Unintended Messages

I read an interesting article by Yong Zhao the other day entitled What Works Can Hurt: Side Effects in Education where he discussed a simple reality that exists in schools and districts all over. Basically, he gives the analogy of education being like the field of medicine (yes, I know this is an overused comparison, but let’s go with it for a minute).  Yong paints the picture of how careful drug and medicine companies have become in warning “customers” of both the benefits of using a specific drug and the potential side-effects that might result because of its use.

However, Yong continues to explain that the general public has not been given the same cautionary messages for any educational decision or program:

“This program helps improve your students’ reading scores, but it may make them hate reading forever.” No such information is given to teachers or school principals.

“This practice can help your children become a better student, but it may make her less creative.” No parent has been given information about effects and side effects of practices in schools.

Simply put, in education, we tend to discuss the benefits of any program or practice without thinking through how this might affect our students’ well-being in other areas.  The issue here might come as a direct result of teachers, schools and systems narrowing their focus to measure results without considering what is being measured and why, what is not being measured and why, and what the short and long term effects might be of this focus!

---

Let’s explore a few possible scenarios:

Practice:

In order to help students see the developmental nature of mathematical ideas, some teachers organize their discussions about their problems by starting to share the simplest ideas first then move toward more and more complicated samples.  The idea here is that students with simple or less efficient ideas can make connections with other ideas that will follow.

Unintended Side Effects:

Some students in this class might come to notice that their ideas or thinking is always called upon first, or always used as the model for others to learn from.  Either situation might cause this child to realize that they are or are not a “math person”.  Patterns in our decisions can lead students into the false belief that we value some students’ ideas over the rest.  We need to tailor our decisions and feedback based on what is important mathematically, and based on the students’ peronal needs.

---

Practice:

In order to meet the needs of a variety of students, teachers / schools / districts organize students by ability.  This can look like streaming (tracking), setting (regrouping of students for a specific subject), or within class ability grouping.

Unintended Side Effects:

A focus on sorting students by their potential moves the focus from helping our students learn, to determining if they are in the right group.  It can become easy as an educator to notice a student who is struggling and assume the issue is that they are not in the right group instead of focusing on a variety of learning opportunities that will help all students be successful.  If the focus remains on making sure students are grouped properly, it can become much more difficult for us to learn and develop new techniques!  To our students, being sorted can either help motivate, or dissuade students from believing they are capable!  Basically, sorting students leads both educators and students to develop fixed mindsets.  Instead of sorting students, understanding what differentiated instruction can look like in a mixed-ability class can help us move all of our students forward, while helping everyone develop a healthy relationship with mathematics.

---

Practice:

A common practice for some teachers involves working with small groups of students at a time with targeted needs.  Many see that this practice can help their students gain more confidence in specific areas of need.

Unintended Side Effects:

Sitting, working with students in small groups as a regular practice means that the teacher is not present during the learning that happens with the rest of the students.  Some students can become over reliant on the teacher in this scenario and tend to not work as diligently during times when not directly supervised.  If we want patient problem solvers, we need to provide our students with more opportunities for them to figure things out for themselves.

---

Practice:

Some teachers teach through direct instruction (standing in front of the class, or via slideshow notes, or videos) as their regular means of helping students learn new material.  Many realize it is quicker and easier for a teacher to just tell their students something.

Unintended Side Effects:

Students come to see mathematics as subject where memory and rules are what is valued and what is needed.  When confronted with novel problems, students are far less likely to find an entry point or to make sense of the problem because their teacher hadn’t told them how to do it yet.  These students are also far more likely to rely on memory instead of using mathematical reasoning or sense making strategies.  While direct instruction might be easier and quicker for students to learn things, it is also more likely these students will forget.  If we want our students to develop deep understanding of the material, we need them to help provide experiences where they will make sense of the material.  They need to construct their understanding through thinking and reasoning and by making mistakes followed by more thinking and reasoning.

---

Practice:

Many “diagnostic” assessments resources help us understand why students who are really struggling to access the mathematics are having issues.  They are designed to help us know specifically where a student is struggling and hopefully they offer next steps for teachers to use.  However, many teachers use these resources with their whole group – even with those who might not be struggling.  The belief here is that we should attempt  to find needs for everyone.

Unintended Side Effects:

When the intention of teachers is to find students’ weaknesses, we start to look at our students from a deficit model.  We start to see “Gaps” in understanding instead of partial understandings.  Teachers start to see themselves as the person helping to “fix” students, instead of providing experiences that will help build students’ understandings.  Students also come to see the subject as one where “mastering” a concept is a short-term goal, instead of the goal being mathematical reasoning and deep understanding of the concepts.  Instead of starting with what our students CAN’T do and DON’T know, we might want to start by providing our students with experiences where they can reason and think and learn through problem solving situations.  Here we can create situations where students learn WITH and FROM each other through rich tasks and problems.

---

Our Decisions:

Yong Zhao’s article – What Works Can Hurt: Side Effects in Education – is titled really well.  The problem is that some of the practices and programs that can prove to have great results in specific areas, might actually be harmful in other ways.  Because of this, I believe we need to consider the benefits, limitations and unintended messages of any product and of any practice… especially if this is a school or system focus.

As a school or a system, this means that we need to be really thoughtful about what we are measuring and why.  Whatever we measure, we need to understand how much weight it has in telling us and our students what we are focused on, and what we value.  Like the saying goes, we measure what we value, and we value what we measure.  For instance:

• If we measure fact retrieval, what are the unintended side effects?  What does this tell our students math is all about?  Who does this tell us math is for?
• If we measure via multiple choice or fill-in-the-blank questions as a common practice, what are the unintended side effects?  What does this tell our students math is all about?  How reliable is this information?
• If we measure items from last year’s standards (expectations), what are the unintended side effects?  Will we spend our classroom time giving experiences from prior grades, help build our students’ understanding of current topics?
• If we only value standardized measurements, what are the unintended side effects?  Will we see classrooms where development of mathematics is the focus, or “answer getting” strategies?  What will our students think we value?

---

Some things to reflect on

• Think about what it is like to be a student in your class for a moment.  What is it like to learn mathematics every day?  Would you want to learn mathematics in your class every day?  What would your students say you value?
• Think about the students in front of you for a minute.  Who is good at math?  What makes you believe they are good at math?  How are we building up those that don’t see themselves as mathematicians?
• Consider what your school and your district ask you to measure.  Which of the 5 strands of mathematics proficiency do these measurements focus on?  Which ones have been given less attention?  How can we help make sure we are not narrowing our focus and excluding some of the things that really matter?

As always, I encourage you to leave a message here or on Twitter (@markchubb3)!

Which one has a bigger area?

Many grade 3 teachers in my district, after taking part in some professional development recently (provided by @teatherboard), have tried the same task relating to area.  I’d like to share the task with you and discuss some generalities we can consider for any topic in any grade.

---

The task:

As an introductory activity to area, students were provided with two images and asked which of the two shapes had the largest area.

A variety of tools and manipulatives were handy, as always, for students to use to help them make sense of the problem.

---

Student ideas

Given very little direction and lots of time to think about how to solve this problem, we saw a wide range of student thinking.  Take a look at a few:

Some students used circles to help them find area.  What does this say about what they understand?  What issues do you see with this approach though?

Example 1

Example 2

Some students used shapes to cover the outline of each shape (perimeter).  Will they be able to find the shape with the greater area?  Is this strategy always / sometimes / never going to work?  What does this strategy say about what they understand?

Example 3

Some students used identical shapes to cover the inside of each figure.

Example 4

Example 5

Example 6

And some students used different shapes to cover the figures.

Example 7

Example 8

Example 9

Notice that example 9 here includes different units in both figures, but has reorganized them underneath to show the difference (can you tell which line represents which figure?).

---

Building Meaningful Conversations

Each of the samples above show the thinking, reasoning and understanding that the students brought to our math class.  They were given a very difficult task and were asked to use their reasoning skills to find an answer and prove it.  In the end, students were split between which figure had the greater area (some believing they were equal, many believing that one of the two was larger).  In the end, students had very different numerical answers as to how much larger or smaller the figures were from each other.  These discrepancies set the stage for a powerful learning opportunity!

For example, asking questions that get at the big ideas of measurement are now possible because of this problem:

“How is it possible some of us believe the left figure has a larger area and some of us believe that the right figure is larger?”

“Has example 8 (scroll up to take a closer look) proven that they both have the same area?”

“Why did example 9 use two pictures?  It looks like many of the cuisenaire rods are missing in the second picture?  What did you think they did here?”

In the end, the conversations should bring about important information for us to understand:

• We need comparable units if we are to compare 2 or more figures together.  This could mean using same-sized units (like examples 1, 4, 5 & 6 above), or corresponding units (like example 8 above), or units that can be reorganized and appropriately compared (like example 9).
• If we want to determine the area numerically, we need to use the same-sized piece exclusively.
• The smaller the unit we use, the more of them we will need to use.
• It is difficult to find the exact area of figures with rounded parts using the tools we have.  So, our measurements are not precise.

---

Some generalizations we can make here to help us with any topic in any grade

When our students are being introduced to a new topic, it is always beneficial to start with their ideas first.  This way we can see the ideas they come to us with and engage in rich discussions during the lesson close that helps our students build understanding together.  It is here in the discussions that we can bridge the thinking our students currently have with the thinking needed to understand the concepts you want them to leave with.  In the example above, the students entered this year with many experiences using non-standard measurements, and this year, most of their experiences will be using standard measurements.  However, instead of starting to teach this year’s standards, we need to help our students make some connections, and see the need to learn something new.  Considering what the first few days look like in any unit is essential to make sure our students are adequately prepared to learn something new!  (More on this here: What does day one look like?)

To me, this is what formative assessment should look like in mathematics!  Setting up experiences that will challenge our students, listening and observing our students as they work and think… all to build conversations that will help our students make sense of the “big ideas” or key understandings we will need to learn in the upcoming lessons.  When we view formative assessment as a way to learn more about our students’ thinking, and as a way to bridge their thinking with where we are going, we tend to see our students through an asset lens (what they DO understand) instead of their through the deficit lens (i.e., gaps in understanding… “they can’t”…, “didn’t they learn this last year…?).  When we see our students through an asset lens, we tend to believe they are capable, and our students see themselves and the subject in a much more positive light!

Let’s take a closer look at the features of this lesson:

• Little to no instruction was given – we wanted to learn about our students’ thinking, not see if they can follow directions
• The problem was open enough to have multiple possible strategies and offer multiple possible entry points (low floor – high ceiling)
• Asking students to prove something opens up many possibilities for rich discussions
• Students needed to begin by using their reasoning skills, not procedural knowledge…
• Coming up with a response involved students doing and thinking… but the real learning happened afterward – during the consolidation phase

A belief I have is that the deeper we understand the big ideas behind the math our students are learning, the more likely we will know what experiences our students need first!

---

A few things to reflect on:

• How often do you give tasks hoping students will solve it a specific way?  And how often you give tasks that allow your students to show you their current thinking?  Which of these approaches do you value?
• What do your students expect math class to be like on the first few days of a new topic/concept?  Do they expect marks and quizzes?  Or explanations, notes and lessons?  Or problems where students think and share, and eventually come to understand the mathematics deeply through rich discussions?  Is there a disconnect between what you believe is best, and what your students expect?
• I’ve painted the picture here of formative assessment as a way to help us learn about how our students think – and not about gathering marks, grouping students, filling gaps.  What does formative assessment look like in your classroom?  Are there expectations put on you from others as to what formative assessment should look like?  How might the ideas here agree with or challenge your beliefs or the expectations put upon you?
• Time is always a concern.  Is there value in building/constructing the learning together as a class, or is covering the curriculum standards good enough?  How might these two differ?  How would you like your students to experience mathematics?

As always, I’d love to hear your thoughts.  Leave a reply here on Twitter (@MarkChubb3)

The Zone of Optimal Confusion

In the Ontario curriculum we have many expectations (standards) that tell us students are expected to, “Determine through investigation…” or at least contain the phrase “…through investigation…”.  In fact, in every grade there are many expectations with these phrases. While these expectations are weaved throughout our curriculum, and are particularly noticeable throughout concepts that are new for students, the reality is that many teachers might not be familiar with what it looks like for students to determine something on their own.  Probably in part because this was not how we experienced mathematics as students ourselves!

First of all, I believe the reason behind why investigating is included in our curriculum is an important conversation!   I’ve shared this before, but maybe it will help explain why we want our students to investigate:

The chart shows 3 different teaching approaches and details for each (for more thoughts about the chart you might be interested in What does Day 1 Look Like). Hopefully you have made the connection between the Constructivist approach and the act of “determining through investigation.”  Having our students construct their understanding can’t be overstated. For those students you have in your classroom that typically aren’t engaged, or who give up easily, or who typically struggle… this process of determining through investigation is the missing ingredient in their development. Skip this step and start with you explaining procedures, and you lose several students!

Traditionally, however, many teachers’ goal was to scaffold the learning.  They believed that a gradual release of responsibilities would be the most helpful.  Cathy Seeley in her book Making Sense of Math: How to Help Every Student Become a Mathematical Thinker and Problem Solver explains the issue clearly:

In the two pieces above Cathy explains the “upside-down teaching” approach.  This is exactly the approach we believe our curriculum is suggesting when it says “determine through investigation,”  and exactly the approach suggested here:

Page 24 of the Ontario Curriculum

---

At the heart of this is the idea of “productive struggle”, we want our students actively constructing their own thinking.  However, I wonder if we could ever explain what “productive struggle” looks / feels like without ever experiencing it ourselves?  How might the following graphic help us reflect on our own understanding of “productive struggle” and “engagement”?

Research Gate, Confusion can be Beneficial for Learning

---

I think it would be a wonderful opportunity for us to share problems and tasks that allow for productive struggle, that have student reasoning as its goal, problems / tasks that fit into this “zone of optimal confusion”.

In the end, we know that these tasks, facilitated well, have the potential for deep learning because the act of being confused, working through this confusion, then consolidating the learning effectively is how lasting learning happens!

Let’s commit to sharing a sample, send a link to a problem / task that offers students to be confused and work through that confusion to deepen understanding.  Let’s continue sharing so that we know what these ideals look like for ourselves, so we can experience them with our own students!

 

 

 

 

 

 

 

 

How to Answer Multiple Choice Questions

It’s that time of year when many start to panic about the inevitable tests that will be given to students all over. And with this panic comes many test-taking strategies that will be told to countless students.  I thought I’d share a few of the “secrets” many students are told about how to ace these tests, specifically how to answer any multiple choice sections:

Tip #1 – Cover up the 4 answers before you read the question.

Many teachers give advice similar to this asking students to cover up their answers with their hand or with a sticky note…  Take a look:

As you can see above, the student covered over the potential answers to help them think through the problem before they start looking for answer.  I believe many might suggest this approach when they notice students guessing, or not taking the time and thought necessary to solve a problem.  However, I’m not sure this strategy is appropriate for all students, nor will it even work for many questions.  How would covering over the answers help here:

Take a look at each of the 6 questions.  Which ones would be helpful if your students covered up the answers?  Which ones would be impossible?  Is this the best strategy for all of your students?  Would you use this strategy yourself all the time?  Personally, I don’t think I’d ever use this strategy!

I wonder what would happen if we told a group of students to do this every time and they encountered questions like #1 or #2 above?  We would be setting them up for failure.  Hopefully you are seeing this method might not be possible for every question, nor will all students benefit from using it at all!

---

Tip #2 – Highlight keywords in the question so you know what is being asked of you to do.

Many teachers might ask their students to highlight pieces of a problem or question in order to help them focus their attention on information that might easily be missed.

For the above question, it is possible that some students might miss some of the final words and even though they understand the question, might get their answer wrong.  Obviously this isn’t ideal!  However, in my experience, many students, even when using a highlighter, miss out on all kinds of information.  In the following question, more than half of the students in one classroom answered “d”, even though all of the students were expected to highlight important information.  Did so many in this classroom get this question wrong because they were highlighting?

In fact, many of the students in this class highlighted nearly every word of every question.  And some of the students who didn’t highlight as many questions as were expected – actually did the best on the test.  I’m not suggesting that highlighting is bad, just that it likely didn’t help any/many of the students in this class, and actually got some students to miss out on information and get the wrong answers.

---

Tip #3 – Eliminate the obviously wrong answer first.

Again, many teachers give advice like this because they know it works for them.  Take the following question as an example.  Would you first eliminate the wrong answers?

Or would you place each of the 4 numbers in the box to see what the answer might be?  Or possibly work out the question without notice to the options, then see if your answer is there?

When I watch students think, I typically don’t see students attempting to find the wrong responses. Would it be a feasible strategy for any/some/all of these questions:

---

Some advice.

If you haven’t already read it, please read my post Quick Fixes and Silver Bullets…  There I discuss some of the many unproductive beliefs and strategies many schools employ as they attempt to improve testing score, followed by more productive suggestions.

Thinking specifically again about multiple choice questions, there are many different tips we can give students to solve a multiple choice question, but because every question is unique, and every student will have their own thinking and strategies, we might be putting too much emphasis on trying to find the quick fixes and easy answers.  Instead of teaching these strategies, I wonder why we don’t just provide students with rich tasks / problems, then encourage more discourse?

If we want our students to do well in our classrooms, we need to make sure we are focusing our attention on providing rich learning opportunities, facilitating meaningful discussions, and consolidating the learning effectively!  However, in some classrooms I wonder how much valuable class time is spent preparing for high stakes testing by “practicing” questions that mimic ones found on the test?  I wonder how much time is spent seeing IF you understand something instead of time spent on actually learning the curriculum standards the way they were intended to be learned?

The problem here is that many of these test questions are evidence OF learning, but they are often not the type of experiences needed TO learn the material!

I think Linda Gojak said it nicely in her NCTM president’s message entitled Are We Obsessed with Assessment?:

---

Final thoughts.

Personally, I’m not a fan of multiple choice questions (see the late Joe Bower’s post for details). In some classrooms there is far too much emphasis on getting right answers on simple questions and far too little emphasis on development of deep understanding of the mathematics!  Too many attempt to raise scores by replicating the form and format of the test instead of focusing on the mathematics itself.  Yet research shows us the more multiple choice tests we give, the worse our students actually perform on standardized tests!

If you have to help your students prepare for these kinds of tests though, please make sure that you remain focused on the mathematics itself, and not expect all of your students to use YOUR strategies. That’s how you kill your students’ relationship with mathematics!  We would never tell our students to pick C for every answer, that will only work 1/4 of the time.  In the same way, many of the strategies we provide for our students will not work for all students and not for all questions… and rarely will these strategies actually help them reach our actual goals for our students:

(unless your goals are misguided – like trying to get a certain percentage of students to pass the test – in which case I’m not sure how helpful I’ve been here!).