Implementing the 5 Practices for Orchestrating Productive Mathematics Discussions

The following is the reflections from Sean Gilmour (’16) after reading and implementing Five Practice for Orchestrating Productive Mathematics Discussions by Margaret S. Smith and Mary Kay Stein in his classroom.

The Importance of Discussion

From this text, I hoped to glean ways to engage my students in activities with multiple entry points thus I can meet students where they are.  I teach 7th and 8th grade students at many different ability levels in each class.  As one might assume, it is common and convenient for me to rely on the most prolific and confident young mathematicians to drive a typical classroom discussion.  These students enable me to cover the most material and get straight to correct answers.

Still, the most ‘productive’ discussion should reach and be digestible for all students.  On a base level, state-testing scores, successful schools are awarded through growth, which means students with the lowest scores can make the largest impact.  Moreover, in math, it is especially important to allow students to struggle with rigorous problems so that they can develop their critical thinking skills, as well as to weigh multiple approaches to solving problems – both the correct ones and those that demonstrate common mistakes.

Two words in the title, “orchestrate,” and “productive,” offer two daunting challenges for a first year teacher.   Any teacher knows that it isn’t difficult for students to lead a discussion on just about anything.  The teacher’s challenge is subtly guiding the discussion that you can meet designated goals by the end of a class.  When students arrive at their own conclusions, it cognitively resonates much more than when a teacher blatantly states them.

Summary of the Five Practices

In order, Smith and Stein’s 5 Practices are:

  • Anticipating
  • Monitoring
  • Selecting
  • Sequencing
  • Connecting

It is crucial for the teacher to prepare, well beyond the typical amount, when planning a class-wide discussion.  Anticipating first necessitates a teacher to set a clear, achievable goal for the task.  The direction of class’ discussion reacts to students’ responses, but the end goal  must be certain.  With this in mind, the teacher must select a task with ‘High Demands.’    These have connections to the underlying concepts at the core of mathematics – and thereby at the core of a person’s understanding of mathematics – rather than something that relies on rote memorization or repetition of a procedure.  On the other hand, ‘low demand’ tasks rely on memorization and repetition.

Here is an example:  In the first couple of exercises, students have a visual representation of the problem – as seems compulsory with Geometry.  However, a clearly predetermined protocol and formula can solve the problem.  Even the ‘real world’ bicycle example gives another image like the earlier problems, making it effectively identical.

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Their counter allows students to actually interact with the shapes, measuring their sides, hands-on.  Students are then allowed to make observations and “conjectures” about the mathematical underpinnings of the exercise.  It becomes more of a ‘science’ experiment than the typical mathematical drill.

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The next steps occur during the class itself.  First, during monitoring, the teacher observes as students work with partners or in groups to solve the problem and prepare to share their methods with the class.  The teacher may use questioning to preemptively draw out students’ mathematical reasoning.

All the while, the teacher reflects on their anticipatory preparation, and then uses these notes to select important information for students to share.   Often, by actively selecting certain groups or students to contribute, the teacher purposefully creates opportunities for students across all levels to contribute to the dialogue.  Typically, teachers (including myself, admittedly) will ask for volunteers to share responses.  While this may streamline discussion, it excludes students who may be most confused and in need of assistance as well as narrowing the rigor and breadth of discussion.  The end goal of selecting is all about specifying ‘what’ needs to be said, not ‘who’ will or wants to say something.

In the penultimate step, sequencing, the teacher imagines the best order for selected students to share.  Ideally, the optimal sequence is like a story ; the class meanders through different strategies before arriving at the climax of the core math concept.  As I said before, the teacher includes incorrect strategies since these allow the teacher to address and engage all levels of understandin g, and then get closer and closer to the solution.  Reaching the final solution is not possible without acknowledging all strategies for approaching a solution.  Mistakes are allowed to be made, since students need to learn from them.

Then, at the end, the teacher connects all of the students’ contributions.  This is the stage where the teacher truly orchestrates everything, but again, everything is led by students’ responses to the problems.  The teacher allows students to consider theirs and their peers’ strategies in their own words before ultimately guiding them to the core mathematical concept.  At two recent Professional Development sessions, I gleaned a lesson that is very fitting here.  Students are mathematicians instinctually.  Whether information is presented through a graph, an algebraic equation, or something else, all students canpick up the most important details of a problem.  However, it is up to the teacher to connect students’ realizations and ground these in mathematics.

Tiling with the Seventh Grade

Smith and Stein use several case studies to show the Five Practices at work.  This format is obviously much appreciated by a teacher hoping to implement them, but each case study was in the context of introducing a concept. At this stage of the year, I would be using this system to review material, standards, and concepts I’ve already taught.  I knew some of my students would be able to remember and fall back on things they’ve already memorized, which would keep me from living up to the true intention of the Five Practices.  However, I hoped this process would help me to flesh out lingering places of confusion and misconceptions in other students.

The first task I took through the 5 Practices model was based on one of the case studies in the book, Tiling a Patio.  In it, students have a set of patterns, where the amount of ‘tiles’ changes at a constant rate from one pattern to another.  Originally, the task was devised to depict slope-intercept form, an 8th Grade standard, but I amended it so that it could address 7th Grade standards surrounding Ratios and Proportional Relationships  (7.RP).  Given the time constraints I have at this point of the year, I had to tackle 3 goals in this task :

  • Recognize Proportional Relationships change at a constant rate.
  • Model Proportional Relationships through multiplication, specifically the formula y=kx.
  • Determine graphs of proportional relationships cross the origin and are straight lines.

Firstly, these is probably far too many disparate big ideas to cover evenin an ideal, student-led math discussion.  However, with Easter Break a clear and present danger at the time, I felt it necessary to multitask.  Still, these concepts are all intertwined as sub-sections of the same standard (7.RP.2.a-d) .

Before the discussion itself, I anticipated two main sticking points.  My task had two examples.  In one set of patterns, the tiles would change by three every time, and students would have to come to the conclusion that ‘Pattern 0’ would have three less tiles than Pattern 1 – that is they would have 0 tiles.

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The second set of patterns would change by two each time, meaning that ‘Pattern 0’ would have 1 tile.

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Many students did indeed fall into a trap when approaching this part.  It was easy for them to continue increasing at a constant rate by simply adding.  But firstly, some gave pause to what to do when they would have to go down a pattern from Pattern 1 to Pattern 0 (i.e. subtract).

Secondly, many students incorrectly assumed that both patterns must start with 0 tiles.  The second set changed at a constant rate of two tiles, but, that meant it should have started with 1 tile, which would mean it wasn’t proportional.  However, many students easily recognized this rate of change, then defied it to make it proportional.

There was a table where students recorded the amount of tiles in a given pattern, and when they picked up the rate of change, they extended the pattern. Thus, at the beginning students could continually add by three or two.  However, for the last two ordered pairs, I broke the pattern, forcing students to use a rule based on the formula, y=kx, with x being the pattern, and y being the number of tiles.

 

Pattern 0 1 2 3 4 5 10   x
Tiles   3           45 y =___x

 

This would necessitate students realizing my second intended goal and using a multiplication equation to represent the changes from pattern to pattern.  Indeed, many students & subtracting at the predetermined rate.

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There were exceptions, as students were willing to ‘break’ the prior rule.

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I suppose it’s always important to teach students to read carefully.  But, those that did correctly multiply in order to find an output, were able to make the connection between the formula, y=kx, and proportional relationships perhaps better so than they did the first time I covered this topic several months ago.

All in all, students liked going through this exercise and were willing to complete it on their own.  While I have often heard complaints about the difficulty of work I give out, my students found this assignment accessible, rigorous, and rewarding.  As I was monitoring students, they were quick to make connections on their own.  When I questioned them, they were comfortable reasoning in their own words, rather than falling back on what a friend said. If and when students did express doubt, they simply didn’t expect the answer to be so intuitive and so easy to find.

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As I wrapped up my lesson during the final stages it was seamless for me to bring up my anticipated mistakes, as students didn’t mind sharing when selected.  All in all, this task seemed to build student’s confidence since the task was accessible even when their answers were wrong.  The tactile nature of adding tiles was easy to grasp, as opposed to earlier in the year when I used concepts that I deemed important and realistic, but maybe too abstract for a middle schooler, such as calculating miles per gallon.  Ultimately, when I connected the change to multiplication and a straight line on the graph, which is caused by constant addition by the same amount, students were clamoring to make the realization themselves.

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Tiling a Patio with Eighth Graders

A few weeks later, I revisited the themes activity with my 8th graders. As I said before, this task was initially designed to depict Slope-Intercept Form.  This concept is central to several standards across two domains in the Mississippi CCSS (8.F.2-4, 8.EE.5).

Rather than adapt it to connect with language and techniques I had taught, like I did with 7th graders, I left the task in the original form I found online. Students had a more complex tile pattern and would have to assign the pattern to an algebraic rule, which would have to be in Slope-Intercept Form.  So, my learning goals were for students to be able to:

  • Recognize the rate of change, or slope
  • Recognize the starting rate of the pattern, or y-intercept
  • Use these to create a rule in Slope-Intercept Form (y = mx+b)

Just like the 7th Graders, I went into this task as a review activity, with a lot of indirect anticipation of how students would address this problem and what challenges they would confront.  Similarly, I knew my students would easily be able to recognize the rate of change, i.e. the ‘slope.’  This made sense given the standards they presumably mastered in the year prior in exercises like the Tile Diagrams.

This prior assumption was met.  Most students quickly picked up on the pattern and (with some guidance) began to make an input-output table recording the numbers of tiles in each pattern.

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Most students realized that the rate of change was two white tiles.  But there was a large degree of variation as some students realized that the pattern didn’t line up with a simple rule of multiplication, as in a proportional relationships.  I could see that some students had second guessed their earlier, correct assumptions regarding the pattern, then tried to make new, incorrect rules to explain the data.

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In its unaltered form, students didn’t have to find a ‘Patio 0,’ which 7th graders did.  This meant that the assignment’s instructions wouldn’t give them the impetus to find a y-intercept.  Therefore, many students used only the rate of change when asked to assign a rule to the pattern.

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A couple of clever, creative students didn’t assign a specific rule, but still were able to extend the pattern.  They may not have realized it, but they bypassed their 8th Grade standards and went directly into Algebra I standards. Obviously, this was exciting for me to see.

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Another point of confusion came up when students compared the patterns of the black tiles and white tiles.  The task’s initial instructions described how each kind of tile progressed but didn’t indicate which pattern would be the focal point.  Given my fixation on the ultimate mathematical goals, I admittedly didn’t consider that this would  become an issue.  However, many students missed my target because they focused on the pattern of the black tiles.  Unlike the 7th Grade task, I was interested to see how students confronted the task in its original form without my guidance.  This way, it required the higher level reasoning of a High-Demand Task.  Looking back, it should have been easy for me to see how the introduction could have confused students about which pattern, or patterns, to analyze.  I caught this about midway through, and stressed my students to only focus on the white tiles.  Still, it was interesting for me to see how students seemed to grapple with comparing and linking the patterns between the two sets of tiles.

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In hindsight, the pattern of the black tiles was interchangeable with the patio number.  If students analyzed the ratios of black tiles to white tiles, they would have reached the same conclusions.  However, when instructions said to analyze the tiles, students decided that they only had to analyze one set of tiles, and naturally, they looked at the simpler pattern.  Unfortunately, they unknowingly didn’t have a chance to meet my goals for the exercise.

Nonetheless, the breadth of strategies used to confront this exercise intriguedigued and pleased me.  Only a handful of students were truly close to the ideal rules for the pattern and the true solution as a result.  Nonetheless, there seemed to be a pretty small correlation between students’ grades and their ability to confront the task.   Perhaps this had a lot to do with my lack of direction, as students were free to understand the problem subject only to their own imagination.  Then, as as I provided a little guidance, many students had the “Ah, ha!” moment as they now had to compromise their findings with an algebraic rule.

As I went about selecting and sequencing class discussion, I was able to draw from numerous perspectives, strategies, and personalities.  All the while, students never seemed to see themselves as part of a hierarchy.  At this point in the year, it obviously would have been ideal for students to understand how this related to specific contents and standards, but the variety of entry points and strategies for solutions led to a lesson and discussion that I’m quite proud of.

Going Forward

After reading Five Practices, and testing them with these exercises, I feel much more prepared for delivering and teaching with high level tasks.  In our Math content group, our mandate has always been to give students rigorous content and let students rise to the occasion.  Accordingly, I, from the start, laid out rigorous tasks for students even though they often complained and yearned for an easily replicated procedure.

After reading Five Practices, I feel much more ready and excited for how I, as a teacher, can manage and orchestrate these tasks.  My experiences, especially the Tiling a Patio, have proven that inspiration can come from all levels of mathematical achievement.  With the pragmatic rules I have learned from this book, I am now eager to recreate and capitalize on my successes so that whole classes can recognize their innate connection with that most daunting and elusive of subjects:  Math.

Implementing Children’s Mathematics

The following post comes to us from Patricia Franklin (’15) who taught 6th grade math in Jackson, MS. She read Children’s Mathematics: Cognitively Guided Instruction and reflects below on her implementation of what she learned from the book in her classroom.

I have always struggled with creating a deep understanding of number concepts for my students consistently finding that my students struggled with their “number sense” especially when applied to integers. In an effort to address this issue, I have completed a book study on Children’s Mathematics: Cognitively Guided Instruction by Thomas P. Carpenter, Elizabeth Fennema, Megan Loef Franke, Linda Levi, and Susan B. Empson. I hope that my findings will be useful to not only lower elementary teachers, but also upper elementary/ middle school teachers.

Purpose of blog post:

  1. Overview and Review of the book  
  2. Show Classroom Applications

Children’s Mathematics focuses on explaining primary children’s intuitive understanding of addition, subtraction, multiplication, and division as well as development of mathematical thinking and classroom applications.

Elementary students choose to use different strategies based question type. The book starts by identifying types of questions for addition and subtraction: join, separate, part- part- whole compare, etc. The authors then show different strategies (such as direct modeling, matching, counting up, counting on to, counting down, pictorial representations, and number facts) to help children develop a deeper understanding of the question. The authors recommend that students be able to flexibly choose which strategy to use. As students advance similar strategies are offered for dealing with multiplication and division (i.e. multiplication, partitive division, and measurement division). The authors then give explanations using the strategies provided including counting, written representations, area representations, and number facts.

Overall I found the explanations of children’s intuitive thinking to be mirrored in my own classroom; however, as a teacher of6th grade math many of the particular strategies were not advanced enough for my students that have already developed addition, subtraction, division, and multiplication concepts. In order to develop more genuine mathematical understanding, I took the same conceptual thinking and applied it more advanced problems involving integers.

Here is an example:

-5 + 3 =?

Matching Strategy: “Students use one-to-one correspondence between two sets until one set is exhausted. Counting unmatched elements gives the answer.”

My students began with counting out 5 red bars for negatives and 3 green bars for positives. They then lined up the numbers and counted that there were 2 extra red bars. Thus, the answer was negative 2.

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Counting On From First: “A child begins by counting on from first addend. The sequence ends when when the number of counting steps that represents the second addend has been completed.”

Me:   You owe Jazmine 5 dollars. You pay her back 3 dollars. What integer represents how much you owe her now?

Caitlyn: 2

Me: You now have two dollars?

Caitlyn: no I owe two.

Me: Can you show me a way to get there using counting?

Caitlyn: -5 … -4, -3, -2… oh -2

My takeaways from the book and its application to middle level math is that students develop their understanding of integers in the same way that they developed addition and subtraction in their primary years.

Once the cognitive processes of the students are understood the book dives into teaching applications, specifically how to help students understand what problems are asking and how to support each other in the solving the problem.

The authors believe that when a class is “unpacking a problem,” the focus should be on the story comprehension before attempting to connect the story to math. Students should be expected to share their thinking, ask follow up questions, and use multiple strategies consistently with other students. The teacher should not impose their ideas. The problem solving needs to occur organically from students.

At this point my thought was… wait… my students do not do this all on their own… The authors agree that ‘organic’ discussion does require some teacher guidance and thus they have given stems to guide this process:

  • Is your way the same or different?
  • How are these two ways the same or different?
  • Can [Student 1] explain [ student 2]’s work?
  • Tell me one thing about your own work
  • Tell me one thing about [student 1]’s work
  • What’s next?
  • Do you want to ask a question to a peer?

The fundamental take away from these questions is that it takes time and practice for students to critique each other. I modeled how I expected them to ask questions and answer questions at first. This seemed to help ease the transition. Additionally, by having higher performing students explain their reasoning and “easier methods” to lower performing students, my lower level students eventually started to solve problems with the same fluency as  my higher performing students. Eventually as students become accustomed to this method the conversation does become ‘organic’.

Overall, I have found that this book has enhanced the way my understanding of student problem solving. Moving forward, I intend to use the questioning strategies and teaching practices in order to enhance my students conceptual understandings. Feel free to ask any questions or comments about the book.

What #Mathissippi Learned at MCTM

Last month I had the opportunity to attend the annual Mississippi Council of Teachers of Mathematics (MCTM) conference in Ellisville, MS. Davis Parker (’15) and Dylan Jones (’15) attended as well, and have shared what they learned below. Take a look, and steal some ideas – specifically…

  • Using Algebra Tiles effectively!
  • Creating “Secondary Circuits” centers-alternative
  • Meaningfully infusing Statistics into Algebra I
  • Using Plickers for quick data collection and sharing!

Dylan’s Findings

My name is Dylan Jones, I am a second year teacher in the Mississippi Delta, teaching 6th – 8th grade mathematics to 50 of the most passionate and energetic students I know. I have had the pleasure of being sent on many professional development experiences on behalf of the Sunflower County Consolidated School District. Recently, I had the privilege of attending the 2016 Mississippi Council of Teachers of Mathematics at Jones County Junior College in Ellisville, MS. At this conference, I made connections with iReady representatives, engaged in many sessions to enhance my classroom, and got a chance to participate in a state wide Professional Learning Community (PLC) with fellow math educations across the state.

The purpose of this blog post is to share some of the information gleaned from these sessions. Since I found most of what I attended to be extremely useful, I have narrowed down my post to include the following sessions: 1) How to use Algebra tiles and 2) “Circuits”.

In my short time as a second year teacher, I have been able to incorporate technology into my classroom, increase the level of confidence and rigor, and fully implement secondary centers. One thing that I have always wanted to figure out is how to increase my use of manipulatives. While at the MCTM conference, there was a session on the fundamentals of algebra tiles. I signed up, made friends with the teacher next to me, and we began our journey together! My partner teacher has used algebra tiles for over 10 years now, so she allowed me to hog all of the tiles as I was learning the rules. Each tiles represents a variable (x, y, xy, x2, and y2) and then there are unit tiles to represent the numbers in the equations. I received a template and the facilitators started to walk us through some equations. Right before my eyes, for the first time in my adult life, I started to visualize the zero pairs when I solved for the variable in the equation. The facilitator took us through over 10 different equations, I was blown away. I couldn’t stop thinking about how I wish I had these when I introduced 7.EE.1-4 to my seventh grades in the first nine weeks. Below I have included some pictures to show you how powerful algebra tiles could be in your classroom! I ordered the set from EAI education, here is the direct link, and I used my EEF card and they were delivered in a week.

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While I was still reeling from the algebra tiles session, I saw a session for “Secondary Circuits”, to which I though was a cooler name for center rotations. When I arrived at this sessions, I was told that it was not about centers, but yet a way to make work more rigorous, engaging, and where students can assess themselves. I was sold! The facilitator starts passing out what look like worksheets, but have two to three columns of boxes with problems in the top. The facilitator asks us to solve the top left box, use that answer to find the next box, so on and so forth. So I am sitting with about 25 other math teachers and there is not a sound. We are all trying to solve these algebra problems and use out answers to advance our progress. I get stuck about 75% through the first circuit. The facilitators looks at my paper and says that I am operating on the target level question because I completed 75% of the circuit. The facilitator then explains that if a students can complete less than 50% of the worksheet, they are not operating on the grade level question target. If a student completes 60%-75% of the worksheet, they are operating on the grade level target question and if a student complete above 75% of the circuit, they are operating above the grade level target question. For better representation, here is a link to the facilitators TeachersPayTeachers website, where some of her circuits are free! I have also included a picture!

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Davis’ Findings

The annual MCTM conference in Ellisville was a unique opportunity to learn and share with other math teachers from the state. Over 300 teachers, representing every corner of the state, were there with the goal of improving their student’s outcomes and to change the way our students see and interact with mathematics.

With nearly 50 different sessions available to attendees, there was a great variety of information to be gathered. The most impactful session I attended dealt with the use of statistics in Algebra 1. The presenter walked us through the process of refreshing students’ knowledge of basic statistical analysis—box and whisker plots, mean, median, and outliers. Often times, statistics is taught as a system of rules and formulas with very little student input. It is rote and boring. The presenter, Dr. Jennifer Fillingim of Madison County Schools, completely flipped this script. Rather than simply copy notes of the board, students were actively engaged in the learning process and contributing to the creation of knowledge.  Through pointed questioning, the presenter was able to draw the needed information out of students, building on prior experiences. Going forward, I am planning a statistical analysis unit for the start of the second semester. We will use statistical techniques to analyze our performance from the previous semester, which will allow us to not only learn the material but also think more critically about how we can improve during the second half of the year.

 

The second most valuable presentation covered the variety of useful technologies available for our classroom. Specifically, it further interested me in using Plickers in my classroom. Up to now, I’ve used a variety of different quiz-type apps such as Kahoot but have always been a bit underwhelmed. For those unfamiliar with Kahoot, it is a quiz app where students answer predetermined questions on their phones and received points for getting them right in a set amount of time. Students enjoy playing Kahoot, but it’s not ideal for teachers as not all students have functioning smart devices, and it does not give useful data on how students are doing. Unlike Kahoot, Plickers requires only the teacher uses technology (reducing tech issues significantly) and gives student specific data, allowing a teacher to more easily identify those students who are struggling. I am in the process of printing and implementing Plickers in my Algebra 1 classes, and I am eager to see how students respond. Certainly, there are issues of students losing their Plickers materials, but I think it will be a much better system than before.

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Kahoot’s ubiquitous loading wheel of death

 

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Plickers only needs one phone and gives useful student data

Overall, the most valuable aspect of the MCTM conference was not the individual sessions but rather the conversations with other math teachers. As a 2nd year teacher, I am always eager to glean wisdom from the veterans who have been in the classroom for many years, and I am hopeful that one day I will be able to provide the valuable insight to others that they have given to me.

 

Joy, Standards, and Relevance: Reflections from NCTM’s Annual Conference

How do we cover the myriad of standards for our grade level while also attending to student engagement and enjoyment? How do we make seemingly abstract math content relevant? The following post comes from Crystal Stone (Mississippi ’15) as she reflects on these and more ideas after attending the National Council for Teachers of Mathematics (NCTM) Annual Conference this past spring. Check it out and steal some resources and ideas! – EPS

I traveled to my first National Council of Teachers of Mathematics (NCTM) Annual Meeting in April. I didn’t know what to expect. I spent my four years in college studying English and attending literature classes only to be thrown into a middle school algebra classroom. Disoriented doesn’t begin to describe my experience. But my first year was almost over and as it was winding down and state tests were on the horizon, I needed to reinvigorate my creativity.

It turns out inspiration wasn’t hard to come by. I learned about the different ways other math teachers across the nation made algebra more relevant and more fun. This was probably most important for me to make happen for my students’ most dreaded unit of the year: functions. When I start flipping through my program booklet, I had just one goal: go to as many sessions on functions as possible.

My favorite session was a project that involved forced perspective photography. Forced perspective photography is an exercise where you create an optical illusion; you change the size of an object you’re capturing by moving the camera different distances. So for example,

The presenters gave task cards differentiated by grade-level and standard that they wanted to explore. I’ve included them here for you to consider:

What I love about this activity is that it not only makes math fun, but allows students to take ownership over their project. They can be creative and autonomous. It has enough guidance that it grounds them in the math behind the activity and forces them to draw on the knowledge that they’ve acquired throughout the year. But the investigation is also a practical one: in an increasingly digital and visual world, it’s important that students are aware of how images are manipulated and how they can manipulate those images themselves. My hope is that when I try this activity with my students next year they will thinking critically about the images they are creating and the messages those images are sending, too. The beauty of activities like these is they allow for teacher creativity. We can make these activities more math-centered or more interdisciplinary.

The forced perspective photography lab clearly isn’t the only activity I found that inspired me to be more creative and more relevant. I attended a session that discussed how to make flags into math problems. I was particularly intrigued by this session because I recently implied of a geometry project I adapted from Nicole Bishop’s classroom on redesigning the Mississippi State Flag. The presenter explained to us that flag typically have very specific proportions and we can resize them and create problems knowing these proportions. He showed us examples of problems ranging from Algebra I through Calculus. For example,

Even without knowing the exact proportions, we can fit these flags onto coordinate planes to analyze length and create equations. Here’s another image he showed us:

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And for those whose students need extra practice using positive and negative integers, there’s always the second quadrant:

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His discussion of flags didn’t stop there. He explained that he takes his students for a walk around places with a lot of flags represented in Canada where he teaches. He takes a field trip and allows culture of different people to be part of the conversation of their flags and creation of the transformations of the flags they view in class. You can view more about his presentation and find his worksheets here.

As a result of these particular sessions, I’ve been making changes to my curriculum for the upcoming year. I am considering how I craft interactive math walks this year. I plan to make my units diverse and interdisciplinary. What locations can we visit? Is there a way to incorporate technology? Can I incorporate social justice? These sessions helped provide me with creative examples of how others make the lessons more fun and will help me to create interactive lessons infused with social justice in more math-grounded ways.

Culturally Responsive Teaching versus Teaching Mathematics for Social Justice

From over at my mathissippi blog.

The idea of “teaching for social justice” holds a long history of discourse in the educational world, and as such holds a myriad of terms and (sometimes conflicting) ideas which describe it. When considering the idea of “teaching mathematics for social justice” (TMSJ), it is important to understand the distinctions in the choice of this wording and its relation to other related fields of study. TMSJ holds many common threads with critical pedagogy and culturally relevant pedagogy, but it is not sufficient to use all of these terms interchangeably.

An immediate consideration of TMSJ is that it is framed as teaching for, not teaching about or teaching with social justice. As Stinson and Wager (2012) summarize of Paulo Freire’s work –

[teaching] mathematics about social justice refers to the context of lessons that explore critical (and oftentimes controversial) social issues using mathematics. Teaching mathematics with social justice refers to the pedagogical practices that encourage a co-created classroom and provides a classroom culture that encourages opportunities for equal participation and status. And teaching mathematics for social justice is the underlying belief that mathematics can and should be taught in a way that supports students in using mathematics to challenge the injustices of the status quo as they learn to read and rewrite their world.” (p. 6, emphasis in original)

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Paulo Friere from http://www.wiisaakodewinini.com/artworks/

As such, TMSJ is not composed of a collection of instructional resources and strategies, but is rather based on the disposition of the teacher and the student. A teacher cannot merely conduct a social justice-themed lesson and consider themselves as teaching for social justice; a teacher must in all their actions be working to challenge dominant and oppressive structures which hold back the full potential of their students. In the words of Gutstein and Peterson (2006) “social justice math is not something to sneak into the cracks of the curriculum” (p. 5).

In this regard, TMSJ holds many similarities with critical pedagogy. Critical pedagogy, which is grounded in “[driving] teachers and students to acknowledge and understand the interconnecting relationships among ideology, power, and culture and the social structures and practices that produce and reproduce knowledge” (Stinson & Wager, 2012, p. 8), also demands a unique disposition amongst educators. Although works of earlier educators such as John Dewey or W.E.B. Du Bois are thought to be grounded in critical pedagogy (Eubanks, Parish, & Smith, 1997, p. 151; Stinson & Wagner, 2012, p. 8), it is the work of Brazilian educator Paulo Friere that might best be said to have originated the movement around critical pedagogy in the United States (Stinson & Wagner, 2012, p. 8). Freire’s Pedagogy of the Oppressed clearly shows its relevance even to this day, as “nearly every contributing author to [the Teaching Mathematics for Social Justice compilation book] acknowledges Freire as having a significant influence on her or his work” (Stinson & Wager, 2012, p. 9). Because of this evolution from critical pedagogy to TMSJ, it is useful to consider the relationship between a more critical pedagogical approach and other related approaches such as culturally relevant or responsive pedagogy.

Culturally relevant pedagogy, which has also been labeled “culturally appropriate,” “culturally responsive,” and “culturally compatible” (Ladson-Billings, 1995, p. 159) shares some fundamental similarities with the more critical approach to TMSJ. As Gloria Ladson-Billings (1995) explains, culturally relevant pedagogy can be thought of as comprising of three categories: academic success, cultural competence, and critical consciousness (p. 160-162). At first glance, these categories might seem well aligned to TMSJ: Ladson-Billings frames meaningful academic success around student culture and identity, she notes cultural competence as “utilizing students’ culture as a vehicle for learning,” and she explains critical consciousness as “a broader sociopolitical consciousness that allows [students] to critique…cultural norms” (p. 160-162). Ladson-Billings even cites Friere in her defense of culturally relevant pedagogy, stating that approaches that achieved academic success for African Americans during the civil rights movement in-spite of were “similar to that advocated by noted critical pedagogue Paulo Freire” (p. 160). Friere and critical pedagogy are clearly an influence on Ladson-Billings.

propagandaHowever, there are some fundamental differences between Ladson-Billings’ culturally responsive pedagogy and Friere’s critical pedagogy, and it is these differences that separate (and at times muddle) the idea of TMSJ. In critical pedagogy, the school system is seen as “a major part of society’s institutional processes for maintaining a relatively stable system of inequality,” or the “hegemony” of a small group of elites over a broader society (Eubanks, Parish, & Smith, 1997, p. 151). This hegemony is counter to TMSJ and strikes out in a myriad of ways. Educational standards frame the purpose of schooling as “merely preparing for professional success,” while standardized tests are both held up as valid measures of success and used as “gatekeepers” which help determine the “intellectual elites” of our society (D’Ambrosio, 2012, p. 202-207). It is not just that students are failing academically; it is that the academic system is set up to sustain hegemony. This idea of hegemony creates inherent issues with the culturally relevant approach to education. While Ladson-Billings (1995) frames culturally responsive teaching  as “[getting] students to ‘choose’ academic excellence” (p. 160), critical pedagogy states that we need to rethink our society’s entire idea of academic excellence (Eubanks, Parish, & Smith, 1997, p. 152). If we consider our educational system to be a structurally unsound house,  culturally relevant pedagogy would have us work to repair and remodel the house, while critical pedagogy would have us demolish the house and start anew. Even though Ladson-Billings (1995) speaks to the importance of infusing cultural competency and critical consciousness in tandem with creating academic success for students, Friere (1970) counters this approach of seeking social justice within the context of the current system, saying –

Unfortunately, those who espouse the cause of liberation are themselves surrounded and influenced by the climate which generates the banking concept [a system of oppressive education], and often do not perceive its true significance or its dehumanizing power. Paradoxically, then, they utilize this same instrument of alienation in what they consider an effort to liberate…[One] does not liberate people by alienating them. Authentic liberation – the process of humanization – is not another deposit to be made in men. Liberation is a praxis: the action and reflection of men and women upon their world in order to transform it. Those truly committed to the cause of liberation can accept neither the mechanistic concept of consciousness as an empty vessel to be filled, nor the use of banking methods of domination…in the name of liberation.” (p. 79)

If the definition of TMSJ is to retain the element of “using mathematics to challenge the injustices of the status quo” (Stinson & Wager, 2012, p. 6), it requires more than a culturally relevant pedagogy. While TMSJ incorporates elements of culturally relevant pedagogy (cultural responsiveness, critical consciousness), it aligns much more with critical pedagogy in its determination to not simply work within the current, unjust educational system, but to rebuild it in a democratic way.

References

D’Ambrosio, U. (2012). A Broader Concept of Social Justice. In Teaching mathematics for social justice: Conversations with educators (p. 201-213). National Council of Teachers of Mathematics.

Eubanks, E., Parish, R., & Smith, D. (1997) Changing the Discourse in Schools. In P. M. Hall (Ed.), Race, ethnicity, and multiculturalism: Policy and practice (p. 151-168). Taylor & Francis.

Freire, P. (1970). Pedagogy of the oppressed. Bloomsbury Publishing.

Gutstein, E., & Peterson, R. (2005) Rethinking mathematics: Teaching social justice by the numbers. Milwaukee: Rethinking Schools.

Ladson‐Billings, G. (1995). But that’s just good teaching! The case for culturally relevant pedagogy. Theory into practice34(3), 159-165.

Stinson, D. W., & Wager, A. (2012). A sojourn into the empowering uncertainties of teaching and learning mathematics for social change. In Teaching mathematics for social justice: Conversations with educators (p. 3-18). National Council of Teachers of Mathematics.

Creating Secondary Math Centers

Centers day is my favorite day of the week. My students have fun, learn from each other, and do most of the talking. On top of that, I uncover tons of relevant information on student mastery and growth. But let me clarify – our centers have not always been this fun and successful. They used to be pretty miserable. Today I’m going to share how I created a meaningful and joyful structure in my classroom using math centers. The first step is organization. Most of the materials are in one packet and students just work on whatever page that corresponds with the Center that they are located at. When designing centers, you want to ensure that everything is ready and waiting for students to engage. Timing and directions are another important part of organization –  consistently behavior narrate expectations to ensure that your students receive feedback on their alignment with the directions and have an opportunity to quickly correct if needed. By having each Center last for 6-8 minutes I also ensure that students have ample and urgent work to complete, cutting down on time to get bored and act out.

Again, my Centers days go well only when I begin with ridiculously clear and explicit directions. I always share our centers rules and make sure to check for understanding in order to ensure that students are listening and have internalized the rules. Here is what I share:

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After grounding ourselves in the overall structure of centers, we clarify the directions for each station. I no longer have to do this for my students,  but it is important the first time you conduct centers to make sure everybody understands what is expected. Below is a breakdown of all directions on the first page of the centers packet. Here is the First Page of my Centers packet that I gave two weeks ago. Also, I think you should follow up the directions page with a page that has a rough outline of your classroom and what students should be doing at each station. The following is an example of a graphic that I display on the Smart Board while students are working at stations. This gives students a graphic reminder of what they should be doing at what Center:

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Centers Directions

Center 1: KABOOM

·         During the Game of KABOOM, students will practice some of the concepts that they’ve learned throughout this review week

·         The Table Manager will serve as the judge for the KABOOM game!

·         Everyone should be having fun at this station! Do not ruin it by getting into petty arguments about the game!

·         Once the timer is up and Mr. JB tells you to transition, the team manager will come to the front and write down the name of the winner!

JUDGE:________________________________________________

Center 2:  Student Work Corrections

·         At this station, students will be responsible for correcting student work with the people at their table.

·         Each question is answered WRONG. Therefore, students must use 2 FULL SENTENCES to explain what is wrong with the student answer.

·         Students must also show the correct answer on the page with the correct work shown.

·         All writing at this station must be done with a RED PEN

Center 3: Match Solutions to Equations

·         Students will match and connect all equations on the left side with the solutions on the right side.

·         Make sure to utilize your white board and marker to break down all equations.

·         There is one solution for every equation, make sure each one is matched up.

Center 4:  Kaboom

·         During the Game of KABOOM, students will practice some of the concepts that they’ve learned throughout this review week

·         Everyone should be having fun at this station! Do not ruin it by getting into petty arguments about the game!

·         Once the timer is up and Mr. JB tells you to transition, the team manager will come to the front and write down the name of the winner!

JUDGE:________________________________________________

Center 5:  Exponent and Square Root Review Work

·         At this station use the Real Numbers  chart to answer all multiple choice questions.

·         Next to each multiple choice question you are responsible for WRITING ONE SENTENCE explaining how you know your answer is correct.

EXTRA SUDOKU WORK IF YOU FINISH EARLY!!!!!

 

Types of centers in my classroom

There is an endless amount of ideas for centers that teachers can use to help develop your students as Math Scholars. Kaboom is an awesome fast paced game for students to focus on correcting common computational errors by incentivizing it with the creation of a board game. This Center is not used to have students produce lengthy justifications for their work. Instead this game exists to give students speedy remediation that will hopefully help with reoccurring computational issues going forward.

The following are directions for the game Kaboom that is huge success within my class! I would tape these directions at the table for Kaboom

Students at table 4 will play a game of Kaboom!

  1. The team leader will serve as the judge during all games of Kaboom!

  2. During Kaboom, students will take turns pulling strips from the KABOOM box

  3. The judge will have the answer key. If a student gets a question right they get to keep their strip. If they get it wrong the strip goes in the middle and nobody gets it.

  4. If Students get the “KABOOM” strip, they must put all of the strips that they have

  5. The student with the most strips at the end of the center time wins!

  6. Judges must keep track of who wins and at the end of each game write the winners name on the board

  7. Each game of Kaboom will last for 7 minutes.

  8. No one is allowed to argue with the judge.

  9. Winners of Kaboom will each win a piece of candy.

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When filling Kaboom Cans with questions it is important that it contains multiple choice questions that are not too intricate. This is supposed to be a fast paced game so I would just fill the cans with questions that are purely remedial  like the four below. I’d fill each can with about 18 questions.

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The judge will be given an answer key as followed. To save you some time, I would just look up multiple choice questions from the internet and just use those questions. The answer key will probably be made available on whatever site you get the questions from.

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The special strips that add an element of luck are as followed. I would include about four of each of the following four strips.

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Kaboom is an awesome game, just make sure you behavior narrate and make your expectations explicitly clear at all times. Things can get dicey during competition, especially when pride/candy is on the line. With that being said, this is a great activity to get student to focus on sources of common mistakes.

Getting students to work on error analysis is really important in my class because it allows students to recognize common patterns and traps that they are falling into! Essentially the student becomes the teacher!  Error Analysis really helps to develop students as metacognitive thinkers. I use this Station get my students to point out some of the common mistakes that they make on Exit Tickets and have discussions with each other about the work. This is a copy of the blank center page.

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And here is a copy of the same student work filled out by one of my students with his corrections on it.

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The last center that I used this week was a matching activity in which students had to match solutions to corresponding equations. I made it up myself and it encouraged great discussion amongst students about the material.

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Also, you should have an extra activity at the end of the Centers packet for higher students who finish their assignment/packet early. This extra activity isn’t just supposed to be something “extra” that kids can do, but instead it serves the purpose of ensuring that students who finish early have something to do to ensure that they are not distracting others. Usually I include more difficult problems at the end, but sometimes I put a Sudoku at the end of the packet as a special challenge to students and if they are able to finish one, I give them a bag of hot chips .Just don’t let students access any computers with the Sudoku or they will look up the answers like my student Lyric did one time.

In conclusion, Centers can be a great way for students to have fun in your class while developing themselves as math learners. The more diverse and flushed out your centers are the higher chance there is for success! Make sure that your directions are incredibly explicit and that you are constantly moving throughout the classroom while behavior narrating. Do not use centers too often in your class because if the appeal wears off then students can get bored/apathetic really quick. I try to pull it off once or twice a month at most!

On Thursday, 11/19 I plan on rolling out a Centers session in Cleveland in which you will go through and participate in some of the Centers that were talked about within this Blog Post! Please register on the course catalog and attend if you need any help developing these, and please reach out with any questions, comments, or concerns about the integrating Centers into your Math Classroom!

Alexa Evans – Implementing Three-Act Math

During my first year of teaching, I constantly thought things like, “I can’t wait until next year so I can try this differently” or “I need to prioritize this more next year.” Honestly, I haven’t followed through on everything, but something that has been very rewarding in my classroom is more regular use of Three-Act Math lessons. These types of lessons allow every single one of my students to engage with the material and leave the classroom feeling like they made a meaningful contribution.

I begin the lesson by showing the students a picture or a short video clip, and simply asking them to write down what they noticed and any questions they have. I make sure to take the time to write down every thought that my students share, because this validation early on in the lesson really gives them the momentum to grapple with the math later on. I love seeing my lower-achieving students, who are normally a bit uninterested during my lessons, shooting their hands into the air to share their thoughts. And even though there is a low entry point for everyone, I always have a handful of students who begin to lead us toward the important mathematical concepts. After considering all of the students’ questions, I present them with the main question we’ll be answering and ask for their answers.

I love this part of the lesson because I really get to see my students start to make sense of the mathematics. It’s also the point where students begin to notice that they don’t have all of the information they need to fully answer our main question. In other words, they start to get those headaches. It’s only after students articulate what information is missing that I fill in those gaps and present the Tylenol. I then allow them to discuss with a partner or two to work toward a solution.

I have found that managing this group work in a Three-Act Math lesson is easier than managing my class in more typical lessons. Because I make sure that I begin with an intriguing stimulus, and take the time early in the lesson to place an importance on the students’ voices, they are very invested and eager to work toward a solution. Again, even my lower students are engaged after experiencing success early on. Three-Act Math lessons are the best way that I have found to foster meaningful student discussion and create an accessible environment for everyone in my classroom.

Grow That Metacognition! Meaningful Feedback on Performance Tasks

Below, Davis Parker (Mississippi ’15, Cleveland High School), shares how he has used performance tasks to build student metacognition – how students think about their mathematical thinking. Davis talks about the intentionality behind his design, execution, and feedback around a recent performance task. I love the feedback activity that he does with students which allows them to not only understand their own rubric rating, but to also use peer responses to grow in their understanding of what great mathematical thinking and communication looks like.  Read on below!

(Note – Davis refers to the QUASAR rubric throughout his post)

The assignment:

  • Students were given a performance task to gauge their ability to work creatively with absolute value functions.
  • In approaching the assignment, I wanted to create a task that gave students significant room for creativity, strategic thinking, and communication. I wanted it to be Math-lite, requiring as little number crunching and slick computation as possible, but allowing students to showcase their reasoning and problem solving skills.
  • The first question set the table for the entire assessment and allowed a low floor for students to enter. It read as follows: “Drake and Meek Mill decide to climb a mountain to settle their differences. Draw a graph that could represent their altitude (y) as a function of time (x). Explain your reasoning in complete sentences.”
    • On this question, students received points for drawing a hill/mountain looking graph and articulating the reasoning behind their graph. While it was hard to differentiate between math and strategic knowledge on this question, I gave students math knowledge credit if they properly interpreted the meaning of x and y-intercepts. With regards to strategic knowledge, students received credit for drawing a graph that could be interpreted (in some way or another) as the trajectory of people climbing a mountain. Ultimately, a student’s strategy was only as good as his/her ability to explain it.
    • If students drew a linear equation but gave sufficient explanation that Drake and Meek Mill never came down (or that they didn’t return on the graph), I gave them points.
    • Students were docked for graphs that didn’t begin at the origin or somewhere along the x-axis.
  • The second question gave students a specific function (y= -2|x-4|+8) to model their path and asked, “how high is [Drake and Meek Mill’s] peak? How long does it take them to reach the peak? How long does it take for them to climb the mountain? Explain your reasoning in complete sentences.”
    • Points were given for an accurate graphical representation of the function as well as correctly identifying the peak height (8), peak time (4), and total climb time (8).
  • The third question forced students to transition from mathematical knowledge (graphing, vertex, etc.) to more strategic knowledge by asking, “At altitudes greater than 6km, Drake needs to use his inhaler 3x/hour. Meanwhile, at altitudes greater than 7km, Meek Mill needs to use his inhaler 5x/hour. Who uses his inhaler more during the hike?”
    • This question forces students to navigate through two distinct measurements (time and distance) on their graph. While the altitude deals the y-axis of the graph, the real question lies on the x-axis with the time. An ideal student would recognize that the number of inhaler uses is dependent on hours, not necessarily kilometers
  • The last question was meant to showcase students communicative and strategic understanding by asking them whose asthma is worse, Drake or Meek Mill?
    • The key aspect of this question is that there is no correct answer. While Drake has to use his inhaler at lower altitudes, MM has to use his inhaler more often. It allows students to approach the question from either side and still come to a strong, evidence based conclusion.

The execution:

  • The most common complaint from students was, “I’ve never done anything like this before! Is this even math?” It seems they’d been trained to think math was 100% worksheets and computations.
    • On a surprising yet positive note, the students outperformed their average quiz score on this assessment, which strikes me as proof that they have underutilized mathematical skills in the way of strategy and communication.
  • The key was keeping students engaged in the problem and having them work through the whole sheet (1 page front-and-back). As the bookend questions required zero mathematical knowledge, even the most struggling student could gain considerable credit if he/she simply finished the quiz. Lastly, including popular culture figures allowed them to feel as if they had an actual opinion on the question at hand. You’d be surprised how many students took the time to confirm that Drake was the superior MC to MM.

The grading:

  • First question (MK: 1, SK: 1, C: 1)
    • The most important aspect of grading question #1 was making sure whatever the student graphed and whatever the student wrote reinforced each other. In the case of student #1 (part A), I docked communication points for a weak explanation. While the slope does in fact increase then decrease, the student failed to connect his reasoning to the original problem. Meanwhile, student #2 (part A) gives sufficient explanation for his graph by relating the vertex to the mountain peak and identifying the variables x and y.
  • Second question (MK: 2, SK: 1, C: 1)
    • This was the most math intensive question of the quiz as it forced students to graph an absolute value function accurately and interpret the results of the graph. The key aspects of the graph were the vertex and x-intercepts. Most students were able to accurately place the vertex, yet many struggled to properly slope their function such that it intersected the x-axis at 0 and 8. Students who failed to do this (student #4, A) lost significant mathematical knowledge credit. From a strategic perspective, students were rewarded for first identifying the vertex then graphing from that point. Additionally, they received credit for properly interpreting the zeros and vertex of the function.
  • Third question: (MK: 1, SK: 1, C: 1)
    • Students received mathematical knowledge points for relating the x-axis on the previous graph to the y-axis. The question forced them to take a y-axis measurement (altitude) and convert it to an x-axis measurement (time). Students who properly did this and performed computations correctly received full points for their work (student #2, B). From a strategic perspective, students received credit for understanding the need to multiply each individual rapper’s rate by the time above a certain altitude. Even if they incorrectly identified these values, they received points for their strategy (student #4, B). Some students had great strategy and math but had weak communication (student #3, B). These students received points for their thinking and computation but very little for communication, as they didn’t effectively explain their thinking or answer the question. Student #1 had a very interesting answer here because he knew to multiply the given rates by some fixed value, but he incorrectly used the altitude instead of the time. Here, he received ½ credit for strategy and close to full credit for communication, as his reasoning is quite clear.
  • Fourth question: (SK: 1, C: 1)
    • Students received credit for clearly articulating an opinion based on results from Q3. Some students received little to credit (student #3, B), while others (student #1, B) received close to full credit, even if they had incorrect conclusions for Q3. The key to Q4 was having a logic-based opinion that was built off the answer to Q3. Even if a student didn’t answer Q3, they received credit for saying that Drake has to begin using his inhaler earlier or that MM has to use his more. There was no right or wrong answer, just good and bad reasoning.

Student responses

  • Student #1
  • Student 1's Task Responses
    Student 1’s Task Responses
    • Across all classes, student #1 was considered to be the worst of the example. What stood out was his poor reasoning and especially poor communication. A common response was, “do people really talk like that?” Fortunately, student #1 didn’t exist and was an answer sheet I filled out with common mistakes. I would strongly encourage other teachers to do something similar if they are trying to show common mistakes.
      • Score: MK: 1.5, SK: 2, C: 1.5
  • Student #2
  • Student 2's Task Responses
    Student 2’s Task Responses
    • Just as student #1 was considered the worst, student #2 was generally considered the cream of the crop. Students appreciated his clear writing style and mathematical computation.
      • Score: MK: 4, SK: 4, C: 4
  • Student #3
  • Student 3's Task Responses
    Student 3’s Task Responses
    • Student #3 was by far the most interesting case. Most students gave him low marks, as they were put off by his poor communication style. What they didn’t recognize was the strength of his reasoning and math skills. For instance, on Q3 he does a terrific job strategizing and drawing a model of when each rapper would use his inhaler (as well as the total # of uses). Student #3 clearly has solid mathematical reasoning, yet it is hidden behind a veneer of sloppy language and organization. He’s like a raw athlete: great potential but very little polish. In fact, this assessment helped identify him as a student who could very well be pushed to not only become a good mathematician but an outstanding one.
      • Score: MK: 3.5, SK: 3, C: 1.5
  • Student #4
  • Student 4's Task Responses
    Student 4’s Task Responses
    • Student #4 received better reviews from her peers than her teacher. Her organized graphs and clear writing were convincing indicators of her knowledge, yet she had significant struggles with her graphs. On all 3, her graphs incorrectly intersect the x-axis, and she misinterprets her previous graph on Q3. To be fair, her errors on Q1 and Q2 were more than likely a result of sloppiness than ignorance, yet she ended up receiving roughly the same score as student #3. What makes that result so interesting is that students #3 and #4 received the score for almost the exact opposite reasons. While one was an effective communicator, the other struggled mightily. While #3 had accurate graphs and strategy, #4 missed the mark. The key is that they both have terrific potential in mathematics, but that they need distinctly different practice going forward.
      • Score: MK: 2, SK: 3, C: 4

Performance Tasks are extremely valuable, if only because they break the monotony that Math class can become. They also allow every student to show off what they can do, as opposed to primarily punishing them for what they can’t. Going forward, I want to keep the PT structure of bookending the assignment with more qualitative, opinion based questions that provide students with low barriers to entry. Additionally, I think exposing the students to the QUASAR rubric is productive because it allows them to see behind the curtain and understand what exactly I am look for as a teacher.  In fact, I don’t plan on taking my upcoming PT up for a grade but rather have students grade their peers and give them valuable feedback. Lastly, I’d encourage other teachers (and myself) to do their best to keep students engaged with the assessment through its entirety. Students often surprise themselves (and their teachers!) with how much they do in fact know.

I hope this review has been helpful, and please leave comments below if you have any questions.

Best,

Davis