Action-Consequence-Reflection Activities for GeoGebra

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

In the Action-Consequence-Reflection cycle, students

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

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

EXPLORING GRAPHS & SLIDERS:

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

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

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

INTERACTIVE VISUALIZERS:      

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

UNDERSTANDING STRUCTURE:

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

INVESTIGATING INVARIANTS:

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

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

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

PLANNING FOR REFLECTION:

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


Notes and Resources:

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

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

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

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Testing Tips: Using Calculators on Class Assessments

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

Testing Tips: Using Calculators on Class Assessments

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

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

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End of Quarter Feedback Is a Two-Way Street

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

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

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

 

My Four Questions

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

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

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

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

What Will You Do?

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


Notes and Resources:

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

Read the full post on the TI BulleTIn Board:

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

Tips for Transitioning to the TI-Nspire

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

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

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

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

test pencil image

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

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

Good luck!

Moving the Needle

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

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

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

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

1. [I will] Talk and write less.

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

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

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

2. [They will] Stock the mathematical toolbox.

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

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

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

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

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

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

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

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

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

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

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


Notes:

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

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

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

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

Table Techniques

ti84ce

How might we use the graphing calculator Table feature to build conceptual understanding and support procedural knowledge?  Here are some ideas…

 

A. Dynamic Tables is one way I use the Action–Consequence–Reflection cycle in my teaching.  We generate a table so that students perform a mathematical action, observe a consequence, and reflect upon the mathematical meaning in order to build conceptual understanding.

In Algebra, when students learn to distinguish linear vs. exponential growth*, we enter the simple equations Y1 = 2x and Y2 = 2x on the TI-84+ family of calculators (TI-84+CE shown here).

To observe the growth of the functions numerically, adjust the Table Settings:  press 2nd WINDOW for [TBLSET] and set the Independent variable to AUTO and the Dependent variable to ASK (see above right).  This will allow students to generate the Y-values one at a time, rather than have them appear all at once.  Press 2nd GRAPH to view the table and press ENTER to generate each value, moving down each column.  View the video here.

How does the Y-value change as you move down each column?  Students should use mathematical language to describe what they observe.  Can you tell where the graphs would intersect?  Which equation grows faster?

Alternatively, set the Independent variable to ASK and the Dependent variable to AUTO.  In this setup, enter the X-value and both Y-values will fill in.  I use this when students are comparing two scenarios to see which grows faster.

Screen Shot 2018-06-24 at 5.41.06 PM

 

Another example of Dynamic Tables is to build understanding of negative and zero exponents.  Enter the following into Y1: equation (the fraction is used to force fractional results in the table; make sure Answers are set to “Auto” under MODE).  Set Independent to ASK and Dependent to AUTO, then enter X-values of 5, 4, 3, 2, and 1 in the table.

  • What mathematical process is happening to the X-value in each new row?
  • Subtracting 1 each time, which means subtracting 1 from the exponent.
  • What mathematical process is happening to the Y-value?
  • Dividing by 2 each time.

Then ask, what do you predict will happen when X = 0?  When X = –1? When X = –2?  Change the base to 3 or 5 or 10 and observe.  Students can now explain what to do when a base is raised to a negative or zero exponent.

Capture 3

Here is what we wrote on the board as we explored the table on the calculator.

IMG_8312

 

B. Noticing Invariants

Tables can also be used to notice Invariants—quantities, shapes or locations that do not change even though other things are changing.  Set up the table starting at 0 with an increment of 15, AUTO-AUTO and use Degree mode.  Examine the table for Y1 = sin(x) and Y2 = cos(x).  What do you notice?  Are any values equal?  Why does this occur?

My students and I don’t just examine the table, we also look at a geometric figure of a right triangle with side lengths that are easy to compute with, such as 3, 4, and 5.  Determine the sine and cosine of each angle and discuss how this relates to the table values.

right tri

Next, add another expression into Y3 as shown below left: use (Y1)2 = (Y2)2 to represent sin2x + cos2x, pressing ALPHA TRACE to access the Y-variables.  What does the table display?  Why is this so?  Again, refer to the right triangle figure—can you explain why this property is known as the “Pythagorean Identity”?

 

C. Aids for Factoring and Simplifying

So far we’ve used the table as a tool for inquiry; now we turn to using it as an aid for computation, number sense, and procedural fluency.  For factoring trinomials and simplifying radicals, students need to determine the numerical factors of a number.  When the number is large, or the student needs some scaffolding support, enter into Y1 the number divided by X and view the table (TblStart=1, ΔTbl=1).

For example, Y1 = 72/X  and the table clearly shows which numbers are factors and which are not, depending on whether a decimal remainder results (note that the “slash” version of the fraction bar forces decimal results).

For students needing assistance remembering perfect squares, cubes, or other powers, enter those functions into Y= and view the table.

When simplifying radicals into exact form, combine the two techniques to find perfect squares, cubes, etc. that are factors of the radicand value.‡

Of course, in both of these examples, students could simply enter calculations on the home screen until they hit upon the “right” divisor.  The table has the advantage of systematically presenting the information in one place.

 

D. Generating Sequences

If an explicit formula for a sequence is known, simply enter it into Y= and set the table to start at 1 with an increment of 1.  For example, the sequence 2, 5, 8, 11, … has explicit formula an = 2 + 3(n – 1).  In function mode, x is used in place of n.

This can also be accomplished in Sequence Mode.  nMin is the starting term number, u(n) is the explicit formula for term un, and u(nMin) is the value of the first term u1.  Note that the symbol n is found on the  x theta t n  key in sequence mode.  Here is the same sequence as above.

Capture 26.png

Although Sequence mode isn’t necessary for explicit formulas, it is very useful to generate a sequence recursively**.  This time, express u(n) in terms of the previous term u(n–1).  The u(n–1) variable is found by pressing ALPHA TRACE (or type it directly with the u above the 7 key).

Capture 27

Back in Function mode, I’ve also discovered that helpful sequences can be created with the Table.  When Precalculus students studied the Binomial Theorem, they often wrote out several rows of Pascal’s Triangle rather than use the nCr values for the appropriate power.  The table comes to the rescue: enter nCx into Y1, with the numerical value of the power for n.  Begin the table at 0 and increment by 1, and the appropriate row of Pascal’s Triangle is displayed.

Whether you use the table to enable investigation and inquiry, or use it to support numerical and procedural fluency, take these Table Techniques to your classroom!


Notes and Resources:

*The complete activity using Dynamic Tables to explore Linear and Exponential Growth is here.

**Recursive sequences can also be generated directly on the Home screen of the TI-84+ family, as an alternative to Sequence Mode.  Simply enter the value of the first term, then perform the recursive operation on the ANS, and press enter for the second term.  Finally, press ENTER as many times as desired to generate the sequence.  Below left is the same sequence discussed above; below right is the sequence based on paying off a $500 credit card bill with 24% annual interest and monthly payment of $75.

‡Thanks to Fred Decovsky for this suggestion.

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 ssddproblems.com, 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:

img_7724.jpg

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 ssddproblems.com 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 ssddproblems.com 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 MakeItStick.net.  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

IMG_7212

IMG_7214

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.

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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.