Wednesday, May 23, 2018

L to J -- Year 1 Reflection

I just wrapped up my first semester of implementing a strategy called "L to J".  (Read more about how and why I implemented it here.)  It's now time to reflect back on how well or not-so-well it went.

Feedback from Students:
As part of the "Mr. Kreie Report Card" I ask students to complete (S/O to The Classroom Chef), students were asked the following question and were able to answer anonymously.

Their responses:

Further, I asked students for more open ended feedback.

Some positive quotes from students:
  • "It helped me remember stuff from early in the year and how to do things from first semester."
  • "I think it was good. The fact that we kept reviewing the same problems and slowly learning the math later in the year really helped me. Plus, once we got to our all time best that was really fun."
  • "I think it helped to review the learning targets throughout the year and that it should be used again next year."
  • "I loved it. It was a great improvement in class."
  • "I think that L to J was kinda fun. I found it a little frustrating when we didn't know the answer, but that was kinda the point. I think we should do it next year."

Some not-so-positive quotes from students:
  • "I did not really like L to J.  It got really boring."
  • "It was the biggest waste of time. Get rid of it."

And some helpful feedback from students:

  • "i liked it but i wish it only applied to the semester we were currently in"
  • "It sometimes made me feel stupid because there were somethings that i didnt know that i probable should have. it was good but not my favorite"
  • "L to J was fun but a lot of people cheat and say they get 8's, 9's, and 10's when they really get 2's, 3's and 4's. Peer pressure is a problem."
  • "You should do it again next year but find a new way of choosing questions so we don't repeat the same ones over and over. Maybe you should do it once a month instead of once a week also."

A quick note about the cheating comment:
There were times that students would see the question and be whispering to each other.  I was not very strict during these quizzes because I knew that they are not graded.  I can do better by not allowing students to converse during the quiz.  However, I feel that sometimes valuable learning can take place within those conversations.

Student Achievement:
Many students showed growth as the semester moved along.  Here are a few examples of two student's progress throughout the semester:

Student A

Student B

As you can see, student B scored a 10 the very first week.  It just so happened that in this particular student's class, all ten of the first week's questions were from the first semester.  The second week, however, a number of questions were from the second semester.  This student's scored dropped from a 10 to a 6, largely due to the randomness of the questions.

Class Achievement:
Each of my classes achieved at least one all-time best after setting the baseline during the first week.  The class shown below set only one all time best.  Two factors really influenced the goal of attaining an all-time best each week:
1)  How many students are absent the day of the quiz, and
2) What questions are selected.

Having only one student absent really hurts the chances of achieving an all-time best.  There were some days that I was missing four or five students in a given section (such as week 9 for this class).

Students did show a very strong interest in the overall achievement of the class each week and showed genuine excitement when an all-time best was attained.

The celebrations for all-time bests were a bit challenging for students to think of and agree on.  Students often tried negotiating for rewards -- extra credit, food in class, skipping homework assignments, etc.  I need to do a better job next year of selling the excitement of a celebration.

This reward that one class chose did draw some attention around school for a day...

Final Thoughts:
I did enjoy doing L to J as something new this semester and I plan to do it for the full year next year.  It took about 25 minutes do complete each week, which led to us not covering as many lessons as we have in the past.  {Some weeks we simply didn't have time to fit the quiz in, as evident in the missing weeks on the graphs above.  A few of those missing weeks were due to snow days and shortened weeks for spring break.}

I believe the benefit of the spiral review and "No Permission to Forget" outweigh the cost of skipping a few lessons along the way.  And most importantly, more than 80% of students said they liked doing L to J this year.  I think that's a pretty high success rate.

Saturday, April 28, 2018

3 ACT Tasks

As I am browsing through my previous blog posts, it appears as though I have never blogged about 3-ACT tasks.  I've been using these types of activities for the past five or six years.  I first found out about them in Dan Meyer's blog back in 2011. 

A large majority of the 3 ACT tasks I use come from Dan or Andrew Stadel (follow the links for a list of their activities).   There are quite a few really good ones to use this time of year in geometry, including Andrew's File Cabinet activity that I used in Applied Geometry class on Friday.

I don't have time to get into the fine details, but let's just say that the activity was a hit for my class of student who usually don't get too excited about doing math each day.  I wasn't fully prepared to record a video, but as we started watching Act 3, I could sense the excitement building.  Here is what I was able to record.

Needless to say, it was a fun day in class.  Thanks, Mr. Stadel, for the awesome activity!

(Last summer, I imported this 3 ACT activity into Desmos.  If you'd like the Desmos activity, here you go: 3-ACT File Cabinet Desmos.)

Tuesday, April 10, 2018

When Conceptual Understanding Fails... My Dilemma with Special Right Triangles

I started writing this blog post over a week ago and have realized that I can't give it enough attention until this summer.  So, I am going to post what I have thus far and resume this summer after a little more thought and with a little more time.

I want my students to leave geometry with a strong understanding of the two special right triangles (45-45-90 & 30-60-90).  I know that the two triangles are foundational when it comes to the unit circle and trigonometric functions & their graphs.  I also strive to connect as many concepts together in my lessons as I can; I believe those connections are critical and lead to true conceptual understanding.

This past year, here is how I approached teaching the section on special right triangles.
For the record, my learning target for these lessons are:
  • I can derive and apply the properties of special right triangles.

On day 1, students worked in pairs with their elbow partner to complete this task, which was handed out on paper:

Task: Find the length of the diagonal. 
Express answer as a decimal rounded to the nearest hundredth and a radical in simplest form.

Each group had a different side length on their square.  Groups had sides ranging from 1 to 12.  No two groups had the same square.

At the same time, I handed each student the following table on a full sheet of paper:

Students have no problems identifying the shape as a square.  Groups had very little problem finding the diagonal length.  We had reviewed how to simplify basic radical expressions the previous week, so very few groups struggled with that.  As I roamed around the room, I made sure each group had the correct answer and had found correct place to fill their answer into the table. After I confirmed their answer was correct, I invited one person from the pair to go to the Smart Board and fill in their row of the table.

As students filled in the table on the Smart Board, I asked that each student fill in the table on their paper.  After about three rows were filled in, I could hear rumblings around the room about a pattern students were seeing.  When all pairs had shared their answer, the table looked something like this:

None of my classes had more than 22 students this day, so the last row was blank each hour.  At this point, I lead a little discussion about what we're trying to derive.

I start by having students share with their partner what they think should go in the last row our table.  Again, this patterns is not hard to see, so I hear a lot of "twelve root 2" being whispered.  I call on a volunteer to share and ask to explain their thinking.

I then go to the wipe board that hangs adjacent to my Smart Board and talk briefly about what it is exactly that we're doing today.  I will draw a right isosceles triangle on the board and ask students to tell me what they know about the angles.  I then introduce the term "45-45-90 triangle".  I remind them that because each group's triangle was a 45-45-90 triangle, all of the triangles we are working with are similar via Angle-Angle.  I will put a random, much larger side length on the leg and again ask them to share with their partner what they think the length of the hypotenuse is.
Tell your partner what you think the value of x is.
Again, I hear a lot of "eighty root two" whispered between partners.

At this point, I ask for someone to summarize the relationship between the lengths of the leg and the hypotenuse.  The response I hear is "the hypotenuse is the leg times the square root of two".   I write the relationship on the board and ask students to write it on the sheet with their tables.

I will ask the group that had the leg of length 1 to share what they found for the decimal value of the hypotenuse.  They respond with "1.41".  I will mention that the rule we derived is saying that the hypotenuse is always about 1.41 times as long as the leg (on a 45-45-90 triangle).  I remind students that each group used the Pythagorean Theorem to find the value of the hypotenuse, and that they can always fall back on that process if they would happen to forget the rule we just derived.

That whole process takes about 15 minutes, which leaves about 30 minutes remaining for the next part of the lesson.

I ask groups to turn the paper with their square over and draw an isosceles right triangle on the back.  Then they are to take the value that was their square side and label the hypotenuse of their triangle the same value.  (So the group that had the square with side length of 4 from above would now have this triangle.)
Find the value of x.
Express your answer as a radical in simplest form.

I also have this table included on the back of the paper that has the previous table:

At this point, student thinking typically takes one of two paths.  Some groups will use the rule that we had previously derived and come up with this answer:

Or, they will use the Pythagorean Theorem and get something along these lines (with sometimes a little guidance):

Once again, I floated around to make sure groups are getting one of the two answers above and are filling in their table correctly.  After I confirmed their answer was correct, I invite one person from the pair to go to the Smart Board and fill in their row of the table.

Each hour, I saw a mixture of solution methods and our class table looked something like this:

Prior to this lesson, we hadn't reviewed how to rationalize the denominator.  I know students see the process briefly in algebra 1, but rarely does anyone remember it.  I took the next five minutes and led a discussion about why "simplest form" doesn't include a radical in the denominator and how we can manipulate these radical expressions and rationalize the denominator.  (There is a good explanation on if you're curious.)  

At the conclusion of the discussion, each hour had a table that now looked like this:

With the expressions in this form, students now had no problem detecting the pattern to our expressions.  I then ask students to generalize the patterns they see, and we arrive at the following two equations:

(Side mini-rant: When will mathematicians decide that it's okay to leave a radical in the denominator?  I vote now.  Each year I have students look at me funny when I try to sell them the idea that three root two divided by two is simpler than three divided by root 2.  The simplification rules are extremely outdated... we have calculators now if we should happen to want to find a decimal approximation of these expressions.)

This concludes the lesson.  Students have a homework assignment that I post online that isn't due for a few days later since the assignment also includes 30-60-90 triangles.


The next day, we do a similar exercise to derive the rules for a 30-60-90 triangle.  Students get some practice working with these rules when they complete the homework, and we assess with a quiz a few days later.

Here is my dilemma and questions for the audience:

One of the quiz questions is to find the missing variables in the diagram below.  Many students did well on this question, but I had a handful of students who claimed y = 8 and x = 4 root 3.  The error the students made was they applied the wrong rule.  But what I struggle with is the idea that these same students would have done just fine with this question prior to us having our lessons on special right triangles.  We had previously learned about isosceles triangles and the Base Angle Theorem Converse.  If they were to analyze the triangle and notice that the two acute angle were congruent, they would have concluded that the legs were also congruent.  Then, with two legs known, the Pythagorean Theorem would have led them home.

It's also like the lesson clouded their thinking.  

My questions for those still reading...
1.  How would you improve the first day of this lesson?  What could I change to do better?
2.  Are there any other resources you'd recommend to use with this lesson?
3.  What other insights do you have about my dilemma?

Thanks for reading!

Friday, March 2, 2018

Guest Blogger

I just wrapped up my second guest blogger post for SDSU's Math 371: Technology for Mathematics Educators.  I wrote about how Twitter has become my Personal Learning Network.

Sunday, January 21, 2018

Something new for 2018: "From L to J"

Last October, our district hired Lee Jenkins to come and train our grades 4-12 staff on a school improvement strategy he calls "From L to J".  I had never heard of Lee Jenkins before, but I left that day knowing that I wanted to integrate "From L to J" into my classroom.

In case you're not aware of what "From L to J" is, here is a very brief overview:

  • Teachers write learning targets for the entire year before the year begins; the learning targets for the year are given to students on the first day of school.
  • Teachers write an assessment question aligned to each learning target.  Questions are typically entered into a PowerPoint / Slides presentation.  This is done at the start of year.
  • Each week, students are given an "L to J" quiz.  Quiz questions are randomly selected each week.  The number of questions is a function of the number of learning targets.

A few more details:
  • Each quiz, students track their own progress by plotting the number of questions answered correctly into a histogram.
  • Early on, students may get very few questions correct.  This is to be expected if the learning targets are things that students haven't previously learned.
  • Yes, quiz questions may be over concepts and topics that haven't yet been covered in class.  Because of this, "From L to J" quizzes do not effect a student's grade.  
  • As the year progresses, [theoretically] students will begin to get more and more questions correct.
  • After each quiz, a class total of number of correct answers is calculated and plotted.  Each time the class total reaches an all-time best, there is a "celebration".
  • [If you're wondering where the term "From L to J" originates, it comes from creating a class histogram after each quiz.  On the x-axis is the number of questions answered correctly and on the y-axis is the number of students.  Each student is a data point in the histogram.  Early on, many students should get 0, 1, or 2 questions correct, creating an "L" shape distribution.  Over time, the histogram begins to take a bell-curve shape.  And by the end of the year, the histogram is hopefully shaped like a "J", meaning a lot of students got most or all of the questions correct.  Take a look at some examples here.]

Jane, Jarrod, and I are the three geometry teachers in our high school.  We spent a day writing our learning targets and creating quiz questions.  We have 72 learning targets and decided to create three different questions for each target.  [As of right now, we have one question for each target and are working on completing the other questions.]

Because we are implementing this at the beginning of the second semester, our histograms shouldn't necessarily ever be shaped like an "L".  Next year, when we implement this starting at the beginning of the year, I expect a much truer "L to J" transition.

What was I drawn to with this strategy?
  • In his presentation, Mr. Jenkins talked a lot about how this idea holds students accountable to remember what they have learned.  Too often students believe that once they take the test over a topic or concept, they can forget about it.  Or maybe it's that we [as teachers] allow them to forget about it.  The "From L to J" quizzes provide a systematic spiral review for students.  There is randomness in which learning targets are reviewed, but my feeling was that this review of previous topics is better than no review.
  • Students are aware of exactly what they are expected to learn (and retain) throughout the year.  We have already been using learning targets with the students since the start of the year; this strategy provides a bit more formality to the learning target goals.
  • Mr. Jenkins also talked a lot about how this strategy helps students who have a tendency to struggle.  Even if a student is getting a D or F for a grade, I can point out to these students that they are still learning something.  [Future potential: Standards-based grading!?!]
  • Mr. Jenkins talked about how this strategy scored high on John Hattie's effect size research.
  • There is a little bit of statistics that gets worked into our classes.

What were my hesitations about implementing this strategy?
  • Time.  We're assuming that the quiz each week will take about 15 minutes.  That's 15 minutes a week that I won't have to do other class activities.
  • Quiz questions need to be DOK level 1 & 2.  The questions need to be able to be answered fairly quickly (< 90 seconds per question) and have one concrete answer.  I can't ask questions where students are asked to explain their thinking because it would be too hard to score.  My hands are tied with recall / skill level questions.

Last week, we took our first "From L to J" quiz.  It seemed to go pretty well with the students.  I found the randomness of the questions fun; my first class of students drew a lot of review questions and I had a lot of high score because of it.  My second and third classes had at least two questions that were over topics that we haven't yet covered.  Needless to say, they didn't score as well.  

We will have another one on Tuesday of this week.  I may post an update midway through the semester and tweet highlights and no-so highlights along the way.  I'm excited to try this out and I'm looking forward to doing it for the full year next year.

Saturday, November 25, 2017

The Positive Effects of Blogging

I received an email notification yesterday morning about a new post on Lisa Bejarano's blog.  It's an excellent post that talks about Lisa's internal thought process when deciding how to lead a lesson on simplifying complex fractions. 

A couple of hours later, I see Dan Meyer tweet he commented on Lisa's post on his blog.  I enjoyed Dan's analysis for a couple of reasons.  He offers the thought that it is impossible to practice the process that Lisa takes with her lesson plan.  I tend to agree with that opinion.  Then, I what I really enjoyed was the nugget he left behind in his "BTW" section.

He links an article called "The Positive Effects of Blogging on Teachers".  I hadn't seen this article before and it really hit home with me.  Many of the items discussed relate directly to me and my professional growth and journey. 

Thank you Lisa and Dan for helping me along my way!

Wednesday, November 15, 2017

Desmos Transformation Golf + What My Assessment Looked Like

My geometry classes just recently completed our unit on transformations.  I was super excited to be teaching transformations this year because a few weeks ago Desmos released one of their coolest activities to date: Transformation Golf: Rigid Motion.  If you haven't checked out this activity, stop reading this post and go take a peek.

I wrestled with figuring out the best time to do this activity with students.  Should I launch the unit with it?  Do I do the activity after the unit as a performance task?  When does this best fit?

I decided to do the activity as a review for the test.  My intent was to help students solidify their understanding as well as to allow students to see that there is more than one composition of transformations that will yield the same result.

So I had students complete the activity in class.  I used quite a bit of teacher pacing and paused students often to discuss some of their thinking during the activity.  It's such a fun day to lead.  Students found a lot of pleasure finding their own ways to complete the tasks.  Take a look at the various ways students did challenge #8.

To wrap up the lesson, I had students complete an exit ticket to summarize their thoughts about the activity.  Here are a few quotes from students:
"It was fun and I liked how we got to make different things different ways."
"It made me think in different ways than I normally would."
"I like how it made me think outside of the box and creatively."
"I liked that it was a little bit of a challenge."

As part of the unit test, I wanted to give students some problems that were similar to the Transformation Golf activity.  I wanted students to have the opportunity to be creative and get the correct solution in more than one way.  At the same time, I wanted the problems to be a bit challenging and I wanted to assess the students' understanding of transformations on the coordinate plane.

So I created six different problems that consisted of a pre-image figure and its image on a coordinate plane.  (A link to the test problems is here.)  Students were required to provide the list of steps needed to map the pre-image onto the image.  The level of precision expected was this: for translations, I needed the translation rule.  For reflections, I needed the equation of the line of reflection.  For rotations, I needed the center and degree of rotation.  Counterclockwise rotations were the default; students could rotate clockwise if they desired, as long as they noted the direction.

I was really stressing out about grading these problems because I knew there were many correct answers.  I wouldn't be able to have one answer as a key; I would need to check each problem with a fine-toothed comb.  With over 80 geometry students, I was worried about how long this task would take me.

Here is a sample of some the student responses.  All of these solutions are for the same problem.

As it turns out, I found great joy in grading these problems.  Yes, it took a bit of time...more time than it would have had I given my students a multiple choice assessment.  But to see the creativity, thinking, and effort that students demonstrated was well worth my time.

I won't lie and say that all students did awesome work on this assessment.  A common error was not being specific about the location of the center of rotation.  [Often times the students intended the center to be the origin, but didn't specify.  These errors led to a good conversation about precision.] 

A few students who struggled mentioned that this assessment was tougher than the Desmos activity for two reasons.  First, checking their work on the assessment was a bit tougher than checking on Desmos.  The Desmos activity provides immediate feedback when a student performs a transformation.  Second, the transformations on Desmos did not require use of coordinates, equations of lines, etc.  I have a handful of students who still struggle with writing the equation of a line.  They are able to draw / sketch the line of reflection when given a pre-image & image, but they are not able to write the equation of that line very well.  These students were able to complete the Desmos activity without too much problem but struggled to complete this assessment correctly.

So, team at Desmos, here is my request.  I LOVE the Transformation Golf activity.  It made teaching transformations incredible enjoyable this year.  I would love to see a "Transformation Golf: Round 2" activity that includes the x- and y-axes on the grid and that requires students to provide translation rules, equations of lines of reflections, and coordinates of centers of rotation in order to perform the transformations.  My thought would be to have students start with the existing activity in order to learn some of the general transformation tools, and then send to the "Round 2" activity that ramps up the precision.  Thanks in advanced!  ;-)