## An Example of Teaching THROUGH problem solving?

Many math resources attempt to share the difference between teaching FOR problem solving and teaching THROUGH problem solving. Cathy Seeley refers to teaching THROUGH problem solving as “Upside-Down Teaching” which is the opposite of a “gradual release of responsibility” model:

And instead calls for us to flip how our students learn to a more active model:

So, instead of starting a unit on Geometry with naming shapes or developing definitions together, we decided to start with a little problem:

Create as many polygons as possible using exactly 2 pattern block pieces. Sort your polygons by how many sides they have.

As students started placing pattern block pieces together, all kinds of questions started emerging (questions we took note of to bring to the whole group in a few minutes):

By the end of a period, students had worked through the definitions of what a polygon is (and isn’t), the difference between concave and convex polygons, defined the term “regular polygon” (which was not what they had been calling “regular” before), and were able to name and create triangles, quadrilaterals, pentagons, hexagons, heptagons, octagons, nonagons, decagons and undecagons. Recognizing a variety of possible ways a shape can look was very helpful for our students who might have experienced shapes more traditionally in the past.

One group compiled their polygons together (with one minor error):

Instead of starting with experiences where students accumulate knowledge (writing out definitions, taking a note, direct instruction), an upside down approach aims to start with students’ ideas. This way we would know which conversations to have with our students, and so our students are actively engaged in the process of learning.

## I want to leave you with a few reflective questions:

• Why might it benefit students to start with a problem instead of starting with the teachers’ ideas?
• Why might it benefit teachers to listen to students’ thinking before instruction has occurred?
• What does it mean to effectively monitor students as they are thinking / working? (See This POST for examples)
• Can all mathematics topics begin with tasks that help our students make connections between what they already know, and what they are learning? Can you think of a topic that can not be experienced this way?
• The final stage in the You-We-I model is where the teacher helps make specific learning explicit for their students. How do you find time to consolidate a task like this? How do you know what to share? (See This POST for an example)
• How might this form of teaching relate to how we view assessment? (See This POST)
• How might this form of teaching relate to how we view differentiated instruction? (See This POST)
• How do you find problems that ask students to actively think before any instruction has occurred? (See This POST for examples)

I’d love to continue the conversation.  Feel free to write a response, or send me a message on Twitter ( @markchubb3 ).

## Spatial Puzzles: Cuisenaire Cover-ups

Foundational to almost every aspect of mathematics is the idea that things can be broken down into pieces or units in a variety of ways, and then be recomposed again. For example, the number 10 can be thought of as 2 groups of 5, or 5 groups of 2, or a 7 and a 3, or two-and-one-half and seven-and-one-half…

Earlier this year I shared a post discussing how we might decompose and recompose numbers to do an operations (subtraction). But, I would like us to consider why some students are more comfortable decomposing and recomposing, and how we might be aiming to help our students early with experiences that might promote the kinds of thinking needed.

Doug Clements and Julie Sarama have looked at the relationship between students’ work with space and shapes with students understanding of numbers.

“The ability to describe, use, and visualize the effects of putting together and taking apart shapes is important because the creating, composing, and decomposing units and higher-order units are fundamental mathematics. Further, there is transfer: Composition of shapes supports children’s ability to compose and decompose numbers”

Contemporary Perspectives on Mathematics in Early Childhood Education p.82, Clements and Sarama

The connection between composing and decomposing shapes and numbers is quite exciting to me. However, I am also very interested in the meeting place between Spatial tasks (composing/decomposing shapes) and Number tasks that involve composing and decomposing.

A few years ago I found a neat little puzzle in a resource called The Super Source called “Cover the Giraffe”. The idea was to cover an image of a giraffe outline using exactly 1 of each size of cuisenaire rods. The task, simple enough, was actually quite difficult for students (and even for us as adults). After using the puzzle with a few different classes, I decided to make a few of my own.

After watching a few classrooms of students complete these puzzles, I noticed an interesting intersection between spatial reasoning, and algebraic reasoning happening…. First, let me share the puzzles with you:

## Objective:

To complete a Cuisenaire Cover-Up puzzle, you need exactly 1 of each colour cuisenaire rod. Use each colour rod once each to completely fill in the image.

Below are the 5 puzzles:

## Assessment Opportunities

Knowing what to look for, helps us know how to interact with our students.

• Which block are students placing first? The largest blocks or the smallest?
• Which students are using spatial cues (placing rods to see which fits) and which students are using numerical cues (counting units on the grid)? How might we help students who are only using one of these cueing systems without over-scaffolding or showing how WE would complete the puzzle?
• How do our students react when confronted with a challenging puzzle?
• Who is able to swap out 1 rod for 2 rods of equivalent length (1 orange rod is the same length as a brown and red rod together)?
• Which of the following strands of proficiency might you be noting as you observe students:

## Questions to Reflect on:

• Why might you use a task like this? What would be your goal?
• How will you interact with students who struggle to get started, or struggle to move passed a specific hurdle?
• How might these puzzles relate to algebraic reasoning? (try to complete one with this question in mind)
• How are you making the connections between spatial reasoning and algebraic reasoning clear for your students to see? How can these puzzles help?
• How might puzzles allow different students to be successful in your class?

If interested in these puzzles, you might be interested in trying:

## Zukei Puzzles

A little more than a year ago now, Sarah Carter shared a set of Japanese puzzles called Zukei Puzzles (see her original post here or access her puzzles here).  After having students try out the original package of 42 puzzles, and being really engaged in conversations about terms, definitions and properties of each of these shapes, I wanted to try to find more.  Having students ask, “what’s a trapezoid again?” (moving beyond the understanding of the traditional red pattern block to a more robust understanding of a trapezoid) or debate about whether a rectangle is a parallelogram and whether a parallelogram is a rectangle is a great way to experience Geometry.  However, after an exhaustive search on the internet resulting in no new puzzles, I decided to create my own samples.

Take a look at the following 3 links for your own copies of Zukie puzzles:

##### Extension puzzles #2

I’d be happy to create more of these, but first I’d like to know what definitions might need more exploring with your students.  Any ideas would be greatly appreciated!

##### How to complete a Zukei puzzle:

Each puzzle is made up of several dots.  Some of these dots will be used as verticies of the shape named above the puzzle.  For example, the image below shows a trapezoid made of 4 of the dots.  The remaining dots are inconsequential to the puzzle, essentially they are used as distractors.

If you enjoyed these puzzles, I recommend taking a look at Skyscraper puzzles for you to try as well.

## How Many Do You See (Part 2 of 2)

A few weeks ago I shared with you a quick blog post showing a simple worksheet at the grade 2 level – the kind of  simple worksheet that is common to many classrooms.  If you haven’t seen the image, here it is again:

As you can see, the task asks students to correctly count the number of each shape they notice.  In my first post (Part 1) I asked us a few questions to start a conversation:

1. Pick one shape (or more if you’re adventurous)
2. Think about what you believe the teacher’s edition would say
3. Count how many you see
4. Share the 3 points above as a comment here or on Twitter

I was quite happy with where some of the conversations led…

Some of the conversations revolved around the issue many have with resources perpetuating stereotypical definitions of shapes:

If we look, there are exactly 4 shapes that resemble the diagram at the top of the page labelled as “rectangle”, however, there are several different sized squares as well (a square are a special case of a rectangle).

Other conversations revolved around actually counting the number of each item:

What interests me here is that we, as a group of math teachers, have answered this grade 2 worksheet with various answers.  Which brings about 2 important conversations:

1. What are we looking for when students complete a worksheet or textbook questions?
2. Are we aiming for convergent or divergent thinking?  Which of these is more helpful for our students?

###### What are we looking for?

Given the conversations I have had with math teachers about the worksheet being shared here, it seems like there are a few different beliefs.  Some teachers believe the activity is aimed at helping students recognize traditional shapes and identify them on the page.  Other teachers believe that this activity could potentially lead to discussions about definitions of shapes (i.e., What is a rectangle?  What is a hexagon?…) if we listen to and notice our students’ thinking about each of the shapes, then bring students together to have rich discussions.

It’s probably worth noting that the Teacher’s Edition for this worksheet includes precise answers.  If a typical teacher were to collect the students’ work and begin marking the assignment using the “answers” from the teacher’s guide, some of the students would have the “correct” answer of 8 trapezoids, but many others would likely have noticed several of the other trapezoids on the page.  If we are looking / listening for students to find the correct answer, we are likely missing out on any opportunity to learn about our students, or offer any opportunity for our students to learn themselves!

I would hope that an activity like this would provide us opportunities for our students to show what they understand, and move beyond getting answers into the territory of developing mathematical reasoning.

###### Convergent vs Divergent Thinking

Again, many of the teachers I have discussed this activity with have shared their interest in finding the other possible versions of each shape.  However, what we would actually do with this activity seems to be quite different for each educator.  It seems like the decisions different teachers might be making here relate to their interest in students either having convergent thinking, or divergent thinking.  Let’s take a look at a few possible scenarios:

Teacher 1:

Before students start working on the activity, the teacher explains that their job is to find shapes that look exactly like the image in the picture at the top of the page.

Teacher 2:

Before students start working on the activity, the teacher tells the students exactly how many of each shape they found, then asks students to find them.

Teacher 3:

Before students start working on the activity, the teacher explains that their job is to find as many shapes as possible.  Then further explains that there might be ones that are not traditional looking.  Then, together with students, defines criteria for each shape they are about to look for.

Teacher 4:

Before students start working on the activity, the teacher explains that their job is to find as many shapes as possible.  As students are working, they challenge students to continue to think about other possibilities.

In the above scenarios, the teachers’ goals are quite different.  Teacher 1 expects their students to spend time looking at common versions of each shape, then spot them on the page.  Teacher 2’s aim is for students to be able to think deeper about what each shape really means, hoping that they are curious about where the rest of the shapes could possibly be leading their students to challenge themselves.  Teacher 3 believes that in order for students to be successful here, that they need to provide all of the potential pieces before their students get started.  Their goal in the end is for students to use the definitions they create together in the activity.  Finally, teacher 4’s goal is for students to access the mathematics before any terms or definitions are shared.  They believe that they can continue to push students to think by using effective questioning.  The development of reasoning is this teacher’s goal.

Looking back at these 4 teachers’ goals, I notice that 2 basic things differ:

1. How much scaffolding is provided; and
2. When scaffolding is provided

Teachers that provide lots of scaffolding prior to a problem typically aim for students to have convergent thinking.  They provide definitions and prompts, they model and tell, they hope that everyone will be able to get the same answers.

Teachers that withhold scaffolding and expect students to do more of the thinking along the way typically aim for divergent thinking.  That is, they hope that students will have different ideas in the hopes for students to share their thinking to create more thinking in others.

Whether you believe that convergent thinking or divergent thinking is best in math, I would really like you to consider how tasks that promote divergent thinking can actually help the group come to a consensus in the end.  If I were to provide this lesson to grade 2s, I would be aiming for students to be thinking as much as possible, to push students to continue to think outside-the-box as much as possible, then make sure that in my lesson close, that we ALL understood what makes a shape a shape.

#### I want to leave you with a few reflective questions:

• I provided you with a specific worksheet from a specific grade, however, I want you to now think about what you teach.  How much scaffolding do you provide?  Are you providing too much too soon?
• Do your lessons start off with convergent thinking or divergent thinking?  Why do you do this?  Is this because you believe it is best?
• How can you delay scaffolding and convergent thinking so that we are actually promoting our students to be actively thinking?  How can you make this a priority?
• What lesson or warm-ups or problems have you given that are examples of what we are talking about here?
• If we do remove some of the scaffolding will some of your students sit there not learning?  Is this a sign of them not understanding the math, or a sign of them used to being spoon-fed thinking?  What do WE need to get better at if we are to delay some of this scaffolding?

I encourage you to continue to think about what it means to help set up situations for your students to actively construct understanding:

I’d love to continue the conversation.  Write a response, or send me a message on Twitter ( @markchubb3 ).

P.S.  I’m still not confident how many of each shape are actually here!

## How many do you see?(Part 1)

A few days ago I had the opportunity to work with a grade 2 teacher as her class was learning about Geometry.  The students started the class with a rich activity comparing and sorting a variety of standard and non-standard shapes, followed by a great discussion about several properties they had noticed.

Shortly after, students started working on following the page as independent work. Take a look:

Take a minute to try to figure out what you think the answers might be.  Scroll up and pick one of the less obvious shapes and count how many you see.

This isn’t one of those Facebook “can you find all the hidden shapes” tasks, it’s meant to be a straightforward activity for grade 2 students. However, I’m not sure what the actual answers are here.  So, I need some help…  I’d love if you could:

• Pick one shape (or more if you’re adventurous)
• Think about what you believe the teacher’s edition would say
• Count how many you see
• Share the 3 points above as a comment here or on Twitter

I’m hoping in my next post that we can discuss more than just this worksheet and make some generalizations for any grade and any topic.

## Targeted Instruction

#### Issues with Individualized / Targeted Instruction

Individualized or targeted instruction makes sense in a lot of ways.  The idea is to figure out what a student’s needs are, then provide opportunities for them to get better in this area.  In practice, however, what often happens is that we end up setting different learning paths for different students which actually creates more inequities than it helps close gaps.  In my experience, having different students learning different things might be helpful to those who are being challenged, but does a significant disservice to those who are deemed “not ready” to learn what others are learning.  For example, in the 3 pathways shared above, it was suggested that the class be split into 3 groups; one working on defining terms, one learning about properties of shapes and the last group would spend time sorting shapes in various ways.  If we thought of this in terms of development, each group of students would be set on a completely different path.  Those working on developing “recognition” tasks (See Van Hiele’s Model below) would be working on low-level tasks.  Instead of providing experiences that might help them make sense of Geometric relationships, they would be stuck working on tasks that focus on memory without meaning.

When we aim to find specific tasks for specific students, we assume that students are not capable of learning things others are learning.  This creates low expectations for our students!  Van de Walle says it best in his book Teaching Student Centered Mathematics:

Determining how to place students in groups is an important decision.  Avoid continually grouping by ability.  This kind of grouping, although well-intentioned, perpetuates low levels of learning and actually increases the gap between more and less dependent students.

Targeted instruction might make sense on paper, but there are several potential flaws:

• Students enter into tracks that do not actually reflect their ability.  There is plenty of research showing that significant percentages of students are placed in the wrong grouping by their teachers.  Whether they have used some kind of test or not, groupings are regularly flawed in predicting what students are potentially ready for.
• Pre-determining who is ready for what learning typically results in ability grouping, which is probably the strongest fixed mindset message a school can send students.  Giving an entrance ticket that determines certain students can’t engage in the learning others are doing tells students who is good at math, and who isn’t.  Our students are exquisitely keen at noticing who we believe can be successful, which shapes their own beliefs about themselves.
• The work given to those in lower groups is typically less cognitively demanding and results in minimal learning.  The intent to “fill gaps” or “catch kids up” ironically increases the gap between struggling students and more independent learners.  Numerous studies have confirmed what Hoffer (1992) found: “Comparing the achievement growth of non-grouped students and high- and low-group students shows that high-group placement generally has a weak positive effect while low-group placement has a stronger negative effect. Ability grouping thus appears to benefit advanced students, to harm slower students.

The original conversation I had about Entrance Tickets illustrated a common issue we have.  We notice that there are students in our rooms who come into class in very different places in their understanding of a given topic.  We want to make sure that we provide things that our students will be successful with… However, this individualization of instruction does the exact opposite of what differentiated instruction intends to do.  Differentiated instruction in a mathematics class is realized when we provide experiences for our students where everyone is learning what they need to learn and can demonstrate this learning in different ways.  The assumption, however, is that WE are the ones that should be determining who is learning what and how much.  This just doesn’t make sense to me!  Instead of using entrance tickets, we ended up deciding to use this problem from Van de Walle so we could reach students no matter where they were in their understanding.  Instead of a test to determine who is allowed to learn what, we allowed every student to learn!  This needs to be a focus!

If we are ever going to help all of our students learn mathematics and believe that they are capable of thinking mathematically, then we need to provide learning experiences that ALL of our students can participate in.  These experiences need to:

• Have multiple entry points for students to access the mathematics
• Provide challenge for all students (be Problem-Based)
• Allow students to actively make sense of the mathematics through mathematical reasoning
• Allow students opportunities to students to express their understanding in different ways or reach an understanding via different strategies

Let’s avoid doing things that narrow our students’ learning like using entrance tickets to target instruction!  Let’s commit to a view of differentiated instruction where our students are the ones who are differentiating themselves (because the tasks allowed for opportunities to do things differently)!  Let’s continue to get better at leveraging students’ thinking in our classrooms to help those who are struggling!  Let’s believe that all of our students can learn!

#### I want to leave you with a few reflective questions:

• Why might conversations about entrance tickets and other ways to determine students ability be more common today?  We need to use our students’ thinking to guide our instruction, but other than entrance cards, how can we do this in ways that actually help those who are struggling?
• Is a push for data-driven instruction fueling this type of decision making?  If so, who is asking for the data?  Are there other sources of data that you can be gathering that are healthier for you and your students?
• If you’ve ever used entrance tickets or diagnostics, followed by ability groups, how did those on the bottom group feel?  Do you see the same students regularly in the bottom group?  Do you see a widening gap between those dependent on you and those who are more independent?
• Where do you look for learning experiences that offer this kind of differentiated instruction?  Is it working for the students in your class that are struggling?

I encourage you to continue to think about what it means to Differentiate your Instruction.  Here are a few pieces that might help:

I’d love to continue the conversation.  Write a response, or send me a message on Twitter ( @markchubb3 ).

## Starting where our students are….. with THEIR thoughts

A common trend in education is to give students a diagnostic in order for us to know where to start. While I agree we should be starting where our students are, I think this can look very different in each classroom.  Does starting where our students are mean we give a test to determine ability levels, then program based on these differences?  Personally, I don’t think so.

Giving out a test or quiz at the beginning of instruction isn’t the ideal way of learning about our students.  Seeing the product of someone’s thinking often isn’t helpful in seeing HOW that child thinks (Read, What does “assessment drive instruction mean to you” for more on this). Instead, I offer an alternative- starting with a diagnostic task!  Here is an example of a diagnostic task given this week:

This lesson is broken down into 4 parts.  Below are summaries of each:

### Part 1 – Tell 1 or 2 interesting things about your shape

Start off in groups of 4.  One student picks up a shape and says something (or 2) interesting about that shape.

Here you will notice how students think about shapes. Will they describe the shape as “looking like a mountain” or “it’s an hourglass” (visualization is level 1 on Van Hiele’s levels of Geometric thought)… or will they describe attributes of that shape (this is level 2 according to Van Hiele)?

As the teacher, we listen to the things our students talk about so we will know how to organize the conversation later.

### Part 2 – Pick 2 shapes.  Tell something similar or different about the 2 shapes.

Students randomly pick 2 shapes and either tell the group one thing similar or different about the two shapes. Each person offers their thoughts before 2 new shapes are picked.

Students who might have offered level 1 comments a minute ago will now need to consider thinking about attributes. Again, as the teacher, we listen for the attributes our students understand (i.e., number of sides, right angles, symmetry, number of vertices, number of pairs of parallel sides, angles….), and which attributes our students might be informally describing (i.e., using phrases like “corners”, or using gestures when attempting to describe something they haven’t learned yet).  See chart below for a better description of Van Hiele’s levels:

At this time, it is ideal to hold conversations with the whole group about any disagreements that might exist.  For example, the pairs of shapes above created disagreements about number of sides and number of vertices.  When we have disagreements, we need to bring these forward to the group so we can learn together.

### Part 3 – Sorting using a “Target Shape”

Pick a “Target Shape”. Think about one of its attributes.  Sort the rest of the shapes based on the target shape.

The 2 groups above sorted their shapes based on different attributes. Can you figure out what their thinking is?  Were there any shapes that they might have disagreed upon?

### Part 4 – Secret sort

Here, we want students to be able to think about shapes that share similar attributes (this can potentially lead our students into level 2 type thinking depending on our sort).  I suggest we provide shapes already sorted for our students, but sorted in a way that no group had just sorted the shapes. Ideally, this sort is something both in your standards and something you believe your students are ready to think about (based on the observations so far in this lesson).

In this lesson, we have noticed how our students think.  We could assess the level of Geometric thought they are currently using, or the attributes they are comfortable describing, or misconceptions that need to be addressed.  But, this lesson isn’t just about us gathering information, it is also about our students being actively engaged in the learning process!  We are intentionally helping our students make connections, reason and prove, learn/ revisit vocabulary, think deeper about specific attributes…

I’ve shared my thoughts about what I think day 1 should look like before for any given topic, and how we can use assessment to drive instruction, however, I wanted to write this blog about the specific topic of diagnostics.

In the above example, we listened to our students and used our understanding of our standards and developmental research to know where to start our conversations. As Van de Walle explains the purpose of formative assessment, we need to make our formative more like a streaming video, not just a test at the beginning!

If its formative, it needs to be ongoing… part of instruction… based on our observations, conversations, and the things students create…  This requires us to start with rich tasks that are open enough to allow everyone an entry point and for us to have a plan to move forward!

I’m reminded of Phil Daro’s quote:

For us to make these shifts, we need to consider our mindsets that also need to shift.  Statements like the following stand in the way of allowing our students to be actively engaged in the learning process starting with where they currently are:

• My students aren’t ready for…