What were many of us MOST looking forward to for Christmas? Food, Fun and Family… Fun in our family over the years has been board games, cards and dice games. And then new health orders came to Saskatchewan and that hope seemed to be dashed… BUT… there was a huge ‘aha’ when I was asked to design and facilitate a “Let’s Play: Math Games Online workshop” this past week. I put my research hat on and found some online platforms for play. There are SO many cool fun platforms that are FREE to everyone. There will be Cribbage for Christmas after all!!

I have always believed that my confidence and competence in math came from the games I played as a child. Cribbage with my Grandpa and Great Grandpa, Monopoly with my friend Tommy, Go Fish with my sister, 31 with my Mom and Dad… So, how might we play these games online with Jori and Michael in Martensville, Erin in Comox, and Adam in Saskatoon? And better yet, how do we play them with the kids’ cousins Paige in Halifax, Amanda and Christian in Winnipeg, and then there is Auntie Sandra in Melfort and Grandma Carol down in Mesa this Christmas? My family is FAR too large to list everyone here, but you get the idea. Here are some games and platforms that might bring some cheer to your homes and families over the holidays.

Each of the following does NOT require an account or sign in. For each, you:

Start a game.

Copy the link and send it out by text or email to your game friends.

Set up a phone call or video call so that you can talk to everyone in the game. (Did you know that your iPhone can create a group call up to 5 people??) Or use Zoom, Google Meet, Facebook Rooms, What’s App… SO many possibilities. If two of you are in the same house on your own devices, consider either using your phone on speaker, or each of you use headphones/ear buds so you don’t get sound feedback.

Each of these is designed for each player to have their own device – they seem to work on computers or tablet/iPad or smart phone.

Our conversations in mathematics teaching are often centered
around the gaps that we observe in student understanding, or how students are
not ready to learn the grade level mathematics that we are trying to teach
them. When we look at our teaching practices in other subjects, we know that it
is important to Activate and Connect Prior Knowledge, and to provide responsive
instruction if there are skills that our students are missing. The same holds
true in mathematics, but how do we do this in a structured, systematic and
efficient way in our classrooms? The Assess-Respond-Instruct framework, developed
with and implemented by teachers from across Saskatchewan, does exactly that.

Foundationally, the Assess-Respond-Instruct framework
provides opportunities for

teachers and students to know whether students
have an understanding and fluency with prior knowledge, and

filling gaps in knowledge and build fluency, and

engage in grade level mathematics.

In order to embark on this way of teaching, some key questions that drive our planning are:

Differentiation vs Modification

A key idea within mathematics is the difference and similarity between differentiation and modification. Working with a school last week, we brainstormed the following key ideas:

Sometimes, a student needs a modified curriculum because
they are unable to grasp mathematical concepts. This determination is made with
much consultation with the education team, parents, and students. Communication
is key between home and school to ensure that parents understand that their
child is not working towards grade level outcomes. Rather, they are on a
modified curriculum with modified assessment expectations. When a child is
working towards a modified curriculum, it should still be differentiated.
Students need to experience a variety of modes and strategies to help them
achieve their unique learning goals.

The difficulty is when a student or class is inadvertently
experiencing a modified curriculum without the pre-thinking and opportunities
to engage in grade level mathematics. This might look like a child being
identified as ‘not being able’ to add and subtract in grade 4, so they only
work on addition and subtraction when their classmates are working towards
multiplication and division. In this example, the child is not given an
opportunity to engage in grade level outcomes, so the gap in their learning is
even larger the following year.

So what is a possible solution? The Assess-Respond-Instruct Framework!

Designing Pre-Assessments

Pre-assessments should focus on mathematics knowledge that
students need in order to be ready to engage in new, grade-level instruction.

Content to Pre-assess

We can identify the pre-skills necessary for a new unit of
study by mapping curriculum and asking ourselves “What might students know before
this grade to help them understand the content at our grade level?”

P6.1 – Extend understanding of patterns and relations in tables of values and graphs.

P6.2 – Extend understanding of preservation of equality concretely, pictorially, physically, and symbolically.

P6.3 – Extend understanding of patterns and relationships by using expressions and equations involving variables.

In this Grade 6 Saskatchewan example, the blue concepts are grade-level, while the yellow concepts are mathematical ideas that appear in curriculum before Grade 6. By mapping curriculum, you can see that new, grade-level instruction is only one small step beyond what students have experienced in the past.

Analyzing the pre-skills from the example above, we can see that they can be clustered in the following way:

It is important to identify the extent to which students might need to understand a concept. In this Grade 6 example, students need to understand and be fluent in addition with single digit numbers, and subtraction of double digit minus single digit numbers. We would not usually expect students to work with larger numbers when we are solving algebraic equations at this grade level. Even though curriculum has students learn and practice addition and subtraction to 10,000 in Grade 5, we do not need students to use these large numbers in THIS unit of study, so we would not pre-assess or respond to those large numbers.

The content of a pre-assessment for this example unit of study
would include:

Addition of single digit numbers and subtraction
of numbers no larger than 100.

Multiplication of single digit numbers and
division of numbers no larger than 100.

Representing relations, including tables of
values and graphs.

Solving one step equations, including balance scale
representations and missing value equations.

Forms of Pre-assessments

You might have assessment data that you have gathered through school-system pre-assessments, through tools like Pearson’s Numeracy Nets, or you can develop your own simple Pre-assessments.

How might we respond to gaps in pre-skills?

We need to consider both the content and structure that we
are using to respond to gaps in understanding. Many teaching innovations focus
on one or the other. I would suggest that we need to consider both the content
of intervention as well as the process, or structure, that we use to have
students interact with that content.

What is Responsive Content? Differentiating Mathematics Intervention

Too often, our mathematics intervention in upper grades
involves symbolic practice of a topic that a student is unsure of. Rather than
only focussing on symbolic practice, we need to differentiate our intervention
– additional practice worksheets are not enough if a student does not
understand.

What does differentiation look like in mathematics? If we consider NCTM’s ways of representing algebraic ideas, and the Theory of Multiple Intelligences, a simple way to look at differentiation for every math concept might be:

Whether we are looking at responsive instruction or new
instruction, it is important that students are given opportunities to learn new
concepts:

Concretely and visually

Video – this can help auditory learners watch and listen to math concepts

Written explanations – a simple and concise description of that mathematical idea

Game – a way to interact with peers and have mathematical conversations

Practice – to build fluency with foundational math ideas

A planning organizer is helpful in identifying the components that you will have ready for students who need intervention in each topic. The concepts that we focus on for responsive instruction are those identified in our pre-skill analysis of our next unit of instruction. The modes of responsive instruction need to be thought out for each concept, or skill. This provides a robust framework for intervention.

A Classroom Structure – Responsive Stations

There are many ways to structure your classroom to ensure
that your students are receiving the instruction that they need. These might
include classroom routines that focus on readiness skills, or rotational
stations like Daily 3 Math. One innovative structure is Responsive Stations.

Implementation

Once you have the pre-assessment data, students go to those
stations that their data on their pre-assessment indicates that they need.

Some helpful organizational hints include:

Colour coding your boards helps students know where they are heading.

Use a tracking sheet to monitor which pre-skills each student needs to address.

Use stickers as rewards to track what stations have been done.

Use a short post-assessment to determine that a student understands the content.

Use bins of materials at each station to help keep organized.

Include an enrichment station for those students who have pre-skills in place. This enrichment station can include games, additional math topics, and ideas such as creating new videos or games based on math concepts.

Once you have provided opportunities for students to be ready for your grade-level instruction, you can then teach new concepts using rich instructional practices that we know help students understand. Through the year, your class will revisit the same pre-skills over time, as many topics repeat as pre-skills throughout curriculum.

If you are interested in learning more about the Assess-Respond-Instruct Framework for building readiness, or would like to bring professional development to your staff in this area, please contact Terry@johansonconsulting.ca

Place value and number sense are foundational concepts on which others build over the years in mathematics. Some of the big ideas within place value include:

Concept Progression Over Time

In Saskatchewan, our curriculum identifies the following ideas:

In Kindergarten, children learn that counting tells us how many. The whole numbers are in a particular order and there are patterns in the way we say them that help us remember their order.

In Grade 1, children understand place value in individual numbers – they look at 17 as a quantity. We can compare and order numbers.

In Grade 2, children understand that the value of the digit depends on its location or place.

In Grade 3, children consolidate their understanding that the place determines a number’s value.

Ideas for Teaching Place Value

Rekenreks, 5 and 10 Frames

Number sense is a foundation of place value. Relating
numbers to ‘friendly’ 5 and 10 are key ideas that can move children past
counting.

Try This – Use a
rekenrek to show the following:

Representing numbers – how might children use
these tools to represent 7? 3? How do they know?

Quick flash – flash a number of beads on a
rekenrek and have children tell you what the number is. How do they know this
is the number? Are they counting? Or comparing to the ‘friendly’ 5 or 10?

Model numbers in a number string – showing 4,
then 5, then 6. Some children will see the pattern of 1 less than 5, 5, and 1
more than 5. You can then repeat with 3, 5, and 7.

Subitizing is a foundational skill and occurs when children
know that a number of objects is present without counting. Subitizing can occur
with random displays of objects or dots, or patterned dots like you would see
on a dice, dominoes or ten frames.

The hundreds chart is an important tool for children to see
patterns in our number system. There are a number of games and activities that
you can try to emphasize different math ideas.

Try This – There are a number of blogs and vlogs that teachers have created to highlight the 100’s chart. Buggy and Buddy does a good job curating ideas from a number of sources. You can also have children try to find the missing numbers on a 100’s chart to emphasize the patterns in our number system.

Base 10 Blocks

Based 10 blocks are a foundational manipulative to help
children understand our number system.

Try This – Go to Hand2Mind website and scroll down to view the lessons provided. These are organized by grade band so that you can find what might fit your students best. Use the base 10 blocks provided to try to work through some of these lessons

Place Value Misconceptions

Misconceptions can be created by a mis-applied pattern, or
incomplete understanding of number concepts. The following are some place value
misconceptions that occur in Early Years, and some possible instructional strategies
to address them.

Misconception: A number is a number, and does not represent a bundle of 10, 100, 1000 etc. objects regardless of its position in a number.

Example: 1 means one, so when it
is placed in a number 17, it still represents one rather than 10.

What to do about it? Use the concrete to abstract continuum to
represent 17:

Place value blocks or other counters, such as coffee stir sticks.

Misconception: Students represent numbers after 100 as they sound.

Example: Students think that the number after 100 is 1001, then 1002, 1003, etc.

What to do about it: Use a chart that goes beyond 100, have children fill in the next numbers after 100.

Misconception: The student orders numbers based on the value of the digits, instead of place value.

Example: 67>103 because 6 and
7 are bigger than 1 and 2.

What to do about it:Have students represent numbers using base 10 blocks and then write out expressions using > and < when comparing.

What to do about it:Have students show numbers on a number line to see which numbers are further from zero to the right.

Misconception: The student struggles with the teen numbers, as they are different from the pattern in other decades.

Example: Students may say “eleventeen”
or may not understand that 16 is ten and six. They may also think that sixteen
is 61 because we say the number six first.

What to do about it: Christina, The Recovering Traditionalist, has curated a number of games and ideas for addressing how to teach the teens.

Having Fun with Math

Mathematics should be playful, and there are a number of
games that can build fluency in mathematics.

Combo-10

This game allows students to see how numbers fit together to
make 10 using domino-like game pieces. It is for groups of 2 – 4 players.

Try This – Play with at least two people or groups. Each group needs 1 set of dominoes. Lay them face down. Each person/group draws 7. The rest are the draw pile.

The player with the highest double (or most dots if there are no doubles drawn) plays first. A piece can be played if the number of dots on one side of the domino adds to 10 with a domino on the table. Doubles can be laid sideways, allowing more arms to grow.

A wild card is a domino whose dots add to 10. If you play a wild card, you can play twice.

Snap

Snap is a game
played with linking cubes. Each pair receives 10 linking cubes. Players may
want to start with the cubes in a stack, alternating colours:

Try This – One player has a stack of 10 cubes behind their back. ‘Snap
off’ part of the stack and show the part that is remaining to your partner.

The partner tries to guess how many were snapped off and hidden from view. The unknown part is revealed.

Variations:

Using more or fewer blocks in the stack.

Breaking the 10 cubes apart and hiding some of them underneath an opaque glass or container.

Race to 100

The goal of this game
is to get to 100 first without going over.

Try This – Play
the Game

Each player starts at 1. The first player uses a spinner or dice to generate a number. They can move up the 100s chart by their number of tens or ones until one player gets to 100 without going first.

Variations:

Each player gets 6 turns. The closest to 100 without going over wins.

Continue playing until a player lands exactly on 100. If the roll takes them over 100, they lose that turn.

Flyswatter math combines the fun of moving and slapping with
the chance to learn number recognition and solving math problems.

Creating the game board: The game board can be as small or
as large as you would like and include the number range and type of numbers
that you are working with in your classroom.

Try This – Play a
Game with two lines of players. Each line has their own swatter.

Counting: swat the numbers in order – in either direction.

Number recognition: say the number and have learners swat the correct symbol.

Counting and 1:1 correspondence: give a number of counters, blocks, etc – they count and then swat the number.

Addition or subtraction facts: give the fact, swat the correct sum.

Addition facts: give the sum and one addend, swat the missing part.

Skip counting: swat the numbers as they count by 2s, 5s, etc.

Using Technology in Mathematics

Technology can be used to enhance mathematics in a number of
different ways:

Place Value Online Games

As you know, not all online games are created equally!
Sometimes, they are just online worksheet with little engagement. Sheppard
Software is a site that encourage practice through play, including flexible
thinking about place value.

Try This – Try playing one of the place value games, Underline Digit Value, on Sheppard Software.

Interactive Whiteboards

These whiteboards all allow you and students to share
thinking. They can include audio, pictures, and mark ups. Some apps are free,
while others require a subscription.

Try This – Log into one of the interactive whiteboards below that you have not used before. Use the username and password provided on the sticky note!

These interactive manipulatives can be used to explore math
ideas. These tools are web-based and do not require a log in or download.

Try This – Go to the Arrow Cards tool in the “Teaching Tools” at ICT Math. You can show the value of numbers using arrow cards along with either rek-n-reks or base 10 blocks simultaneously. Show the value for 3299. What happens when you add one more ones digit?

QR Code Scavenger Hunt

This teaching idea comes from Kristin Kennedy and is available free on Teachers Pay Teachers. It would be relatively easy to create your own based on this idea.

This blog post is a work in progress! Be sure to come back and visit in a few weeks, as I will be adding to it over time…

It can sometimes feel overwhelming when we look at all of the individual and group needs of our mathematics learners. Building readiness to learn, along with ensuring that we meet the individual needs of students might give us the impression that we need to create an individual lesson plan for each and every person in our classrooms. That sounds exhausting…

But what if we can create structures and use a variety of math instructional strategies within those structures? What if we can create diverse learning experiences that encourage mathematical thinking and growth over key concepts? This is an idea worth investigating!

Our elementary and middle years math curricula in Saskatchewan cover a number of topics, from number to patterns to shape and space and statistics. Ironically, when you look at the skills needed for students to be READY to engage in these grade-level concepts, there are only a handful of pre-skills. These pre-skills are the math concepts that are applied and used in new learning.

For example, when we are learning about adding and subtracting fractions, we need to know about:

addition and subtraction

multiplication and division, multiples and factors

what a fraction is, finding equivalent fractions, improper fractions

So, how do we teach and reteach each of these key concepts in our classrooms? You can find a large number of curated resources in this Google Drive which contains folders of resources

We can also pull out specific concepts and see how they grow. The following concept trajectories were created by a province-wide math leadership group a number of years ago, and show the language, strategies and concepts over time. Each continuum has four instructional strategies listed.

Math fluency is the ability to perform mathematical operations quickly and accurately. Math automaticity with basic facts is part of fluency. John Munroe (2011) indicates that we as learners have a finite amount of working memory. It is important that student working memory is available to learn new math concepts, solve complex problems and think creatively in mathematics, rather than being used to recall basic math facts.

So, how do we promote automaticity of basic math facts without endless worksheets, mad minutes and text-book assignments? Math games, puzzles and routines related to grade level concepts allow for flexibility in thinking, practice and student engagement and fun. The following documents can help us connect practice with curricula:

There has been a significant amount of research in the area of mathematics routines to enhance learning, building automaticity and fluency. There are a number of key resources that are helpful to teachers from grades 1 to 10.

Productive Mathematics Discussion

Margaret Smith (2011) has created a structure for planning for and implementing classroom discussion in a mathematics classroom. Discussion and sharing mathematical thinking is the key to most mathematics routines.

This sequence of thinking can be applied to a teaching strategy, Learner Generated Examples (Crawley, 2010), to create a powerful way of sharing student thinking in mathematics.

Learner Generated Examples:

Brian Crawley, a teacher from Saskatchewan, did extensive research on learner-generated examples. Overlaying the Five Practices, you create a cycle that can be applied to many number routines:

Move through this cycle three times. This allows students to push beyond the knowledge that they find easy to access and move to more and more complex ideas.

Examples of Math Routines

There are several powerful math routines gathered from many Math Routine Resources that can be adapted to different concepts over grade levels. The following is a short list:

The key to math routines influencing math fluency in your classroom is to choose appropriate concepts and numbers related to your curriculum. For example, in the routine “Today’s Number”, if you are in early primary, you might choose a number between 1 and 10. If you are teaching ideas around skip counting, perhaps choosing the number 6. If you are working on perfect squares, perhaps choosing a number like 36.

Math Puzzles

Math puzzles allow learners to explore mathematical ideas and practice thinking flexibly. Well-designed puzzles are engaging and logical, with most of the time focussing on solving the puzzle mathematically. As with routines, puzzles can be adapted to the range of numbers and concepts appropriate for a given grade level.

Extensions can include puzzles that go above the 1-10 line and below the 91-100 line on a hundreds chart. What numbers exist below 0? What numbers exist above 100?

Place value puzzles can help students overcome misconceptions. They were originally created by Naoki Inaba from Japan, and then adapted to decimal numbers on the “Embrace the Challenge” blog.

When choosing and setting up games in your classroom, along with considering the math concepts you are emphasizing, it is also important to consider student grouping and classroom norms.

By creating homogeneous groupings, it is possible that some groups will use tools to help them calculate or identify numbers, such as:

In heterogeneous math game groupings, it potentially frustrating for both the stronger student and student who needs more time to compute or recall math facts.

There are many excellent math games sites that include Math online games. As well, there are many games that require few materials and are engaging for students.

The key to finding and using math games in your classroom is to know the math that you would like to build fluency in, and then search for those concepts. There are literally thousands of great sites that you can access both online and paper copy games.