Category Archives: Teaching Practices

Where Are You? I’ll Meet You There


I was so proud.  I had created a great technology activity to use in my Algebra 2 class, complete with a well-thought out lab sheet for students and their partners to work through and document their learning.  It was an exploration of slopes of parallel and perpendicular lines, with students being guided to “discover” the concepts involved.*

Questions for assessment involved different levels of cognitive demand, including creating their own sets of equations, paying attention to mathematical structure, and writing an explanation of their process.  The graphing calculators were ready and the students worked diligently through the class period.  The lesson was a success—everyone demonstrated their understanding of the mathematical objectives.

So what was the problem?  I asked a few students on their way out of class if they enjoyed the calculator lab activity since it was different from our “regular” routine.  They told me: “It was fine.  But Mrs. Campe, we all already knew about slopes of parallel and perpendicular lines.”

I had failed to properly pre-assess my students’ understanding of the concepts. I had wasted a full class period to cover something they had already mastered, when, instead, I could have been moving forward or exploring some other problem more deeply.  I didn’t check where my students were in their understanding before launching into my “great” activity.

Similar things can happen in my one-on-one work with students.  Since I am not in their classroom with them, when students arrive for a work session, I have to rely on them to tell me what their lesson and unit topics are.  Sometimes I go down a path that veers away from what they have done in class.  Some students resist conceptual explanations, wanting only the quickest route to the answer.  I have to push them to realize that learning the “why” behind a procedure helps them understand when and how to use it, and the conceptual background makes their learning more durable and leads to more success in math class.**

So what have I learned from these situations?

1.  It is vitally important to pre-assess and utilize formative assessment to know where my students are.  Class time is at a premium and I want to use it wisely.

2.  Don’t rely on students’ self-report of their understanding; require them to demonstrate their capabilities by doing problems, explaining a process, and answering “why” questions.

3.  Don’t use technology just because I have it.  It must further the lesson objectives and enhance student understanding.  The same warning goes for “fun” or “cool” lesson activities.

4.  Reflect on your lessons: ask yourself what went well and what needs improving so mistakes don’t get repeated.  And discuss with your colleagues, local and virtual. You will find lots of support in the MTBoS; one teacher commented to another on Twitter just last night: “Thx! I am always looking to improve my teaching!”

Mistakes, obviously, show us what needs improving. Without mistakes, how would we know what we had to work on?     

Peter McWilliams


Notes & Resources:

The technology lab activity on Parallel and Perpendicular Lines is here.  It was written for the TI-84+ family of calculators, but any graphing technology may be used.

*This lab activity is a “Type 1” investigation structure in that it guides students toward the desired mathematical knowledge, in contrast to a “Type 2” inquiry which encourages more open exploration.  Both types of lesson structures are effective, so match the level of exploration with your objectives.  More about this in McGraw, R. & Grant, M. (2005).  Investigating Mathematics with Technology: Lesson Structures That Encourage a Range of Methods and Solutions. In W. J. Masalski & P. C. Elliott (Eds.), Technology-Supported Mathematics Learning Environments: 67th Yearbook (303-318). Reston, VA: NCTM.

Another dimension useful in analyzing a lesson is type of teacher questioning.  “Funneling” questions guide students through a math activity to a predetermined solution strategy, while in “Focusing” interactions, the teacher listens to students’ reasoning and guides them based on where they are and what strategies they are employing, rather than how the teacher might solve the problem.  More in Herbel-Eisenmann, B. A. & Breyfogle, M. L. (2005). Questioning Our Patterns of Questioning.  Mathematics Teaching in the Middle School, 10(9): 484-489.

**Connecting new knowledge to what you already know (elaboration), building conceptual structures (mental models) and practicing what to do when (discrimination skills) are among the strategies for successful learning discussed in Make It Stick (Brown, Roediger & Mc Daniel, 2014).  See this website for more.

As a final thought, my title is misleading, because I don’t just want to meet the students where they are and stay there, I want to plan for appropriate challenges to take them beyond their current understanding.  There is great value in productive struggle, and choosing lesson components within the students’ “Zone of Proximal Development”.

Setting the Stage

How can we prepare our students for a successful school year?  I work with students mostly one-on-one, and with the school year under way, I’ve been thinking about this issue a great deal.  Whether your perspective is classroom teacher, coach, tutor, or parent, there are valuable preliminaries that can help enhance the learning environment.  I try to set the stage with a few important messages I want my students to hear.


First and foremost, I tell students to “Bring Your A Game” to their math learning.  This means that they can’t just let the math class happen TO them, but instead they have to actively ENGAGE with the material in class and for homework.  Take down the teacher’s notes and examples, and if something seems confusing, mark it to ask about later.  Put in full effort to make sure you’ve done the work (even if the teacher doesn’t check it*).  Work slowly and mindfully through the problems, and watch out for “avoidable errors” (my preferred term for “silly mistakes”).  Know what things tend to trip you up: for example, double- check negative signs and units on answers.  Don’t wait for the test; when you get a quiz back, figure out what went wrong so you can get it right next time.

Students tend to think that each new course brings all new math, but I remind them that they can rely on the fact that all of the math they have ever learned is still true! “Use what you know” about the + and – signs of the four quadrants to help with the unit circle in trig, and for calculus, build the difference quotient definition of the derivative on the formula for slope.  This also helps when you don’t know where to start on a word problem or a geometry proof.  And making connections between mathematical concepts deepens understanding.

Just as an athlete might train by doing different types of workouts (rather than more repetitions of the same exercise), math students can learn something better by trying to “find more than one way” to a solution.  I try to highlight alternative methods of solving by posing questions:

  • What would happen if you didn’t distribute first  to solve this equation? –15 = 3(x – 8)
  • Is this student’s reasoning correct?
  • Find the error in this work.
  • Can you solve this (geometry problem) another way?**

I constantly tell my students to “show your mathematical thinking.” One obvious reason for this is so you can get partial credit for wrong answers, if the teacher grades that way.  But the main purpose is that showing your steps makes it more likely that your work will BE correct.  You can reflect on your thinking process through a problem, which helps improve retention and understanding, and enables you to explain it to others when needed (and study for a test later).  If you have poor handwriting, make it readable (enough).  If you are using a calculator or doing some arithmetic in your head, sketch a graph or jot down what you are computing so your process is clear.  A friend’s daughter summed it up this way: “I feel it is important to show my work. My teacher and I can see what I’m thinking. It’s hard to see it if it’s only in my head.”

Another key to success is to “know your resources.”  Does your teacher have a website?  Is there an online textbook?  Who is in your class you can study with?  Don’t forget to search Google, YouTube, and Khan Academy if you are stuck on something and can’t proceed.  I remind students that getting help from a tutor isn’t the only way to get assistance.  Go to your teacher for extra help because they can see your effort and they can focus you in on the key concepts.

A final message is to “persevere even when it gets hard.”  This is more than just SMP #1 talking, this is about being confident that you can do it and being willing to grow your mathematical skills.  One teacher at my son’s Open House night spent time coaching us parents that the class was going to be challenging (and explained why), and gave strategies students could use to facilitate their understanding.  She said “if you aren’t getting at least a few things wrong, then you aren’t learning something new.”  I try to reinforce a growth mindset in my students and a belief that they can achieve the goals they have set for themselves.

I think it’s going to be a very good year…


* Students often don’t put in effort if there isn’t a grade attached. As a teacher, if you want to promote habits of doing homework, showing mathematical thinking, checking results, extending to more than one solution etc., try to consider this in your grading scheme.  See Wilson, Linda. “What Gets Graded Is What Gets Valued.” Mathematics Teacher 87, no. 6 (September1994): 412–14.

** This summer I worked for awhile on the Art Of Problem Solving Geometry course with an online group.  I was impressed by the focus on finding more than one solution: “Doing a problem two different ways is an excellent way to check your answer” and “When you can’t find an answer right away, try finding whatever you can—you might find something that leads to the answer! Better yet, you might find something even more interesting than the answer.”

These messages about approaches to learning are one version of Classroom Norms. Read more here about one teacher’s use of Jo Boaler’s suggested list.  And here is a consideration of how to change the norms that students may have become accustomed to, in order to increase student engagement.

Summer Assignment!

With the academic year winding down (and already finished for some), take a moment to think about what you will be doing this summer.  I’m sure you are planning to relax and refresh, but don’t neglect recharging your professional batteries…

READ A BOOK.  Is there one on your teaching bookshelf that you’ve been meaning to read?  Here are a few that I’ve read recently or have on my “Pedagogy and Learning” list for this summer:

  • Make It Stick: The Science of Successful Learning by Brown, Roediger & McDaniel
  • 5 Practices for Orchestrating Productive Mathematics Discussions by Smith & Stein
  • Embedding Formative Assessment by William & Leahy
  • Mathematical Mindsets by Boaler

If you have the opportunity, find a “buddy” or group and read together.  Try one chapter a week and discuss in person or via email.  Our Teachers Teaching with Technology cadre of instructors did book “discussion chats” this past year.  Here are some ideas and prompts to organize your comments on Embedding Formative Assessment and 5 Practices (thanks Jennifer!)

And since it is summer, I’m also planning to do some fun math reads.  Consider these, or maybe there are others you have your eye on (tell me in the comments!). The last two have the advantage that each chapter is a stand-alone essay, which is especially good if your attention span is shortened by summer distractions.

  • How to Bake Pi by Cheng
  • The Man Who Knew Infinity by Kanigel (now a major motion picture!)
  • Here’s Looking at Euclid by Bellos (also a math columnist for The Guardian)
  • The Joy of X by Strogatz (originally an essay series for The New York Times)

One more suggestion for your reading list is to catch up on an NCTM journal article you meant to read this year but didn’t have time to.  There are free previews of some articles on the website if you aren’t yet a member.

LEARN SOMETHING NEW (anything! Doesn’t have to be math-related.)  If you come back to school in the fall and share your experience with your students, they will see you as a learner and it may encourage their efforts.  Here are some ideas:

Watch a webinar.  Did you miss one this year you meant to join?  On-demand recordings are easily paused so you can take notes or try a problem yourself.  For example, Texas Instruments has an archive of their free webinars here.

Go to a workshop/class/conference. There are plenty of these available in person and online.  Your department or district may have local offerings.  The TI Teachers Teaching with Technology PD workshops are listed here, and include “virtual” workshops as well.  Jo Boaler’s YouCubed organization offers an online course for teachers “How to Learn Math” [info here].

For my Connecticut and New England colleagues, two great opportunities are nearby.  The T3 Northeast PD Summit is June 22 & 23 in New Britain, covering both the TI-84+ and TI-Nspire [info here and sign up here].  And the Geogebra Institute of Southern CT is holding their 4th annual conference in New Haven on August 16 [info here], including a pre-conference workshop August 15 for beginners.

MAKE A PLAN.  One of the best things for me about being a teacher is the chance to revise and improve my teaching practice on a regular basis.  Some years I taught the same course to more than one class, so each lesson got two or three tries in the same day (or week).  I reflected on how it went with the first group and made adjustments and improvements for the next class.  Remembering the details for the next school year is harder to do, so I would make quick notes right in my lesson plan to capture the changes I’d like to try in the future.  Here is the reminder I used as the last item on my “Lesson Plan Template”:border self eval

If you have notes from this year about lessons you’d like to modify, find them before leaving the building for the summer.

In 1994, Steve Leinwand wrote an article in Mathematics Teacher  “Four Teacher-Friendly Postulates for Thriving in a Sea of Change”.  One of them has resonated with me ever since then:  “It is unreasonable to ask a professional to change much more than 10% a year, but it is unprofessional to change by much less than 10% a year.”  While many of us try to change and improve our teaching each year (or are asked/mandated to implement new practices), change is challenging and daunting.  Leinwand suggests that teachers consider changing about 10% of what they do each school year, a very reasonable amount (just one new lesson every two weeks, or one unit out of the ten you teach).  What will be your 10%?  What topic/class/unit needs work?  Get a jump start this summer on something you’d like to teach differently than you’ve taught it before.

So, take this as your summer assignment: PICK SOMETHING you intend to do to learn and grow professionally this summer, along with your plans to decompress and have fun.  Let me know how it goes.  Have a wonderful summer!


Leinwand, Steven. (1994).  Four Teacher-Friendly Postulates for Thriving in a Sea of Change. Mathematics Teacher 87(6):392–393. [Reprinted in 2007 during 100th anniversary of MT, with commentary by Cathy Seeley: Mathematics Teacher 100(9):580-583.]

More on  Recreational math books here: Math-Frolic.

Problems With Parentheses

I have been noticing lately that my students are making mistakes involving the use of parentheses.  Sometimes parentheses are overused and other times they are missing, and errors are also made while using calculator technology.  Using symbols and notation correctly is part of SMP #6, “Attend to Precision”, and is also a component of mathematical communication, since so much of math is written in symbols.  I want my students to be efficient and accurate in their work, and I hope their notation supports their conceptual understanding. So I’ve been contemplating the purposes of parentheses…

Purpose #1: To Provide Clarity with Negative Integers.  Negative integers can be set off with a pair of parentheses for addition and subtraction, as in these examples, but the expression’s value is unchanged if the parentheses are not used:

  1.    (–4) + 6 = 2
  2.    6 – (–4) = 10

With an exponent on a negative integer, however, the parentheses are essential.  We are working on sequences and series in Algebra II.  When a geometric sequence has a negative common ratio, the explicit formula has a negative number raised to an exponent:

  1.    The sequence  2, –6, 18, –54, … has explicit formula  An = 2·(–3)n-1

To convince my Algebra II students that the parentheses are required, consider  –32  vs.  (–3)2  on the TI-84+ calculator:

Capture 1

The calculator executes the order of operations: exponents are evaluated before multiplication. Since the negative sign actually represents –1 times 32, the 32  is evaluated first.  Although I prefer students to focus on conceptual understanding and not merely procedural rules, I say to “always use parentheses for a negative base”.

Purpose #2: To Specify the Base for Exponentiation.  Another class is studying exponents and logs, and students notice that using parentheses has mathematical meaning for the result.

  1.    (2x)3    vs.   2x3
  1. Each component of the fraction within the parentheses gets raised to the power; these are all different (and the TI-Nspire CAS handles them nicely):

04-28-2016 Image004

Attending to precision is essential for students, and by doing three similar but different problems as a set, they get practice analyzing how the notation changes the results.

Purpose #3: To Properly Represent Fractions.  Fractions generally don’t need parentheses when written by hand, and I’m direct with students about my strong preference for a horizontal fraction bar rather than a diagonal bar when writing fractions on paper or on the board.

Complications can occur when students try to enter the fraction into a calculator without using a fraction template.  Pressing the DIVIDE button to create the “slash”, as in 3/4, has the advantage of connecting a fraction with the operation of division but the drawback of the diagonal bar.  For anything more complex than a simple fraction, parentheses are needed to “collect” the numerator and denominator so that the fraction is computed correctly.   For example:

  1. Find the mean of these three test scores: 85, 96, 77.

Capture 4

  1. Graph a rational function

Capture 3

Thankfully, fraction templates are readily available, so errors using parentheses are avoided.   On any TI-84+, set the mode to “MathPrint” and press ALPHA and Y= to access the template.  On a TI-Nspire, press CTRL and DIVIDE or select the fraction from the template palate.

This was especially useful for finding the sum of the following geometric series; notice the error on the first try due to missing parentheses, and then the corrected version:

And the calculator comes to the rescue! I encourage students to enter complicated expressions all at once.  Making separate entries for each part is taking a risk:

Capture 7

Purpose #4: To Indicate Multiplication. Probably the area in which I am observing the most “overuse” of parentheses is for multiplication.  At some time before students reach me in high school, they have been taught that in addition to using the × symbol to multiply, they can also use • , a raised dot. A third alternative is to use parentheses to indicate multiplication, especially for negative integers or to distribute multiplication over addition:

  1.    (–4)( –6) = 24
  2.    2x(x + 5) = 2x2 + 10x

I’ve seen some students “over-distribute” if they rely on parentheses instead of the raised dot for multiplication:

11.       (–4)(x)(3x2)  should be –12x3 ; however what if a student “distributes” the –4?

over distribute

[One more pet peeve of mine: when students utilize the × symbol for multiplying even when using the variable x.  I strongly suggest that once they are in Algebra I, students should “graduate” to the raised dot  •  to symbolize multiplication.]

When using the Chain Rule in Calculus, students sometimes make the error of “invisible parentheses” and then lose them entirely in their subsequent algebraic simplification.

  1. Find the derivative of (3x2 – 4x + 5)–2

invisible parenth

Notice the missing parentheses for (6x – 4) and how the error carries through.

Purpose #5:  Operator Notations.  My final category of parentheses usage is as part of the notation of certain function operators.  Students are familiar with using parentheses in function notation f(x), where the independent variable x is the input for the function expression.  Other functions such as logs and trig functions can use parentheses to set off their “arguments”, and the calculator supports this use by providing the left parenthesis.  Entering the right parenthesis is optional on the TI-84+, but a good practice for students:

Capture 5

If students get in the habit of using the parentheses, it enables them to correctly apply the “expand to separate logs” and “condense to a single log” rule.  Here the parentheses are not “needed” to indicate the argument of the log, but helpful for this student.

  1. Solve each equation:

And in these last two examples, the parentheses helps the student get the correct result:

  1. Expand to separate logs:


  1. Condense to a single log:


One final note: I want my students to harness the power of parentheses to support their conceptual understanding and mathematical accuracy.  Being precise about notation is not about “doing it my way” but instead about doing it in a way that helps them grasp the purpose of the symbols they use to clearly communicate their mathematical thinking.


For more about the “loss of invisible parentheses”, ambiguous fractions and other common math errors, see this site.

For one teacher’s approach to using parentheses to evaluate function values, read this blog post: An Algebraic Oath.

And here is one teacher’s elegant and simple definition of parentheses: Parenthetically Speaking.