The Time for Inquiry - Where should it be in the learning process of a math classroom

Scrolling through my google reader I saw a quick, short post from David Wees asking how do we encourage more questions and the following image:



I am a teacher that believes in inquiry learning and also the power of a student driven project or product, but usually these occur as a summary or summative of learning. However, after my recent experiment of completing the Barbie Bungee activity at the start of my unit and in a non-worksheet format, I have seen and appreciate the power of exploring ideas and concepts through the unknown. This activity had my students exploring the ideas of linear equations without knowing what a linear equation was.

They asked brilliant questions including:

• Does the height of the Barbie matter?


• Should I find out how much she weighs?


• Are all the elastics pre-stretched? How can I make sure that they are all pre-stretched the same amount?


• Can I cut my elastics up into smaller parts?

Now, only 9 classes after that activity where students didn't know what they were looking for, they are now appear to be automatically programed into trying to find a constant rate of change and a starting point (y=mx+b) in each of the problems we look at. As they have gained a greater understanding of their topic they are not asking the insightful questions from before but instead are wondering why I didn't leave the space on their table of values to include the zero starting value, or if they need to capitalize their variables in an equation. I don't know if I should be celebrating or concerned.

I also just finished being apart of a webinar by Roger Schank, author of Teaching Minds who shared his view of how he feels that Algebra's sole purpose is as a method to easily test students, and make benchmarks for university entrance exams. He feels that the multiple steps and multiple areas to have an error do not benefit a student, but only the adults and institutions who require an easy way to grade and place students by how they solve for a variable or express a pattern in the appropriate standard formula.

I know that my students participating in Barbie Bungee activity didn't need to know the point-slope form or y-intercept form in order create a table, graph the data and find a pattern but they did it anyways. I also know that they used their algebra skills to create a formula or relationship to determine how many elastics they needed to use, all with levels of success. All of this great learning happened, but was followed up with 9 in class days to ensure that they understood what slope meant, how to calculate it in the 3 ways it could be presented in a standard question and how it connects to a graph of a line.

At the end of the day, I am left questioning... do my students know more now that they have been provided the vocabulary and structure to work with linear relations, or have I put a stop on their creative application of the concepts. As a class we will be doing more STEM-like activities and I will be looking to see what my students go to first as their method to solve these new challenges. Will they begin by breaking down the big problem into smaller ones? Will they play with the materials and ask questions to determine if they can find any pattern? Or will they think in what way a linear pattern could be created and what the start and change values should be. Will their questioning level increase or have they reached a level of understanding where they have no desire to explore and think through the problems thoroughly.

 

Back to the textbook - teaching reading literacy in math class

I am a "no-textbook" teacher. I feel that as a teacher it is my job to understand my students and then develop the classroom materials, activities and practice questions that are reflective of where they are in their progress and understanding. A textbook does not allow this flexibility and uses terms that may not make sense or encourage a student to take ownership over their understanding.

However, after spending the past few years not using a textbook I was wondering if my students were missing any specific skills that their textbook reading counterparts were gaining. But then it hit me, they were not able to develop the skill of reading and understanding a mathematical text. Everything I had created was based on what they had previously learned and was all hands on or discussion based. At no point were they being asked to read and explore a text that was developed for the purpose of being another resources for them to use. So I decided to have a literacy day in my math class.

After looking through the Math and Literacy materials created by the OAME I created a template that I felt would guide my students through their first textbook based reading. My goal was to have the students learn how to read a textbook or resource by first scanning headings, identifying words they already knew and making some judgments about what material they would be reading about. I asked them to use a chunking method to break up their reading and guide their notes. They use the worksheet that I have linked to here.

As a class we have already looked at slope, but by using the counting method using the rise/run - right triangle start. I photocopied a chapter from a textbook that explore the concept of slope through examples, explanations and a summary of key concepts. Their goal was to take notes using the guided worksheet provided and then assess their understanding by completing 12 questions that had been placed around the room as different stations. They could only use their notes to answer each one.

I was amazed with the positive feedback that I got from the class. They all, contrary to what I thought, went straight into the reading portion and followed the guided steps on their own. I did create a mini "book club" of students that I felt may need more support to go through the process with me, but they became independent after only a slight nudge in the right direction. Since they were able to move onto the 12 stations at their own pace, no one felt rushed or that it was a test. In the end of it all they asked that we could do this activity again at some point in the future, all identifying it as a skill that they feel is necessary for them to have but something they have not had the opportunity to practice.

As a math teacher I realize that I don't prepare my students in how to read a mathematical text. We go over how to read and decipher word problems, but never the additional resources that are available to them as students. How can I expect my students to become the independent learners if they are unable to use the online and print resources available with any success. As my teaching practice is mainly based on inquiry and hands on activities and group discussion, I have cut the literacy aspect from my curriculum without noticing. How can I expect my students to learn how to read a math text in their English class? As a Grade 8 they won't make those connections on their own, but be provided with adequate guidance to do so.

Moving forward, I want to do this activity again after we have learned another key concept. I plan to use another textbook so that my students can see a variety of printed sources and next time focus on how to properly read and learn from the examples in the text. In our first attempt I saw that they were unable to properly read and learn from the  examples.

I found it interesting to see that with a change in teaching philosophy (no-textbook and inquiry), my students had challenges with skills I took for granted in my textbook based education. What else am I assuming they must be able to do that without practice or direct instruction they are finding more difficult. Next challenge is checking their answers with those posted in the classroom.

No Structure Barbie Bungee - Reflection

Over the past 3 days, my Grade 8 class have been working through the popular Barbie Bungee activity. However, as I have shared in my previous post, they did it without any worksheets or guidelines. It was very coincidental that John Golden of mathhombre wrote a post challenging the worksheet approach of the activity and if an unstructured method would be more beneficial. So after 3 days and the big drops completed, I am writing the results of this first time attempt to the activity and doing it in this manner.

To start off, this activity was used as an introduction to linear relations. The goal of spending 3 days on this activity was to allow the students to generate a solution that without realizing it took into account the rate of change, independent and dependent variables, data collection and organization of data. They will compare their methods and solutions by posting them in the room. As we move through the unit the students and I will point out that they already know many of the new concepts. For example, pointing out that they already know how to calculate the rate of change, but only called it the number of centimeters each rubber band would stretch. I was worried that this final goal would not be possible due to students creating guess and check methods and estimating rather than using the necessary data. I also worried that they had no place to start, or could determine that they needed to do test trials or even consider the height of the Barbie. However, I let go and they struggled, but also came out with valuable results.

The first day was just a day of struggling (or teacher terms of practicing rigor). Given a Barbie, ruler, meter stick and 10 rubber bands my students struggled in where to start, understanding what the goal was and how they could actually get there. This was also the day of deciding what factors were important to measure. Some groups were unmotivated to do anything and decided that finding just how far 10 rubber bands would take the Barbie would be enough. I had prepared a list of probing questions and a worksheet just in case. I only gave the worksheet to 3 out of the 24 groups, but at the end of the day it was in the recycling bin as well.  The level of struggle differed between classes (I teach 4 grade 8 classes of 22 students). There were some that didn't know where to start to others that were debating the need to measure the weight of the barbie and wanting to determine how gravity was affecting the stretch. To allow for these great discussions I now know that groups needed to be small, no larger than 3 people and with something to get them started.

To get everyone started there were guidelines to create a shared document using Google Docs and also to pick up the necessary materials. Each group identified a group leader for these tasks. By doing this it made it easier for every group to get started and everyone to have a job.

To help guide their discussions and questions I asked them to use a Question Ladder which helped them outline a potential plan of attack. By the end of the day, all but three of the 24 groups had an idea of what was going on and a plan. At the end of the day I did go home wondering if this was going to be a long few days. If I had given them the outlined instructions I would have known what step they were on and no one would have been off track, but it wouldn't have allowed for the insight found on day 2.

Day 2 was the test drop day. By giving the groups a test height to work from also provided the struggling groups a goal and motivation. They were told the height at the start of the class and told that they could go to the drop site as many times as they wanted to test their method. To leave the room and test their method they needed to complete a proposal that had them describe their reasoning behind the number of rubber bands and also the evidence to back this up. This day was the magic day. Students left the room excited to test their ideas and methods, to return shocked that either their Barbie crashed to the floor or didn't come close. They headed back to the drawing board to identify what they felt was the issue and come up with a new plan or make changes to their old one. Groups came back into the room sharing how close or far they had got. By doing this the groups not only engaged in friendly competition, but also they were sharing with the other groups that it was possible to get close and at least one method worked. Since no one new each others methods or plans, everyone took ownership and pride over their Barbie and calculations. Groups went back two or three times to the test drop to alter their proposals. Each time they returned to the test drop area they needed to create a new proposal and explanation using data. This made the "just add a few" or "take a few off" not an option since they needed data as evidence to back it up. By the end of the day students came back being within 2 centimeters of the floor and ready to move on. There were groups that struggled, but did not stop. They also were motivated to make changes and improve.

Day 3 looked much like day 2. The class came in, excited and eager to know their final height. The proposals from the previous day were handed out and groups retrieved their barbies. I wrote the final height on the board and handed out their final blank proposal sheets. They hurriedly got to work, writing out their methods and completing the calculations and collecting their rubber bands. I set a clock on the board indicating when the drop time would occur. The energy in the room was great as each group felt that they could be the closest to the floor. At the end of it all, the groups were working together to divide up the responsibilities and go to the drop zone. It was a great period.

Here is an example of all of the materials one group created over the three days including their final reflection sheet. 

At the end of it all, I had Barbie's be as far away as 1.5 meters to multiple coming under the 5 centimeter goal. The final group that won was about 1-2 centimeters from the floor.

The next day (Day 4) I had the girls compare their methods, make a table of values and graph their work. The goal with this was to identify that the methods each group took were very similar, but the cause for differences resulted in how precise they collected their original data. The scatter plots of the data and showed that there was a relationship between the number of rubber bands and the distance dropped. It was at this point that I introduced the linear relations unit, the goal of collecting data and making an equation to represent that data. I also explained to them that this could have been a final test or assessment at the end of the unit and that they all have the skills necessary to do well in the unit. The goal moving forward is to ensure that they understand how the math relates to the data and the vocabulary that we use to describe it.

Looking forward, I would do this activity again next year as the introduction. Since the last day, the students have had no challenges grasping the concepts and demonstrating a good understanding of slope and rate of change. We will be preforming many more of these non-structured experiments throughout the unit and it will be interesting to see what they learned and will improve on each time.

For my IEP students I feel that they would have done better in this type of activity than in a structured one, contrary to my Spec Ed. training. I think that by giving more structure and steps, students can see it as being more places that they could mess up on and feel that they can not continue without the support of a teacher. In this method it looked like just one problem that I only had to give them support at the start and the rest was just a continuation of this.

If you have other great activities for linear relations I would love to hear of them and try them out in a similar manner. I also want to improve on how I am using the results of the activity moving forward. We did the Day 4 summary, but as we have gone ahead, I am finding it challenging to see how I can connect the material back to the activity without the standard "Look, rate of change is just like the number of elastics". This might be the extent to it, but any suggestions would be great!

Freestyle Barbie Bungee - No Steps Required

After completing day 1 of the infamous Barbie Bungee activity I look on my twitter to see the the following tweet from @ddmeyer:

How many different ways can a popular lesson (Barbie Bungee) go wrong? Extremely provocative post from @mathhombre. bit.ly/ULssnB

— Dan Meyer (@ddmeyer) February 2, 2013

Even though it is only Day 1 of the experience of our 3 day activity, I want to respond to @mathhombre and share what is happening in my Grade 8 class.

Goal:
In 3 math classes, you and your bungee company will take your first customer to the drop point, letting them fall over the side and experiencing the most exhilarating moment of their plastic doll life. Your job is to determine the right length of the bungee cord that will create the best fall. Good Luck!

With only the above goal, my class has started their 3 day (no worksheet) activity to get their Barbie closest to the floor. The students were given only two things to guide them. One is the outline of what is happening and the other is a graphic organizer to help them break down their large question into smaller, simpler ones that they are able to answer. Divided into groups of 3 or 4 with a barbie, ruler, meter stick and 10 rubber bands they have a day to create a plan that will help them determine how many rubber bands they will need to create the best fall for their barbie. As explained in the outline, they have no idea what that height is until the drop day and they aren't allowed to drop their barbie if they don't have a logical and well thought out explanation that includes data to support their ideas. To help those that may get stuck I created scaffolding questions that could be given depending where they were being challenged and needing support. I also created a worksheet just in case groups became too lost and frustrated that the overall goal was not being met.

The three days are broken down as follows:

Day 1:

• Collect your materials (barbie, 10 elastics, ruler, metre stick)


• Create a plan in how your group will determine the final number of rubber bands for the drop day.


• Ensure that you are thinking of how to collect DATA as evidence for your proposal.

Day 2:

• Write a proposal for your company answering the question of How do you know the approximate number of rubber bands needed to drop the barbie from a height.


• The proposal will explain why you know the number of rubber bands based on the data you have collected in the previous class.


• When given the test height, use your proposal to determine the number of rubber bands. Make the cord and test it in the secret location. You are not allowed to guess and check for the test drop day.

Day 3:

Drop Day

My students will test their ideas and are required to write a proposal on the test drop day, alter their plans accordingly and drop them next Wednesday. After this we will be sharing how each group created their plan and explain their method with the class. My hope is that each group will write out their plans/formulas/calculations/tables on chart paper and post them around the room. As a class they will do a gallery walk to find similarities and differences between their own methods and others. We will also explore which methods worked better than others and ask why that was.

From this I plan on launching into the linear relations unit by associating vocabulary with their methods . For example:

• Some students were creating a table of values but didn't know that is what it was called or various ways to create one.


• Another group was determining how much further the Barbie would drop with each added rubber band but didn't know that this is the rate of change.


• Other groups were debating which central tendency (mean,median,mode) would be the most valuable one to use in the situation for their test trials (which they determined they needed).

The students bring with them and understanding of percentages, finding patterns and creating an equation to represent a pattern but all within structured standard questions. I am hoping (fingers crossed) that this unstructured activity will give them a starting point to explore and identify what is important in a problem, how to measure it and if there is anyway that they can use their results to make a prediction. We also will be exploring what makes a prediction valid and how to question if it will work in various situations.

I have no idea if any of the groups will save their Barbie's brains or allow them to have a great fall, but already with the lack of structure of history of working through similar problems they automatically directed themselves to the steps and processes a worksheet would have done. I just feel that a structured worksheet directions would not have made them ask the following questions which were heard today:

• Should we measure the weight of the Barbie? I know that in Science we have to do that thing with gravity and weight.


• If there are 6 centimeters from the Barbie's head to the floor with one meter, will it double when we double the height and number of rubber bands or will it stay the same?


• Are all the elastics the same size and thickness and pre-stretched the same amount like my jeans?

I wasn't planning on posting until the end of the process, but after @mathhombre's comments, I wanted to share that we are testing this very idea over the next few days.

I hope that you will check in and see how things progress next week. I will also be thinking about where to take the group next and posting their created proposals to show the diversity of what they were able to come up with on their own and without coaching.