons.

The preservice teachers looked forward to these centers at the end of each unit and were grateful for the instructional modeling of mathematics learning centers. Many had never experienced anything but traditional mathematics instruction in their K-12 experience. Still, I am not sure that one or two courses focusing on the Process Standards is enough to provide a more balanced perspective on what it means to do mathematics. After all, the process is as important as the product.

Are you ready for "something completely different?"

My wife gave me The Complete Monty Python's Flying Circus for my 48th birthday. I just got around to watching the first episode and saw the following sketch. It demonstrates perfectly what happens when we fail to use formative assessment to inform planning and instruction.

It is important to understand that formative assessment does not need to be in the form of a test. We can gather data simply by observing. Imagine how this Italian class might have looked if the teacher had really seen the students during the lesson and adjusted his instruction accordingly. In the Design Thinking methodology, this is a part of the first phase - Empathy.

This reminds me of an article by Dr. Rheta Rubenstein titled, "The Learning of Mathematics: Perspectives of Different Researchers." (It comes from our text, Teaching and Learning Middle Grades Mathematics.) Under the section on Vygotsky, she includes the following diagram attributed to Dr. Lauren Resnick.

Vygotsky in Venn Diagram Form

As Terry Jones's character discovers, when teachers use scripted lessons that do not take learners' current level of understanding into account very little learning actually occurs.