Category Archives: Teaching Practices

Summer Math Refreshments


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).


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

The second event is the return of the Big Internet Math-Off organized by the folks at  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


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 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 , 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  Printing is also an option.  Warning, these puzzles are habit-forming!

Reading List


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.


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!


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.


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


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.


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.


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.


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.


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.


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.



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.


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.


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


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


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


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


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.

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

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!


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 and (check out Craig’s podcast page and recommended research papers list).  Two other helpful websites are and

**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  and in the book Mathematical Mindsets (2016).

Conference Conversations…

Last month, at the T3 International Conference* #T3IC, I was one of the speakers at the “Seven for Seven” session**.  In an Ignite-style setup, each of us seven speakers spoke for seven minutes on topics that we were passionate about.

I spoke on “The Power of the Action–Consequence–Reflection Cycle”, in which

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

I believe that the REFLECTION piece is the most important component of the process.  I talked about strategies for questioning and how to make reflection part of your classroom practice.

Screen Shot 2018-04-18 at 5.35.11 PM

The video of my talk is here. In the alloted seven minutes, there wasn’t time to fully discuss classroom examples, but these activities fall into several categories:

  1. Graphs and Sliders (Transformation of Functions)
  2. Visualizers
  3. Understanding Structure
  4. Looking for Invariants
  5. Dynamic Tables, Lists & Spreadsheets
  6. CAS Capability

I wrote about two examples in this post that made use of graphs/sliders and invariants.  Some examples of dynamic tables are described in this post.

Next week, I will be presenting a workshop on the same theme at the NCTM 2018 Annual Meeting in Washington DC.  If you are attending #NCTMannual, join me at Session 36, Thursday April 26 in Convention Center 207A from 9:45 to 11 am for “Action–Consequence–Reflection Activities: Using Technology to Make Math Stick!”.  We will explore dynamic activities using TI Graphing Calculators, GeoGebra, or Desmos that leverage the Action–Consequence–Reflection Cycle to promote conceptual understanding and enable student success.  I will post some additional lesson ideas here after the conference is over.



Notes and Resources:

*All of the video highlights of the Teachers Teaching with Technology T3 International Conference are here.  The other six speakers were wonderful and inspiring—check them out to see what they had to say.  Next year, T3 will take place in Baltimore March 8–10, 2019, so save the date!

**Jill Gough’s summary sketchnote from the 7 for 7 session is here.  Thanks, Jill!

Looking Below the Surface

icebergThis week, I came across the treasure trove of problems available at, created and curated by Craig Barton.  SSDD stands for “Same Surface, Different Deep” and each set of problems contains four questions that have a similar presentation (a common image, shape, or context) but where the deep mathematical structures of the problems are very different¹.

SSDD problems are the essence of “Interleaving Different Practice”, one of the techniques that enhances learning from the book Make It Stick: The Science of Successful Learning². Page 85 explains how interleaving fosters conceptual understanding:


The trouble with interleaving is that it can actually impede initial learning, which is one reason teachers might not gravitate to the technique; however, the research “shows unequivocally that mastery and long-term retention are much better if you interleave practice than if you mass it.” (p. 50) [massed practice is focused, repetitive practice of one thing at a time.]

Craig’s SSDD problems are based on extensive cognitive science research showing that this strategy of interleaved practice is beneficial to learning by helping students discriminate between problem types and choose appropriate strategies to solve based on the underlying deep structure.  He says this is in contrast to the “auto-pilot” approach that can be common with students: they see a right triangle and jump right to the Pythagorean Theorem, whether or not it is useful.  Here is one of his question sets involving right triangles:

Screen Shot 2018-03-15 at 9.28.25 PM.png

Students working on “auto-pilot” is one consequence of how we often teach math topics: practicing similar types of problems in a given lesson, then moving on to another type, then another.  When students are faced with cumulative tests or spiral review problems, they might not remember what they had “mastered” earlier.

Of course, I wanted to get in on the fun!  Craig tweeted yesterday that in the prior 5 days, his website grew from 80 question sets to 200 sets.  He has a helpful template available for creating an SSDD powerpoint slide. I scanned the list of topics on the website, looking for something fruitful that could also capitalize on technology and connect various representations, and didn’t yet have several sets posted.  I developed this question set on Quadratic Graphs:

Screen Shot 2018-03-15 at 9.29.58 PM

My process was this: I first brainstormed the types of questions that could get asked about a quadratic graph. By the time Algebra 1 or Algebra 2 students are done with a unit on quadratics, they have covered a lot of ground, and can be confused by different forms of quadratic functions or when to use various solving techniques.

I decided that my focus would be on examining features of the graph to write different forms of the quadratic function, or use the graph’s features to solve equations and real-context problems.

It was challenging to choose one graph that could work for all these purposes:

  1. Determine Vertex Form from a graph (and with a ≠ 0 to increase cognitive demand)
  2. Determine Factored Form from a graph (required relating zeros of the function to its factors, again with a ≠ 0)
  3. Use a graph to Solve an Equation (either find points of intersection with technology, or solve using quadratic formula; including simplifying radicals to exact form or finding decimal approximations)
  4. Interpret a Graph in a Real Context (with different scales on horizontal and vertical axes)

I took screenshots from TI-84+CE SmartView Emulator software³.  The color and graph grid makes it possible for students to gather information by inspection and not have to rely on the calculator to analyze the graph’s features.  I’m pleased with how the problems turned out; any feedback is welcome.

You can find my set on under the topic “Equation of a Quadratic Curve” or this direct link and the original powerpoint is here.  One caveat to keep in mind about the problem sets on the SSDD site is that Craig works in England; there are some differences in vocabulary between American math and British maths, with both represented on the site.

I am excited about diving in to the rich resource at and looking below the surface for deeper learning!  Join me…

Notes and Resources:

1. There is much more background explanation about the SSDD problems and their research base on Craig’s website. They were introduced in his book “How I Wish I Taught Maths: Lessons Learned from Research, Conversations with Experts and 12 Years of Mistakes” (2018).

2. The book “Make It Stick: The Science of Successful Learning” by Brown, Roediger & McDaniel (2014) has a companion website  The authors advocate for the strategies of spaced and interleaved retrieval practice, elaboration, generation, reflection, and calibration in order to optimize learning.  Another helpful blog post by Debbie Morrison about using these techniques in your teaching is here.

3. More information about the TI-SmartView CE Emulator Software for the entire TI-84+ family can be found on the TI Website here.  Currently there is a 90-day free trial available.  Take advantage of color demonstrations for your class, save work in progress, and insert screenshots and keystrokes in your handouts and assessments.

4. Michael Pershan wrote this blog post with his thoughts on the SSDD problem sets. He focused on geometry diagrams, for which I think the SSDD sets are extremely well suited.



Great Thinking!

light bulbOne of the bonuses of working with students one-on-one is that I can get a window into their mathematical thinking by asking questions and having them “narrate” their work as they proceed.  Several of my recent conversations demonstrate unique and flexible thinking that helped my students work through computations effectively.

How might we enable this type of visible thinking in our classrooms?  I tried these prompts:

  • Tell me what you were thinking.
  • What would you do to compute this if you didn’t have a calculator?
  • What math operation did you do first?
  • Can you explain the steps you did in your mind?

What follows are some observations of “great thinking” by my students, who used flexible number sense for success.

A. Using number sense for easier multiplying or dividing



B. Fantastic Fraction Flexibility

Here, noticing the multiplier is easier than solving the proportion directly:IMG_7207

In this question about slope, the student noticed that the first denominator must be 2 if the fractions are to be equal.IMG_7210

C. Simplifying with Radicals

It is hard to find perfect square factors of large numbers; this strategy employs factors and the student never found the original product.IMG_7208

Here, the student used an alternative to the traditional process of rationalizing the denominator.IMG_7213

D. Factors and Multiples

Rather than set up proportions, this student noticed multiplesIMG_7205

I’ve avoided the term “cancel” for some time, so here we talked about “dividing away common factors”. IMG_7206

E. Other interesting choices for computation and order of operations

The formula for area of a trapezoid can be thought of as “height * average of the bases”.  This student also applied binomial multiplication to compute without a calculator.IMG_7215

Here the typical order of operations would do the distributive property first.  Instead, the student saved a step.IMG_7211

Also on my mind as I write this are three things I have read this week.  First, James Tanton addresses a multiplication mistake made by an education official in this blog post, forgetting what 7 x 8 is.  Tanton suggests that the “best answer” in this case would have been “This is a tricky one. But I do have in my head that 7×7 is 49, so 7×8 is seven more than this: 56” because it highlights the thought process and downplays the memorization aspect of number facts.  His discussion is thought provoking (and he applies the same idea to trig identities too!).

The second item I was contemplating was in Sarah Carter’s Monday Must Reads post; she and I both liked David Sladkey’s “No eraser allowed” technique of insisting students leave their mathematical thinking available for the teacher to see.


The third concept informing my thinking is the book study I’ve been doing with colleagues from Teachers Teaching with Technology (#T3Learns) on “Visible Learning for Mathematics” (2017) by Hattie et al.  Hattie describes a spiral relationship between surface learning, deep learning, and transfer learning that enables students to achieve.  He notes that surface learning is NOT shallow learning (p 29) but is instead “made up of both conceptual exploration and learning vocabulary and procedural skills that give structure to ideas” (p 104) that “sets the necessary foundation for the deepening knowledge” (P 131) on the path to understanding. Techniques such as number talks, guided questioning, worked examples (including accurate work and work with mistakes), and highlighting metacognitive strategies all enhance the process of surface learning for students, according to Hattie.

I constantly remind students to “Show Your Mathematical Thinking”—which is the updated version of “Show Your Work.” By asking them to “narrate” their thinking, I am focusing on my students’ surface learning to build their foundation of skills and tools on their learning journey.



Action-Consequence Advantage!

Using Technology to Make Math Stick

How might we enable students to grasp mathematical concepts and make their learning durable?  One approach is to use the sequence of Action-Consequence-Reflection in lesson activities:

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

The ACTION can be on a graph, geometric figure, symbolic algebra expression, list of numbers or physical model.  Technology can be used in order to have a quick and accurate result or CONSEQUENCE for students to observe.

The REFLECTION component is the most important part of this sequence; without this, students might not pay attention to the important math learnings we intended for the lesson.  They might remember using calculators, computers, ipads, or smartboards, but not recall what the tech activity was about.  And if they did learn the concept in the first place, the process of reflection helps make the learning stick—it is one of the cognitive techniques shown to make learning more successful.*

Students can reflect in many ways: record results, answer questions, discuss implications with classmates, make predictions, communicate their thinking orally or in writing, develop proofs and construct arguments.  The intended (or unexpected) learnings should be summarized either individually or as a class in order to solidify the concepts, preferably in a written form.**

In my one-on-one work with students, we often fall into the “procedural trap” in which my students just want to know “what to do” and don’t feel that the “why it works” is all that important (I’ve written about this before here). Also, our time is limited with many topics to cover.  But this past week, I was able to sneak in some Action-Consequence-Reflection with two students because they had mastered the prior material and were getting ahead on a new unit. It was a great opportunity to have them discover a concept or pattern for themselves, far better than simply being told it is true.

For each of these, I used a simple REFLECTION prompt:  What do you observe?  What changes?  What stays the same?

Student #1: Polynomial Function End Behavior, Algebra 2 (or PreCalculus)End Behav QWe used a TI-84+CE to investigate the polynomials.  We began with the even powers on a Zoom Decimal window, and my student noticed that higher powers had “steeper sides”.  I asked “what are the y-values doing to make this happen?” and we noticed the y-values were “getting bigger faster”.  We then used the Zoom In command to investigate what was going on between x = –1 and x = 1, and noticed that higher powers were “flatter” close to the origin because their y-values were lower.


Why did this happen? Take x = 2 and raise it to successive powers, and it gets bigger. Take x = ½ and raise it to successive powers and it gets smaller. We confirmed this with the table:

Setting the Table features to “Ask” for the Independent variable but “Auto” for the Dependent made the table populate only with the values we wanted to view. Fractions can be used in the table, and we used the decimal value 0.5 for number sense clarity. Another observation was that the graphs coincided at three points: (1, 1), (0, 0) and (–1, 1).

In fact, none of this was what I “intended” to teach with the lesson, but it was mathematically interesting nonetheless, and my student already had a deeper appreciation for the graph’s properties. We moved on to the odd powers, and the same “steep” vs. “flat” properties were observed:

AC end beh 6


Then I asked my student to consider what made the graph of the even powers different from the graph of the odd powers, and what about them was the same, finally getting around to my “lesson”.  He noticed that even powers had a pattern of starting “high” on the left and ending “high” on the right, while odd powers started “low” on the left but ended “high” on the right.  We made predictions about the graphs of x11 and x12, to apply our understanding to new cases.† Then we moved on to the sign of the leading coefficient: when it is negative, our pattern changed to even powers “low” on the left and “low” on the right, with odd powers “high” on the left and “low” on the right.

All of this took just a few minutes, including the detour at the beginning that I wasn’t “intending” to teach.

Student #2: Interior and Exterior Angles of a Triangle, Geometry

Int Ext 1

We began with a dynamic geometry figure of a triangle which displayed the measurements of the three interior angles and one exterior angle.  I asked my student: “What do you notice?” and “what do you want to know about this figure?”  We dragged point B around to make different types of triangles.

This motivated my student to wonder about the angle measures; he was familiar with Linear Pair Angles, so he noticed that ∠ACB and ∠BCD made a linear pair and stated they would add up to 180°.  I then revealed some calculations of the angle measures, which dynamically update as we changed the triangle’s shape (the 180° in the upper right is the sum of the 3 interior angles).  What changes? What stays the same?

Int Ext 6

We noticed that both sums of 180° were constant‡ no matter how the triangle was transformed, but the sum of the 2 remote interior angles kept changing, and in fact, matched the measure of the exterior angle.  My student recorded his findings in his notebook, and then I asked, “can we prove it?”.  It was easy for us to prove the Exterior Angle Theorem based on his previous knowledge of the sum of the interior angles and the concept of supplementary linear pairs.  This student loved that he had “discovered” a new idea for himself, without me just telling him.

Even though these concepts are relatively simple, I feel that using a technology Action-Consequence activity made the learning more impactful and durable for my students, and I believe it was more effective than just telling them the property I wanted to teach. It took us a few more minutes to explore the context with technology than it would have to simply copy the theorems out of the book, but it was worth it!


Notes and Resources:

*For a full elaboration of cognitive science strategies for becoming a productive learner (or designing your teaching to enhance learning), see Make It Stick: The Science of Successful Learning by Peter C. Brown, Henry L. Roediger & Mark A. McDaniel, 2014.  Website: .

**Written summaries allow students to Elaborate and Reflect on the learning, two more of the cognitive strategies.  In addition, by insisting that students record the results of the work, the teacher sends the message that the technology investigation comprises important knowledge for the class.

†Making predictions is a way to formatively assess my students’ understanding.  It also is a form of Generation, another cognitive strategy that makes the learning more durable.

‡This is an example of an invariant, a value or sum that doesn’t change, which are often important mathematical/geometric results.

Here are the two activities discussed in the post. Note that the Geogebra file for Power functions has the advantage of having a dynamic slider, but students won’t view the graphs at the same time, so won’t notice the common points or the graph properties between –1 and 1.