TeachingTweaks

After I read a recent education blog titled “What Exactly Is ‘Understanding?’ And How Do We Assess It?”, I was left wondering if either question had been answered.  The author reiterates the ambiguity of the term and the difficulty in assessing it, but neglected to provide an example.  Instead, there were just more terms:  apply, reflect, metacognition, interpret, synthesis.  Geez, Louise!  What is it, already?

In my last post, I encouraged readers to examine their test and quiz questions to determine what was being rewarded.  Were students rewarded for recall? for innovation? for showing understanding? (Yikes! It’s hard to do this without throwing out even more educational-ese.  Apologies to the aforementioned blog author.)

Maybe a way to define “understanding” is to be explicit about what students should “understand”.

I’m in favor of writing an understanding goal for lessons.  A quick example for adding fractions would  be: Students should understand that to add fractions we need to add same size pieces.  This statement contains the answer to “Why do we make common denominators when adding fractions?”  Another quick example for solving equations using properties of equality: Students should understand that adding the same value to both sides of an equation maintains equality and does not change the solution to the equation.  This statement explains why we need to do the same thing to both sides.

To illustrate how we might assess understanding, I’ll give an example of a quick assessment I used in a classroom I visited during a PD project.  The class was working on similar figures, and finding and using scale factors (this day we were focusing on the relationship between corresponding sides, the next lesson was about relationships of sides within each figure).  The teacher and I had talked about what students should understand about similar figures and scale factors. We wanted students to understand that the multiplicative relationship has to be the same for each set of corresponding sides.  We wanted to check if students would fall for an additive relationship.  We wanted to know if students rejected a non-integer scale factor.

We asked these three questions:

1. What does it mean for two figures to be “similar”?
2. Are these figures similar?  How do you know?
3. Are these figures similar? How do you know?

We posed these three questions at the end of the lesson, and students submitted their work on the way out the door.

The results were so cool!  The students’ papers fell into 4 piles based on the understanding shown in the answers.

Pile 1:  There were students who answered all three questions completely and correctly.  We accepted this as an adequate answer to #1:”figures with the same number of sides and angles; angles are the same; you can use a scale factor to make one shape from the other”.  Some students used division to answer #3, and some used a guess and adjust strategy to narrow down the scale factor.

Pile 2:  Other students answered #1 and #2 correctly (“5×2 is 10, but 8×2 is not 13″; or, “adding 3 to each side doesn’t make them similar”), but wrote something like this for #3: “there is no number times 6 that makes 15″.

Pile 3:  The next pile was for students who struggled with #1, fell for #2, and said “no” for #3 with the same 6 x ___= 15 is impossible”.

Pile 4:  No clue how to answer any of the questions.

The teacher and I were pretty pleased with ourselves that we got lots of information out of about 5 minutes of work from students.  It was clear that the next day we needed to reinforce the multiplicative nature of similar figures.  We also needed to talk about whether or not there is a number that you can multiply by 6 to get 15.  And if you believe there is such a number, how do you find it?

Did we find out if students can find a missing measure given a pair of similar figures and three of the four relevant measures?  Nope.  Would being able to do that give us as much information about understanding? I don’t think so.

Those who know me well know I’m a crusader against using cross-multiplying with students until they show understanding of proportional relationships (other than if a/b = c/d, then ad=bc).  Of course, students could answer #2 and #3 using cross-products, but I’m not convinced that introducing this early on does anybody any favors in the understanding department.

Assessing for understanding is tough.  Heck, teaching for understanding is tough.  Gosh, learning with understanding is tough.  But it’s important.

What test or quiz or formative assessment questions have you tweaked to get at this thing called “understanding”?

It’s test time.   The time of year that prompts two things:  1) wondering where the time went, and 2) wondering how to motivate students between the end of testing and the end of the school year.

Test time also prompts lots of discussion about standardized testing and the stakes for schools and districts.  Many say the scores on the tests do not adequately reflect progress students have made.  Others argue that a multiple choice test does not allow students to show what they know.

Some released or practiced items I’ve seen require students to think about a concept in a deep way, but most call for remembering and applying the appropriate procedure.

I’m currently working on a couple of projects for which I need to design assessments.  The projects are about understanding fractions, and we want to learn about how students think about and understand concepts.  It’s very difficult to write test items that reveal students’ understanding.  Even a simple comparing fractions item may be answered using a procedure students may or may not understand.  Assessing for understanding is tough work because you can’t accept execution of a procedure as evidence of understanding.

So, what do the standardized test question in your state or district reward?  Remembering the right thing at the right time or showing understanding?  I can probably guess the answer.

Now, what do the test and quiz questions you write for your students reward? Do you routinely tell students that thinking and solving problems different ways is important?  Do you tell them the process is as important as reaching the right answer?  Is this communicated in the rewards or penalties students see on their graded papers?

Giving grades was one of my least favorite activities as a teacher.  I enjoyed “grading” because I learned what my students thought and needed help with, but assigning a letter grade was never fun.  I wanted to present students with tasks and problems that provided opportunities to learn new things, but in the end I needed to grade performance and skill.

For this week’s Tweak, think about “grading”.  Take out an old quiz or test.  What were students rewarded for? What were they penalized for?  What did those rewards and penalties convey to students?

Last week I attended the Research Presession for the 2012 NCTM Annual Conference.  I stuck to my goal of attending sessions around the topic of understanding fractions.  I heard about how fractions are addressed in the Common Core State Standards, and about the development of fractional thinking.

My favorite quote from the conference is from Pat Thompson, a professor at Arizona State University.  He said this:  Fractional thinking is an intellectual achievement, not a behavioral objective.

Wow.  This really hit home for me.  I’ve worked on fractions for several years now.  When I had my own classroom I was curious about why working with fractions was so difficult for me to teach and for students to learn.  I wasn’t satisfied in accepting executing procedures for doing stuff with fractions as understanding. I’m still not.

In designing assessment items for my current research project, I’ve tried to design items that can only be solved if students use fractional thinking.  This  is hard to do, but really interesting.

The upside is that I get to look at lots of student work and interview students as they share their thinking about things like whether or not there is a number between 5/6 and 1. Students often believe that such a number doesn’t exist because 5/6 is a fraction and 1 is a whole number.  I also hear from students that there is no such number because the number just after 5/6 is 6/6, which is the same as 1. There’s obviously a difference in understanding between students who believe there is a number between 5/6 and 1, and those who don’t.  There is also a difference between students who can name such a number  and those who can’t.

This brings me to this week’s Tweak:  Take a look at the assignments or quiz questions you’ll give students this week.  Is there a question that is based on “thinking” rather than “remembering”?  What “thinking” will be revealed in student responses?  Tweak one of your questions or prompts to elicit thinking.

Share your experiences here.  What was the original question and how did you tweak it?

This week I’m attending a conference for math education researchers. It’s the Research Presession for the annual National Council of Teachers of Mathematics (NCTM) annual conference.I’ve presented a few times during the teacher part of the conference, but this will be the first time for me during the research part.

It’s a conversation for another day, but the research part is held concurrently with the NCSM (the S is for supervisors) conference, and just before all the teachers arrive.  I know all three groups have different needs, but in the spirit of Venn, I would like to see some more deliberate overlap.

For this week’s Tweak, I want to suggest giving yourself a “focus” goal for conferences or presentations you attend.

When I started teaching I attended a few conferences and wore myself out running from session to session garnering materials, listening to ideas, aspiring to be a presenter myself.  I came back to school after each conference simultaneously exhausted and refreshed, but without a clear plan of how to utilize the information I received.

What I learned to do was to choose a focus for the conference. One year my teaching goal was to look for every opportunity I could to find, mention, and use the distributive property with my 6th, 7th, and 8th graders.  When I attended a conference, I looked for sessions where the distributive property would be a topic.  At any other sessions, I listened with a goal of finding a use for the distributive property.

This may sound tedious and boring, and I struggled with the idea of missing the best presentation ever on fractions.  For me, though, this focus helped me actually use what I learned. I found that I made more connections to my own classroom, and was able to incorporate the ideasand activities.  This was much more useful and effective than attending a smorgasbord of sessions.

As you attend conferences or professional development this year or over the summer, choose a focus and stick to it.  Deliberately think about how you will incorporate what you hear and learn.

My focus this year is to learn about assessing understanding of fractions. I want to know about how other researchers do this, and how the Common Core will affect their work.

What is your focus for an upcoming conference?

I started teaching in the mid-90′s.  Around that time it seemed like concrete manipulatives exploded onto the scene.  We were told that students need a tactile connection to the math they were learning.  Then and now I hear teachers remark to their students that it helps them understand the math they’re doing when students use materials such as pattern blocks or fraction circles or algebra tiles or any of the other widely marketed and purchased manipulatives. I told my students this.  I used these materials.  And then I became dissatisfied.

Over the last twenty years, the manipulatives mantra has persisted despite little evidence that manipulatives are helpful for teaching and learning.  Research that finds manipulatives use, well, useless is shamefully inaccessible to most teachers. What is accessible through a google search is many opportunities for purchase with claims of helpfulness and fun.  I did some digging and found this summary of research with lots of references.  If you’re interested in gaining access to actual research findings, please contact a math education professor you know, or one you don’t know at a local university.  Believe me, they will be excited to hear from you.

The Standards for Mathematical Practice we’ll look at this week are pertinent to the discussion of math manipulatives.  The Standards are:

4.  Model with mathematics.

5. Use appropriate tools strategically.

Through my own teaching and work in professional development and research, I have rarely encountered use of these materials with a goal for understanding.  I’ve seen (and done) lot’s of demonstrating, that’s for sure.  But for the most part, I haven’t witnessed what I would consider math being done with this stuff.  In the worst cases, the demonstration with materials does not even match what the teacher eventually wants the students to do on their own.

Here’s an example:  With fraction circles (which, by the way, I think are an extremely limiting representation of fractions) it is easy for students to agree that two of the 1/3 pieces cover the same amount of area as four of the 1/6 pieces.  But, when it comes time to represent this relationship with numbers, we talk about multiplying by 1 or multiplying the numerator and denominator by the same number.  What does this have to do with an area covered by fraction circle pieces?

Here’s the missing connection:  The 1/6 pieces are each half the size of the 1/3 pieces, so it takes twice  as many of them to cover the same area.  When we multiply the denominator by 2, we are actually making the pieces half the size.  When we multiply the numerator by 2, we are accounting for this size difference.

Now, what does this have to do with Practices 4 and 5?

When we choose to model mathematics or use a tool, concrete or otherwise,we should think about how the model or tool interacts with the math we are trying to teach.  We might ask ourselves these questions:  Does the model or tool

• organize information?
• demonstrate a concept?
• parallel a common algorithm?
• reveal a relationship?
• fall short?

The answers to these questions might help in choosing, using, revising, or rejecting particular models or tools.  All the questions are not relevant all the time, but if they are relevant to your lesson’s understanding goal, you want to be sure you can answer them.

Don’t get me wrong.  I love me some pattern blocks.  However, I spend a lot of time thinking about what I want students to understand and how use of pattern blocks will serve that by answering the above questions every time I design a lesson.

This week’s Tweak: If you plan to use a particular model or tool in your lessons this week, spend some time messing around with it.  Get clear on how the model or tool might be beneficial for students’ understanding.

Share your experiences and insights in the comments!

This post is not about how we feel returning to school for the final push toward the end of the school year.

Rather, it’s about breaking down the lesson parts we’ve worked on tweaking.  We’ll continue the Standards for Mathematical Practices series next week.

Take some time this week to reflect on tweaks you’ve implemented to decide if the tweak has taken hold.  Have you deliberately changed some small feature of your teaching and noticed effects?

Here’s a break down of tweak suggestions related to a classroom lesson.

• Beginning the class
• Beginning the lesson
• Launching the Big Idea
• Model sense making and reasoning
• Communicate with precision

This is the second in a series of posts about the Standards for Mathematical Practices found in the Common Core State Standards for Mathematics.  There are eight practice standards, and each week I’ll address two of them.

My suggestion through this series is to take a few days to notice the presence or absence of the practices in your classroom, and then take a few days to deliberately implement one of them.  My intention in this space is to give a quick overview and one example. The Standards document provides an extensive list of ways to demonstrate the practices, so be sure to check those out.

I’ve organized the practices into pairs, which places some of them out of numerical order.  This week we’ll work on these two:

3.  Construct viable arguments and critique the reasoning of others.

6.  Attend to precision.

These two go together because they describe how we should communicate with others about mathematics.  We want to explain what we’ve done clearly and correctly.  This is true for teachers, as well as students.

3.  Construct viable arguments and critique the reasoning of others.

For me, this standard for mathematical practice represents an opportunity to give students a taste of what doing math really means.  It does not mean remembering what to do all the time.  Doing math means making moves that are mathematically allowable.  There are allowable moves for combining and scaling and solving.  This is a very different perspective than a focus on remembering rules.  A viable argument is one based on meaning, not on referring to a list of steps.

In a previous post I wrote about using a few more words to communicate the reasons behind the math decisions we make.  That’s applicable here.  One of the examples presented in that post is about adding fractions with unlike denominators.  It’s pretty weak to only say “we have to find common denominators in order to add fractions.”  That statement might lead students to think this constitutes a mathematical argument: “We do it because it’s what we’re supposed to do in order to get the right answer.”

Rather, if we say “When we add, we need to add like stuff:  like place values or same size pieces,” we’ve communicated mathematically about addition and the numbers we’re adding.  This statement is more meaty, and focuses attention on math, as opposed to simply remembering what to do.

6.  Attend to precision.

This one applies to more than just precision of language.  It’s also about using mathematical symbols and notation correctly. The equal sign is specifically mentioned in the Standards document.  As students move through middle school and onto more abstract math, we need to be diligent about correct use of the equal sign.  One of the common misuses of the equal sign that I’ve seen is a string of equal signs used to show the work for simplifying an expression such as this:

This post is a continuation on the series of lesson tweaks we began a few weeks ago.  We’ve been working on starting class and starting the lesson.

This week we’re moving on to the “guts” of the lesson.

If you haven’t yet, you might want to get yourself familiar with the Common Core Standards for Mathematics, including the Standards for Mathematical Practices.  Whether your state, district, school, or homeschool group adopts the Standards, the fact is that many resources will be developed with them in mind.  Knowing a little about the Standards will go a long way toward using these resources the way they were intended.

This post will focus on the Standards for Mathematical Practices.  I think this is a good list of practices for students and teachers.  As the lesson progresses and math happens, it’s important to thinking about both who is doing the work and the nature of that work.

There are 8 practices suggested in the Common Core.  This post and each subsequent one will focus on only 2 at a time.  Remember that we’re in the business of “tweaks” meaning that we don’t try to change or implement too much at one time.  (I think maybe 2 at a time is too much, but hey, I want to give people options).

As you plan your lessons, including writing a goal for understanding and planning how to launch the big idea, also plan which mathematical practices might be featured in the work on the big idea. Additionally, the way the practice standards are written, there are several options within each standard for introducing tweaks that promote the practices.

As usual, for the first couple of days of the week, go about your business as usual, and then mid-week introduce your tweak.

1.  Make sense of problems and persevere in solving them.

At the beginning of the week plan for and carry out how problems will be solved during the lesson.  If  you solve problems at the board, are you sharing with students how you made sense of the problem?  Is it clear why this problem requires division?    Does this answer make sense based on what we are given in the problem?

Teachers and students should share what’s in their brains about why a specific operation or strategy was chosen, and should talk out loud about how they know the answer makes sense.  It takes practice for all of us to clearly communicate why we worked on a problem in a particular way and how we knew when we were finished.

2.  Reason abstractly and quantitatively.

One thing I’ve been thinking about lately is how when I was teaching, I’m not sure I consistently pointed out the difference between the math we were doing that was related to the problem, and the math we were doing that was based on mathematical properties and conventions.

Students are not able to discern this.  Math is math for them.  I believe that if we don’t point out the math that naturally arises from a problem context, students might think it’s all abstract and rule based.  We need to be clear about how the problem situation is connected to math we need to do to solve the problems.

When you write an equation to use in solving the problem, do you make it clear how the components of the equation match the situation described in the problem?  Do you refer to those components and match them back to the problem throughout the solving process?

This week’s Tweak:  For the first couple of days, notice how you and your students

• communicate about making sense of problems OR
• talk about how the problem situation is connected to the math manipulations needed to solve it

Look for opportunities to be more explicit and direct about these two Standards for Mathematical Practice, and share what you notice here.

Over the past few weeks we’ve been thinking hard about the beginning of class and the beginning of the lesson.  The beginning of the class involves day-to-day housekeeping and the beginning of the lesson involves launching the math lesson of the day.

We went even further to think about the beginning of the lesson as two different launches (both done at the same time because we multi-task like that).  One aspect of planning the beginning of the lesson is to plan for the logistics of the lesson, and the other is to plan how to launch the math concept of the day.

Before we think about tweaks for the “guts” of the lesson, let’s revisit the beginning tweaks we’ve implemented.

Is there a small and simple change you implemented for the beginning of the class period?  Pay close attention to how that’s working.  Ask students or ask a colleague to give you some feedback about how it’s going.

Have you planned differently how to introduce the lesson?  Did you try having students talk to each other, or write some thoughts or notes?  Is that going smoothly?  Is it beneficial?

Have you changed the way you introduce the math content, task, or prompt for the day?  What are the specific strategies you tried?

It’s important to take a step back and reflect on how effective your tweaks have been.  Before we move on to the guts of the lesson, we want to be sure that any small changes are effective.

It may be the case that you need to devote another week to getting the class and lesson started.  That’s cool.  We want to make sure that a tweak has been implemented and is working.  If it’s not working, try something else!

Share your experiences here so we can all learn from your efforts