Honored to teach Honors Precalculus

A little over a year ago, I outlined a curriculum for a regular precalculus course for use by two other teachers in our department.  I hoped to help better align student readiness for AP Calculus as prior students arrived deficient in trigonometry skills, specifically, as well as knowledge and skill with functions, in general.  Students who enter my AP course this fall will be the first cohort through the revised, regular precalculus curriculum.  The students who come from honors precalculus will bring their experience with a curriculum developed separately by another teacher who is no longer with our department; although over a three-year period the BC teacher and myself, from time to time in response to requests for our thoughts, helped guide the honors curriculum she developed.  I so look forward to a better prepared cohort this fall.

Happily for me, nearly 100 students will arrive in one of my three, new honors precalculus sections in slightly over two weeks.  In preparation for their arrival, I outlined a new curriculum for the course leveraging my earlier efforts for the regular precalculus course.  In doing so, I intend for students to experience: 1) trigonometry and a variety of functions in the depth and breadth needed to succeed in AP Calculus, whether AB or BC; 2) several prerequisites for success in BC such as vectors, parametrics, conics, sequences and series; as well as 3) topics in probability and statistics in preparation for AP Statistics.

The following figures illustrate my plan for both semesters.  Excited to teach this new course, doubly so as many of these students will take AP Calculus with me in the following year, I look forward to kicking off the semester soon.

2014-2015 Honors Precalculus Sem1 Plan 2014-2015 Honors Precalculus Sem2 Plan

If anyone has any comments or suggestions related to my plans above, please feel free to share them with me.

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Transcendental Darts (Vi Hart)

Amazing discussion by Vi Hart about the significance of transcendental numbers and their enormity relative to the rational numbers.

 

 

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A Summer Course to Remember

I recently finished teaching an accelerated algebra 1 summer bridge course.  The following images best depict and contrast my experiences teaching the recent summer course and teaching my regular course this past year, in reverse order.

Dry parched desert

 ***Regular School Year Experience***

 

Lush oasis

***Summer Course Experience***

This post captures the nitty-gritty details of the summer course.  A later post will capture my emotions and personal experiences in a more readable fashion.  For now, I needed to describe the summer course in a more anthropological fashion.

Background

Earlier this year, around mid February, minutes after I submitted an application to teach a gifted and talented mathematics summer course hosted at a local community college, my principal asked me if I would teach a summer bridge course for rising eighth graders across our district using the ALearn Catalyst program.  [1]

I agreed to do so with the understanding that an existing curriculum and materials would be provided; in my credential program, I student taught with a similar program, the Silicon Valley Education Foundation (SVEF) Stepping Up to Algebra (SUTA) program, which I enjoyed immensely.  Following up with the principal in early April, I learned the district decided not to go ahead with the Catalyst program due to cost constraints; they planned to create their own blended-learning program.  When May rolled around, I inquired once again about the curriculum and materials as I wanted to get my arms around everything well in advance of the course commencing.    To my surprise, the district administrator overseeing the program pointed out that there must be a misunderstanding: the course needed to cover 160 hours of algebra 1 instruction following our district’s Common Core State Standards for Mathematics (CCSSM) algebra 1 framework and unit plans.  My reply, below, went over like a lead balloon.

After working with the CCSS units and assessments, it is my professional opinion that they are not matched to the vast majority of our rising ninth-grade students’ skills, knowledge, or understanding.  I hope the district recognizes this mismatch, at least for a large segment of our students who are below basic, or far below basic, in their understanding of mathematics.

Given my opinion, the district questioned whether I should teach the course and whether someone else should teach the course.  Fortunately, my school administrator believed in me, and between the three of us, an agreement on how to proceed with the course was reached.  While I had liberty with the curriculum and pacing, I needed to commit to teaching the entire district framework and to using the district’s unit assessments.  [2]

Most importantly, the rising ninth grade students selected for the course would be required to have a B+ or higher in their Math 8 course.  This last point was critical.  Without a certain level of readiness, students would not be able to succeed with the accelerated nature of the algebra 1 course where we cover 160 hours of instruction in twenty school days of six hours each.  In fact, the instructional part of the course was constrained to five hours per day as one hour was set aside for either online use of Khan Academy (three times per week) or college and career counseling (twice per week).

The following table outlines the specific topics covered in the Math 8 course, which served as the foundation upon which students would build their understanding of algebra.  From my perspective, it is a fine foundation upon which to build algebraic understanding.

Math 8 CCSSM Framework

Note that in the table above, for unit 5: Pythagorean Theorem, the unit overview states “[students will] explain the proof of the Pythagorean Theorem and its converse.”  As such, to occupy students at the outset of the course on the first day while we awaited stragglers (nearly a third of students did not show up), I spontaneously drew the following figure on the board asking students to use it to derive a proof for the Pythagorean Theorem.  After waiting five minutes or so, I could see that not a single student knew what to do.  Sadly, this remained the case even after I directed their attention to the four triangles and smaller square inside the larger square.  It was not until I pointed out how to compute the areas of each that one student arrived at work approximating a proof.  While I was not surprised that some students had not seen this proof, I was surprised that none of them had seen it, or perhaps recalled it, as these students came from three different middle schools representing six math teachers.

Pythagorean Theorem Proof

 

The Curriculum and The Plan

As mentioned, the district created a specific curriculum built around the new CCSSM standards.  [3]  The curriculum consisted of five units spanning the traditional concepts in an algebra 1 course sprinkling exponential relationships, function behavior, complex numbers, and recursive notation among a few of the units, and adding an entire unit on descriptive statistics.

Alg1 CCSSM Framework

As an initial plan, I allocated the five units across the available days for the course.

RUTG Original Schedule

Comparing the breadth of the CCSSM standards across all units, assuming coverage in any depth, it is clear the course definitely provides the acceleration desired by students, and the district.  I point this out as I learned of the importance of acceleration in response to a question I posed to the district official overseeing the course where she stated the following, shown in bold with my response following in italics.

The students are interested in acceleration and we are there to support them in doing just that.

In terms of acceleration, I believe this will be an exciting experience for students in this compressed timeframe.  If they are up to the challenge, I definitely will do everything in my power to help them learn and develop mastery with the mathematical concepts, procedures, and processes.

I believe the course delivered in terms of acceleration.  In fact, looking at the summary data further below, it seems the acceleration was more than some students could handle, which did not surprise me.  I believe my eldest son, who struggles at times with rapid instruction, would not have succeeded in the course given how fast we needed to go.

Curricular Adjustments

The initial plan worked well until day 3 when, after grading the students’ first test, I realized they required more instruction on linear relationships to include solving and graphing linear equations and inequalities; ostensibly, students covered this material in their math 8 course well enough to receive a B+ in the course.  After a “day’s” worth of re-teaching, students scored higher on a retake of the test, as shown in the following box and whiskers plots.  [4]

Re-teaching also clearly demonstrated that teaching to the test, especially if the form of the test is known a priori, greatly improves student outcomes, as would be expected; whether it reflects their understanding or not remains to be seen, in my opinion – more on that later.  Lastly, rather than create new assessments and other instructional material for the accelerated course, I decided to leverage existing material I created for the past school year.  Doing so enable me to compare outcomes between the students who just completed the course and summer students.  As such, the names used for units changed those I used this past year.

  • Unit 1: Linear Relationships and Systems
  • Unit 2: Functions
  • Unit 3: Quadratics

Unit 1: Linear Relationships and Systems

The plot on the left is the initial take of test #1 and the right is the retake.  Note the significant compression in the range of scores as well as the increase in the median score.

Test #1 Linear Relationships

 Test #1 Linear Relationships: Initial and Retake Score Distributions

A similar situation occurred partway through unit 2 after an assessment on systems of equations revealed student understanding lower than desired.  The plot on the left is the initial take of test #2 and the right is the retake.  Note the significant increase in the median with a slightly less level of compression in the range of scores.  Again, students were to have studied systems of equations in their math 8 course.  Many of them mentioned that they only spent a day or two on the topic.  Student testimonials like this are highly subjective, typically off by a few days or more, and vary depending upon whom each had for their instructor.  Nonetheless, students had not mastered the material and required extended re-teaching.

Test #2 Linear Systems

 Test #2 Linear Systems: Initial and Retake Score Distributions

Once we completed linear relationships and systems, we moved on to the new content in the course starting off with functions followed by quadratic relationships; we did not have enough time to cover the unit on descriptive statistics.

Unit #2: Functions

Function coverage included function notation, evaluating functions, and graphing functions all spanning linear to quadratic to exponential.  Results on test #3 follow.  There was not enough time to give a whole-class retake on functions.  However, it did not seem necessary as score distributions were very reasonable with the median score approaching 80%.

Test #3 Functions

Test #3 Functions: Score Distribution

Unit #3: Quadratics

The unit on quadratic relationships included solving and graphing quadratic equations written in standard, factored, or vertex forms using the traditional methods: factoring, square roots, complete the square, and quadratic formula.  Explorations of quadratic applications made extensive use of word problems spanning projectile motion to economic models to area problems.

As any algebra 1 teacher will tell you, the second semester of the course exceeds the first in terms of difficulty as quadratics are more challenging for students than linear relationships as many more manipulations and characteristics are involved.  Additionally, the inclusion of more context-specific exercises (spelled “word problems”) creates greater challenge.

This course was no different.  Results for an initial test #4 followed by two whole-class retakes follow.  While aspects of student scores improved each time, improvement with these concepts and procedures did not follow the same pattern as with linear relationships or linear systems primarily due to the increased complexity of the topic and inclusion of a larger set of word problems.

Test #4 Quadratics

Test #4 Quadratics: Initial, Retake #1, and Retake #2 Score Distribution

For the quadratics retakes, while I used a similar test each time, I removed select questions and replaced them with different ones to make sure students were challenged to think about how to approach a question and not simply apply a rote procedure with minimal to no thinking.  These questions did not constitute a large enough part of the test to make the difference between passing or failing the test.  However, they did help highlight the increased understanding of higher performing students.

Example of these types of question follow.

Thinking Question 1

“Thinking” Question: Example 1

Thinking Question 2a Thinking Question 2b

“Thinking” Question: Example 2

 

Exit Exam

On the last day of the course, students completed a thirty-four multiple choice question exit exam that was created by the district.  Formally known as the algebra 1 HE exam, it is used to test middle school students who passed a middle school algebra 1 course to decide if those students should receive high school credit for the course.  Ironically, as a high school algebra teacher in the district, I never saw the exit exam until recently when I asked for a copy for possible use with my summer students.  It is graded on a pass-fail basis when used by the district.  I used it as a comprehensive indicator of student understanding weighting it 30% of the course.

I do not have access to any data about typical student results on the exit exam.  As the test covered all the material in a traditional algebra 1 course spanning linear relationships to linear systems to quadratics, I do know that it challenges many students in the district as those who do not pass the HE exam end up repeating algebra 1 as a freshman.  Nearly 90% of my algebra 1 students during the regular school year fit this situation.

Exit Exam

Exit Exam Score Distribution

No student with a passing score in the course before the exit exam ended up failing.  Likewise, no student with a non passing score in the course ended up passing after the exit exam.

Instructional Activities

I employed a variety of instructional activities throughout the course ranging from the occasional, but periodically necessary, direct instruction to discovery-based learning, both group and individual mostly linked together using a Socratic method of inquiry to check for student understanding as well as strengthen student understanding.  While I believe the “sage on the stage” is necessary in mathematics coursework, especially when covering new topics, it should not be the primary mode of instruction or the majority of instruction.  At the same time, the “guide on the side” approach can be confounding to students, especially when students are so accustomed to the direct instruction method.  I believe one can use aspects of each wrapped within a Socratic-esque method: this is my preferred approach to teaching mathematics.

Group Activities

For the new content (functions and quadratics), I relied heavily upon activities created specifically for the CCSSM by The Charles A. Dana Center at UT Austin.  I discovered their material trawling the internet for Common Core related projects, activities, lessons, etc.  I found their CCSSM Toolbox to be most beneficial.  Students enjoyed these group activities.

The following figures illustrate some of the activities I used from the Dana Center.  All in all, I used over a dozen of their activities in the summer program as a means for students to engage with the material in a method akin to discovery-based learning in quasi-heterogeneous groups; I sorted students periodically by their cumulative scores ensuring a balanced set of skills in each group.  While all students had a B+ in the math 8 course coming into the summer program, student scores quickly distributed themselves along the continuum from F to A.  The pace of the course demanded students master the material quickly.

UTA Dana Center - Nested Rectangles

 

Group Activity 1

 

UTA Dana Center - Calculating Cost

 

Group Activity 2

 

UTA Dana Center - Insects in the Water

Group Activity 3

 

Student Work

The  “Insects in the Water” group activity provided a great opportunity for students to illustrate a poster for group presentations and discussions.  We did not have enough time for students to illustrate or to present each activity in as much detail as these.  Nonetheless, students spent a large part of their time working through discovery-based activities such as these.  I am most grateful to the Dana Center for publishing their work free for download and use by teachers like me.

Insects in the Water 1

Student Group Work 1

 

Insects in the Water 2 Student Group Work 2

 

Individual Activities

While much of our time was spent in group work, there were times where students worked individually on various tasks such as the following activity I created the previous summer for use with my algebra 1 students last school year.  At the time, those students reacted negatively to the task as they felt it was too difficult and too different from their earlier math coursework.  A few complained that I did not give them worksheets similar to what they used in all earlier math classes where they repeated a procedure over and over, what some call a “drill and kill” worksheet.

Fortunately, my accelerated algebra 1 students embraced activities like the following, which I called “Race Car Rally.”

Race Car Rally 1

Race Car Rally 2

Individual Activity Worksheet

 

Perhaps closer to the spirit of a “drill & kill” worksheet, for my summer course, I also leveraged various task sheets similar to the following, which was created by another math department, who shared work within their department.  I wish I had access to this material during the school year, for it would have saved me an immense amount of time and effort creating my material.  I believe this math department’s format, material, and approach is excellent and should be available to every algebra 1 teacher in the district.  It is not.  Why I simply do not know.  This is one of the mysteries of teaching I do not fully understand.

Translations

Student Worksheet on Translating Function

 

Assessment Data

As with all of my courses, students took diagnostic tests the first day of class.  The following series of graphs depict various aspects of the data from the diagnostic exams.

The first graph shows the diagnostic scores for each student using a color coded diamond.  Green diamonds are scores for which I estimated a student would pass the course with an A or a B grade.  Yellow diamonds correspond to scores where students might receive a C or a D grade.  Finally, red diamonds represent scores for students who were likely not going to pass the course.  While some may be reluctant to predict possible outcomes for students based on a diagnostic exam, for fear of prejudicing instruction or how one relates to a student, I find it critical to have an idea of who may struggle, or excel, in the course to help direct differentiated supports to the extent they may help.  [5]

Additionally, the mean (average) and standard deviation for each diagnostic are plotted as a black four-pointed star.  Note that all the green diamonds are at or above average on both diagnostics all the yellow diamonds are below average on one or both of the diagnostics.  The red diamonds are far below average on both diagnostics.

RUTG Diag Scores & P-F Estimates

Diagnostic Scores Coded by Estimates of Course Grades

The second graph overlays three cross-like symbols on the data where each symbol represents the average, minimum, and maximum score for students from the three middle schools from which students were selected for the course.

RUTG Diag Scores with MS Bars & P-F Estimates

Diagnostic Scores with Middle School Symbol Overlays

Note the differences in the means and the range of scores for the middle schools.  Let’s call them middle school B (blue), green (G), and purple (P).  Students from middle school B consistently scored above 80% on each diagnostic with the highest average on each diagnostic of 94%; middle school B exhibited the least variance in student scores.  Students from middle school G scored next highest in terms of average score with a wider variance in student scores.  Students from school P scored close behind middle school G, about 7.5 percentage points radially, but higher in terms of score variance.  Middle school G

The third graph codes each diamond to correspond to the student’s middle school using the same color code as used for the overlays.

RUTG Diag Scores by MS Bars

Diagnostic Scores Coded by Middle School

School B is a traditional, neighborhood middle school that feeds into the top performing high school in the district.  School G is a magnet school that feeds into the top two performing high schools in the district although as a magnet, students may come from outside the local neighborhood and likewise return to their local high school upon graduation.  School P is a traditional, neighborhood middle school that feeds into a local, traditional, neighborhood high school.

School B’s student makeup consists primarily of white and Asian students with less than 20% of students representing underserved populations.  School G, as a magnet school, draws from across the district with greater than thirty-percent of students from socioeconomically disadvantaged families.  School P’s composition exceeds forty-percent socioeconomically disadvantaged with its underserved student population exceeding fifty-percent.  I point these out as a clear correlation exists between student outcomes and poverty.  See here, here, here, here, here, here, and here.  There are many, many more with similar conclusions.  While poverty is not an excuse for low outcomes, and never, ever a reason to lower expectations for a student to learn at their highest, possible level (which may be nowhere near the state- or district-mandated grade level standards), it absolutely impedes student progress.  [6]

Outcomes

On day 1, my first roster listed thirty-six students.  Only twenty-six students showed up that day.  The district added six more students over the next few days.  Over time, as tests were returned, and the course complexity increased, attrition arose leading to a final headcount of twenty-two students.  Of those, thirteen students passed the course with a recommendation for high school credit and placement in high school geometry.  [7]

The figure below includes the final course score for all twenty-two students color coded by:

  1. who passed the course and received high school algebra 1 credit,
  2. who passed the course and did not receive high school algebra 1 credit,
  3. who failed the course, and
  4. who dropped, or withdrew, from the course.

RUTG Final Scores and Diag Scores

Final Course Scores and Diagnostic Scores Coded as to Pass or Fail Status

While I would have preferred that more students passed the course with high school credit, I believe it would be a disservice to any student to move them forward with an insufficient level of understanding of algebra 1.  They most likely would not suffer in a geometry course, depending upon how they performed with linear relationships, and evaluating functions.  However, they definitely would experience difficulty in algebra 2, whether accelerated / honors or not.  I could not with clear conscience place students into this situation.  I do believe there should be other means for students to prove proficiency using trusted, online resources where motivated students may overcome any real or perceived setback in their mathematical placement.  Hopefully, more methods will exist shortly given the explosion of investments in the educational software and online services sector.

Student Feedback

As I will blog about in the future, I survey my students at the end of the course to collect detailed feedback to improve my teaching.  The survey I created most recently for use with last year’s students in algebra 1, algebra intervention, and AP Calculus spanned five or six pages with 75 different questions covering different aspects of themselves (as student), me (as teacher), the course content, classroom environment, etc.

Here is a collection of responses to a few of the open-ended questions on the survey.

COMMENTS RELATED TO ACADEMIC CONTENT

  • I just want to say that i am very thankful or the wisdom of algebra that you gave me.
  • You are great teacher because you make your students think harder instead of just memorizing the content you have to actually learn to pass
  • Please tell more Star Trek jokes.
  • Nothing he is by far the best math teacher i have ever had and it was only five weeks so i can’t imagine how it will be to be with him for like 40 weeks it would be awesome

RECOMMENDATIONS

  • It is good to try to make us think, but try to ask problems that we have seen before more often.
  • Provide more fun group activities and let students talk to each other for worksheets more often.
  • Make sure the people who don’t know the subject well get extra help.
  • Better seat arrangements in a way where everyone is facing forward but still in groups.

FINAL COMMENTS

  • you are a great teacher
  • You are fantastic and you can help many students excel in life. 
  • KEEP DOING A GREAT JOB HOPEFULLY YOU WOULD GET RECOGNIZE AT THE TEACHERS RECOGNITION AWARDS
  • I enjoyed taking this course and it has helped me a lot. Thank you Mr. Math Teacher
  • Thanks for being a really fun teacher!
  • Mr. Math Teacher was a great math teacher and a great friend, he was awesome to see while going to school. 
  • Thank you for everything that you have taught us taking away a month of precious sumer from you. I am very grateful and throughout the course, you have encouraged me and helped me through and I tried my very best. I didn’t study for tests in math 8 but i still did well, but I studied a lot in this course but I didnt get the scores i wanted for every test. i have very high expectations for myself and I wanted a 85% or more but i couldn’t even if i tried really hard. But even though this course was really stressful and hurter my brain a lot, I learned a lot of things and made my knowledge much stronger so thank you and have a wonderful rest of the summer!

Structured Responses

The following figures show student responses to a specific question shown in bold at the top of the bar chart.  These are four of the 75 or so questions asked.  Detailed analysis of the feedback is time-consuming.  Nonetheless, a quick review of the results yields directionally important information to consider for immediate improvements.

Student Feedback 1 Student Feedback 2

 

The quintessential question to ask in a product or service survey is whether the respondent would recommend the product / service.  Seeing as I am the product / service as a teacher, I ask that question at the end of most student surveys.

Student Feedback 3

I rarely receive a 100% recommend result as some students misunderstand the question, note the explanation above, and some truly did not enjoy their experience with me as their teacher.  This situation reminds me of the following saying.

“You can please all people some of the time and some people all of the time but you can never please all the people all the time.”

 

ENDNOTES

[1]  By mid February, burnout had taken full hold of me, sapping me of my passion to teach as a few too many of my algebra 1 and algebra intervention students in each of my three periods were simply too much for me to handle after spending nearly six months attempting to harness their energy in more positive directions.  I saw the gifted and talented program as an opportunity to recharge my passion while making needed additional income as the financial cushion I built up before transitioning to teaching depleted faster than expected.

[2]  Ironically, and to my perceptible relief, the district recognized that teaching the entire curriculum in the five-week period was not realistic, and left it up to the teacher to decide how best to make sure students knew algebra 1 sufficiently to receive credit.  More importantly, the district had no requirements that their assessments be used in the course.  They were surprised when I mentioned that I was told I must use all of the assessments.  It made me wonder how that directive was determined in the first place.  The disconnect between district intent and individual teacher understanding varies considerably.

[3]  A couple of years earlier, I spent a significant amount of time and energy helping our district team develop a framework for algebra 1 that led off with a month’s worth of prerequisite review / relearning.  It is my firm opinion that nearly all students who took algebra 1 in our district required significant intervention immediately upon starting algebra 1 to make sure they could engage with the course content with any hope of success.  Someone at the district decided students did not require the intervention and redesigned the entire framework with the assumption that students who entered the course were proficient with all prerequisites; assuming all students are proficient with prerequisites, the framework should serve students quite well, however, this is not the case for a significant percentage of students taking algebra 1 in high school in our district.  Additionally, the new framework added topics pushed down to the algebra 1 team by the algebra 2 team making the framework content a tremendous stretch for our students.  While advanced students who mastered mathematics up to the point of entering the redesigned algebra 1 course could succeed with the content as designed, most remaining students would likely struggle to reach a basic level of understanding, much less proficiency with the material.  Common Core will not improve educational outcomes without adequate preparation leading to student skill with prerequisite material, which has been the Achilles heel of algebra courses nationwide.

[4] Note that in each of the box and whisker plots that the vertical scale starts at 0% and rises to 100% in intervals of 10%.  The green, horizontal line in each plot represents the minimum passing score for each assessment, which is 55%.  The following figure from our online grade book software helps explain other aspects of the box and whisker plot.

IC BWD Help

 

[5] In many ways, I am stunned at how little information we collect and use in school systems to aid us in providing students with the utmost of service.  It seems the fear of recreating the (disgusting) discriminatory practices of the past, and some would posit present, prevents school systems from collecting and utilizing information rich data that could benefit student learning tremendously.  In this day and age, it is a shame we cannot develop a system that enables the pros of this approach while preventing the cons.

[6]  Too many ignorantly skirt around the contribution poverty and its companions present to student learning.  Until we accept reality, and develop processes and systems to facilitate learning at what Vygotsky termed the “zone of proximal development (ZPD)”, students living in poverty will continue to fall behind in their educational outcomes.

[7]  Of note, except for one anomaly,  no student who scored below the class average on both diagnostics passed the course.  It is clear to me that proficiency, and ideally mastery, of course prerequisites are required before students are placed into the next course in any mathematics sequence.

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