Category Archives: Formative Assessment Lessons

Reflecting on Feedback

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Funny how things kind of pile on to hit you in the face!  @druinok and I are revisiting Wiliam & Leahy’s Embedding Formative Assessment;  The current issue of ASCD Express is filled with articles focused on feedback and our first day back with faculty this semester- we had a PLC about Formative Assessment & Feedback.  Though this post was more about success criteria – there are several comments concerning feedback.

Chapter 5 in EFA2 was a bit frustrating.  Initially it felt like it was saying so much of the research on feedback was not useful…for several reasons.  But as I read and later watched this presentation (while sansone walking for my cardio!) – there were some big ideas that stuck out to me…well, hit me in the face.

How when done incorrectly, feedback can have a negative influence on learners.  Some things were obvious, but others were definitely worth noting.

How we should not be expected to give thorough feedback on every single thing.  He suggested the 25% idea.  25% of the work is self-assessed, 25% of the work is peer assessed, 25% of the work is skimmed by teacher, 25% of the work received thorough feedback.  Hmmm.  This feels doable.  I have felt so overwhelmed at times in recent years.  And I also wondered if by giving too much written feedback, does it become common and expected, therefore losing some of its ability to drive student achievement forward?

The article we read during our first day back AND Wiliam in this book both said without any follow-up action, formative assessment is essentially useless.  The article said – “it is not fair to students to present them with feedback and never give them the opportunity to use it.”  In his book, he said, IF its important enough for students to use the feedback, then you must find the time to allow them to do it in class.  Ouch.  But when?  We can’t possibly get everything in!!!

This is the pie in my face.  As I was planning the FALs for my classes, I realized – that giving students feedback on their pre-assessments…being intentional with the wording, expecting them to do something with it…either answer a question, extend a pattern, redo a part of the problem, look at a specific piece of their work, sketch a new picture…

Oh my goodness.  That’s it!  When we pass back the pre-assessments… usually a few common things happen…

  1.  The student is given a few minutes to revisit their work and read the feedback, then attempt to use the feedback and make their response better..  then
  2.   The student is paired or in a small group and they all use their feedback to create a group response to the task.  OR
  3.   After the lesson, students are given an opportunity to revisit the initial task and/or a similar but different task.  I usually copy these front/back – this allows me to flip over and see their initial work, feedback and see if they were able to clarify misconceptions and correct mistakes.

How might I use this idea to implement into my other tasks/lessons?  The time to “ACT ON the FEEDBACK” was embedded into the lesson.  Lightbulb!

Generalizing Patterns: Tiling Tables

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Last fall after instructional rounds, one of the observers asked me if I would mind having some folks visit my classroom.  Sure.  They were most interested in questioning, interactions with students and use of Formative Assessment Lessons (FAL).

When they emailed to set up a date, we agreed on January 10.  Oh, wait.  This will be the beginning of a new semester with new students.  I won’t really know them.  They won’t really know me.  Great.  Now, I’m scared.  Oh well, let’s look at the positive – this will give me a chance to try out a new lesson.

I printed off 3 lessons to look at the evening before students returned to school.  I liked all three.  Building complex equations seemed perfect, so I began to prepare for it.  We were out for weather our second day back.  As I began looking over my lesson plans, it seemed the Tiling Tables was a better fit for the upcoming unit, so I switched gears.

I had done this lesson a couple of years ago, but never taught it in class.  As I began to revisit the task, I knew I liked it.  I knew it would offer some good discussion on ways to extend the patterns.  But wait.  These students barely know what a parabola is.  Would they have a clue as to how we would write an expression for a quadratic relationship?  Would I have a clue as to how to introduce it, this early in the semester?  No.

So I pondered for a while.  I would simply use the task as a way to say, we have the knowledge and tools to do parts A and B, but part C, well – that’s what we will be learning later in the semester.  It would give us a reason to learn it later, right?  Goodness.  What a canned comment.  By now, we had another snow day, so our visitors would be in our classroom on the 4th day of instruction.  I was stressing just a bit.  What was I thinking?  Starting off a new class with a FAL I had never used before?  We needed time to build some rapport.  Too late.  Let’s go with it.

I gave students the pre-assessment:

table tiles 1table tiles 2

The class was divided pretty much 3 ways – Those who doubled the number of tiles, after all – a side length of 20 is doubled to get 40, right?  The second group had sketched the designed on the the grid paper which had been provided, however, they wrote answers for the 30 cm table instead of the 40 cm.  And finally, several had the correct number of tiles by extending the pattern on the grid paper.  But I ask how efficient this strategy would be for, say 300 cm table?  Hmmm.

We began the lesson the following day by giving 3 samples of work.  Last school year, I figured out, I could save paper by having them use the shop ticket holder sleeves to hold the sample work – allowing them to draw, sketch, etc with dry erase.

These instructions would help their discussions:

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The first sample was Leon:

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After some small group time, we shared our thinking with the whole class.  There was one student in particular who had confusion all over their face.  I encouraged them to ask the person sharing for clarification (using our starter stems).  I believe this is important to model and have them do early in the semester, so they become more comfortable with it.  Even with more explanation, they were still not seeing the pattern.  So another student shared.  Still no help.  Finally, a third students explained how they saw the pattern.  The confused student nodded and said, “Okay, I got it.”

Now, years ago, I would have said – great and moved on.  But I’ve learned…ask them to explain it to you.  They may say they’ve got it – just so you will move on, but how do you know they understand?   This student, however, could explain their thinking and were correct – they could even extend it to the next table size.

The next student sample was Gianna:

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So many more of the students picked right up on Gianna’s approach.  The confused student – smiled stating they liked / could see this one better.  For me, it was listening and watching the students discussing – that brought me an a-ha!  This is the example we will use to generate the quadratic expression I was worried about!  The total whole tiles would equal (step x step) + (step – 1)x(step-1)  Yay!

Finally, we had Ava’s sample:

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Many of the students could not make the connection with the side lengths on Ava’s sketches in the beginning.  Then they began going back and looking at their own sketches to verify the numbers Ava recorded in the table.  They noticed the same patterns and agreed with them.

After this final discussion – we went back to see if each student had answered the task fully.  We quickly realized though there was some good, correct thinking going on in their work – they had not addressed the questions completely.  The class agreed that Ava’s was the most complete with her table.  And it was interesting to hear their discussions of how they would explain to the other students how they could expand their responses to be better and more thorough.   One student even brought up it was challenging trying to figure out their thinking since there was no written explanations of what they were doing.  (I thought – yes, this is what I feel like sometimes too.)

As we continued discussing having thorough answers – I shared Ava’s data in a graph…  they were quick to see the quarter tiles always remained four and the half tiles being linear, a focus from 8th grade.  But what about the total tiles.  How can we write an expression to model that data?  And I took them back to the slide with Gianna’s work to look for patterns between the table size/step number and the total whole tiles.  We test our thinking with different sizes and it worked.  We tested our expression in Desmos…and what?  It hit all of the data points!

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They had some experiences with the visual patterns – and good feedback to me about liking them, but still having to think.  This task reinforced some of those ideas.  IN their reflections – though many may have preferred someone else’s sample work – they “saw” how Gianna’s work led us to a more efficient expression or even Ava’s approach to orgaznizing the data in a table was pretty helpful to see the patterns so we could find describe the expressions.

Total Tiles = 4 quarter tiles+ 4(n-1) half tiles + n^2 + (n-1)^2  whole tiles.

I will definitely be using this lesson in my future.  It brought just enough confusion, but great opportunity for sharing and discussion.  And the observations were great.  Students were not shy.  At the end of the day – I was amazed we had only been together for 3 or 4 days… wow, this is going to be an outstanding semester!

Interpreting Distance Time Graphs

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On the 3rd day with a new group of students, I had visitors from some other districts in our classroom.  I was nervous – I really didn’t know these students yet and they certainly didn’t know me.  I had chosen Interpreting Distance Time Graphs lesson from MARS to begin our semester.  Although this is listed under 8th grade, it leads to some great discussions and uncovering of ideas and misconceptions.   The Keeley & Tobey book also lists “Every Graph has a Story” in the Formative Assessment Strategies.  This was the ideal lesson to introduce our first unit on functions, while trying to be intentional with planning FAs.

Pre-Assessment

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Telling students it is only for feedback, not for a grade seems to drive most of them to really share their thinking.  After reading their responses, I had some ideas of how I wanted to change the lesson up a bit from times past.  The first time I ever used this lesson was around 2011-2012.

Let the Lesson Begin

We began our actual lesson with only the graph in this picture.  I asked students to jot down 3 things they noticed about the graph.   Pair share.  I called on students randomly with my popsicle sticks, then allowed for a volunteers (this was something @druinok and I had read in EFA2, which allows everyone to be heard).    We then read the scenarios aloud and at the table groups, they discussed which story was model by the graph.

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Next I took one of the scenarios we didn’t choose and asked them to sketch a graph on their whiteboards to model it.  We had about 5 different overall graphs – I drew on the board and let them discuss at their tables which they agreed/disagreed with.  Then we shared our thinking.  Some very good sketches and great discussions.

Open Card Sort

Many years ago, a colleague shared the idea of open sorts, something she had learned from a John Antonetti training.   I instructed students to remove only the purple graphs from their ziploc bags.  (Side note suggestion- use different colors of cardstock and this allows them to quickly grab the cards they need, ie the purple graphs, green scenarios OR blue tables.  I used to have all the same color and we wasted a lot of time sorting through which cards we needed).  In pairs, they were sort the graphs any way they wished, the only requirement, was they must be able to explain why they sorted them as they did.  Again, sharing whole class led to seeing some details we had initially noticed.  If you’ve never done an Open Sort – let go and let them show you their thinking.  You might will be amazed and wonder why you’ve never done this before.  They love to think.  We should let them.

List 3 Things

A couple of years ago, I began asking students to list 3 things they noticed or knew about their graphs – anytime we were interacting with a graph.  IF you ask them to do this enough, it eventually becomes habit.  I also like this approach because it gives them a chance to survey the information in the graph before they start worrying about / answering questions.  Today, I asked pairs to label their whiteboards A – J and I set the timer.  They had to share/discuss/jot down 3 things about each graph.  Once again, I used popsicle sticks to randomly call on a few students.

Graph & Scenario Matching

Using the “rules” listed in the lessons powerpoint, students were then given time to discuss and match graphs to the scenario.  This went so much quicker than times I’ve done this lesson before.  I believe it was because they had already interacted with the graphs twice…they were not “new” to them.  I will definitely use the Open Sort and Name 3 Things before matching tasks in the future.

I gave them a chart to record their matches.  We then shared out our matches.  Each time, I neither confirmed or disputed their matches, but rather would call on a couple of other students to agree/disagree.  After some discussions, I came back to the original student to see if they agreed / disagreed with their original match.

One of my favorite graphs is this one –

not possible graph

And our final sorts…  And again – Scenario 2 is always up for some debate.  It reads: Opposite Tom’s house is a hill.  Tom climbed slowly up the hill, walked across the top and then ran down the other side.

distance-time-matched

Though every student did not get every match exact, there were several a-ha’s during the lesson and questions asked.  I look forward to reading their post assessment.

I’ve used this lesson as written many times with much success.  However, just making some adjustments prior to the matching made a vast difference in the amount of time students needed to complete the task.

Let me know how this lesson has gone / goes for you if you use it.

Modeling Systems

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Sort of a rambling post. But trying to make some sense of my thinking…

I always appreciate posts from @emergentmath.  This particular post made me pause, I had just completed the MARS task, Boomerangs, he references.  We are in the midst of our systems unit.

I used Mary & Alex ‘ s suggestions with beginning systems without the algebra.  Conversations were great, students’ strength in reasoning was evident.

I plan to use Geoff’s suggestion for a matching/sorting activity this werk for students to see the benefits of each type of tool to solve systems.

But where I struggle is with this standard:

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I am experiencing some pushback from a handful of students who are able to reason and solve a system without actually modeling it algebraically.

Their reasoning is correct.  They verify their solutions and interpret them correctly.  They can sketch a graph yet “refuse” to model as a system of equations.  I struggle because “their math” is right on.  I realize places where algebraic models can help but I honestly can’t tell them my way is better…yet the standard says…

It feels almost like I am punishing them if I make them model it algebraically.

Then I have others who are not sure where to start.  The equations model provides them a tool, yet they will not embrace it.

How do others handle this situation in your classrooms?

I use graphical, alongside a numerical table of values, with solving/verifying with the equations, letting them see their own connections eventually.

My biggest goal for systems is to provide enough modeling for students to actually “see a context” to connect/make sense of a naked system of equations.

This is where I believe skill/drill has ruined the power and beauty of math.  Finding an intersection point but what in the world does in mean?  It’s a point on a graph. Whoopee.  Why isn’t it all taught in context as a model?

Get to Know Your Students pt 2 #julychallenge Post 12

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 Get to know your students, especially how they learn and think.

Taking my lead from this post, my intent is to consider how I can improve or implement the 14 ways discussed.  In my last post, I shared how important I feel it is to know our students as real people.  This one is to share #5things that impacted my classroom and helped me know how my students learn and think.

My 3 years with Kentucky Leadership Network and my experiences with #MTBoS have changed my mindset.  The work with KLN introduced me to a new set ideas and #MTBoS allowed me to explore with others and develop a new frame of reference as I seek to grow as an effective educator.

I cannot be grateful enough to all those who have challenged me and help me grow.  But as I think of the experiences that have opened my eyes to see better ways I can consider my students as learners, these are the ones that first come to my mind.  #5things for getting to know how my students think and learn…

Wait Time II
I learned about this routine from 75 Practical Strategies for Linking Assessment, Instruction and Learning (Keely, Tobey 2011).  A simple adjustment.  Yet it forced me to really listen to my students.  You can read more on a previous post, here.  Basically, it allows  the students AND teacher to process a student response.  We were all told in undergrad to wait 3 seconds after asking a question before calling on a student.  Some people actually think this deters the class flow.  I disagree. The idea with Wait Time II is to wait again, after the student response.  It allows the responder to consider what they said, the classmates to process what was said and the teacher to consider next steps, questions, etc.  A bit uncomfortable in the beginning, but once I explained the rationale to them, they got it, as did I.  Waiting and listening adds value to what students are saying.

What Makes You Say That?
Making Thinking Visible, (Ritchhart, Church, Morrison, 2011)
A chat with Liz Durkin challenged me to consider ways I could implement these routines into my high school math classroom.  It was the question “What makes you say that?”  that helped me begin drawing out student thinking.  What were they seeing? What evidence supported their statement?  With this routine, I began learning new ways of seeing problems myself.  Students’ ideas, strategies and approaches are way more intuitive than my own.

Notice and Wonder
I was first introduced to Notice & Wonder with Max Ray’s Ignite talk sharing The Math Forum’s simple, yet impactful strategy.  You can read more in Powerful Problem Solving (2013) as well.  When I pose a problem, scenario, graph, students may not readily know where to start.  But they can tell me what they notice.  Its a starting point.  Everyone can share something.  When we listen to what others are saying, that ignites other ideas as well.  And they begin sharing their “I wonders” which are great transitions to explore more.  Its great.  Its simple.

This routine carries over to standardized tests as well.  Students shared how they didn’t know how to approach certain problems on ACT or their EOCs, but they looked at it, thought about what they noticed, connected it to something they knew and was able to at least make an educated guess. 

Friendly Class Starters
After reading What’s Math Got to Do with It? and completing the Jo Boaler How to Learn Math course last summer, I knew I needed to find ways to invite students to think differently about math in my classroom.  Some major a-ha’s and sad realizations as to why so many kids are down on math.  I began with things like Number Talks she presnted during one session.  Amazing how many different ways students can see / approach a single problem.  When I invited them to share their thinking, they owned the math.  This past year, I implemented Counting Circles, Estimation 180, Visual Patterns as well.  These resources were primarily used as bell ringers to get students in math mode. However, there were days it lead to deeper, richer discussions and I was flexible enough to go with it.  My students’ confidence began to grow.  Their number sense was developing.  They were sharing their reasoning without me asking them to.  I saw some big gains on benchmarking and standardized testing for several students and I attribute them to these “friendly” and accessible resources.

Small Groups and Discussions
When I completed my initial National Board Certification in 2002, I quickly realized small group discussions provided a definite means to seeing student thinking.  It was a chat last summer, that made me realize I needed to quit butting-in.  I would hear a misconception and jump to add my 2 cents rather than allowing them to reason out if they were correct or needed to adjust.  I was stealing their learning opportunities! Yikes.  I began listening more-offering questions rather than telling them the direction they should go.  It was frustrsting for some students.  They despised me answering their questions with questions.

5 Practices for Orchestrating Productive Mathematics Discussions (Smith & Stein, 2011) is a quick read that offers samples to incorporate into your classroom. The 5 practice provide structure to help you develop discussion based tasks rather than step-by-step inquiry lessons.

Another valuable resource for me are the Formative Assessment Lessons provided by Mathematics Assessment Project.  Most lessons follow a similar format to the #5pracs.  I used to struggle offering questions that would move learners forward.  Though some disagree with scripted lessons, this resource supported me with sample questions for specific student misconceptions.  As a rssult, I began asking better questions on my own.

Another aspect of the FALs is the way they suggest grouping students, not by ability, but similar thinking – whether it be similar misconceptions or approaches to a problem.  This supports what I have been reading this summer with Ilana Horn’s Strength in Numbers (2012).  She presents how social status in the classroom may actually hinder student learning and achievment.  I believe grouping students homogenously by approach and thinking puts them on equal playing fields to share and build their ideas. 

By observing student responses and listening to their discussion, I am able to select and sequence ideas for them to share that will allow more engagement from the class as a whole.  Students are able to listen and view strategies similar to their own, but also consider new approaches which in turn builds their own skill set and toolbox for thinking.

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The common thread is to not to do all of the talking, but to sincerely listen to my students and their thinking.

Gallery Walk #ppschat Challenge

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A common theme in many chapters of Powerful Problem Solving is Gallery Walks. Several techniques are offered throughout the book, but the common goal is to allow students to view their classmates’ approaches to problems.

One of my faults with online book chats is lack of follow-through. I can sometimes use an extra nudge of accountability. There are often so many great ideas and strategies in the books we are chatting that I get overwhelmed and not sure where to begin. Advice: pick 1 thing. Try it. Reflect. Revise. Try it again.

So here is my attempt at a gallery walk. I simply cut apart a pre-assessment for a Formative Assessment Lesson and each pair of students taped it to a large sticky note, discussed and responded. I was confident in many of the questions, but my goal was to identify the few some students were still struggling to understand completely, mostly questions involving transformations.

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1. The large majority are fine with creating a possible equation, given the x-intercepts.

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2. Initially these students tried -6, -4 and 2 as their intercepts. I asked them to graph their equation then reread the instructions. Oh. They had read write an equation, looked at the graph for possible intercepts and failed to read the y-intercept of (0, -6). One quickly stated the connection between y-intercept and factored terms and was able to adjust their response with ease. I believe it happens often to see a graph skim question and think we know what we’re supposed to do, only to realize skimming sometimes results in miseed information.

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3. Within the lesson, many students quickly realized when a factor was squared it resulted in a “double root” and the graph would not actually pass through the x-axis at that point.

The 4 transformations seemed to causes the most disagreements. These were the ones we discussed folowing our gallery walk. However, it was during the gallery walk most students were able to adjust their thinking.

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4.i. Listening to students as they were at the poster helped me realize there was not a solid understanding of the reflection across x-axis and maybe we needed to revisit. Possibly, they are confusing with across y-axis?

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ii. A few students disagreed initially, but the convo I overheard was addressing that changing the x-intercepts was not sufficient, they looked at the graphs, then said, the functions needs to be decreasing at the begiining, that’s why you have negative coefficient.

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iii. Horizontal translations always seem to trick students up. One disagreement actually stated ‘they subtracted and did not add.” Of course, we definitely followed up with this one.

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iv. This pair of students argued over which one was right. The expaned version or factored form. Simple, graph the new equations and compare to see which one translates the original up 3 units.

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A1 & A2 I believe they’ve got this one.

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B1 & B2 some confusion here due to the extra vertical line in the graphic. This student was also interchanging graph & equation in their statement.

I thought the gallery walk was a good task to overview some common misconceotions. It was not intimidating, students were able to communicate their ideas, compare their own thinking to others. I truly tried to stand back and listen. They were on task, checking each other’s work. Each station allowed them to focus on one idea at a time. They were talking math. Most misconceptions were addressed through their discussions or written comments.

Having a moment to debrief the following day highlighted the big ideas students had addressed the previously and reinforced the corrections they had made. This was so much more valuable than me standing in front of the room telling them which mistakes to watch for. Their quick reflection writes revealed majority have a better ability to transform the functions, which was my initial goal for the gallery walk. A few still have minor misgivings that can be handled on an individual basis.

Forming Quadratics lesson from MAP

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I have planned to share this lesson for several weeks but time has gotten away.  My students were not where they needed to be with quadratics, so I pulled together some tried and true tasks-framing quadratics, Wylie Coyote, et al and a new one from Mathematics Assessment Project called Forming Quadratics. You can download lesson, domino cards and assessment in that link.

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No big surprises on the pre-assessment, but I did use it to place students in pairs based on similar thinking/reponses. There are 4 equations students are asked to match to 4 graphs and explain their matches.

I like this lesson for a lot of reasons. Discussion of how different forms give us different information. Allows students to seek key features from graphs, connecting them to parts of different but equivalent forms of equations. Students work in their pair but also must visit other groups to confirm/dispute their responses. The lesson outlines its goals:

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This lesson should be used after students are familiar working with different forms of quadratics. This is not an intro lesson, but one I see being successful about 2/3 way through unit or as a follow-up/review activity. They will encounter standard, factored and completed square/vertex forms.

I followed the lesson pretty true to outline, changing only minor things based on my classes. After the whole class intro, pairs worked at matching dominoe-style cards including sets of functions and graphs. I was adament about them taking turns explaining their matches. Some cards had all equation forms, some had only parts. They recorded their matches on a card for the next round.
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Following the initial round, one person stayed and another person moved to a different group. In the new groups, they were asked to compare responses, then discuss any differences. This took only a few minutes. Upon returning to original partner, they now had to fill-in missing information on the equations. Again, upon completing their equations, one person stayed and the other traveled to a new partner to compare. Some a-ha’s came about during this part as they maneuvered between the different forms, such as the last term in vertex form does not necessarily correspond to the y-intercept as in standard form. So if and when would they be the same was a nice question for discussion.

As an exit slip this day, students were asked to fill-in front side of this foldable for their INBs.
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The following class, I pased back their foldable and gave them a few minutes to respond to my feedback. They received smaller copies of the dominoe cards to cut apart and match inside their foldable. They were asked to write any missing equations, and Color With Purpose different parts of equations and graphs.
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I used the same cards and was able to offer some feedback on simple mistakes, but in the future maybe I should have a fresh set and use it as a true formative assessment to see they are able to match new sets & write new equations.

A panel on the trifold was a place to record/review other important info concerning quadratics.

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Classes were still somewhat split in the post assessment. 1/3 were right on track, 1/3 had trouble with writing the equations, 1/3 seemed almost clueless- I was like “what happened?” They couldn’t correctly identify key points from the graphs. Before passing back their work, I asked what made it difficult? Even if their matches were correct, they failed to give correct coordinates of key points. Their responses all very similar “there were no values on the graphs.” I passed them back and allowed them to talk over feedack with nearby students. After speaking with them individually, I was convinced all but a couple were now moving in the right direction.

Hmm. Maybe next go, I should scaffold the assessment. Part I, very similar to their practice, including labels on graphs. Part II, similar to current assessment with no labels on graphs. Part III writing equations for given information or identified points from graphs.

All in all, I was satisfied with the discussions students were having; How they had to explain their reasoning for matches made. I see these conversations prooving valuable as we continue in our next unit on other polynomial functions.

A post of the same lesson from Ms. Rudolph.

Equations of Lines FAL

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So this is from a Formative Assessment Lesson from MARS site a couple of weeks ago. 

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As I think back, the pre-assessments were very lacking, some even left blank or only minimal scribbles.  Their post-assessments were much better.  They were more confident in manipulating equations to a similar form so they could more easily compare, picking those that were parallel and those perpendicular.

However, a handful really struggled with the given graph in the lesson.  It had 3 lines without the x- & y-axes. 

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Part of the task asked them to place & label the axes on the graph.  Some actually drew the graph and all lines forming the rectangle outside the given graph, then transferred their work to the graph.  Interesting.  It seemed easier for them to graph the entire thing than to simply add the missing information.  I wonder why?

Several a-ha’s were noted throughout the lesson.  Students thinking opposite slopes would be perpendicular, how to find the x-intercept, in the beginning naming equations like y+4x=3 and y= 4x+5 as parallel.  It was definitely a task where I had to bite my tongue, let them struggle a little, then ask questions without telling them how I did it.

As I look over the first sort, I recall several having trouble getting started simply because the equations were in different forms.  Once they realized putting them in similar forms would allow for easier comparisons.  I gave them the categories for the sort, but I wonder how they would have sorted them had I chosen an open sort?  One reason I chose to use the lesson’s headings was because a couple served as quick reviews of checking to see if a point was on the line and how to find the x-intercept.

Would a better assessment be to create equations (not in slope-intercept form) to fit it given categories?

Purple Circle Card Sort

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Last spring, I placed equations of circles after distance between 2 points.  The idea came from a mini-investigation in my Discovering Geometry book (formerly Key Curriculum, now Kendall Hunt). 

Earlier in the semester a new colleague shared the success her students experienced with the Formative Assessment Lesson Equations of Circles 1.  I decided to use this lesson…

In my early geometry class yesterday, we literally stared at circles.  It felt like a wasted class.  No matter what example I referred back to, or what question I asked, it just didn’t work.  Thankfully, I had planning immediately following and I was able to reflect very quickly.  For my last geometry class of the day, I adjusted my sequence of leading examples.  Reviewing our previous work from last week. 

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The remainder of the lesson went smoothly.  A quick white-board quiz at the beginning of class today allowed me to address some small errors.  Once again, I had them create their own notes/examples in their INBs.  Yes, a few are still lacking, but the majority are very thorough in what they are including.  Asking questions about specific what-ifs, like one student brought up none of our examples today had a center located at the origin, so I asked the class if they could remind her.  Several went on to include a similar example on their page.

The lesson continued with a collaborative pair.  They were given 12 equations to sort by center and radius.  There were 4 blank blocks in their grid that required them to create their own equations.  At the beginning, some were “cheating” so I stopped them to remind them 1 person picked a card, explained why they were placing it, the other person had to agree and understand before taking their turn.  They are getting better at disagreeing and telling why when their partner is making a mistake.

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Their assignment was to create an artistic picture incorporating 5 different circles and listing their equations on the back. Short, sweet, simple.  Can’t wait to see them.

Addressing Questions about Formative Assessment Lessons

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Rather than go through a gazillion tweets, thought I would share my thoughts here.

The Formaltive Assessment Lessons I have shared I  the past come from the MARS site.  You will find tasks, lessons even sample assessments.

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If you are just visiting the site for the first time, I would encourage you to spend a bit of time in the Professional Development modules.

     “Module 1 intoduces the model of formative assessment used in the lessons, its theroetical background and practical      implemention.Modules 2 & 3 look at the two types of Classroom Challenges in detail. Modules 4 & 5 explore two crucial pedagogical features of the lessons: asking probing questions and collaborative learning.” MARS site description.

The assessment tasks are shorter, but still allow for some amazing mathematical discussions, especially when implemented using the format of #5pracs model.  Tasks are organized by levels with the expert involving a wider range of Mathematical Practices, less structured and requires more problem solving /centent knowledge.  Where as the novice seem to be more straight forward, provide a bit more structure.  Which task to choose? Well, it depends on your students and your purpose of assessment for a particular standard.

The classroom challenges (FALs) are much more lengthy.  Usually 2, even 3 day total for complete implementation of the lesson.  I am okay with ‘sacrificing’ this time when students are engaged, having mathematical conversations.  The productive struggle they may experience causes the ideas to stick with them.  For example, this year, some of my Algebra 2 students referred back to a lesson from their 9th grade year about “Tom” which was a lesson on time-distance graphs…one of the first FALs I ever attempted.  How many other lessons have I taught over the years that truly stuck with them?  Monster Trucks @mathprojects, definitely, but my lecture, notes, worksheet practice…never.

FALs are either problem-solving based, usually students attempt a problem individually, then in a pair or small group, then they analyze student samples of the same task.  The concept development often uses cards sort activities.  Using these have impacted how I present other lessons as well.  I see the value in student discussions and sharing, allowing them to create their own ideas rather than me telling them every single step.

Like any other resource, FALs can be modified to fit your learners. However, I have seen greater impact on learning when I follow the layout of the lesson closely.  Teachers have tested these lessons, anticipated student strategies/misconceptions and even outlined possible questions you may ask to move a learner forward.

Each FAL is outlined to show intended learning goal, along with mathematical practices that will be evident in the lesson. There is prep time involved. Don’t think you can download the night before, make copies  before class the nest day and begin. 

Ideally, the FALs would be placed about 2/3 through the corresponding unit of study.  I have found them to be very eye-opening to my students’ thinking.  Some FALs require some pre-requisite skills, so you must go through the lesson in order to see what the students will be doing.

Also, if you are an Algebra 1 teacher, dip back into 7th & 8th grade for some great lessons, especially if you are in the transitioning phase of CCSS.  I especially like Increasing Decreasing Quantities by a Percent, Interpreting Distance Time Graphs, Modeling Situations w Linear, Representing and Combining Transformations Equations from middle school lists.

If you have specific questions, please share in comments.  I am no expert, but I have implemented enough of the lessons in the past 3 years to know they have a place in my classroom.

Pam Wilson, NBCT
Currently Reading
5 Practices for Orchestrating Productive Mathematical Discussions, Smith & Stein
Teach Like  a Pirate, Dave Burgess
From Ashes to Honor, Loree Lough