Personalized Learning Can’t Trump Content & Pedagogy

The problem I’m seeing with personalized learning (overall and especially as it pertains to math instruction) is the common understandings about what it is, what it can look like, what it shouldn’t look like, and how it works as related to our own learning experiences are fragile at best.

Many school systems, including my own, are looking at personalized learning as a means to improve math instruction, raise math test scores, and increase student engagement. These goals are great and many systems have them in some form or another. However, when personalized learning forces teachers into using sweeping generalized practices that often trump solid content pedagogy, something is drastically wrong.

I don’t think this is necessarily the fault of personalized learning as a concept,  but I do think it is problematic when common understandings become compromised.  These compromised understandings lead to sweeping generalized practices like:

  1. No whole group instruction – ever
  2. Students should be on a self-paced computer program for personalized learning
  3. Teachers have to create new groups of students every day/week to make sure learning is personalized
  4. Teachers should do project based learning several times per unit to engage learners
  5. Teachers need to use choice boards for every standard they teach.

This is not a definitive list – just what I’ve heard from within my own district over the last few years.

I may not have a response to each of these, but I can point out a few sources in addition to my thoughts:

  1.  No whole group instruction – ever – Dan Meyer’s post: http://blog.mrmeyer.com/2014/dont-personalize-learning/  my favorite idea from this is from Mike Caufield: “if there is one thing that almost all disciplines benefit from, it’s structured discussion. It gets us out of our own head, pushes us to understand ideas better. It teaches us to talk like geologists, or mathematicians, or philosophers; over time that leads to us *thinking* like geologists, mathematicians, and philosophers. Structured discussion is how we externalize thought so that we can tinker with it, refactor it, and re-absorb it better than it was before.”

2.  Students should be on a self-paced computer program for personalized learning Personalized learning is not something you get get from the App Store or Google Play  or from any ed tech vendor.

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Some other comments from Dan Meyer:  Personalized Learning Software: Fun Like Choosing Your Own Ad Experience  and from Benjamin Riley:  “Effective instruction requires understanding the varying cognitive abilities of students and finding ways to impart knowledge in light of that variation. If you want to call that “personalization,” fine, but we might just also call it “good teaching.” And good teaching can be done in classroom with students sitting in desks in rows, holding pencil and paper, or it can also be done in a classroom with students sitting in beanbags holding iPads and Chromebooks. Whatever the learning environment, the teacher should be responsible for the core delivery of instruction.”

3.  Teachers have to create new groups of students every day/week to make sure learning is personalized – I’m not sure this is the case.  If teachers really know where their students are in their mathematical progressions (lots of ways to do this – portfolios, math journals, student interviews (GloSS and IKAN from New Zealand, etc.)  These types of data are much more effective that computer testing programs because teachers are able to see and hear students’ thinking as well as their answers.  In my opinion, you can’t get more personalized than that!

4.  Teachers should do project based learning several times per unit to engage learners – anyone who has had PBL training knows that 1 per year is a good start!  PBL takes time – to plan, and plan some more (most often with other content areas).  If anyone expects more than one per year or semester initially, it’s time to have some Crucial Conversations!

5.  Teachers need to use choice boards for every standard they teach – student voice and choice does not have to be a choice board.  And really, how much of a choice do students have if we’re giving them all possible choices with no input from them?

To sum up: In order to really improve those goals of improving math instruction, increasing student engagement, and raising math test scores one thing is certain – an investment to increase teacher content and pedagogy knowledge must be at the forefront.  There is no other initiative or math program that will help districts reach these goals more effectively than this!

 

 

 

 

 

 

 

NCTM Nashville – Twitter, Modeling, and Desmos, Oh, My!

OK, so my timing on this is not great.  This was actually written back in December (still a little tardy) and then the holidays ran over me.  Blah, blah, blah.  Nevertheless, everything below is still relevant.

Having attended and presented at conferences before, I have to say some conferences are good, and some not so good.  NCTM Nashville 2015 was, in my opinion, the best I have attended – hands down!

Here’s why:

On Wednesday evening, in the opening session, Graham Fletcher, Robert Kaplinsky, Laila Nur, Andrew Stadel and Cathy Yenca set the tone for the conference.  They spoke about their personal experiences of improving mathematics teaching and what they use to continuously improve their practice, they all spoke about how accessible and personalized PL for math teachers’ needs can be with a Math Blogs, Twitter, the #MTBoS (Math Twitter Blog-o-Sphere) that links them all together, and Web 2.0 tools that are not only changing the ways we think about teaching mathematics, but also the ways students engage in mathematics in their classrooms.  One word:  Powerful.  And as I said before, it set the tone for the rest of the conference.

The rest of the sessions, at least the sessions I attended, all connected to the opening session.  In the Desmos sessions I attended with Michael Fenton and Christopher Danielson, the presenters were able to take novices through the simplicity and beauty of this free graphing calculator (which is really much more – see my post on this here) and those of us who are just above the novices had plenty to learn as well.  I even had a Desmos special tutoring session from Cathy Yenca and Julie Reulbach in the back of one of these sessions.

The twitter sessions I attended were always full and the session facilitators, as well as many attendees, lent a hand to those who wanted to get on board “this Twitter math train.” In addition, LOTS of people stopped by the MTBoS booth and were given some “small group” lessons on how to use Twitter, who to follow, and were given some general tips to make the whole experience low stress!  Michael Fenton and John Mahlstedt were the facilitators of the Twitter sessions I attended.  In each of these sessions, attendees were eager to learn more about Twitter and how it could help them become better math teachers.  Even some not so eager people were asking questions near the end of these sessions!

The rest of the sessions I attended (I even co-presented one) had to do with modeling with mathematics – SMP 4.  These sessions were probably the most valuable to me for two reasons:

  1. We got to really dig in to some math and have some great mathematical discussions!
  2. I got to experience more modeling in secondary mathematics which is great since I have just rejoined the secondary math world.

Ashli Black‘s session:  Selecting and Using Tasks to Develop MP.4: Model with Mathematics was all about investigating characteristics of modeling tasks and working with pitfalls.  I recommend following Ashli on twitter: @Mythagon.  She really knows what modeling with mathematics should look like in the secondary math world, she’s a great presenter, and I’m thankful that she took the time to fill out the speaker form last year.  

Michael Fenton‘s session on modeling provided a one-two punch – Modeling WITH Desmos!  This was an incredible session.  Michael’s presentation combining Desmos with mathetmatical modeling was.  I was making sense of mathematics through the models created.  I wish I had learned math this way, initially! While I can’t go back in time to learn this way for the first time, I can make sure that the students in my district have the opportunity.  And it’s one of my goals for this year.

Andrew Stadel’s session: Model with Mathematics using Problem Solving Tasks.  I have to admit, I’ve been using Andrew’s resources from his blog for a few years, but it was a real treat attending his session.  He engaged us in a three-act task: Swing Wraps.  This problem solving task engaged us in mathematical arguments, modeling, and sense making and a few other SMP’s.  Mr. Stadel also did some modeling of his own through the types of questions he asked to the whole group and small groups, through his guiding of the discussion, and through his commentary about the importance of doing these types of problems.

So, in conclusion, here’s what this all boils down to:

  1. Join Twitter and become a part of the #MTBoS
  2. Allow students to model the problems they solve with mathematics.
  3.  Take a look at Desmos – a long hard look – one that allows you to see it for more than just a free online graphing calculator that students can use to model with mathematics (that should be enough-but there’s oh-so much more to it!)

 

 

Desmos Math Addiction

Hi, my name is Mike… and I love using Desmos with students.

This is not a bad thing at all.  I’m not giving up time with my family to spend on Desmos. It’s just that whenever I think I’ve exhausted all of the ways to use this fantastic tool with students, the Desmos team adds a new activity or game that I can and want to use right away!  These people know how to keep us wanting more!

Here you can find out what Desmos is all about!

Now, for all of you teachers out there that haven’t engaged your students in this amazing math tool, let me move from a user to a pusher.  4 reasons why you should use this amazing tool with your students:

crazy about math

  1. It’s completely free!  (not just this first time – all the time)
  2. It’s a graphing calculator that works beautifully online or as an app for students to Model with Mathematics – SMP 4.

This is a screenshot of how my son, Connor, used the Desmos Calculator to make sense of transforming quadratic functions.

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3.  When you sign up as a teacher (again, for free) you can assign activities and games (yep, they’re all free to use, too) to your students and you can check their progress from your teacher page.

So, beyond the graphing calculator – which is amazing on its own – as a teacher you can assign an activity to your students based on the content they are investigating. Try Central Park  – it’s my favorite activity.  (If you like, you can go to the student page and type in the code qqbm.  I set this up for anyone reading this post. Feel free to use an alias if you like).

And as far as games go, check out Polygraphs.  It’s like the Guess Who? game for math class. Trust me, your students will love it and there are polygraphs for elementary as well as secondary. The polygraphs are all partner games, so students will need to work in pairs.  I’ve even made a few:

Polygraph: Teen Numbers

Polygraph: Inequalities on a Number Line

Polygraph: Geometric Transformations

4.  As you get sucked in to this tool, you may begin to think to yourself, “Boy, I really wish there was an activity for ______.  If only knew how to create an activity for my students to use on Desmos.” That’s taken care of, too, with Activity Builder and Custom Polygraph (and, yep, you guessed it – they’re free to use, too)

And before you begin to doubt whether you can create an online activity or polygraph, the Desmos team has already taken steps to make this extremely teacher friendly.  Before you know it, you’ll have your own Desmos activity published!

Finally, as a great end of year gift, Dan Meyer blogged about the latest from Desmos – Marbleslides.  If this doesn’t get you to use Desmos with your students. . . well, I’m sure they will think of something else, soon. But seriously, try this out.  I have re-learned and deepened my own understandings of mathematics by trying and reflecting on many of these activities and games, and then having my own kids do them (and then they ask me why their teachers aren’t using them – “Can you talk to them, Dad?”).  The conversations will be happening this semester for sure!

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But the best part about all of this is that students get to use the calculator to investigate graphs and compare graphs and equations/functions.  They get to notice and wonder about what matters and what changes a graph’s slope, and y-intercept for linear functions and what changes the vertex and roots of parabolas.  They get to investigate periodics and exponentials and rationals and so much more.  They get to engage in activities and games that have components that ask them to reflect on what they’ve learned in the games and activities themselves.  The students are doing the mathematics.

Then, in class, we get engage students in talking about the math they’ve investigated!  How sweet is that?

You see, as great as Desmos is, it can’t take the place of great teaching.  It’s a tool that can help us become better at our craft and help our students gain a deeper understanding of mathematics!  Sounds like a win-win!

So, I guess I don’t have a Desmos math addiction.  Addictions have adverse consequences and I see none of that here!  I just have – as we all do thanks to Desmos – access to a powerful mathematical learning tool!  Thank you Desmos.  I can’t wait to see what’s coming next!

Filling Gaps: Buy a Program or Help Teachers Grow?

This post actually started as a rant as I was sitting through meeting after meeting with really nice people trying to sell products to “Fill the Gaps.”  So, if it has a rant-y feeling, just know where I’m coming from.  If no one really likes this, that’s ok.  At least it’s out of my system for now.  You see, when you’re “invited” to attend meetings to raise student achievement, you really need to show up, or who knows what will  happen.  So, in the effort to stand up for teachers and students, I attended all of them.

Man shouting, pulling hair

These were really nice people presenting to us, and they were very passionate about their products.  I even largely agree with several of them on their basic philosophy.

At least one of the people listening with us in the room was sold on many of the ideas before we even started these meetings.  Every slide or picture shown was met with a “That’s good!” or a “That’s really good!”  I think if they showed us a shiny, new penny, this person would have said, “This is what our students need!” with the same reaction!  The pictures of bulletin boards showing concept maps and vocabulary word walls and even students working may be good – or may not.   Really, there’s no way to tell – especially with the picture of the students working.  What were the students saying?  Were they discussing mathematics?  Were they using the vocabulary on the bulletin board?  Were they making connections to the concept maps?  Did they give and receive feedback about their work?  Let’s see some video, so I can see how this is really working.

expresspicture_slider

Again, philosophically I agree with their framework of instruction.   However, the product is not really necessary if the PL these companies are willing to provide is effective.

Now, on to the PL.  Lots of good strategies offered here.  And more pictures of students “engaged.”   My question:  what are the students engaged in?”  Are they engaged in the mathematics or the product?  My initial response to this self-posed question was:  Does it really matter?  The students are working.  After  thinking about this for just a few seconds, though, I can say without a doubt that it does matter!

Engaging students can be tricky.   A passerby, seeing students working silently in their seats, might conclude student engagement in a task.  A passerby, seeing and hearing students discussing a task, may conclude non-engagement in a task as well as lack of classroom management.  Really it’s hard to tell, in either case, whether there was any engagement or what kind of engagement there was.

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Students in the sixth-grade Harlequin Team from Paris Elementary School work on a math problem. Clockwise, from front left, are Abby Steeves, William Dieterich, Annie Choi, Katerina Crowell, Halie Page and Sebastian Brochu.

So, what does engagement mean?  It depends on what you want.  One of my goals year after year is to engage students in the mathematics they’re studying.  When I first started teaching, I wanted students to just be engaged, no matter what.  As I think back, they were engaged – probably in my educational “performance.”  I was the “fun” teacher that did crazy math lessons.  As I grew professionally, my lesson focus evolved to take the students’ engagement away from me and toward the mathematical content.  So, why is it so important?  If students are engaged in creating the product (creating a poster, making a presentation, etc.)  they may be learning mathematics, but how do we know.  I’ve seen students engaged in creating beautiful products and walk away with little mathematical understanding.  I’ve also seen students engaged in mathematics and creating not so beautiful products, but beautiful understandings and mathematical connections.

So, for all of the professionals in the room thinking this (or any of the other presentations we’ve seen) is the silver bullet. . . It’s not.  The only silver bullet out there that’s going to raise student achievement is teacher PL grounded in  understanding mathematics conceptually and building teachers’ pedagogical understandings and strategies.  If we want high achieving students, we have to help teachers achieve their greatest potential.  No program out there will do that, but if you really want to become a better math teacher, Twitter and the #MTBoS are a great place to start!

Change vs. What Worked in the Past

So, I’m at this Standards Setting meeting in Atlanta this week.  I’m working with people I’ve never met before.  As we settle into our assigned seats, we begin the small talk:

I introduce myself (since I seem to be one of the last ones to arrive).

Others at the table introduce themselves as well and before long we’ve found some common ground (many of us are in a coaching role) and start building a professional relationship.  I love this part of attending professional learning sessions at a state (and national) level.  “All of us are smarter than one of us.”  By day two of our work, you would think we worked at the same school.  Our conversations, while still mostly professional, are much more relaxed.

During one of our breaks, we begin talking about some of the teachers we work with who are “stuck in their ways.”  The question bouncing around (at least in my head) is “Why?”  Why are they so stuck?  As we talked at our table, the “reason” that seemed to dominate the conversation was one that many of us have heard before:

The teachers “reason” is that teaching this way has worked for the past “y” years so why should I change now?

Our conversation then takes an interesting turn.  A “what if” turn.

What if Apple thought the same way.  How would our world be different?  Would we still have iPads, iPhones, Apple TV, etc.?  It’s unlikely.  We’d probably have something that looks like this:

First Apple Computer

because what worked in the past should be good enough now, right?

What if Ford Motor Company thought the same way.  How would our driving experience be the same?  I doubt we’d have radios, or even seat belts.   Our new ride may look like this:

1910Ford-T

because what worked in the past should be good enough.

What if we wanted cataract surgery?  How would that look, if the surgeons of today had the same attitude about what works best?  Did you know that cataract surgery goes back to ancient Egypt?  Would you rather have Lasik or have a surgeon come at you with one of these?

Ancient Eye Surgery

If the only thing keeping you from changing is because it’s the way you’ve always done it, then it can’t be the best.  We’ve been growing and changing the way we do things because we are always searching for the most efficient way, or the more cost-effective way, or the safer way, or the way that will improve our lives.  Have we done that for students.  Is the way you’re teaching mathematics what’s best for your students?  Is your pedagogy guided by what’s proven through research or just what you’ve done for years?

Our conversation ended abruptly because we had to get back to work on standards, but as I met and reconnected with others at the workshop, this same conversation came up multiple times.  My thoughts on this are below, but I hope others chime in here with their own thoughts on this.

Steven Leinwand wrote something several years ago that I think relates well to this.  What he wrote was (and I’m totally butchering this, I’m sure) that we shouldn’t expect more than 10% growth/change per teacher per year.  On the flip side of that, he also said that teachers should strive for more than 10% growth/change per year.

This is something I’ve really tried to work on in my coaching role with teachers.  When learning something that seems daunting to a new or veteran teacher (moving toward a standards-based, student-centered approach to teaching mathematics for example), I suggest teachers choose one thing, one piece of what we’ve discussed that they think they can become really good at over several months, rather than trying to make everything fit at once.

Letting teachers know they are not expected to become experts all at once is great, but following through is even more important.  Without constructive feedback, teachers will likely fall back to their comfortable habits.  Just like teachers need to really listen to students, coaches need to listen to teachers.  We need to model what we expect.

If we don’t, we may end up with this 20 years from now:

The Best Part about Blogging

This is super exciting!  I love it when teachers keep thinking – especially when I stop!  What you’re about to read is truly the best part of blogging!

Readers of Under the Dome have been terrific commenters and questioners of my posts over the last 2 years and you all just keep getting better.  Recently, Sharon Wagner, a teacher I met during a three-day summer institute in June visited my blog and reached out to share her ideas about the Olympic Cola Display 3-act task.

Sharon’s words:

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Through the course of a few emails over the summer and a lot of my time spent doing things outside of the MTBoS (my lovely wife got some of her honey-do’s completed and I got some of my Mike-do’s finished) I have Sharon’s extension and am now posting it with her blessing!  Please take a look.  Her idea is a natural extension and allows students to design their own display using the colors of Coca-Cola twelve packs (which she most helpfully added to her document).  Any Pepsi fans out there?

Sharon’s idea also ups the rigor by providing an audience (the merchant).  This, again, is a part of that natural extension (of course someone designs these displays for the merchants).  As for the Standards for Mathematical Practice . . . let’s just say your students will be engaging in multiple SMPs.

Again, this is super exciting.  I love to share my ideas here, but when someone else takes it and makes it better – in this case by adding to it – everyone wins.  Especially the students in our classrooms.

Thank you Sharon.

Sharon’s Display Extension:

coca cola display project extension

 

 

A Further Discussion of “Funny Math”

Georgia’s new state school superintendent, Richard Woods, recently wrote a column about teaching mathematics. “Funny math methods” was the catch-phrase taken from the article and sent out through the media.  This was not unexpected.  Frankly, I’m surprised it took this long.  This was part of his campaign platform.

Though his column has prompted some emotional responses from math educators, it is imperative that this significant dialogue he has opened, continue.  The best thing we can do for the students of Georgia is to keep this discussion going in order to come to a common understanding about the mathematical terms, strategies and ideas presented by Mr. Woods in his column.  We can truly help the students of Georgia by making sure we are all speaking the same language.

Since I am unable to respond directly to Mr. Woods’ column, I would like to continue the dialogue here.  I welcome any and all comments that keep this discussion moving forward in a positive light.  I encourage all viewpoints, since one-sided dialogues don’t tend to be very productive.

Mr. Woods talks about hearing from parents unable to help their children with their math homework.  I, too, have heard this from parents.  My response to this is:  If students are not able to do their homework independently, perhaps it should not have been assigned.  This is difficult for many to hear.  If you think about it, though, it really makes sense.  If we want students to build their understanding of mathematics based on what they have learned, we have to make sure they have learned it before they can build on it.  That said, I look at homework as falling into one of three categories:

  1. Practice – students use understandings learned in class to practice and build a more solid understanding at home.
  2. Preview – students are given a few problems to get them thinking about a new concept that is related to what they already know.
  3. Extension – students take a problem or task they worked on in class and are asked to extend their understanding. For example, in middle school, students may discover a growth pattern and as an extension, they may be asked to create a growth pattern that grows twice as fast.

Notice that each of these types requires students to have an understanding before they begin.  Understanding in mathematics, as in reading, is crucial for student success.

Mr. Woods also mentions the need for students to have a firm understanding of the fundamentals of mathematics.  He goes on to say that basic algorithms, fact fluency, and standard processes for addition, subtraction, multiplication, and division contribute to building that strong foundation for student achievement.

This is interesting.  First, algorithms have gotten a bad rap.  But, there is a place for algorithms in the big picture of how students learn mathematics.  An algorithm is just a mathematical term for a series of steps that can be followed to determine a solution to a mathematical computation.  Problems occur when algorithms are taught just as a series of steps to memorize, rather than facilitating an understanding of the computation(s) first.  Without understanding, the steps often don’t make sense and one or more of three things may happen next:

  • Students may complete algorithmic steps out of order.
  • Students may skip one or more steps of the attempted algorithm.
  • Students may confuse the steps of one algorithm with another.

These may seem like easy fixes -“just tell the students again”.  Telling them where their errors are and having them practice more problems does not work.  Without a conceptual understanding of what the computation means, students will continue to make these errors.  Though students may be able to show some success in the short term, over the long term they will revert back to one or a combination of the error patterns above.

Completing algorithms incorrectly doesn’t even compare to one of the worst side effects of this procedural teaching: students who don’t realize their answers are unreasonable.  For example, a teacher recently sent me the email below:

No understanding anonymous

This student has some major misconceptions.  With a conceptual understanding, this student could have reasoned that 1/3 of a pound is less than a whole pound, so the answer should be less than $5.25.  Without conceptual understanding, the student is attempting to recall and use procedures they do not understand, is confusing procedures, and is unable to determine whether or not the solution they have found is reasonable.  This is only one piece of numeracy that is lost in the procedural mathematics instruction that Mr. Woods seeks.

Fact fluency and the standard procedures for the four basic operations is next.  I don’t think there is a math teacher anywhere in the world that doesn’t think fluency is important.  In order to be clear though, memorization and fluency are not the same thing.  Not even close.  To keep this short and sweet, with the focus on students, I have copied the excerpt below from the GA DOE frameworks for mathematics.  I think this sums everything up nicely (no pun intended).  However, if you would like to learn more about fluency, click the links below.

Fluent students:

  • flexibly use a combination of deep understanding, number sense, and memorization.
  • are fluent in the necessary baseline functions in mathematics so that they are able to spend their thinking and processing time unpacking problems and making meaning from them.
  • are able to articulate their reasoning.
  • find solutions through a number of different paths.

For more about fluency, see: http://www.youcubed.org/wp-content/uploads/2015/03/FluencyWithoutFear-2015.pdf  and: http://joboaler.com/timed-tests-and-the-development-of-math-anxiety/

The fundamentals that Mr. Woods should have mentioned are actually reasoning and sense making.  This is what it takes to learn and do mathematics.  So, again, understanding must take place. However, understanding cannot take place through the memorization of algorithmic steps alone.  This is not just what I think.  It’s what I know from years of teaching students mathematics. This is also backed by research, papers, & videos.  The building of understanding is also fostered through a passionate, grassroots movement of mathematics teachers #MTBoS (Math Twitter Blog-o-Sphere).  This is our place to collaborate, share, and work to improve our teaching of mathematics.

Next, Mr. Woods discusses teaching “funny math methods.”  He specifically mentions “the lattice method” and he correctly states that this method is not state mandated and not required for students to achieve on state tests.  Mr. Woods is absolutely right!  This is not state mandated because it is a ridiculous strategy for multiplying multi-digit whole numbers.  to be fair, it works – every time. If you follow the steps of this algorithm by making the grid correctly, and placing the digits of the numbers correctly in the grid, and placing the products of the digits in the right places, and drawing the diagonals correctly, and adding the digits along the diagonals correctly, and copying the product correctly from the grid written in standard form.  You think that’s ridiculous?  Here’s something even worse – it’s often used for those students who have trouble remembering the steps of the standard algorithm.  This method is the definition of “funny math” and math teachers should not use this since it is does not make sense to students (or teachers) and it does not align with any of the standards for multiplication.

Mr. Woods says that mathematics has become over complicated.  It hasn’t.  It is only as complicated as it has been for centuries and that complication is exacerbated through teaching without sense making.  We can teach students to think mathematically on their own.  We can support and help them grow through their own understandings of mathematics.  We can help students make sense of mathematics and learn to use this to make informed decisions, rather than listening to others make these decisions for them.  We can do this because what we know now is how students learn mathematics.  It is not through memorization. It is through sense making and reasoning.  What we know now is that teaching students to think mathematically, through problem solving by building conceptual understanding provides students experience and allows them to make connections to algorithms they create and those created by others.  What we know now is that this works best for all students.  Not just average students, or above average students, or below average students.  All students.

Finally, just to be clear:

  • We (mathematics teachers) are most likely more current than most on research-based, best practices in the mathematics classroom.
  • There is a place for the algorithms you wish to see in the classroom, and they are found in the appropriate grade level standards. However, using an algorithm is not the end-all, be-all for learning mathematics.  There is always a need for students to be flexible, efficient, and accurate in their computations.  Multiple strategies, based on student understanding, must be explored.

At the end of the column, Mr. Woods states that “Offering choices and clarification are some of the steps we are taking to address the concerns surrounding mathematics in our state.” I applaud these steps.  Choices are always a good idea.  Clarification is even better!  Let’s work together to clarify the misconceptions about best practices in mathematics instruction.  Armed with these common understandings, Georgia can lead the charge as a state united to raise student achievement in mathematics.

I look forward to all comments and continuing this dialogue to help build these common understandings.