Category Archives: Teaching Practices

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 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: http://makeitstick.net .

**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.

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Searching for Structure

Recently, I read on Twitter some teachers’ frustration with students who just want to know the quick procedures to do the math at hand and don’t have much interest in the meaning of the underlying concepts.  I often come across this dilemma in my one-on-one work with students; in this tutoring role I especially feel the pressure to teach the “how-to” for an upcoming test and don’t always have the time to explore the “why” with the student.  I wrote a bit about this tension before in this post.

Another conversation on Twitter was specific to Algebra 2, about how to build on past knowledge even when some/all of the students seem not to remember that past knowledge.  How might we deepen students’ understanding and not simply retread the procedures?

I was faced with these dual dilemmas when I worked with a student this week reviewing Complex Numbers and Quadratic Equations for an upcoming test.  My approach: pay attention to mathematical structure.

A. Fractions involving imaginary numbers:

imag numbers

These three examples, examined together, allowed us to explore how to handle a negative value in the radicand (“inside the house”) and also how to handle a two-part numerator* with a one-part denominator.  Once the imaginary  i  was extracted and the radical simplified as much as possible, we took a look at when we could and couldn’t “simplify” the denominator.

I wanted to help my student avoid the common mistake of trying to “cancel”** when you can’t.  We used structure to explain.  When there is a two-part numerator and a one-part denominator, you can do one of 3 things:

“Distribute the denominator” to make two separate fractions.  I find this is the most reliable routine to avoid mistakes.

distribute-denominator-e1507861388968.jpg

Divide “all parts” by a common factor.

Divide all

Factor a common factor (if any) in the number, then simplify with the denominator.

factor-common.jpg

{*I wasn’t sure if the numerators qualify as “binomials” since they are numeric values, but my student and I discussed how they have two terms on top and one term on bottom, which can be challenging to simplify.  This structure will be encountered later when solving equations using the quadratic formula.}

{**I have avoided the use of “cancel” since I became more familiar with the “Nix the Tricks” philosophy of using precise mathematical language and avoiding tricks and “rules that expire”.  See resources below for more on this.}

B. Operations with complex numbers:

Again, we looked at a set of three problems to examine structure, which leads us to the appropriate procedures:

operations2

What is the same and different about #10 and #11?  What operation is needed in each?  Which is easier for you?

What is the same and different about #11 and #12?  What do you call these expressions:  4 – 5i and 4 + 5i ?  If you notice this structure, how does the problem become easier?

This led to a fruitful discussion of combining “like terms”, what is a “conjugate”, and whether it mattered if the multiplication was done in any particular order.  My student had been taught to always list answers for polynomials in order of decreasing degree, as in x2 – 2x + 1, so he was writing any  i2  terms first.  This isn’t wrong, but the rearranging of the order of the multiplication could have caused a mistake, so we talked about whether  is a variable or not, and when might it be helpful to treat it like a variable.

By noticing the structure of conjugates and why they are used, we got away from merely memorizing math terminology and instead added to conceptual understanding.

C. Using the discriminant

The discriminant is one of my favorite parts of the quadratic equations unit!  Students must pay attention to the structure of a quadratic equation (is it in the standard form ax2 + bx + c = 0 ?) before using the discriminant to give clues about the number and type of solutions.

discriminant

Rather than memorize what the discriminant means, look at where it “lives” in the quadratic formula.  It is in the radicand, which is why a positive value yields real solutions and a negative value does not.  The radical follows the ± , which is why nonzero discriminants give two solutions (either a real pair or a complex pair).  And the two solutions are conjugates of each other, something that I hadn’t really thought about when I got real solutions using the quadratic formula.  (And there is a nice surprise when you examine the two parts of the “numerical conjugates” and relate them to the graph of the quadratic. See note below for more on this.)

conjugates

I recently read this post about one teacher’s success having students evaluate the discriminant first, then tackle the rest of the quadratic formula.  Her strategy integrates the use of the discriminant with quadratic formula solving, instead of making it a stand-alone procedure.


Calculator Note: when evaluating the Quadratic formula on the TI-84+ family of calculators, use the fraction template  Untitled  to make the calculator input match the written arithmetic.  Press ALPHA then Y= for the fraction template, or get it from the MATH menu.  Then edit the previous entry for the second solution (use the UP arrow to highlight the previous entry and press ENTER to edit).  Here is #26 from above:

Capture 1

Another Calculator Note: the TI-84+ family in a+bi mode can handle the addition and multiplication questions #10-12, so if you are assessing student proficiency on these skills, have them do it without using the calculator.  The color TI-84 Plus CE can operate with an imaginary number within the fraction template, such as questions #4-6 and anything with the quadratic formula.  You can use either the  i  symbol (found above the decimal point) or a square root of a negative number.

Capture 2 arrow

However the B&W TI-84+ can’t use an imaginary number within the fraction template.  Use a set of parentheses and the division “slash” for this to work:

Capture 1 BW


Notes and Resources:

Nix the Tricks website and book: nixthetricks.com.  And lest you think that “Distribute the denominator” is yet another trick, consider this:  The fraction bar is a type of grouping symbol (like parentheses) and it indicates division.  Dividing is equivalent to multiplying by the reciprocal.  So the “distribute the denominator” work for #4 above is also this: Capture

Three articles about “Rules That Expire” have been published in the NCTM journals.  Currently all three are available as “FREE PREVIEWS” on the website.

“Look for and make use of structure” is one of the Standards for Mathematical Practice (SMP #7) in the Common Core State Standards found here: http://www.corestandards.org/Math/Practice/

The “nice surprise” about the solutions to a quadratic equation written as “numerical conjugates” and their relationship to the quadratic graph was pointed out to me by Marc Garneau.  His post here gives more detail and a student activity to go with it.

Thanks as always to the #MTBoS and #iTeachMath community on Twitter for great conversations!

That Voice In Your Head

question 2When I work with students one-on-one, I get a unique window into their thinking.  Everyone has a test this week, including several students who are taking the AP Calculus exam.  As we are preparing, I’ve noticed the constant push-pull of the conceptual vs. procedural  debate, because students need to finish by the end of the hour with me, and go away knowing “how to do it.”  I know that giving them conceptual background will help make their learning more durable, but some of them resist going beyond the procedure and don’t welcome the “why does it work that way” explanation.

I’ve found myself with students referring to “that voice in your head” in order to get them to communicate their mathematical thinking, to connect the new knowledge to past related topics, to think about the underlying concepts for each process and help them build structures to support their understanding.

Here’s what I want that voice to be saying:

1. What Does It Look Like?

Knowing what the graphs of various function families look like allows for easy transformations using parameters.  This week we transformed graphs of log functions and rational functions:

In addition, students found limits of functions without technology, based on what they knew of the nature of the graphs.

To find this limit capture limit

it is helpful to know these graphs:

To find this limit, think about the end behavior and how to determine horizontal asymptotes for rational functions:

Screen Shot 2017-05-09 at 9.50.12 PM


2. What Am I Looking For?

A colleague noted recently that math is all about the verbs: solve, simplify, evaluate, and so on.  When students pay attention to what they are being asked to do, the process follows easily.

For example, while solving equations, students are looking for the variable, which is located in different places in linear, quadratic, exponential, logarithmic and rational equations.  Finding where the variable is now can guide students to a process for solving: inverse operations, factoring and zero product property, converting between exponential and log forms, condensing to a single log, finding a common denominator to clear fractions, etc.

This is an example of an exponential function in a quadratic format; relating prior knowledge of quadratics and “looking for x” enabled the student to solve successfully.

IMG_5386


3. What Are The Tools In My Toolbox?

When faced with a problem, think about what tools are available.  With rational expressions and equations, students begin by factoring, and then often (but not always) find a common denominator.

One of my students could easily add these fractions with an LCD: rat expr

But had trouble solving this equation: rat eq

She had learned one strategy for the expression, and then there was a “new” strategy for the equation that involved multiplying through by the LCD to “clear fractions”.  She couldn’t keep track of which factors remained and was suceptible to errors.

Instead, we built on the strategy of creating common denominator fractions; once all the fractions have the same denominators, she can work with only the numerators and solve successfully:

IMG_5396

We also used this strategy to simplify a complex fraction; we created a single fraction in both the numerator and denominator, then remembered that a fraction means DIVIDE:

IMG_5397

The AP Calculus students also think about their toolbox when faced with an integral problem: what are the integration strategies they can use as tools?

  • Do I know an antiderivative? (Can I simplify algebraically to make one)?
  • Is there a known geometric area I can find?
  • Is part of the integrand the derivative of another part? (U-substitution)
  • Can a trig relationship help me rewrite the integrand?
  • Does the integrand contain a product? (Integration by parts)
  • Is this a definite integral on the calculator section? Use the calculator!

4. Where Are The Trouble Spots?

When finding the domain of a function, focus on the numerical values that “cause trouble.”  Where should we look for trouble in these?

domain functions

Finding limits and derivatives of piecewise functions also puts students on the hunt for trouble, and certain integrals need special treatment due to discontinuities:


5. What Should I Write Down?

File May 09, 11 50 40 PMWrite enough to show your mathematical thinking to a teacher/reader who doesn’t know you.  Write enough to be clear and get it right.  No bonus for doing it in your head on multiple-choice, and there is definitely a penalty for doing too much in your head and getting it wrong.  And on free-response questions on the AP (and most questions on teacher-created tests), you need to show work that supports your conclusion.


6. Does My Answer Make Sense?

Even if calculators aren’t available, students can estimate square roots, logs, and other results.  For example,

estimate

In word problem situations, does the answer make sense?  If Amy can do the job in 4 hours and Josh can do it in 6 hours, together they should take less time than either of them working alone.  And don’t forget appropriate units if a problem is situated in a real context.


7. How Are These The Same/Different?

Analyzing the small differences between examples helps students home in on important features.

What intercepts and asymptotes will these functions have in common?

IMG_5399

(Calculus) What are the different requirements and results of the Intermediate Value Theorem and the Mean Value Theorem?  What is the difference between average rate of change and average value of a function?


8. How Do I Feel About This?

Finally, I ask my students how they are feeling about the material:  Are you finding this unit easy or hard?  What parts are more difficult for you?  If you find this to be challenging, you need to put on your thinking cap.  Saying “I can’t do it” gets in the way of your understanding; instead, say “I can’t do it YET, I’m learning” and focus on the MANY things that you do know.  Don’t overthink the easy things or overlook the tough details.  Being confident is an important ingredient for your success.

You’ve got this.


NOTES & RESOURCES:

Two Geogebra applets for transformation of functions are found here (multiple parent functions) and here (rational functions).

More on “easy” and “hard” labels and their impact on students in these blog posts: “Things Not To Say” and “The Little Phrase That Causes Big Problems”.

In The Zone

I attended the Teachers Teaching with Technology (T-cubed) International Conference this past week in Chicago, and my four days were chock-full of thought-provoking sessions and conversations.  As expected, there was much talk about technology in math and science classes, but there was also a great deal of discussion concerning thinking about teaching and learning in math and science classes.  Here are some of my take-aways:

index

1. Get out of your comfort zone into the “learning zone.” In order to learn something new, get out of established routines (about doing math, about teaching math) and try another technique or perspective.  I pushed myself to learn about topics that are not my specialty, things I might have otherwise avoided.  I tried coding some calculator programs, tackled STEM/engineering design projects, learned a bit of 3D graphing, and brushed up on statistics.  Some things came easily, and other new skills were quite difficult.  At one point in a STEM session, I declared that the project was “really hard”, and sat back to watch others.  I was urged to continue by the presenters and fellow participants, and we worked together to complete the task.  I gained confidence because they had confidence in me, and was able to achieve something I thought might be beyond my abilities.  I was really glad I persevered in the task, and had acknowledged that it was “hard” but not “too hard.”

2. Productive struggle takes time. One of the eight Mathematical Teaching Practices in NCTM’s Principles To Actions (2014) is to “support productive struggle in learning mathematics.” In one of the sessions, presenters Jennifer Wilson and Jill Gough had us work on a difficult geometry problem, asking us to:

  • Allow for “individual think time.” Please don’t steal another student’s opportunity to think by talking too soon.
  • Go beyond getting the answer. What can you show about how you make sense of the problem?
  • Work with it some more. Can you find another way? Can you relate your solution methods to each other, or to another person’s method?

I was surprised by two things that happened in this session.  First, as the presenters circled around the room reviewing our progress, they didn’t evaluate our work or tell us if we were on the right track.  When Jennifer suggested that I keep working—which I interpreted to mean that I had made an error in my reasoning—my solution was actually correct.  I realized later that taking more time to work had been advantageous and it had spurred me to find alternative methods.  A teacher saying to a student “work on this some more” is not the same as “you’re doing it wrong.”

The second big surprise was how long the process of struggling toward a solution took us.  I thought perhaps we had been working for ten or fifteen minutes (we are math teachers, after all, so I expected we would find solutions quickly).  In fact, we had been working for 35 or 40 minutes, which was astounding to me.  Time had seemed to slow down, and there wasn’t any race to finish up. I was able to really make sense of my multiple solution methods, and enjoyed pursuing more ways to solve the problem.

So if “promoting productive struggle” is a goal for your classroom, consider how to encourage it with your messages to students.  Think carefully about implementation: you need to invest ample time, but you will reap rewards in terms of the variety and depth of math concepts your students will uncover.

3. Be responsive to your professional peers. In his keynote, Timothy Kanold spoke about the professional community that exists in a school, and how a robust collaboration enhances the success of all students.  When teachers engage with each other, sharing best practices, kids experience similar (high-quality) learning environments even if they have different teachers.  When teachers operate independently, there can be substantial inequity between what students in different classrooms receive.

How might we apply this in our school settings? If there is a great activity you use in one of your courses, perhaps share it with colleagues teaching the same course.  If you employ technology to build mathematical understanding, try to make sure others have access to it as well.  You will benefit too:  discussing a lesson or teaching practice with a colleague will make your teaching repertoire more robust.  Attempting something new and different is easier when you have a like-minded peer who can give feedback (or possibly observe your lesson in action).

So what are my next steps? Returning home after an exciting professional conference always involves catching up and getting back to my regular teaching routines. But Jennifer and Jill commented that “closure” is not simply packing up at the end of class; instead I need to reflect on what I’ve learned, and thoughtfully consider how to put it into my practice.  I invite you to join me: take a risk, plan to do something new in the next few weeks, and find a colleague to collaborate with you as you move into the learning zone.


NOTES & RESOURCES:

For more about the T³ International Conference, search #T3IC on twitter or go to the Texas Instruments website.

To register for T³ summer workshops on Promoting Productive Struggle, STEM Projects, Coding, and Teaching Strategies for Success with Technology, go to the registration section of the TI website.

There is a full set of coding lessons and teacher guides available on the TIcodes section of the TI website.

The “Learning Zone” idea is related to Vygotsky’s “Zone of Proximal Development” (1978).  For more on scaffolding and ZPD, see Middleton, J. A.  & Jansen, A. (2011). Motivation Matters and Interest Counts: Fostering Engagement in Mathematics. Reston, VA: NCTM, chapter 9 “Providing Challenge: Start Where Students Are, Not Where They Are Not”.

More on “Promoting Productive Struggle” on Jennifer Wilson’s and Jill Gough’s websites.

Where Are You? I’ll Meet You There

guy-magnifying-glass

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.

bringyouragame

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…


NOTES & RESOURCES::

* 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!


NOTES & RESOURCES:

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:

Expand

  1. Condense to a single log:

Condense

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.


NOTES & RESOURCES:

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.