Tuesday, March 21, 2017

What Do You See?

A video from Dan Finkel, from http://mathforlove.com/ on the number chart above:

Pedagogical Documentation
Video: Young Mathematicians: Pedagogical Documentation for Planning and Provoking Math Learning

Ministry of Education on Pedagogical documentation for K-2:

Also, "Pedagogical Documentation Revisited K-12"

Saturday, February 4, 2017


Taking a Yoda approach to a Growth MindSet

Possible Bulletin Boards

Other Jedi Mindset

Saturday, January 7, 2017

Wednesday, November 23, 2016

Image result for website Under construction image

Nya:weh to Mr. Restoule-General for letting me update his numeracy blog for Six Nations Schools. In the middle of reorganizing, checking links and adding links.

~ Miss. Graham :)

Tuesday, September 3, 2013

New Site for mrrgteacher

I've migrated all the Six Nations Numeracy blog posts to my new blog: @mrrgteacher

I will be keeping this site (Six Nations Numeracy) online so teachers and students can access all the links and content on the sidebars.

To access my new blog, click here or visit www.mrrgteacher.blogspot.ca or click the link on the right.

I will put a link on the new @mrrgteacher blog that will direct you back to this site in case you want to revisit this blog and forget the address.

In the meantime, please update your favourites, or make my new blog your homepage to see what's new with @mrrgteacher each day.

Friday, August 30, 2013

Moment of Transition

Hello faithful readers and purveyors of this blog!  I appreciate the comments and kind words concerning this site and its support as a resource for teachers and students.  I'm currently deciding whether to continue on updating this blog or migrating elsewhere now that I am no longer in the Six Nations District Numeracy Teacher position.

Rest assured, I do not plan to wipe this blog off the interwebs, but may open up a new url/blog/address that more accurately fits my interests as an OCT professional.  These include current math initiatives and progressive practice, the effective and fluid use of technology for education for the 21st Century (not old models on new devices), and cultural topics relating to First Nations, Social Justice and Equity.  You may find the odd topic outside of these areas, but that's what I plan to continue sharing in the future.

If you have an ideas or comments that may help inform the future of my digital presence, please leave a comment below.  In the meantime, this blog is about to go dark with few if any updates until a decision is made.

Wednesday, August 28, 2013

I Have a Dream

As I'm sure you've probably already seen or heard in the news that today marks the 50th anniversary of Dr. Martin Luther King Jr.'s "I Have a Dream" speech.  Why not take a moment to use our district's BrainPOP resource account with your class to learn a bit about the historical event, by clicking here?

Tuesday, August 27, 2013

Monday, August 26, 2013

Good Questions made Greater

Welcome to another school year, Six Nations schools!

Glad to have you back.  Perhaps you might try something a little different with those Pearson Math Makes Sense textbooks that we all have.  Consider doing a little reconstruction with a question to open up the learning with your students.  Fawn Nguyen provides an excellent example of how to do this on an amazing math blog.

Not only is this blog post that I'm sharing with you, (found here), entirely brilliant in the way it takes a textbook problem and explodes it (or deconstructs it) to provide a richer learning experience for the students, it is also extremely funny and entertaining to read.

I have added the website to our list of awesome math websites on the sidebar.  Enjoy!

Friday, August 23, 2013

Talk Moves

Here's a great chapter of a book from Math Solutions that details the importance of certain "talk moves" and how to establish them in your class.  It can be found here.

Thursday, August 22, 2013

Educational Value of Math Puzzles

Recreational and Educational Value of Math Puzzles | Edutopia
The figure above shows a puzzle called OkiDoku. Prof. Dani Novak & Prof. David Rosenthal, Ithaca College
Many people enjoy working on grid puzzles as small, quick challenges of their mathematical and logical skills. Here is one you may not have seen, the OkiDoku. How does it work? Looking at the grid above, try to find four different numbers and put them in these 16 squares in a way that will satisfy the following two conditions:
  1. Each of these four numbers must appear exactly once in each row and in each column.
  2. The blocks with thick borders are called cages. Each cage shows a target number and a mathematical operation. The operation applied to the numbers in the cage should produce the target number. For example, there is a cage in the first row with a target number of 20 and a mathematical operation of multiplication. The puzzle solver should put three numbers in the cage so that the product of these numbers is 20.
Give it a try.
The most famous of all grid puzzles is the Sudoku, a logic puzzle found in a vast majority of newspapers. Some of you may have encountered a mathematical grid puzzle called KENKEN® that appears in more than 100 national U.S. publications. KENKEN® was invented by Japanese teacher Tetsuya Miyamoto and introduced in the United States by NexToy, Inc.
Professors Dani Novak and David Rosenthal of Ithaca College have created a similar puzzle called OkiDoku and used it to make learning math an enjoyable experience. Several other grid puzzles appear at a popular puzzle website called ConceptIsPuzzles. On the average, a whopping 20 million puzzles developed by this website are solved every day by adults and kids around the world. Clearly, there are many people who choose to solve grid puzzles as a recreational activity.

Motivation to Learn

Because many students enjoy working on these puzzles, they can be easily motivated to adopt learning strategies that will improve their puzzle-solving skills. Grid number puzzles provide strong intrinsic motivation to solve for unknown numbers from a handful of clues. As many math problems have a similar form, students who enjoy solving these puzzles can develop positive attitudes toward other forms of math in non-puzzle contexts as well. I have taught puzzle math to students in grades three to six. In these classes, I have found that students show a significant positive change in their attitudes toward math after a year of recreational math activities. Professor Harold Reiter of University of North Carolina, along with Professor Novak of Ithaca, also found that teaching puzzle math has a positive influence on student attitudes towards math.

Educational Value of Puzzles

Even without supervision, students can learn to be creative and persistent after working on many hard grid puzzles. In my classes, I augment self-exploration of grid puzzles with a guided exploration that teaches problem-solving, reflective learning and algebra techniques.
Let me illustrate this with the puzzle shown at the beginning of this post. Even though we can use the creative problem-solving and logical reasoning approaches that we usually use in solving puzzles, we can progress a bit faster on the above puzzle with some help from algebra.
Suppose the top number in the 11+ cage is x and the bottom number in the same cage is y.
We know that each row has to have the same four numbers. So the product of all numbers in each row is the same. As the product of first three numbers in the first row is given to be 20 and the fourth number is x, the product of all numbers in the first row is 20x. As the product of the first three numbers in the second row is given to be 35 and the fourth number is y, the product of all numbers in the second row is 35y. As the product of all numbers in the first row is the same as the product of all numbers in the second row, we know that 20x= 35y. We have also been given the clue in the 11+ cage that x + y = 11.
So what we have is a set of equations:
20x = 35y
x + y = 11
This is now an algebra problem. The use of algebra to solve a problem that students are deeply engaged in allows them to appreciate the power of algebra and also provides a strong motivation to study algebra.
I have taught a yearlong extracurricular class that covers a variety of math problem-solving techniques and algebra in the context of grid puzzles. These techniques include:
  • Making a table
  • Divisibility rules
  • Multiplication tricks
  • Sets
  • Venn Diagrams
  • Factors
  • Logical charts
  • Logical reasoning
  • Working backward
  • Arithmetic sequences
  • Case-based reasoning
  • Algebra
I have written three books that can be used to learn problem solving with grid puzzles. Additional techniques on grid puzzles are described at the Math Olympiad site.

Effectiveness of Puzzle Math in Improving Student Performance

In my classes, I teach math problem solving in the context of puzzles, letting students practice their skills with puzzles and finally encouraging them to ensure their mastery by applying their skills in other math word problems. I have found that students in my class show remarkable improvement in their math problem-solving and learning abilities. However, skill improvement among students varies significantly depending on other factors such as the amount of effort students put into doing homework, as well as student aptitude. In a national Noetic math contest, a majority of students in my puzzle math class showed significant improvement in math skills. Two students were perfect scorers, and a significant percentage of students won national recognition.
While puzzle math is valuable for achieving certain educational objectives, it may not be the best choice in all educational situations. For example, if students and teachers have short-term objectives to achieve the best results on particular tests with specific types of questions, then practicing questions similar to those on the tests can help them achieve those objectives -- not the approach described above. Similarly, teachers may want to emphasize the relationships of math concepts to real-world applications. Again, puzzle math would not support this objective.
In short, teachers will find the puzzle math approach useful to help achieve certain objectives but not others. However, there is no question that teaching math problem solving in the context of grid puzzles, long regarded as a recreational activity, is effective in cultivating students' interest in math and in improving their problem-solving, reflective learning and algebra skills.

Wednesday, August 21, 2013

3 Ways to Not Screw Up the Curriculum Mapping Process

3 Ways To Not Screw Up The Curriculum Mapping Process (view the original source here)

The school year isn’t a series of sprints, but the way you forge your curriculum can make it feel that way.

The most common way of structuring how you teach is by first assembling standards into units, then those units into lessons.

You may use a backwards-design process (popularized by UbD and Grant Wiggins), where you start with what you want the students to understand, then decide what can act as evidence of that understanding, then finally design an assessment that provides the best opportunity to uncover what students know.

This is an entirely rational response to the huge workload of surveying, harvesting, bundling, and distributing the breadth of academic standards in your content area.

This is especially true if that content area is English-Language Arts, which has no less than six separate sets of standards in Common Core menu. That means dozens and dozens of standards ranging in complexity from “spell correctly” to “Analyze multiple interpretations of a story, drama, or poem (e.g., recorded or live production of a play or recorded novel or poetry), evaluating how each version interprets the source text. (Include at least one play by Shakespeare and one play by an American dramatist.)”

Organizing makes sense. Sequencing is a type of organization—thus Scopes and Sequences and Curriculum Maps based on given academic standards. These documents function in a lot of ways, primarily in ensuring all of the content gets covered, and creating the possibility of a common experience for teachers so that data and instructional resources can be shared.


The Problem

The challenge comes when that organization creates artificial barriers and awkward pathways through content. In pursuit of “getting through it all,” it’s easy to encourage bad thinking habits, and worse, provide misleading data about what students actually understand.

Imagine for a moment each of your units. Whether you use genre-based units (a “poetry” unit, an “linear equations” unit, etc.), thematic units (where learning experiences are planned around themes and thematic questions), project-based learning, or some mix of these approaches and others, not all content is equally important.

So how can you keep from screwing up your curriculum map? Or revise the one you were thoughtlessly handed 3 days before school started, taking lemons and making lemonade?

1. Prioritize

Recently, the concept of “power standards” has surfaced, recognition that not all academic standards are created equal. Some standards are naturally more interesting than others, more enduring than others, more complex, or can be leveraged to aggregate other standards and related content together. For example, in English-Language Arts, a standard involving explicit and implicit themes might be considered a power standard due to its ability to involve other standards or content, including: author purpose, audience awareness, tone, text structure, theme vs thesis, and others.

2. Use Spiraling

It makes sense then that if certain content is “more important”—for any number of reasons—that content should be “spiraled.” We’ll get more into spiraling in a separate post, but essentially spiraling in curriculum is the process of embedding critical content throughout the year. This usually means that at the beginning of the year, this content is delivered at lower levels of Bloom’s taxonomy. So if we take the aforementioned example of “theme,” in August theme will be defined, examples will be given, and early analysis will occur. By mid-year, students will be analyzing themes of simple texts more closely, and begin analyzing the themes of more complex texts (and digital media), and by March, students will be analyzing complex themes of complex texts.

3. Diversify Assessment Forms

The process of responding to the personalized learning needs of your students likely begins with assessment. Even if you don’t differentiate the content, processes, or products of learning, simply altering how you assess understanding can go a long way towards truly personalized learning for students.

Whether you simply offer students choice in assessment forms—multiple choice versus concept-mapping, short response versus student conferencing, an exit slip versus a journal entry—the more variety and choice you can build into assessment, the better you can protect students from the problematic curriculum mapping practices than can sabotage student academic performance.
Testing is not engaging, but assessment can be.


In seeking to cover all of the content, do all that you can to avoid the phenomenon of sprinting through unit after unit. The content that will serve both you and students most powerfully likely needs to be prioritize and spiraled throughout the academic year.

If you can also diversify assessment in your classroom—offering student voice, choice, and the ability for students to prove what they understand—and the depth at which they understand—in a variety of ways, the better the chance you are able to meaningfully respond to the personalized learning needs of your students.