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.
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:
Adding It Up, 2001
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?
I’d love to continue the conversation about how we can use these puzzles to further our students’ spatial/algebraic reasoning. Leave a comment here or on Twitter @MarkChubb3
If interested in these puzzles, you might be interested in trying:
A few weeks ago a NYTimes published an article titled, Make Your Daughter Practice Math. She’ll Thank You Later, an opinion piece that, basically, asserts that girls would benefit from “extra required practice”. I took a few minutes to look through the comments (which there are over 600) and noticed a polarizing set of personal comments related to what has worked or hasn’t worked for each person, or their own children. Some sharing how practicing was an essential component for making them/their kids successful at mathematics, and others discussing stories related to frustration, humiliation and the need for children to enjoy and be interested in the subject.
Instead of picking apart the article and sharing the various issues I have with it (like the notion of “extra practice” should be given based on gender), or simply stating my own opinions, I think it would be far more productive to consider why practice might be important and specifically consider some key elements of what might make practice beneficial to more students.
To many, the term “practice” brings about childhood memories of completing pages of repeated random questions, or drills sheets where the same algorithm is used over and over again. Students who successfully completed the first few questions typically had no issues completing each and every question. For those who were successful, the belief is that the repetition helped. For those who were less successful, the belief is that repeating an algorithm that didn’t make sense in the first place wasn’t helpful… even if they can get an answer, they might still not understand (*Defining 2 opposing definitions of “understanding” here).
“Practice” for both of the views above is often thought of as rote tasks that are devoid of thinking, choices or sense making. Before I share with you an alternative view of practice, I’d like to first consider how we have tackled “practice” for students who are developing as readers.
If we were to consider reading instruction for a moment, everyone would agree that it would be important to practice reading, however, most of us wouldn’t have thoughts of reading pages of random words on a page, we would likely think about picture books. Books offer many important factors for young readers. Pictures might help give clues to difficult words, the storyline offers interest and motivation to continue, and the messages within the book might bring about rich discussions related to the purpose of the book. This kind of practice is both encourages students to continue reading, and helps them continue to get better at the same time. However, this is very different from what we view as math “practice”.
Below is a chart explaining the role of practice as it relates to what Dan Finkel calls play:
Take a look at the “Process” row for a moment. Here you can see the difference between a repetitive drill kind of practice and the “playful experiences” kind of practice Dan had called for. Let’s take a quick example of how practice can be playful.
Students learning to add 2-digit numbers were asked to “practice” their understanding of addition by playing a game called “How Close to 100?”. The rules:
Roll 2 dice to create a 2-digit number (your choice of 41 or 14)
Use base-10 materials as appropriate
Try to get as close to 100 as possible
4th role you are allowed eliminating any 1 number IF you want
What choice would you make??? Some students might want to keep all 4 roles and use the 14 to get close to 100, while other students might take the 41 and try to eliminate one of the roles to see if they can get closer.
When practice involves active thinking and reasoning, our students get the practice they need and the motivation to sustain learning! When practice allows students to gain a deeper understanding (in this case the visual of the base-10 materials) or make connections between concepts, our students are doing more than passive rule following – they are engaging in thinking mathematically!
In the end, we need to take greater care in making sure that the experiences we provide our students are aimed at the 5 strands shown below:
What does “practice” look like in your classroom? Does it involve thinking or decisions? Would it be more engaging for your students to make practice involve more thinking?
How does this topic relate to the topic of “engagement”? Is engagement about making tasks more fun or about making tasks require more thought? Which view of engagement do you and your students subscribe to?
What does practice look like for your students outside of school? Is there a place for practice at home?
Which of the 5 strands (shown above) are regularly present in your “practice” activities? Are there strands you would like to make sure are embedded more regularly?
I’d love to continue the conversation about “practicing” mathematics. Leave a comment here or on Twitter @MarkChubb3
A few days ago I had the privilege of presenting at MAC2 to a group of teachers in Orillia on the topic of “Making Math Visual”. If interested, here are some of my talking points for you to reflect on:
To get us started I shared an image created by Christopher Danielson and asked the group what they noticed:
We noticed quite a lot in the image… and did a “how many” activity sharing various numbers we saw in the image. After our discussions I explained that I had shared the same picture with a group of parents at a school’s parent night followed by the next picture.
I asked the group of teachers what mathematics they noticed here… then how they believed parents might have answered the question. While we, as math teachers, saw patterns in the placements of utensils, shapes and angles around the room, quantities of countable items, multiplicative relationships between utensils and place settings, volume of wine glasses, differences in heights of chairs, perimeter around the table….. the group correctly guessed that many parents do not typically notice the mathematics around them.
So, my suggestion for the teachers in the room was to help change this:
I then asked the group to do a simple task for us to learn from:
After a few minutes of thinking, I shared some research of the different ways we use fractions:
When we looked at the ways we typically use fractions, it’s easy to notice that WE, as teachers, might need to consider how a focus on representations might help us notice if we are providing our students with a robust (let’s call this a “relational“) view of the concepts our students are learning about.
Data taken from 1 school’s teachers:
We continued to talk about Liping Ma’s work where she asked teachers to answer and represent the following problem:
Followed by a quick story of when a student told me that the following statement is true (click here for the full story).
So, why should we focus on making math visual?
We then explored a statement that Jo Boaler shared in her Norms document:
…and I asked the group to consider if there is something we learn in elementary school that can’t be represented visually?
If you have an idea to the previous question, I’d love to hear it, because none of us could think of a concept that can’t be represented visually.
I then shared a quick problem that grade 7 students in one of my schools had done (see here for the description):
Along with a few different responses that students had completed:
Most of the students in the class had responded much like the image above. Most students in the class had confused linear metric relationships (1 meter = 100 cm) with metric units of area (1 meter squared is NOT the same as 100cm2).
In fact, only two students had figured out the correct answer… which makes sense, since the students in the class didn’t learn about converting units of area through visuals.
We wrapped up with a few suggestions:
And finally some advice about what we DON’T mean when talking about making mathematics visual:
You might recognize the image above from Graham Fletcher’s post/video where he removed all of the fractional numbers off each face in an attempt to make sure that the tools were used to help students learn mathematics, instead of just using them to get answers.
I want to leave you with a few reflective questions:
Can all mathematics concepts in elementary school be represented visually?
Why might a visual representation be helpful?
Are some representations more helpful than others?
How important is it that our students notice the mathematics around them?
How might a focus on visual representations help both us and our students deepen our understanding of the mathematics we are teaching/learning?
I’d love to continue the conversation. Feel free to write a response, or send me a message on Twitter ( @markchubb3 ).
If you are interested in all of the slides, you can take a look here
My wife Anne-Marie isn’t always impressed when I talk about mathematics, especially when I ask her to try something out for me, but on occasion I can get her to really think mathematically without her realizing how much math she is actually doing. Here’s a quick story about one of those times, along with some considerations:
A while back Anne-Marie and I were preparing lunch for our three children. It was a cold wintery day, so they asked for Lipton Chicken Noodle Soup. If you’ve ever made Lipton Soup before you would know that you add a package of soup mix into 4 cups of water.
Typically, my wife would grab the largest of our nesting measuring cups (the one marked 1 cup), filling it four times to get the total required 4 cups, however, on this particular day, the largest cup available was the 3/4 cup.
Here is how the conversation went:
Anne-Marie: How many of these (3/4 cups) do I need to make 4 cups?
Me: I don’t know. How many do you think? (attempting to give her time to think)
Anne-Marie: Well… I know two would make a cup and a half… so… 4 of these would make 3 cups…
Me: OK…
Anne-Marie: So, 5 would make 3 and 3/4 cups.
Me: Mmhmm….
Anne-Marie: So, I’d need a quarter cup more?
Me: So, how much of that should you fill? (pointing to the 3/4 cup in her hand)
Anne-Marie: A quarter of it? No, wait… I want a quarter of a cup, not a quarter of this…
Me: Ok…
Anne-Marie: Should I fill it 1/3 of the way?
Me: Why do you think 1/3?
Anne-Marie: Because this is 3/4s, and I only need 1 of the quarters.
The example I shared above illustrates sense making of a difficult concept – division of fractions – a topic that to many is far from our ability of sense making. My wife, however, quite easily made sense of the situation using her reasoning instead of a formula or an algorithm. To many students, however, division of fractions is learned first through a set of procedures.
I have wondered for quite some time why so many classrooms start with procedures and algorithms unill I came across Liping Ma’s book Knowing and Teaching Mathematics. In her book she shares what happened when she asked American and Chinese teachers these 2 problems:
Here were the results:
Now, keep in mind that the sample sizes for each group were relatively small (23 US teachers and 72 Chinese teachers were asked to complete two tasks), however, it does bring bring about a number of important questions:
How does the training of American and Chinese teachers differ?
Did both groups of teachers rely on the learning they had received as students, or learning they had received as teachers?
What does it mean to “Understand” division of fractions? Computing correctly? Beging able to visually represent what is gonig on when fractions are divided? Being able to know when we are being asked to divide? Being able to create our own division of fraction problems?
What experiences do we need as teachers to understand this concept? What experiences should we be providing our students?
Visual Representations
In order to understand division of fractions, I believe we need to understand what is actually going on. To do this, visuals are a necessity! A few examples of visual representations could include:
A number line:
A volume model:
An area model:
Starting with a Context
Starting with a context is about allowing our students to access a concept using what they already know (it is not about trying to make the math practical or show students when a concept might be used someday). Starting with a context should be about inviting sense-making and thinking into the conversation before any algorithm or set of procedures are introduced. I’ve already shared an example of a context (preparing soup) that could be used to launch a discussion about division of fractions, but now it’s your turn:
Design your own problem that others could use to launch a discussion of division of fractions. Share your problem!
A few things to reflect on:
How do you use contexts and visuals to help your students make sense of concepts?
Forty years ago, Richard Skemp wrote one of the most important articles, in my opinion, about mathematics, and the teaching and learning of mathematics called Relational Understanding and Instrumental Understanding. If you haven’t already read the article, I think you need to add this to your summer reading (It’s linked above).
Skemp quite nicely illustrates the fact that many of us have completely different, even contradictory definitions, of the term “understanding”. Here are the 2 opposing definitions of the word “understanding”:
“Instrumental understanding” can be thought of as knowing the rules and procedures without understanding why those rules or procedures work. Students who have been taught instrumentally can perform calculations, apply procedures… but do not necessarily understand the mathematics behind the rules or procedures.
“Relational understanding”, on the other hand, can be thought of as understanding how and why the rules and procedures work. Students who are taught relationally are more likely to remember the procedures because they have truly understood why they work, they are more likely to retain their understanding longer, more likely to connect new learning with previous learning, and they are less likely to make careless mistakes.
Think of the two types of understanding like this:
Students who are taught instrumentally come to see mathematics as isolated pieces of knowledge. They are expected to remember procedures for each and every concept/skill. Each new skill requires a new set of procedures. However, those who are taught relationally make connections between and within concepts and skills. Those with a relational understanding can learn new concepts easier, retain previous concepts, and are able to deviate from formulas/rules given different problems easier because of the connections they have made.
Skemp articulates how much of an issue this really is in our educational system when he explains the different types of mismatches that can occur between how students are taught, and how students learn. Take a look:
Notice the top right quadrant for a second. If a child wants to learn instrumentally (they only want to know the steps/rules to solve today’s problem) and the teacher instead offers tasks/problems that asks the child to think or reason mathematically, the student will likely be frustrated for the short term. You might see students that lack perseverance, or are eager for assistance because they are not used to thinking for themselves. However, as their learning progresses, they will come to make sense of their mathematics and their initial frustration will fade.
On the other hand, if a teacher teaches instrumentally but a child wants to learn relationally (they want/need to understand why procedures work) a more serious mismatch will exist. Students who want to make sense of the concepts they are learning, but are not given the time and conditions to experience mathematics in this way will come to believe that they are not good at mathematics. These students soon disassociate with mathematics and will stop taking math classes as soon as they can. These students view themselves as “not a math person” because their experiences have not helped them make sense of the mathematics they were learning.
While the first mismatch might seem frustrating for us as teachers, the frustration is short lived. On the other hand, the second mismatch has life-long consequences!
I’ve been thinking about the various initiatives/ professional development opportunities… that I have been part of, or have been available online or through print that might help us think about how to move from an instrumental understanding to a relational understanding of mathematics. Here are a few I want to share with you:
Phil Daro’s Against Answer Getting video highlights a few instrumental practices that might be common in some schools. Below is the “Butterfly” method of adding fractions he shares as an “answer getting” strategy. While following these simple steps might help our students get the answer to this question, Phil points out that these students will be unable to solve an addition problem with 3 fractions. These students “understand” how to get the answer, but in no way understand how the answer relates to addends.
On the other hand, teachers who teach relationally provide their students with contexts, models (i.e, number lines, arrays…), manipulatives (i.e., cuisenaire rods, pattern blocks…) and visuals to help their students develop a relational understanding. If you are interested in learning more about helping your students develop a relational understanding of fractions, take a look at a few resources that will help:
Tina Cardone and a group of math teachers across Twitter (part of the #MTBoS) created a document called Nix The Tricks that points out several instrumental “tricks” that do not lead to relational understanding. For example, “turtle multiplication” is an instrumental strategy that will not help our students understand the mathematics that is happening. Students can draw a collar and place an egg below, but in no way will this help with future concepts!
Teachers focused on relational understanding again use representations that allow their students to visualize what is happening. Connections between representations, strategies and the big ideas behind multiplication are developed over time.
Take a look at some wonderful resources that promote a relational understanding of multiplication:
Each of the above are developmental in nature, they focus on representations and connections.
So how do we make these shifts? Here are a few of my thoughts:
Notice instrumental teaching practices.
Learn more about how to move from instrumental to relational teaching.
Align assessment practices to expect relational understanding.
Goal 1 – Notice instrumental teaching practices.
Many of these are easy to spot. Here is a small sample from Pinterest:
The rules/procedures shared here ask students to DO without understanding. The issue is that there are actually countless instrumental practices out there, so my goal is actually much harder than it seems. Think about something you teach that involves rules or procedures. How can you help your students develop a relational understanding of this concept?
Goal 2 – Learn more about how to move from instrumental to relational teaching.
I don’t think this is something we can do on our own. We need the help of professional resources (Marian Small, Van de Walle, Fosnot, and countless others have helped produce resources that are classroom ready, yet help us to see mathematics in ways that we probably didn’t experience as students), mathematics coaches, and the insights of teachers across the world (there is a wonderful community on Twitter waiting to share and learn with you).
I strongly encourage you to look at chapter 1 of Van de Walle’s Teaching Student Centered Mathematics where it will give a clearer view of relational understanding and how to teach so our students can learn relationally.
However we are learning, we need to be able to make new connections, see the concepts in different ways in order for us to know how to provide relational teaching for our students.
Goal 3 – Align assessment practices to expect relational understanding.
This is something I hope to continue to blog about. If we want our students to have a relational understanding, we need to be clear about what we expect our students to be able to do and understand. Looking at developmental landscapes, continuums and trajectories will help here. Below is Cathy Fosnot’s landscape of learning for multiplication and division. While this might look complicated, there are many different representations, strategies and big ideas that our students need to experience to gain a relational understanding.
Asking questions or problems that expect relational understanding is key as well. Take a look at one of Marian Small’s slideshows below. Toward the end of each she shares the difference between questions that focus on knowledge and questions that focus on understanding.
I hope whatever your professional learning looks like this year (at school, on Twitter, professional reading…) there is a focus on helping build your relational understanding of the concepts you teach, and a better understanding of how to build a relational understanding for your students. This will continue to be my priority this year!
Thinking Mathematically 