An Open Letter to Those Who Understand Depth of Knowledge

Dear Teacher Who Incorporates DOK in your Classroom,

First of all I would like to thank you for all of the help and support you have given me this year.  I have learned so much from the resources and ideas you are posting on your websites, blogs, and Twitter.  As I work to improve my students number sense and move them beyond algorithms, focusing on depth of knowledge and reworking some of my questions in class has greatly improved the work we do in class.  Sometimes a simple change to a question makes all the difference.

If you have ever read my blog before you know that I have far more questions than ideas to share.  So here are the questions I am still struggling with about DOK that I am hoping you can answer.  Many teachers talk about tasks that have low floor, high ceiling.  Is that how I should be viewing DOK?  For students who struggle with math do I start them at DOK 1 and that is all I expect from them?  And students who seem ready for more depth can get more complex tasks in DOK3?  Perhaps you think this sounds absurd.  I certainly do, but this is how I see some teachers using/interpreting DOK.  I feel like we should be giving all students DOK2 and DOK3.  The idea of differentiated instruction is sometimes interpreted as give struggling students basic tasks they can be successful at (or in my opinion memorize some stuff and do without truly understanding).  I understand that if a student isn't ready for grade level skills it does not benefit them to give them those task.  So what does this actually look like in a classroom?  Do all students get the same task and the differentiation is in the conversations that stem from it?  Or am I giving different students different tasks?

I am also curious about assessment.  We do tests at the end of each chapter in our textbook.  Typically we ask DOK1 (maybe some DOK2) questions because we are assessing skills.  We have a letter based grading system, and when creating our tests we discuss what mastery of each skill looks like.  In the end we have many bare number problems and very little depth.  In class we use higher level thinking but somewhere along the way it seems to have become the belief that not all students can do higher level thinking independently and therefore we should not assess them on it.  I would love to know how you use DOK in your assessments.  At this point I feel like I can't address these issues (perhaps they are not issues and it is just my perspective that needs to change) if I don't fully understand where we need to go.  Once we have a better understanding of DOK, how can we use that lens to improve our assessments?

I appreciate your time in reading through this letter and welcome any and all responses and comments.

Adrianne Burns
Grade 7 Math
A little lost and hoping for guidance

Math That Keeps Me Up At Night

I have started many blog posts that I have not been able to finish.  Many of them center around the same questions.  I can't seem to pull my thoughts together.  Sometimes the thoughts are sparked by something I read on Twitter.  Other time a colleague says something that seems to resonate with me or perhaps is strikes a nerve.  Yesterday was no exception.  Whether I am able to complete any of these thoughts or simply babble, I am committing right now to publishing this post.  So here goes:

This year I have focused on building conceptual understanding of math.  I want students to make sense of the math and see the number 7 as more than just a 7.  Think about it.  It does mean a certain number (in this case 7) items, but what are those 7 items?  7, 700, 879, 7(4), 3 to the 7th power.  Students do not understand that these 7s while looking the same represent different things.  This is not an easy undertaking as a teacher.  I believe that if I can deepen their understanding all of the pieces of confusing math will start to come together for them.  Is this a pie in the sky thought?  Am I misguided in this thinking?  Or is it developmental?  At what age are they truly ready to understand this?  Is my expectation too high in 7th grade to think that students would be able to slow down and make sense of this?

One statement that continues to stick with me is a statement about meeting students where they are at.  I don't remember the exact quote but it was something to the effect that not all students are going to be able to understand math, so we need to give them the rules to memorize so they can do the grade level math that is being asked of them.  What is my goal as a teacher?  Depth of knowledge takes a lot of time to build.  In 7th grade students know the algorithm and (while I like to think it was built on number sense) no longer remember any of its meaning.  These students can more or less get As and Bs in math by using what they have memorized.  But what does that mean for higher level math?  Am I really doing my job as a math teacher if they don't really know math, but have simply memorized it?

Enter the models and visual representations.  These are great tools for understanding algorithms.  If a student already knows the algorithm do I care that they can show it on a model?  I do if they are able to do it on their own.  If I have shown a model and now these students are just memorizing how to show it on a model what is the point?  But after spending great amounts of time trying build deeper knowledge of decimal and fraction operations I feel my students would be scoring better on assessments if I had just given them rules to memorize and follow.  They aren't ready for the assessments because they need more time to work with numbers and make sense of it.  How long is too long?  I need to move on to the other skills.  These are actually review skills in 7th grade and I feel like I am going back to the beginning to get them to understand place value and multiplication.  Perhaps my approach is wrong.  I am trying to build concepts and they have the wrong background knowledge.  Maybe I need to look at it a different way.  I don't want to just explain their memorized rules.  I want them to understand it.  Is that a different approach than if they don't know anything about decimals (does this kid even exist?  Probably not).

Part of me has thought that I should be working in an elementary school to get a better understanding of this.  But my heart breaks for my middle school students who are so far below grade level.  Who is going to help them if I am not there?  Someone with a passion for numeracy and building number sense must be in the middle schools to help these kids see math as more than just memorized rules.

Yesterday I was talking with our computer literacy teacher.  The students were working in Excel and unable to write a formula to find a percent.  They knew to take one number divided by the other but then wanted the computer to move the decimal (or drop it) but did not know what math this was.  AAAAAAAHHHHHHH!  Our 7th graders do not know that moving a decimal is dividing by 100?  This is a travesty.  We have failed our students.

Part of the problem is our assessments and grading system.  A colleague brought up at a math team meeting yesterday that perhaps we are not asking the right types of questions on our assessment.  If students are able to memorize rules with no understanding and pass our tests we are sending the wrong message.  I love all my coworkers, but this one made my heart happy.  She suggested adding even just one question on the test that asks a little bit more than computation.  We used to have these on our tests.  Students weren't able to get them correct or found them confusing and so slowly they were removed.  One teacher made a comment that we would need to change how we teach if we put questions like that on the test.  I agree completely and think it is time for that.  We do need to change our teaching.  We seem to have simplified everything and are now teaching to the lowest common denominator (did I use that phrase right?)  We are setting the bar low so everyone can be successful, instead of helping students be successful and truly understand math.

Standards based grading versus letter grades plays into this too.  Our job as teachers is to help students be successful.  In a letter based grading system that means getting As.  In a standards based grading system that means helping students reach the standard.  Our assessments would ask deeper questions so we could figure out where they are on the spectrum of learning that skill.  Students and parents would be asking for more than rules to follow to help their kids grow in their learning.  In our letter grade system, we just simplified the test and asked, what is the most basic thing students need to do to with this skill.

There is still so much more swimming in my brain, but it did feel good to get that out.  Now I'm off to the store to buy Thanksgiving foods and try to relax my math brain so I can spend time with my family.
I truly am grateful for my wonderful coworkers, Twitter friends, and PLN who put up with rants like the one above as I continue to learn how to teach math.
Happy Thanksgiving.

If I were a math coach - Part 2

This is part of a series I write when I feel like I can't do it all. For more of an intro. read Part 1

My list of things I would do (or do better) if I had more time is piling up.  I thought I should get some of them out of my head.

If I were a math coach...

Grant Writing:
There are so many great professional development grants out there.  The best way to close the gaps in math is to train teachers on research based strategies and deepen teachers' number sense.  If I were a math coach I could work on getting some of the money that is available to help teachers have more time to delve into these topics and improve their teaching practices or perhaps just give them more time to do the things they already know but don't have time to organize or discuss.

Parent Connections:
I mentioned in the Part 1 of my post, but after parent teacher conferences I realized that there really should be someone available during this time to help educate parents on how to help their child.  There are parents who want to know and with only 10 minutes per conference, I am not going to be able to show them.  If I were a math coach I could sit down with those parents and show them some games they could use at home and teach them how to talk math with their kids (#tmwyk).

Reading and Understanding the Research:
There are so many things I want to read: blogs, books, dissertations, etc.  With all the research out there on children's development of number sense it is a shame that math teachers are not equipped with that knowledge.  Here is a great example: Paying Attention to Mathematics.  It is not easy reading and is hard to dive into during a prep time when there are 10 interruptions and 5 other people I need to see.  I love this stuff.  I am fascinated by the ins and outs of teaching mathematical concepts.  I want to know more about what pieces have to be in place before others and what concepts work side by side.  Why might my student be able to add fractions without understanding that if I cut 1/4 off the whole it is still 1/4?  I want to know more and I can't get enough.  While I am not alone in this quest, not everyone gets as excited about this as I do.  As a math coach I could spend time learning more and helping teachers to understand it.  Who is facilitating these conversations if no one on the team as this type of understanding?  When push comes to shove this is what gets lost.  Here is a small list of what I would read (or reread) if I were a math coach:

• All the books by US Math Recovery Council
• Paying Attention to Mathematics
• Math Progression Documents
• Anything (and everything) by Catherine Fosnot

Concrete Models - Are they actually helpful?

I am not sure what manipulatives and concrete models look like at an elementary level, but here is an example of what I see in middle school.  Last week we were working on this problem:

 6 - (-2)

This was our first introduction to subtracting integers, but we had done lots of work with addition.  Using our background knowledge we put out 6 positive integer tiles.  Like so:

Then we discussed our understanding of subtraction.  From an early age students understand subtraction as taking away.  So we looked at taking away 2 negative tiles.  But wait, I don't have any negative tiles to take away.  What should we do?

Some students wanted to add 2 negatives in order to remove them.  This led to some good discussion about how that would change the problem.  Then it was brought up by a student that we could add zero pairs (a positive tile and a negative tile), which we learned about in our addition problems.  This is what that looked like:

From this point we were then able to remove our 2 negative tiles, leaving 8 positive tiles.

After this, the lesson called for more building of expressions with the tiles, removing tiles, adding zero pairs when necessary, in order to find the result.  As the students worked I walked around helping teams understand the idea of adding zero pairs if it was needed.  

By the end of class students were in one of two places:

• Some students had no idea what they were doing with the tiles and were copying their teammates.
• Some students learned to add zero pairs in order to have what they needed to remove.  

Unfortunately there were no students who had reached the lesson objective outlined in the textbook:

Develop an understanding of integer subtraction using + and - tiles.

Or the mathematical practice goal:

Make sense of subtraction with signed numbers.

Here is the problem as I see it.  

As math teachers we have made the shift to helping students have a deeper understanding of math.  We are using more representations, models, and manipulatives to help students explain what is happening with the numbers.  Unfortunately too many teachers (myself included) are just using these models to explain the math rules.  Students do not own them.  Students are now just memorizing the steps of the manipulatives.  Is this a step in the right direction?  To quote Big Bang Theory, "Way to think outside but pressed up against the box."

If the purpose of the representation is to give students deeper understanding we need to give them time to explore with it.  The challenge for me comes with the question, "How?".  What does that even look like at a middle school level?  

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So let's take a look at the lesson above, this time allowing the students to play with the numbers a little more.  

Use the tiles to show the number 5.  (No pics of this.  I'll let you create the visual)

Now show 5 another way.  (Blank stares, processing time, have students circulate to see how others have created 5)

Now show 5 another way. (More students are able to do it.)

Should we do it one more time? Why not.  Show me 5 one more way that you haven't used yet.

How long can we play this game?  How many more ways do you have to show me 5?

Think about how this shifts the conversation.  Students are developing an understanding of 5 that they may not have had before.  5 is a constant, so how can it be represented in different ways?  This is a powerful idea.  It could be a time to introduce vocabulary like the Additive Identity Property.  

Now let's move to the subtraction, but frame it in a way that is still allowing students to explore these numbers.  

Use the tiles to show the number 6. 

Have different students draw their representations on the board.  (Verify that each one does have a value of 6.  Perhaps a good time to ask again, how we can have so many representations with a value of 6.)

I want to create some problems where I will be removing tiles.  For each representation of 6, I want you to list numbers that I could remove based on the given tiles.  

I had students do this in teams.  It created lots of great discourse.  I brought the class together so we could discuss after. Some questions we discussed:

• What were some numbers that you listed for each representation? 
• How did you decide what could be removed? (students realize you can only remove what you have)
• Can you name a number that I can remove from one of these representations of 6 that I cannot remove from another?  Why is that?
•  So if I want to remove -2, which representations of 6 could we use?

We eventually worked our way to writing the subtraction expressions.  The focus was always on which representation of the starting value is needed.  Keep in mind there is not one right answer for this.  As long as students have what they need to remove, extra zero pairs are just fine with me.  They will be more efficient when they are ready.  

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A side rant:

There are many important mathematical concepts found in using integer tiles for subtraction.  However, for the objective of understanding integer subtraction, I personally prefer the Hot Air Balloon model. I find it much more intuitive for students and the relationship between addition and subtraction is much clearer.  Here is a Desmos activity I created for addition with the hot air balloon.  Desmos is a great tool for exploration because you can use sliders to change numbers and actually see what happens.  Unfortunately my Hot Air Balloon does not do this. What I really want is a tool where students can drag puffs of air or sandbags to the balloon and it moves.  I also want it to create the equations as they click.  Plus students should be able to write the equation and the hot air balloon moves.  I am not sure it is exploration if they have to move the hot air balloon themselves, because they have to know how it will move.  Do they really understand or have they just memorized it?

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I may not be where I want to be, but at this point I feel like I am now outside the box and making steps in the right direction.  I want my students to explore their understanding of math each day, so I will continually ask myself how I can provide opportunities to explore the numbers, not just model them.  I encourage you to do the same, and I would love to hear about it.