Tag: pdf

Math Games – building a foundation for mathematical reasoning

Mark Chubb7 Comments

In 2001, the National Research Council, in their report Adding it up: Helping children learn mathematics, sought to address a concern expressed by many Americans: that too few students in our schools are successfully acquiring the mathematical knowledge, skill, and confidence they need to use the mathematics they have learned.

Developing Mathematical Proficiency
The potential of different types of tasks for student learning, 2017

As we start a new school year, I expect many teachers, schools and districts to begin conversations surrounding assessment and wondering how to start learning given students who might be “behind”. I’ve shared my thoughts about how we should NOT start a school year, but I wanted to offer some alternatives in this post surrounding a piece often overlooked — our students’ confidence (including student agency, ownership and identity). If we are truly interested in starting a year off successfully, then we need to spend time allowing our students to see themselves in the math they are doing… and to see their strengths, not their deficits.

[The] goal is to support all students — especially those who have not been academically successful in the past — to develop a sense of agency and ownership over their own learning. We want students to come to see themselves as intellectually capable and competent — not by giving them easy successes, but by engaging them as sense-makers, problem solvers, and creators of meaningful and important ideas.

MathShell – TRUMath, 2016

When we hear ideals like the above quote, what many of us see is as missing are specific examples. How DO we help our students gain confidence becomes a question most of us are left with. Adding It Up suggests that mathematical proficiency includes an intertwined mix of procedural fluency, conceptual understanding, strategic competence, adaptive reasoning and productive disposition. Which again sounds nice in theory, but in reality, these 5 pieces are not balanced in classroom materials nor in our assessment data. Not even close!

Adding It Up: helping children learn mathematics, 2001

So, again we are left with a specific need for us to build confidence in our students. There is a growing body of evidence to support the use of strategy games in math class as a purposeful way to build confidence (including student agency, authority and identity).

To be helpful, I’d like to share some examples of possible strategy games that are appropriate for all ages. Each game is a traditional game from various places around the world.

*The above files are open to view / print. If you experience difficulties accessing, you might need to use a non-educational account as your school board might be restricting your access.

How to Play:

Each link above includes a full set of rules, but you might also be interested in watching a preview of these games (thanks to WhatDoWeDoAllDay.com)

A few things to reflect on:

  • Some students have missed a lot of school / learning. Our students might be entering a new grade worried about the difficulty level of the content. Beyond content, what other aspects of learning math might be a struggle for our students? How might introducing games periodically help with these struggles?
  • How do you see equity playing a role in all of this? Pinpointing and focusing on student gaps often leads to inequities in experiences and outcomes. So, how can the ideas above help reduce these inequities?
  • One of the best ways to tackle equity issues is to expand WHAT we consider mathematics and expand WHO is considered a math person. How might you see using games periodically as a way for us to improve in these two areas?
  • If you are distance learning, how might games be an integral part of your program? How do you see including games that are not related to content helpful for our students that might struggle to learn mathematics? (building confidence, social-emotional learning skills, community, students’ identities…)
  • If you are learning in person this year, but can not have students working together, how might you adapt some of these strategy games?
  • What might you notice as students are playing games that you might not be able to notice otherwise?
  • How might we see a link between gaining confidence through playing strategy games and improvement in mathematical reasoning?
  • Why do you think I choose the games above (I searched through many)? Hopefully you can see a benefit from seeing mathematics learning from various cultures.

If interested in more games and puzzles? Take a look at some of the following posts:

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

Skyscraper Puzzles – printable package

Mark Chubb6 Comments

An area of mathematics I wish more students had opportunities to explore is spatial/visualization. There are many studies that show just how important spatial/visual reasoning is for mathematical success (I discuss in more depth here), but often, we as teachers aren’t sure where to turn to help our students develop spatial reasoning, or now to make the mathematics our students are learning more spatial.

One such activity I’ve suggested before is Skyscraper Puzzles. I’ve shared these puzzles before (Skyscraper Puzzles and Skyscraper Templates – for relational rods). With the help of my own children, I decided to make new templates. The package includes a page dedicated to explain how to solve the puzzles, as well as instructions on each page.

For details about how to solve a Skyscraper Puzzle, please click here

New Puzzles can be accessed here

*The above files are open to view / print. If you experience difficulties accessing, you might need to use a non-educational account as your school board might be restricting your access.

You’ll notice in the package above that some of the puzzles are missing information like the puzzle below:

Puzzles like these might include information within the puzzle. In the puzzle above, the 1 in the middle of the block refers to the height of that tower (a tower with a height of 1 goes where the 1 is placed).

You might also be interested in watching a few students discussing how to play:

A few thoughts about how you might use these:

As always, I’d love to hear from you. Feel free to write a response, or send me a message on Twitter ( @markchubb3 ).

One-Hole Punch Puzzle Templates

Mark Chubb48 Comments

Recently Lowrie et al. published an article in the Journal of Experimental Education where they looked at the effects of a spatial intervention program for grade 8 students. This study followed the 876 grade 8 students across 9 schools as they received 20 hours of spatial interventions (as well as other grade 8 students in schools that conducted “business as usual” mathematics classes). Their findings were quite interesting. They found students who received spatial intervention programs achieved:

  • significantly better on spatial tasks (13% higher)
  • significantly better on Geometry – Measurement problems that included material not discussed in either class
  • significantly better on Number – Algebra problems
  • equally well on Statistic – Probability problems

While this study supports many studies showing a link between spatial abilities and mathematics performance (Mix and Cheng, 2012), and others showing how students’ spatial abilities are malleable at any age (can be learned with the right experiences) (Uttal et al., 2013), there seems to be a definite need for teachers to have at their fingertips good examples of experiences that will help our students develop spatial reasoning. At the end of this post are possible examples for us to try.

A recent meta-analysis of 217 studies, representing more than two decades of research on spatial training, found that a variety of activities improve spatial reasoning across all age groups. Not only did the authors find that spatial training led to improvements on spatial tasks closely related to the training task, but improvements were also seen on other types of tasks that were not part of the training.

Taking Shape, 2016

A focus on spatial reasoning, from my experience, has helped the students in my schools make sense of connections between concepts, it has been the underpinning of new learning, and has been the vehicle for so much of OUR learning as educators.

One-Hole Punch puzzles

You might be familiar with various cognitive tests that ask students to think through mental rotations, 2D/3D visualization, paper folding or other tasks where students’ abilities to visualize are measured. Spatial tasks like these are very predictive of a students’ math success, however, few resources are aimed to specifically help our students develop their spatial reasoning (see Taking Shape as an ideal K-2 example).

Below is an example of one of these cognitive test questions. As you can see, students here are asked to mentally fold a piece of paper and then punch 1 hole through the folds, then imagine what the paper would look like once opened.

Templates

Instead of trying to measure our students’ abilities by giving sample problems like this, it is far more productive to offer experiences where our students can learn to think spatially. This is why I have created a few sample experiences. Below are sample templates that you can use with your students. They will need squares of paper (linked below), as well as the puzzles they would be aiming to solve. Take a look:

How to Solve a One-Hole Punch puzzle:

Directions for solving the puzzles are only written on the first page:

“These are One-Hole Punch puzzles.  To complete a puzzle, take a square of paper, fold it using as many folds as needed so that if you punch one hole and unfold it, you will match one of the puzzles.  Complete the following puzzles in any order.  Be prepared to discuss your strategies with others.”

Students will need several blank squares of paper and a single hole puncher. Student will need to visualize how to fold their paper and where to punch their one hole to match the puzzle they are working on. These puzzles are tricky because only 1 hole can be punched to create several holes.

Pro Tip: Make sure you have a lot of squares of paper for every student.

A few thoughts about using these:

  • How will you introduce these puzzles to your students?  How much information about strategies and tips will you provide?  Will this allow for productive struggle, or will you attempt to remove as much of the struggle as possible?
  • Would you use these as an activity you give all students, or something you provide to just some.  Why? (Something for students who finish early or something for everyone to try?)
  • How would giving a page of puzzles to a pair of students be different than if you gave it to individuals?  Which were you assuming to do here?  What if you tried the other option?
  • How will you orchestrate a conversation for your class to help consolidate the learning here? How might this be helpful before you give the next set of puzzles?
  • What will you do if students give up quickly?  What questions / prompts will you provide?
  • Who is successful at solving these puzzles? Sometimes, those who are often left out of mathematical discussions can become more involved when given spatial tasks. How might you capitalize on increasing your students’ interest in mathematics? Do you see tasks like this helping improve your students’ agency, identity and authority in mathematics?
  • How might you see intentional decisions to focus on spatial reasoning as helping your students with mathematics in other areas? This post might be helpful.

A belief I have: Teaching mathematics is much more than providing neat things for our students, it involves countless decisions on our part about how to effectively make the best use of the problem / activity.  Hopefully, this post has helped you consider your own decision making processes!

As always, I’d love to hear your thoughts. Feel free to write a response here or send me a message on Twitter ( @markchubb3 ).

Spatial Puzzles: Cuisenaire Cover-ups

Mark Chubb5 Comments

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:
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: