# Making Graphs Anecdotal Records Form

Knowing what my students know is important to me, but remembering who is struggling with what aspect of a given concept from day to day is impossible.   There are just too many students and too many variations.   I absolutely have to write down what they know if I want to act upon it the next day.    Unfortunately, there isn’t a lot of time in a given class period to take those notes.   As I result, I have started creating anecdotal record forms that are specific to a big idea that I am addressing in class.    On these anecdotal records, I have a single box for each kid.   In each box, I have specific items of which I want to ensure mastery.  I use one sheet for each period.   I put a different student’s initials in each box.  Then, I make copies of the sheets so that I have enough to use for more than one day.   Alternatively, I use a different color pencil each day.  I keep them on a clipboard for ease of use.   As I circulate around the room, I circle an item in a student’s box if it is an area that I need to address with him or her.   The next day, I simply look at the sheet from the previous day and I know exactly what I need to address with specific students.

I have just started a unit in which students will be representing and analyzing data in tables and graphs.   During this unit, I will be using the anecdotal records shown below.

I use the” Ind/Dep” category to indicate whether students are correctly selecting the correct axis for the variables (independent variable on x, dependent variable on y).   The “Intervals” category indicates whether the student is maintaining uniform intervals on the axis (a very common error in the early days of making graphs).   The “Continuous/Discrete” category indicates whether a student is correctly determining whether or not to connect the points on the graph.   The “Plot points” category indicates whether a student is correctly plotting points (x,y vs y,x).  The remaining categories are more minor errors, but errors that I want students to clean up.

graphing-anecdotal-recordspdf

# How far can you ride in a day? – A Lesson Introducing How to Make A Line Graph

What twelve year old wouldn’t want to spend a few days bicycling along the ocean front, spending their days amid sunshine and ocean breezes and their nights under star-filled skies?   Along the way, they get to swing through Cape May, a lovely ocean-side town filled with beautiful Victorian buildings, visit Chincoteague Island to see the annual auction of wild ponies who swim to the island from Assateague Island, and swim in the ocean.   This is the context for my students’ exploration of different ways to represent  and analyze data  (tables, graphs,  and eventually equations).   The series of (Connected Math) lessons center around a set of college students who are setting up a summer bicycle tour business to earn money for school.    In the series of lessons, they explore the question of how long each day’s ride should be, whether the planned route is feasible (they test out the route and collect data), where they should rent bicycles for the tour, finding the perfect price point to maximize their income, how long the drive back from the final destination will take at various different driving speeds, and the cost of taking the tour participants on a side outing to an amusement park.

Yesterday, I started the unit by introducing the problem  context.   I began by showing a short video clip of someone on a bicycle tour through Great Britain.    I chose to begin with a video clip in order to support my English Language Learners and students from lower socio-economic households, in order to bridge language and economic divides that might make the problem context difficult to grasp.   By seeing a bit of a bicycle tour, they would have better access to the problem context.

After students watched the video clip, I introduced the problem – five college students setting up a summer bicycle tour business.    The first question the college students were considering was how long each day’s ride should be.   I asked my students what they thought would be reasonable.   This led into a nice discussion of some of the factors that might influence the answer to that question – the relative incline (uphill, downhill, flat), the terrain (pavement vs sand or gravel), the weather (riding into a wind, no wind, wind at one’s back).   I then asked them if they thought it was reasonable to expect the riders to maintain a constant rate for the entire day.    Some students thought not, but some students thought the riders could pace themselves.    This laid the groundwork for the first part of the lesson.

In order to explore the idea of pacing, I had students conduct a jumping jack experiment.   At each table group, one member of the group would perform jumping jacks for 2 minutes.   Another member of the group would be the timekeeper, marking the time in 10 second intervals.   Another member of the group would be the counter, counting the jumping jacks.   The final member of the group would be the recorder.   (I assigned tasks to group members by their seat position within the group.   If a group had only three members, I had the jumper also do the counting.)   In order to speed the process, I provided the recorder with a pre-made table so that he or she would  spend less time copying out a table and could instead just fill in the table entries.

At the conclusion of the experiment, I had groups collect data from their recorder and complete their own data tables.  Then, I asked students to describe what happened to the rate of jumping jacks as time progressed.   There were a few groups throughout the day that maintained a fairly steady pace, but most groups experienced a steady decline as time passed.   Some groups had a jumper who stopped completely part way through the experiment and then resumed their jumping after a short break.   As the class discussed this, I asked them how they saw these changes in the tables that they had created.  This gave them the chance to  explore the idea of how a change in the jumping jacks compared to a constant change in time. (I had not yet introduced dependent/independent variables.)  At this point, my goal was to begin to tie this lesson back to work they had done with ratio tables in a previous unit.   I wanted them to see that this was not in fact a ratio table because the rate was not constant.

I told students that I wanted them to look at the trends of the jumping jack data in a graph.   In order to do that, they needed to learn how to correctly make a graph.   I began by introducing the concept of independent and dependent variables in a table and talking to them about the mathematical conventions.   Then, I introduced the process for translating data into a line graph using a Flow Map (this is a Thinking Map used for sequential processes).   I provided students with the  advance organizer to complete their notes on creating a graph.  I had a regular version of the organizer and a modified version of the organizer (that is more of cloze activity) to support students with learning disabilities.

howtomakeacoordgraphpdf

howtomakeacoordgraphmodifiedpdf

After students completed the Flow Map, I had them use the data from the jumping jack experiment to make a graph.   As they worked, I circulated among them, taking anecdotal records on their work.   As they finished, I selected graphs to share with the class.   I was very intentional in selecting graphs with errors.   I then asked the class to examine the graph to see if there were any errors.   This forced the students to think more deeply about the work they had been doing.   When someone found an error, I gave a piece of candy to both the student who allowed us to look at his or her work and to the student who found the error.   I explained that the person who let us see the mistake did as much to help us grow mathematically as the person who found the error did.   (This is a common practice in my class).   I made sure to share multiple graphs with errors in each period to ensure that no one student felt like he or she was the only one still learning how to do this.

After analyzing several graphs, I asked the students to explain how they saw the rate of jumping jacks changing in the graph.   Here, I was laying the ground work for upcoming lessons in which they will be analyzing data in graphs.

After summarizing the lesson, I had students complete an exit card in which they had to find the error in a graph.   After some thought, some of the students were able to see that the independent and dependent variables were on the wrong axis.   For those students who were having difficulty, I told them to go back to their Flow Map and work through each step to see if they could find the error.   Eventually, everyone successfully completed the exit card.

Following the lesson, I posted an anchor chart on the wall of the classroom that corresponds to the Flow Map that they used in their notes.

# Pondering the GCF & LCM

Some ideas are a little bit like raindrops that have splashed onto my puppy in a summer rainstorm.    They land lightly on her but are quickly shed as she shakes her body in a happy dance.   They touch her for only a moment and are gone.   Some ideas, though are more like a piece of sticky gum on the sidewalk on a hot summer day.    They seem to be permanently attached.    For some reason, that is how Mark’s post on exit cards has been for me these last few weeks.   I keep pondering what exactly I should want to know about my students.

This question has been dancing around in my head as I have been planning the first unit for my sixth graders.   This particular unit incorporates the ideas of the Greatest Common Factor and the Least Common Multiple (among other things).   For some reason, a lot of students seem to lump the two ideas into a single muddy mess.   I think part of the problem is the language – the words are so similar and yet so different (greatest/least, factor/multiple).    Even parsing out just the factor/multiple piece is tricky.   So many students confuse the two words.    They have trouble making them their own.  I also think part of the problem is that students have some measure of difficulty determining just what a Greatest Common Factor or a Least Common Multiple is.   I don’t mean the question of how you find them, I mean the question of exactly what are they and why should anyone care.

As we explore this larger question in the unit, we will do so in a problem context using Connected Math.

• We’ll start with a scenario in which two siblings are going to ride two different Ferris Wheels, one larger and one smaller, at a carnival.   Given the amount of time for a single revolution for each Ferris Wheel, students will consider how long it will take for the two siblings to both be at the bottom of the revolution simultaneously and how many rotations each wheel will have made.    I usually see a fairly wide range of strategies as students tackle the question.    Some students will start making a list of the times for each sibling to return to the bottom and then find the common time in each list.   Some students will use this strategy but will organize the information in a table (which makes it so much easier to make sense of the information).   Some students will really struggle to make sense of it and we’ll need to talk about what kind of strategies they might use to help make sense of the situation.   Eventually, they try something like drawing a picture or modeling the scenario using some different colored cubes.    A few will immediately see that they need the LCM.
• Next, we’ll move onto an exploration of the life of two different cicada species, one of which emerges every 13 years and one of which emerges every 17 years.   Students will explore the question of how often both species emerge simultaneously.    Generally, the thinking will be similar to that in the Ferris Wheel problem but it will come together a little more readily because of the discussion that ensued after the Ferris Wheel problem.
• Our next exploration will consider a scenario in which a girl is planning snacks for an upcoming hike.    She has a specified number of apples and a specified number of bags of trail mix and needs to determine how many snack bags she can make if each snack has the same amount of food and there is no food left over.    Initially, some students want to apply that same hammer – they want to find the LCM because that worked on the last two problems so it must be the way to go on this one as well.   Then, they get an answer that doesn’t make sense – how can you have more snacks than you do apples?   I like when this happens because it makes them step back and try to actually make sense of the problem – to realize that sense-making is what math is really all about.   Some students will reach for some blocks, using one color for apples and another for trail mix.   After they play around with it for a few minutes, they usually realize that they are dividing into groups and move forward without actually modeling the whole problem.   Some students will do the same kind of modeling just sketching or tallying and reach the same end result.   A few students will just seem to see it and will find the GCF.    I always try to bring out the different strategies as we debrief the solution and ask a lot of questions about why they chose that strategy and what that strategy showed them.   We follow that up with a scenario in which the canary ate some of the trail mix and the numbers suddenly get messier.   The idea remains the same, though.
• Finally, we consider a scenario in which someone has donated some cans of juice and some crackers to the school for snacks on a field trip.    Students need to find out how many kids can be served by the donation if each kid is going to get the same amount.   This problem seems to come together pretty quickly since it is the second GCF problem and follows a pretty solid debrief of the first problem.

Following this sequence, I expect some of the students will have made sense of the concepts of GCF and LCM, but some of them will not have.    Some of them will still struggle on the next novel scenario to figure out how to solve it.   I am wondering if my question (exit ticket) needs to be something like this.

What kind of problem requires you to find the LCM?   What kind of problem requires you to find the GCF?

I want to know if they have been able to generalize the concept.    Do they realize that LCM problems are really problems of alignment and that GCF problems are really problems of grouping (dividing)?   This is a shift for me.   In years past, my exit card would have been another problem and I would have evaluated their mastery based on their ability to solve that problem.   With this question, I am asking them to think on a different level.   I want them to consider what is the same about LCM problems, what is the same about GCF problems and how are the two different.

I’m not sure if this will be my exit card question, yet.    I have a little more time to ponder.   Am I asking the right question?

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# Proportional Relationship Anecdotal Records

You just can’t hurry some things.   They take time.    In middle school, grasping the big ideas of proportional relationships seems to be one of those things.    Students need to build an understanding of a constant rate of change and how it presents in tables, graphs, equations, and verbal representations.   They need to build an understanding of what it means to have no “start-up” value in each of these representations as well.   They need to explore these ideas in a little of different ways in order to make sense of it.

Because there are so many pieces to this idea, there are a lot of different places that understanding can break down.   In order to know where that breakdown is happening for each student, I like to take anecdotal records.   At the end of the day, I can go back and see who knows what and make decisions about what I need to do the next day to help each of them move forward.   The thing is, I don’t have time to write a paragraph about each kid as I walk around the class looking at their work and listening to their discussion.    I need something that just takes a second or two for each kid at any given time.    To that end, I decided to create an anecdotal record form specific to proportional relationships.

I started with what I want to know my students can do.

• Recognize a proportional relationship in a table
• Recognize a proportional relationship in a graph
• Recognize a proportional relationship in an equation
• Recognize a proportional relationship in a verbal representation (word problem)
• Be able to connect proportional relationships represented in a table and a graph.
• Be able to connect proportional relationships represented in a table and equation.
• Be able to connect proportional relationships represented in a table and word problem
• Be able to connect proportional relationships represented in a graph and equation
• Be able to connect proportional relationships represented in a graph and a word problem

If students can’t identify a proportional relationship in a given representation, I also want to know if the break down is the rate of change or the y-intercept.

I came up with this Proportional Relationships Anecdotal Records form.    I have one box for each student (I can use more than one page for a given period).   I put the student’s initials in the box and then circle the place where a breakdown is happening.   If the student is not recognizing a proportional relationship, I can write in k if the breakdown is the rate of change or the letter b if the breakdown is at the y-intercept.

I will probably use this to drive a brief review activity the following day.    I will pair students to do a card sort or to use the cards to play rummy.   Initially, I will pair them so that a student who has the idea down is working with a student who doesn’t.  After working with the idea for several days, I may place them so that kids who are struggling with the same issue are working together.     At that point, I can work with that small group to address whatever disconnect is still in place.   You can get the card sort here.

# Getting to Know Students – A Survey on Mathematics Affect

It’s  perfectly normal to encounter people who will very readily tell you that they “can’t do math”.   They say it almost as a badge of honor.   I think, though, that they are really saying something else.

My question is, what are they really saying?     I’m afraid of math (or of failure)?     Math is hard (I don’t want to work that hard)?   I had a bad experience in a math class (name your reason)?   Someone told me I’m not good at math?   Someone told me I’m stupid?

As I am getting to know my students during the first week of class, one of the things that I want to know is how they feel about math.   So I ask them.   I also ask them some other questions that might help me get to the why behind those feelings.   I use the survey shown in the photo.  It is taken in part from a survey in the NCTM Assessment book and in part from a survey that a friend gave to her students.   You can download the survey by clicking on the text below the photo.

# Necessity Is The Mother Of Invention – #ILookLikeAnEngineer

Necessity is the mother of invention.   Unfortunately, the “necessity” can be all too easily forgotten as an essential component in education.    I teach what I teach, in part, out of necessity but it is my necessity not that of my students.   I need to teach the curriculum that I teach because it aligns with the standards set forth by the state but it is not a burning necessity for my students no matter how many times I tell them the essential questions and how they will use it in the future.   Knowing something only becomes a burning necessity in the mind of an eleven year old when they see a need to know it so they can do something they want right now.

So how do we create that need to know?   I think we give kids real problems that they really want to solve.   It’s not something that I can do every day, but I try really hard to find time and space to do it every year.   To do this, I  compact lessons and I accelerate where I can.   This year, I managed to squeeze out almost a month at the end of the year to do an engineering project with my students.

Request For Proposal

Students were presented with a Request For Proposal (RFP) from a fake toy company.    The proposal indicated that this fake toy company was seeking to expand market share to include more girls in their customer base for motorized toys.    The toy company wanted those bidding on the contract to conduct market research and build a toy to meet that need.    The toy company indicated that the toy must meet one of three different criteria:  travel 3 m in 3 s, climb 1 m at a 15 degree slope in 2 s, or climb 1 m at a 30 degree slope.

Creating a Team and Conducting Market Research

Students were assigned teams and formed mini-companies that would bid on the RFP.   They created a team name, logo, and slogan.   Then, they conducted customer surveys with both adults and children in the target age range.    They analyzed the data and determined the type of toy the customer was seeking.

Building Technical Knowledge

During the same time-frame, students built knowledge of how gear trains work.   They began by building gears on a frame and exploring relationships between the rotations of the gears and the number of teeth on the gears (gear ratios, teeth ratios).   Next, they added a motor and wheels so that they could calculate the rate on a 3 m course and measure the rim force on the wheel.   They repeated this process with gear ratios ranging from 1:3 up to 225:1.   As they did this, they were building important skill in construction as well as an understanding of the different kinds of performance they might expect from different kinds of gear ratios.    From there, they measured rim force on the tooth of a gear connected to the motor.    They did so for different sized gears and then learned how to calculate torque.    With this knowledge, they could explain why certain gear ratios would not move and why certain gear ratios would be well-suited to climbing.   At this point, they had built sufficient knowledge to answer the first stages of that burning question of how to build a toy that would meet each of the criteria.

Making a Prototype

Each team began construction of a basic prototype to meet their desired criteria.   This amounted to attaching the motor and the desired gear train along with the wheels on the frame structure.   Students then tested their motorized frame to see if it met the criteria.   Once they had a basic working prototype, they started constructing a body to give the toy the desired aesthetics.    As they constructed the body, they continued to test the toy to make sure the additional weight did not place them out of compliance with the criteria in the RFP.    They repeated tests multiple times and used median values in order to eliminate outlier trials resulting from poor testing technique.

Sealing the Deal – Writing a Written Proposal and Giving an Oral Presentation

When the toy was completed, each team wrote a written report in response to the RFP and prepared an oral presentation.   The final stage of the project required each team to present their toy to a panel of judges representing the fake toy company.   I recruited 3 engineers and a soon-to-be lawyer to represent both the technical and business interests of the company for the panel of judges.   (I am lucky enough to have Sandia National Laboratories nearby and willing to provide this kind of support to encourage excellence in math and science.)   The judges selected a winning team based on the presentation and a demonstration of the toy.   (The winning team members each got a gift card to Cold Stone Creamery).

While this last stage is not “math”, it is very much a part of what engineers do and I wanted my students to appreciate the importance of being able to communicate effectively as an engineer.  Reading, writing, and speaking are just as much essential skills for an engineer as are math and science mastery

Why It Mattered

• Students got to experience the engineering process, which is so much more powerful than hearing about it.
• Girls had to learn how to make something and how to make it work.   It’s not that they are any less adept, but many of them are much less experienced.   This results in a certain amount of hesitancy, initially,   Having to make it work pushes them past this hesitancy and they discover just how good they are at it.    Giving girls this experience and confidence is important in leveling the playing field when it comes to engineering.
• Students used the math that they have learned this year to do something real that mattered to them (finding unit rates, conducting surveys, making data representations, analyzing data to make decisions, finding medians, using equations to calculate torque, measuring radii).
• Students had to find ways to work together – teams could not shift part way through the month long project.
• Students who lacked confidence as speakers learned that public speaking is a learned skill and that you get better at it with practice.   (I made each team do a dry run of their presentation in front of their classmates and get feedback the day before the final presentations.  They took the feedback and were so much better the second day.)

Gallery of Toys

Presentations

# Fighting a Fixed Mindset

The importance of establishing a growth mindset is clearly critical for students who think they aren’t good at math.   Helping those students to see that perseverance pays off can change their outlook and their outcomes.   How much does it matter, though, for those students who believe they are good at math?  This is a question that I have been pondering for the last year or so.

In considering this question, I have to acknowledge that I do think there are varying levels of ability when it comes to math.   That does not in any way mean that I think some people can’t “get math”.   Quite the opposite.   I think everyone can and should be “good” at math.  I think there are a lot of different ways to see and do math.   Often, when someone thinks they aren’t good at math it is because they haven’t had the chance to see and do it in a way that is meaningful to them.    Given the right opportunity and the right environment, I think everyone can be very successful at math.  I just think math comes more easily for some people than for others, just as is true of all things in life.    This seems to be counter to a lot of the conversation that I am hearing.     I’m not trying to debate anything here, just pondering ideas and trying to make meaning out of them in the context of my work, which currently is teaching math to students who have been identified as gifted.

This brings me back to my question of how much it matters to establish a growth mindset in kids who already think they are good at something.   Most of these kids come into my class with a very fixed mindset that they are “smart”.   Most of them also come having had an experience in which everything has always come pretty easily to them.   They just “get it” without really having had to work very hard.   I know that is not going to last forever, though.     I wonder how they will cope when they have to really work to figure something out for the first time.   I worry that they will just quit because they have not developed coping skills, they have not learned how to study, they have not learned how to persevere.   I worry that they will decide they are not “smart” because in their minds “smart” means things comes easily.

I talk to my students a lot about the importance of perseverance in the face of challenge and try to make sure that I stretch the content far enough that they all experience the need for perseverance.    I would rather that they face it for the first time now when the stakes are low rather than later when the stakes become much higher.

This year, I tried using a self-assessment with my students to measure their perceived growth.   I wanted to give them something concrete to help them see the value of perseverance.   I made up a simple triple bar graph structure with categories for each of the broad topics we would address over the year.   I had students complete a bar graph showing their perceived level of mastery in each category in September.   They then added a second group of bars for each category in December and a final group of bars for each category at the end of April.   In September, I wanted them to see that no one was “good” at everything, that each one of them had strengths and weaknesses.   In December, I wanted them to see some growth in whatever area they found challenging.   In April, I wanted them to each see that they had gained mastery of all the material.

As I look back on how this played out, I think helping these “smart” kids shift to a growth mindset has been equally as important as if they were kids who struggle.    It was good for kids to see that no one was “perfect” during those early days.   It was good for kids who were used to getting an A and were suddenly getting a B to see the growth in their understanding in December.   They got to see that learning is so much more important than a grade.    It was good for kids to see that perseverance paid off in April.

I’m not done pondering these ideas, but I am feeling a little “growth”.