All posts by kdcampe

Summer Math Refreshments

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In my previous post, I set out an ambitious Reading List of math and education related reads.  So far, I’ve made only fair progress; because of daily life but also because of other opportunities for fun and enjoyment with math online and in person.  Read on for some diversions and refreshments to include with your summer pursuits.

Online “Events”:

Two online opportunities that I enjoyed immensely last summer are back again for 2019.  First is the #MathPhoto19 weekly photo sharing on Twitter.  Erick Lee (@TheErickLee) is hosting weekly prompts asking for photos on all sorts of math-y topics, such as Circles, Estimation, and even Beauty.  “The Coffee Porch” is my favorite summer location for my math diversions (and this week’s entry for #Lines).

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Stay up to date on twitter with #MathPhoto19 and check out the archive of previous years at mathphoto19.wordpress.com.

The second event is the return of the Big Internet Math-Off organized by the folks at Aperiodical.com.  Last summer, sixteen mathematicians shared a fun math(s) pitch in a short blog post and/or video format.  Every topic was captivating, from origami and hexaflexagons, to airplane seating, phantom parabolas, mathematical modeling, and more.  The only downside was that every face-off resulted in an interesting mathematician going home, so there is a new format this year.

This time around, there is a “group stage” so every participating math person can give three presentations. Then on to the semi-finals and finals.  The full list of sixteen “players” and schedule is here, and the fun begins on July 1.  Follow on twitter using #bigmathoff  and @aperiodical.wallchart-2019-1-1024x724-border

Podcasts:

Listening to math conversations via podcasts is another way to enjoy math on-the-go, whether you are walking the dog in the early morning like me, traveling to a vacation spot, or even while cooking or working around the house.  I’m catching up on some that I’ve missed from Mr. Barton Maths Podcasts with Craig Barton (@mrbartonmaths) and Estimation 180 with Andrew Stadel (@mr_stadel) and his math minions.

I’m not alone looking for good listens; this thread from @JennSWhite on Twitter gives some more suggestions (too many to list here, so click through!).  Consider loading up a Global Math Department webinar podcast, or Make Math Moments That Matter makemathmoments.com/podcast and enjoy.  And Craig Barton has recommended the Odds And Evenings podcast for “cracking puzzles and 100% math goodness”.

Blogs (Reading &/or Writing):

Summer is a great time to reflect on the teaching year that has gone by, and one way to do this is to dust off the neglected blog and write about some great teaching and learning experiences you meant to share along the way.  What will you do again?  What needs changing?  What difficulties did you face and/or overcome?  Many teachers in the #MTBoS* community have commented on Twitter that they plan to catch up on their writing (and the challenge of remembering what happened during the academic year!).  And even if you aren’t writing, catch up on reading the blogs you bookmarked during the year when there wasn’t time to process.

Chats & Discussions:

Take part in the many chats and discussions happening on Twitter… about books you are reading or how you solved a math problem.  Here are some to check out:

#MathStratChat with Pam Harris @pwharris every Wednesday evening.

#NecessaryConditions and #MathRecessChat for slow chat discussions on those books with a weekly schedule this summer.

inf powersThere is also a summer reading group chat on the book Infinite Powers by Steven Strogatz (@stevenstrogatz).  Learn more here.

The hashtag #EduRead is being used as well for discussing educational books by teachers on Twitter.  Or just search #MTBoS or #iTeachMath to find others thinking about the same things as you; use #ElemMathChat #MSmathchat #GeomChat #Alg1chat #PreCalcChat etc. to specify your audience.

Puzzle Play:

I love puzzles and I spend many hours engaged with them, especially when they have a math or logic angle needed to solve.  I recently discovered the daily MindGames puzzles from The Times UK [website here] along with their book collections.  Cell Blocks is a great visual brainteaser, whose object is to divide the grid into square or rectangular blocks, each containing one number and made up of that many cells (image shows a solution; puzzle starts without any of the dark boundaries).CellblockMy current favorite is Suko**, which is a 3-by-3 array for the digits 1 through 9.  The number in each circle must equal the sum of the four surrounding cells, and the total of the colored cells must match the color totals given outside the array.

For example, in this puzzle from www.transum.org , the three blue squares add up to 17, and the four lower left squares add up to 23.  Thus the value of the green square in that section must be 6.

The two red squares sum to 12, and the four lower right squares sum to 24, so the two blue squares in the middle column must also equal 12.  Since the three blue squares add up to 17, that leaves 5 for the lower left blue square.  And so on.

And if this isn’t enough to keep you going, one more puzzle idea is to try the Daily Math & Logic Challenges from @brilliantorg.

Whatever math you teach, summer is a wonderful time to reflect, refresh, and recharge.  Hope you enjoy these math entertainments!


Notes and Resources:

#MTBoS stands for “Math Twitter Blog-o-Sphere”, the online community of math teachers on Twitter.  Ask questions, find resources, & discuss issues about teaching math, and follow @ExploreMTBoS for more.

**  Suko, was created by Jai Kobayaashi Gomer (@kobayaashi2018), of Kobayaashi Studios, in late 2010. Website with information on Suko & other puzzles is here.

I’ve found these websites with interactive Suko puzzles, listed above and here again: TIMES Mindgames, the Sunday TIMES Brain Power, and www.transum.org.  Printing is also an option.  Warning, these puzzles are habit-forming!

Reading List

books

Summer is around the corner (and already here for some educators) and it is a great time to dive into some reading that you have been meaning to get to throughout the school year.  Here is my list:

 

Teaching Practices:

The first two books on my list will be discussed this summer in Twitter Slow Chats.  This is is a great motivator for me to get reading and join a thought-provoking discussion with other educators.  This chat format is very accessible since you can be part of the discussion anytime during the week; there is not a particular time of day or week that you must be available.

Necessary Conditions by Geoff Krall (@geoffkrall) will be discussed starting May 27, with one chapter each week through August.  Use the hashtags #NecessaryConditions and #StenhouseMath to join in.

Math Recess by Sunil Singh (@Mathgarden) and Christopher Brownell (@cbrownLmath) will be on a faster pace, starting May 29 and finishing up in early July.  Use the hashtag #MathRecessChat.

Routines for Reasoning by Grace Kelemanik (@GraceKelemanik), Amy Lucenta (@AmyLucenta), & Susan Creighton is my third book focused on teaching practices. This past winter Connecticut’s ATOMIC math organization did a book study on the book which I facilitated.  The prompts and discussions were open to all and still available here. You can download a sample chapter from this link. I am overdue writing a blog post containing my thoughts on the book and our discussion, but I will link it here as soon as I get to write it.

routines

In case you missed them, some other wonderful books on teaching practices include Becoming the Math Teacher You Wish You’d Had by Tracy Zager (@TracyZager), and Motivated by Ilana Horn (@ilana_horn).

Cognitive Science:

I am extremely interested in the power of applying cognitive science research to education practices, and I’ve been trying to use these ideas in my work with students. I’m looking forward to the upcoming release of Teaching: Unleash the Science of Learning by Pooja Agarwal (@PoojaAgarwal) and Patrice Bain (@PatriceBain1). Here is a link to their planned summer book discussions.

powerful teaching

Others in this genre that I’d recommend are Make It Stick: The Science of Successful Learning by Brown, Roediger, & McDaniel; Why Don’t Students Like School by Daniel Willingham and How I Wish I’d Taught Maths by Craig Barton (@mrbartonmaths). Also check out the great cognitive science resources and podcasts on his website.

Equity & Race

It is becoming increasingly clear that issues of social justice and equity cannot be sidelined away from discussions of pedagogy and math teaching practice.  We teach our students within the context of their lived experiences, background, and perceived privilege, and it is urgently important that we consider this factor in our teaching. Therefore, I have several books on my list this summer that consider equity, privilege and race.

  • This Is Not A Test by Jose Luis Vilson (@TheJLV)
  • Multiplication is For White People by Lisa Delpit
  • White Fragility: Why It’s So Hard for White People to Talk About Racism by Robin DiAngelo
  • Building Equity: Policies and Practices to Empower All Learners by Smith, Frey, Pumpian, Fisher (ASCD)
  • Blind Spot: Hidden Biases of Good People by Mahzarin Banaji & Anthony Greenwald

If you are diving into some of this work, check out the Twitter account @ClearTheAirEdu, and use the hashtag #ClearTheAir.  Val Brown (@ValeriaBrownEdu) has posted a complete discussion guide for White Fragility here, and there are other discussions and resources on the Clear The Air website.

The Joy and Influence of Math:

This category includes a range of books, all touching on the power of math and its influence on us as individuals and on society.

  • Infinite Powers by Steven Strogatz (@stevenstrogatz)
  • The Art of Logic in an Illogical World by Eugenia Cheng (@DrEugeniaCheng)
  • Hello World: Being Human in the Age of Algorithms by Hannah Fry (@FryRsquared)
  • Mind and Matter:A Life in Math and Football by John Urschel (@JohnCUrschel)
  • Math with Bad Drawings by Ben Orlin (@benorlin)

Puzzles and Fun:

I am a huge fan of number and geometry puzzles for entertainment, and I buy way too many books in this category. Some of these book authors post puzzles on Twitter (for free!), and people post their various solutions and methods.

So, as school winds down and summer days stretch out ahead of you, please join me in picking a few books for reading, learning, and enjoyment.  There’s something for everyone on the list!


Notes:

My post “Summer Assignment” from 2016 also has book recommendations.

Another list of math-related books is from Math Frolic here.

I could not fit this into another category, but wanted to mention Adding Parents to the Equation: Understanding Your Child’s Elementary School Math by Hilary Kreisberg (@Dr_Kreisberg) and Matthew Beyranevand (@MathWithMatthew).

 

Grading Guidelines

On Twitter last week, Kristen asked, “What are your grading categories and what percent of the students’ final grade comes from each?”  Kristen is a 7th grade math teacher and was looking for suggestions because she wants to change her approach.

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When I was in the classroom full time, I used a grading system that I had revised and honed over several years, that worked for me and my students.  Before I describe its details and justifications, let me first say that each school and classroom has its own considerations, so every teacher should do what works for your situation.  Also, I mostly taught high school math, so some of this would be different for middle school and 9th grade.¹

POINTS NOT PERCENTAGES:

In my math classes, every test, quiz, lab activity, or hand-in assignment was worth a certain amount of points.  Quizzes were usually 10–40 points, Tests 50–100, Labs/Hand-ins were 20–50.  To find a student’s average, add up all the points and divide.

clipboardStudents need to show the mathematical thinking (“show your work”) in order to get full credit.  If they show good work, but have the wrong answer, they might earn 4 out of 5 points.  If they have the right answer but without supporting work, they only earned 1–2 points.  Any mistake that is carried through to later parts of a problem without making new errors does not get new deductions (akin to the AP exam free response question grading).

Each question on a test or quiz is worth whatever the mathematics warrants, from 1 to 10 points.  Thus every similar question throughout the marking period has similar weighting, it doesn’t matter if it is on a “test” or a “quiz”.  This is one reason I changed from “Tests 50%, Quizzes 30%” type system; in that setup, the same question on a test can be worth much more than on a quiz.  Also, I sometimes only had two tests in a marking period, which didn’t seem worthy of half of the student’s grade.

Lab activities and hand-in assignments meant that a student’s grade did not just depend on timed “higher-stakes” assessments.  Students with test anxiety could demonstrate their knowledge in another way, with less stress and time pressure.  I usually had one of these every week or two throughout the marking period.

HOMEWORK:

I checked homework daily, for completeness (not accuracy). During the few minutes I took to get around the room, students were discussing the homework with partners or small groups, checking answers from a key, and resolving misunderstandings.  I marked down if homework was complete, incomplete, or not done.

The student’s homework results moved their grade up or down from their class average in a range from +2 to –5 percentage points.  My reasoning was that doing homework consistently helped their class average be as good as it could be, and homework was essential to success.  If a student did their homework all the time, their grade was increased by 2 percentage points; if they missed many assignments, they were penalized by losing up to 5 points. Many students counted on those two added points and worked hard to earn them.

I had felt that other systems for grading homework weren’t equitable.  For example, if the homework was worth 100 points (perhaps 3 points per night), students who had been running a 90% might have their grade go up 3 points, but students with a 70% might get a 10 point boost.  With the +2 to –5 range, every student got the same impact for doing (or not doing) their homework.

OTHER CATEGORIES:

As for Class Participation or Notebooks, these seemed to be hard to capture with a grade, and often created extra record-keeping work for me.  Students might have viewed them as easy ways to bring their grades up, but I generally did not attach any grades to them.  If I valued notebooks or taking notes for a particular class, I might grade it as a lab, especially in middle school or 9th grade when I was trying to build habits for success for the rest of high school.  In some classes, we did a quarterly portfolio as a way to summarize, consolidate, and reflect upon the learning.²

Rating clipboardOther commentators in the twitter discussion pointed out that a teacher might value engagement in discussion, or seeking help, or collaboration with other students. Consider using a rubric (shared in advance with your students) to promote the student habits you desire.  Here is one on “Class Participation” along with a record sheet for students to analyze their contributions (thanks to Carmel Schettino @SchettinoPBL) and here is one on “Student Work Habits”.

Other types of Formative Assessment don’t fall into my grading scheme, because they are formative… the information being gathered helps steer my teaching and gives the student feedback on their learning progress.  Nearly everything that happens in the classroom is part of formative assessment, helping all of us calibrate where we are on the learning journey.³

The decision whether to do Test Corrections or Retakes is a much larger discussion, but basically I did not give retakes or give points back for corrections.  My experience while teaching high school was that if students expected a guaranteed option for a retake, they didn’t always take responsibility for being prepared in the first place.

There were some times when everyone bombed an assessment, and usually that means I didn’t do the job as the teacher.  We would reteach, review, reflect, and then take a second version of the test that was averaged with the first.  I wanted to send the message that the first (poorer) grade doesn’t go away.

EXTRA CREDIT/BONUS QUESTIONS:

When a good opportunity arose, I would put a bonus question on an assessment or give extra credit for an optional part of a lab activity.  The points earned for these things accumulated as a separate “bonus quiz” for each student, rewarding them for doing more math extensions on our current work.

geometry working at deskIf a bonus was worth 5, you got 5/5 in your bonus quiz.  By the end of the quarter, students had bonus quizzes worth anywhere from 1/1 to 35/35, and some had none.  The bonus quiz didn’t fill holes of points lost elsewhere, but helped boost your average on the margins.

I’ve seen teachers who have successfully added ON bonus points (or included a grade such as 5/0).  This method allows bonus knowledge to make up for mistakes, which I have tried when coursework is very difficult and/or class averages are very low.  But if class averages are doing well, the 5/0 method results in averages greater than 100%.  It also might imply that you can make up for not knowing/doing some math last month by doing an extra project this month, and I wanted to make the point that all our work is important and students can’t avoid some work while still earning top grades.

MISCELLANEOUS:

I did NOT give pop quizzes, because I felt that to be a punitive practice (kind of like, “Gotcha! You’re not prepared”) that could be used by teachers to combat other issues, such as poor behavior or not doing homework.  Students always knew in advance what a test or quiz would cover, and most classes had designated review time in the day(s) prior.

Whatever your school’s grade scheme (letter grades, numerical grades up to 100%, 4.0 GPA, etc.) decide in advance what your cut-offs and rounding routines will be.  I had a firm “.50 and higher rounds up, .49 and lower rounds down” policy, which meant that a student with an 89.48 did not get the A– for the quarter.  If this feels unfair to you, decide in advance what you would do; options are to “borrow” the .02 from the coming quarter or to be lenient if you feel that particular student has earned the higher grade level.

MOST IMPORTANT REMINDERS:

stopwatch checkboxWhatever your grading system is, perhaps the most critical thing is to be prompt returning graded items with feedback.  The learning process is a partnership between me and my students, and if I delay or deny feedback, I’m not doing my half of the job.  When students wait days before getting a quiz back, they cannot learn from mistakes on concepts that are the foundation for new material.  Often, the same topics will be on the upcoming test, and I want my students to benefit from having the quiz to study from.

Be transparent about your grading system and keep students informed of their current grade and progress.  This gives students agency over their performance and the grade they have earned.  Back in the day (before online grading portals), I would print grade slips from my spreadsheet or grading software and hand them out several times during the marking period.  I had a physical piece of paper to hand to the student so there was no grade mystery and no surprises at the end of the quarter, and I could give them a verbal or written comment if something specific needed to be addressed.

If you decide to change your grading system part way through the year, be honest with students about the changes and your rationale for making them.  Discuss with them the incentives you want your system to provide.  Linda Wilson wrote a 1994 Mathematics Teacher article, “What Gets Graded is What Gets Valued” and that is true to a large extent, for better or worse.  I found that if I didn’t check or grade homework, my students wouldn’t do it; so if I valued that practice, I needed to include it in my grading structure.

Ralph Pantozzi (@mathillustrated) notes that whenever people are given a metric that they will be judged upon, they behave so that they perform well against *that standard*.  His advice is to “make your system revolve around students doing the math you value” so that they will work to achieve those goals.  Well said.

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Notes:

¹For younger students, I used +3 for homework (which I also did for HS when material was very difficult).  I required test corrections in some middle school classes as a homework assignment, to place clear value on understanding what went wrong and what can be done differently to avoid errors in the future.

²Portfolios took some time, but were worthwhile in both Algebra 2 and PreCalculus.  A good resource for portfolios is Mathematics Assessment: Myths, Models, Good Questions, and Practical Suggestions edited by Jean Kerr Stenmark (NCTM, 1991). These same classes also did writing in Math Journals with a few prompts each marking period. A nice summary of how to use writing is Marilyn Burns’ article “Writing In Math” in Educational Leadership (ASCD, October 2004).

³Thanks to Steve Phelps @giohio and Martin Joyce @martinsean for these points about Formative Assessment.  I have also found the book Mathematics Formative Assessment by Page Keeley & Cheryl Rose Tobey (Corwin, 2011) to be helpful.

 

How Else Can We Show This?

question 2What I love about using calculator technology in my teaching is the “Power of Visualization” and the opportunity to examine math through different lenses.  The multiple representations available on TI graphing calculators—numeric, algebraic, graphical, geometric, statistical—allows me to push my students to approach problems in more than one “right way.”  By connecting these environments and making student thinking visible when we dig into a mathematical situation, we support students in productive struggle and deepen their understanding.*

Read my post on the TI BulleTIn Board Blog for two scenarios in which my students and I pursue multiple pathways to show and make sense of the mathematics at hand (with demonstration videos!)

How Else Can We Show This?

Read the entire post at the above link, and here is a quick summary:

1. Riding the Curves and Turning the Tables: studying quadratic and polynomial functions.

VIDEO 1  Using different forms of quadratic functions to reveal graph features.

VIDEO 2  Using the graph-table split screen to see numerically what is happening at key points.

2. Absolute Certainty: solving absolute value equations and inequalities.

VIDEO 3 Using the graphical environment to support an algebraic solving procedure.


*Connecting mathematical representations and supporting productive struggle are two of the high-leverage mathematical teaching practices discussed in NCTM’s Principles to Actions: Ensuring Mathematical Success for All (2014).

Action-Consequence-Reflection Activities for GeoGebra

When I choose to use technology in my math teaching, I want to be sure that the technology tool supports the learning, and helps students to develop conceptual understanding.  The Action-Consequence-Reflection cycle is one structure that I use towards this goal.  I’ve written about Action-Consequence-Reflection activities before, in this post and this post, and I recently had an article published in the North American GeoGebra Journal, “Using Action-Consequence-Reflection GeoGebra Activities To Make Math Stick.”

In the Action-Consequence-Reflection cycle, students

  • Perform a mathematical action
  • Observe a mathematical consequence
  • Reflect on the result and reason about the underlying mathematical concepts

The reflection component is, in my view, the critical component for making learning deeper and more durable.  The article includes the following six activities that use the cycle to help “make the math stick” for students.  Each of the GeoGebra applets is accompanied by a lab worksheet for students to record their observations and answer reflective questions.

EXPLORING GRAPHS & SLIDERS:

The first two activities use dynamic sliders so that students can make changes to a function’s equation and observe corresponding changes on the graph.

In Power Functions, students control the exponent n in the function f\left(x\right)=x^n , and can toggle between positive and negative leading coefficients. Capture power funct border

In Function Transformations, students investigate the effects of the parameters a, h, and k on the desired parent function.Capture funct transf border

INTERACTIVE VISUALIZERS:      

Using the power of visualization to deepen understanding, the Domain and Range applet highlights sections of the appropriate axis as students manipulate linear and quadratic functions.Capture DR lin quad border

UNDERSTANDING STRUCTURE:

In the Rational Functions activity, students explore how the algebraic structure of functions relates to important graph features. The handout includes extensions allowing investigation of other rational function scenarios not already covered.Screen Shot 2018-11-29 at 9.52.52 PM

INVESTIGATING INVARIANTS:

The last two activities have students looking for invariants—something about the mathematical situation that stays the same while other things change.

In Interior & Exterior Angles, students investigate relationships among the angles of a triangle and form conjectures about the sums that do and don’t change as the shape of the triangle changes.Capture int ext border

In Right Triangle Invariants, the applet links the geometry figure to a numerical table of values, and students discover several invariant properties occurring in right triangles.Capture right tri border

PLANNING FOR REFLECTION:

Simply using these robust technology activities will not guarantee student learning and conceptual understanding; it is imperative that we as teachers plan for reflection by including focusing questions, discussion of students’ mathematical thinking, and clear lesson summaries with the activity.  Use the provided lab worksheets or adapt them for your needs.  Capitalize on the power of the Action-Consequence-Reflection cycle to make the math stick for your students’ success!


Notes and Resources:

This post contains excerpts from the full article (pdf available here) from Vol 7 No 1 (2018): North American GeoGebra Journal.

The North American GeoGebra Journal (NAGJ) is a peer-reviewed journal highlighting the use of GeoGebra in teaching and learning school mathematics (grades K-16). The website for the NAGJ is here.

My GeoGebra Action-Consequence-Reflection applets are in this GeoGebra book, or they can found by entering “kdcampe” into the GeoGebra search box.  Thanks to Tim Brzezinski, Marie Nabbout, and Steve Phelps for their assistance with some of the GeoGebra applets.

Testing Tips: Using Calculators on Class Assessments

If you’ve been using TI graphing calculators in your teaching, you may have contemplated how to implement the calculators for in-class testing.  Whether you are giving a short quiz, a chapter test, or end-of-term exam, read my post on the TI BulleTIn Board Blog for some tips for how to use TI calculators successfully on class assessments.

Testing Tips: Using Calculators on Class Assessments

There is much more in the full post, but here is a summary:

  • Determine the Objectives: decide which math skills and problems you will assess with and without the calculator.
  • Separate the Sections: separate the calculator and non-calculator problems into two sections.
  • Set up the Handhelds: to be sure the calculators are useful tools for students and don’t interfere with assessing their math knowledge, set up the handhelds for security and equity.
  • Electronic Quizzes with TI-Nspire CX Navigator: take advantage of electronic quizzes if your classroom has the TI-Nspire Navigator System.

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End of Quarter Feedback Is a Two-Way Street

[Note: this is an excerpt from my blog post on the TI BulleTIn Board.]

With the first marking period winding down here in the northeastern US, teachers and students are focusing on the grading process.  How might we make end-of-marking period evaluations into a constructive tool for the teacher AND the students?  Here is one idea…

Rating clipboardAt the end of a marking period, students’ grades indicate their progress and achievement in math class.  It is also a great time to encourage reflection and feedback on what teaching and learning practices have played out in the classroom and what changes can be made so the class is more productive in the future.  Here is how I have turned my end-of-quarter evaluations into valuable conversations about how to make math class better for all of us.

 

My Four Questions

My students answer these four open-ended prompts.  Names are optional.

  1. Tell me something specific you did well or are proud of this quarter.
  2. Tell me something specific you want to improve for next quarter.
  3. Tell me something you think I did well.
  4. Tell me something you want me to change or improve.

I give students time to reflect and write, and the ground rules are that they can’t say “nothing” and can’t propose major changes like “stop giving homework/tests”.  Because I require them to be specific, they have to find some details about their learning and my teaching to discuss.  Most of the time, students write about things that are actionable in their evaluations.

I feel that this process makes evaluation a two-way street, since students are commenting on me and my teaching but also on themselves.  By asking them to name what they are going to do differently for the coming quarter, I place the responsibility on their shoulders for making changes in their class performance.  The set of four questions opens the door for us to communicate constructively about improving our math class experience for everyone.

What Will You Do?

I’m interested in what other teachers find useful for end-of-marking period feedback.  Let me know what works for you and your students here in the comments or on Twitter (AT KarenCampe).


Notes and Resources:

Some helpful blog posts about End-of-Quarter/Semester feedback are here and here from Sarah Carter (twitter AT mathequalslove) and here from Jac Richardson (twitter AT jacrichardson).  Thanks so much for sharing!

Read the full post on the TI BulleTIn Board:

End-of-Marking Period Feedback Is a Two-Way Street

Tips for Transitioning to the TI-Nspire

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Are you transitioning to TI-Nspire™ CX graphing calculators from TI-84 Plus family models in your classroom? The TI-Nspire CX graphing calculator is a powerful tool with many features, yet it is easy to perform familiar operations like calculating and graphing. Read my post on the TI BulleTIn Board Blog for some suggestions to get you started.

Tips for Transitioning to the TI-Nspire CX from TI-84 Plus

These tips should get you started on your transition, but there is much more to explore about the TI-Nspire CX graphing calculator. Check out the on-demand webinars, product tutorials, and free activities at education.ti.com.

Testing Tips: Using the TI-84+ on the SAT

test pencil image

The fall dates for the SAT and PSAT tests are around the corner.  The TI-84 Plus CE and the entire TI-84 Plus family of graphing calculators are approved for use on the Math–Calculator section of these College Board tests. Read my post on the TI BulleTIn Board Blog for tips on how to leverage your TI-84 Plus for success on test day…

Six tips for using the TI-84 Plus CE on the SAT®

Good luck!

Moving the Needle

I’m late to the party—the Virtual Conference on Mathematical Flavors—but I’ve been pondering for awhile the questions asked in the prompt: “How does your teaching move the needle on what your kids think about the doing of math, … what math feels like, or who can do math?”

I work one-on-one with students and while preparing for the start of the academic year I’ve reflected on what has gone well and what changes I’d like to make going forward.

In my sessions with students, my goal is to help them master skills and concepts taught in the classroom setting.  I’m remediating, coaching, explaining, test-prepping, and only a little bit extending beyond what their main teacher has taught.  I seem to get good results: students do well on their in-class assessments, grades improve or are steady, and students and parents are happy.  However, I worry that while my students are learning some mathematical skills in the short or medium term, they still view themselves as needing the support of a tutor to succeed; that they are “not good at math”.  I’m concerned that my teaching practice doesn’t push them that much farther into confidence, agency, or deep understanding as I would like, that I haven’t “moved the needle” enough.

Therefore, I have three goals for the coming year, in order to meet the needs of my students AND to better equip them to experience the mathematical flavors they encounter elsewhere in their academic careers.

1. [I will] Talk and write less.

notebookIn the past, I routinely guided students through the concepts of a math unit, making summary notes for them to keep, and doing much of the talking.  I generally avoided the “do you get it?” self-report trap that can occur in a classroom, because the individual setting necessitates that my students demonstrate their understanding by doing examples, showing me their steps, and fixing their own mistakes.

But I’ve learned from my reading in cognitive science* that if I make students write things themselves, generate their own questions and examples, reflect on their results, and practice self-explanation, their learning will be deeper and more durable.  That my students need to engage in their own processing time, in order to solidify and build their framework of math concepts and procedures.  This takes more time, but the idea is that it makes the teaching episode much more productive; instead of an hour racing through lots of topics and examples, we can study in a way that ensures each math component we discuss is synthesized with prior knowledge and able to be retrieved for future use.

In the past I’ve shared some cognitive science research with my students, particularly about spaced retrieval practice and the limits of working memory (e.g. why you should “show your work” as you do your math).  For the coming year, I will make this a key part of our curriculum, so that students learn how to build their own effective study practice.  And I’m planning to talk and write less, taking the time to have my students talk and write more, because it is their learning that is the focus of our work.

2. [They will] Stock the mathematical toolbox.

indexStudents often say to me, “I don’t know anything about” the math topic they are facing.  I typically have jumped right in, explaining the topic from the beginning, providing scaffolding and support for the learning.

But in truth, the students probably do know something about the topic, and if I insist that we begin there, I will enhance their learning in two ways.  First, by meeting them where they are in their conceptual development, I can build on what they know, expanding and strengthening their math expertise, rather than starting from nothing at all or repeating techniques they have already mastered.  Second, by having students do the cognitive work of retrieving concepts, connecting to other knowledge, and applying prior skills to new problems, they are engaging in productive struggle that will ultimately make the learning deeper and stronger.  That work might be hard work, but their active participation in doing the math is a critically important ingredient.

So I plan to explicitly help my students stock their mathematical toolbox—fill it with strategies, vocabulary, big ideas, and “things to try” when faced with a math exercise.  Things like these:

  • Draw a diagram & label what you know.
  • Use inverse operations to do and undo.
  • Try a different representation to get more information (graph, table of values, algebraic expression, etc.).
  • Look at structure to figure out how to solve or how to graph (What does this equation look like—variables, powers, coefficients, fractions, etc.).
  • Explain it in your own words (before using mathematical terms).
  • Try a simpler problem first.
  • Test out with numbers in place of the variables.
  • Think about the computation before grabbing the calculator.
  • And more…

Of course, the mathematical toolbox includes resources for help when you are really stuck: search Google, Khan Academy, or YouTube for help, or find a friend or classmate who is willing to show you how to do it, and gives you enough explanation so that you can do it yourself afterwards.

The final component of the toolbox is for students to actively engage in their main math class: taking notes, working examples, making sketches, and thinking about why and what makes sense.  When something seems confusing, mark it to ask about later.  Come to the tutoring session prepared with some questions to ask or trouble spots to work on, so we can target our efforts together.

3. [We will] Build self-confidence and a positive outlook.

This one is a perpetual challenge, with students who haven’t consistently succeeded in their math classes, or who have only achieved with supports.  I have always begun the year asking students how they feel about math, or when did it first become difficult, or what part of it feels straightforward.  As the year goes on, we discuss what topics are “hard for everyone” (so don’t worry if it feels hard for you) or what is “easy to do” (so once I explain it, you will get it too).

But there are dangers in labeling topics as “easy” or “hard” because each individual student experiences it in a unique and nuanced way.**  That saying “this is easy” can backfire in the event that the student still feels lost, and then feel anxious because they should have been able to understand, even though my message is intended to be “you’ve got this”.  And saying that something is really tough can make it feel like an insurmountable mountain.

So I’m planning to explicitly work on promoting a growth mindset*** with my students: that anyone can be good at math, that mistakes and challenges help their learning, and their efforts and practice will strengthen their understanding.  That if they say “I’m not good at math” they need to add on “yet” to the sentence.  And that everyone experiences difficulty some of the time, they are not alone, they have to believe they can do it (and I believe they can).

images Finally, we need to approach our work with a positive attitude, tackling tough math with our full efforts.  We will find accessible entry points and break the material into manageable pieces.  We will build your self-confidence.  We want to move the needle on how you view tutoring: moving towards the view that it is a helpful opportunity and an occasional safety net, and away from the view that it is something that you can’t do without, or (worse) something that releases you from responsibility to do your work.  We’re in this together, but we’re working towards you being able to go it alone.  I believe in you!


Notes:

The Virtual Conference on Mathematical Flavors is a wonderful set of blog posts well worth your time to read and reflect upon.  Thanks to Sam Shah for hosting, compiling, and cheerleading.

*Two books discussing cognitive science research and its implications for teaching and learning are Make It Stick: The Science of Successful Learning by Peter C. Brown, Henry L. Roediger & Mark A. McDaniel, (2014) and How I Wish I Taught Maths: Lessons Learned from Research, Conversations with Experts and 12 Years of Mistakes by Craig Barton (2018).  The accompanying websites are http://makeitstick.net and www.mrbartonmaths.com/teachers/ (check out Craig’s podcast page and recommended research papers list).  Two other helpful websites are www.retrievalpractice.org/ and www.learningscientists.org.

**Tracy Zager discusses the trouble with “this is easy” in this post.

***Lots about a growth mindset and how to foster it among your students from Jo Boaler and her team at www.youcubed.org/resource/growth-mindset/  and in the book Mathematical Mindsets (2016).