Archive for the ‘math’ Category

investigating numbers with the Kindergarten class at Garden City

Posted on: March 20th, 2017 by jnovakowski No Comments

I visited the kindergarten class at Garden City twice over January and February, introducing different routines to develop number sense and to investigate numbers. Teacher Lori Williams had initially asked me to come to her class to introduce counting collections to her students and after that lesson, I suggested some other routines or practices she might try with her students.

To introduce counting collections to the class, the students and I sat in a circle together. The class’ “special helper” and I counted a collection in different ways, taking suggestions from the suggestions. I intentionally modelled working together as a “team” – taking turns, taking on different roles (one of us moving the items, the other counting, etc) and having each of us support each other when we were unsure or “stuck”. We counted a collection by 1s in different ways – each of us placing an item in a container taking turns while counting, putting the items in a line and counting them together, moving the items from one pile to another taking turns counting as we moved the items one by one. I asked the students if they could think of any other ways they might count their collections and they had some new ideas as well as some suggesting that they count by 2s or 10s. Pairs of students then went off to choose a collection to count, with the expectation that they count it in at least two or three different ways.

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The students and I came together after about 30 minutes of counting and I invited some pairs of students to share what they counted and how they counted their collections. I encouraged the students to listen and make connections in their mind as to how they had counted their collections.

For my next visit, I introduced the clothesline and explained that it was another way to investigate counting, particularly ordering numbers.

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The students took turns placing different representations of numbers on the clothesline – they were asked to explain their placement decisions. We followed this routine with an invitation to investigate ordering and sequencing numbers using a variety of materials.

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The students were creative in their use of materials and the inspiration of the materials often nudged them beyond their familiar counting range and what the curricular expectation are for kindergarten in BC (number concepts, including counting from 0-10).

For the classroom teacher, this was a time to notice her students engaging in new routines with different materials and to think about how she might incorporate them into her classroom. It is always a conundrum for kindergarten teachers – there are always more materials to add to the classroom but we also have to let things go and put things away, even if temporarily, to create open access to the materials students will use regularly and purposefully.

~Janice

investigating patterns with the grade 3s at Garden City

Posted on: March 18th, 2017 by jnovakowski No Comments

I visited the grade three class at Garden City Elementary twice in February, focusing on ways to teach mathematics through the big ideas in BC’s new curriculum.

Teacher Stella Santiago asked that we do some work around patterning together and we began with a class discussion around the question: What makes a pattern a pattern? The students shared their developing ideas about patterns, which included many examples of patterns, and then the students were provided with a choice of materials with which to investigate different types of pattens with. We asked them to push their thinking about what patterns were and to investigate different types of patterns and what makes them patterns.

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The students mostly began with repeating patterns but used different formats for their patterns such as going around the circumference of a table instead of just in a straight line. Blank grids were provided for students to investigate and some students engaged with those.

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Students also explored using different shape frameworks to create patterns with.

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And they played with seeing patterns in three-dimensions, from different perspectives.

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About half an hour in to the work with materials, we asked the students to pause and walk around and notice what other students were doing – to be inspired, to capture an idea, to make connections.

As the students created their patterns, I recorded the questions I was asking them during their investigations, meant to provoke their thinking about what makes a pattern a pattern.

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It was these questions that I came back to in the end as students shared their findings about patterns. We were able to come to consensus as a group that a pattern is predictable and generalizable, that there is regularity in it. Big words for a big mathematical idea.

On my next visit to the class we connected the idea of patterns to the students’ current study of multiplication. We began with a number talk, using a grid to support students in visualizing the patterns in multiplication. We played with the idea of decomposing numbers to support us in calculating multiplication questions. After our number talk, the students were provided a choice of materials and tools with which to investigate the focus question: Where do patterns live in multiplication?

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One of the tools provided to students was a 100 chart and students used gems to cover multiples of different numbers on the chart to investigate what visual patterns might emerge.

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Some students started with their understanding of multiplication – equal groupings of objects and then used the materials to create different visual patterns with these groupings.

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Some students connected the idea of growing patterns with multiplication and used different materials to represent this connection.

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The students were very curious about the geoboards, as they are not a regular item in their classroom. One student wasn’t sure where to start with the geoboard and I showed him how to stretch a band to make a square. He made another square and then I added a third. By then, he was making his own connections and began an investigation of square numbers.

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Another student used the geoboard to create arrays. She played with the idea of halving and doubling the arrays, including using diagonals to create triangles.

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A group of three students chose to use the magnetic grids to play with patterns and multiplication, using alternating colour patterns.

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The students again had an opportunity to walk around the class and see what other students were doing to investigate patterns and multiplication and then go back to their own materials, adding new ideas if inspired to do so.

We met together as a class in a “math congress” to report out our findings and make connections between what we had found out about patterns and multiplication. Students used the terms growing and increasing but identified the regular-ness of the patterns involved in multiplication. Some students focused more on the spatial relationship of arrays and how those change as factors increase or decrease.

As the students continue to study multiplication and division, I am looking forward to hearing what relationships and patterns they find.

~Janice

investigating mathematical big ideas at Hamilton

Posted on: March 9th, 2017 by jnovakowski No Comments

In January, I spent some time in the two grades 4 & 5 classrooms at Hamilton Elementary. Coverage was provided to teachers so that they could observe and take part in math lessons in another teacher’s classroom. Teacher were then able to teach this lesson to their own classes, having seen and heard how another class responded and thus, anticipating and planning for their own students. This form of “adapted lesson study” is a common structure we use in professional learning in our district, with time to plan together, observe and discuss and then enact and debrief. The teachers at Hamilton had requested a focus on teaching through the big ideas in the curriculum.

For both classes we focused our planning around these big ideas:

Development of computational fluency and multiplicative thinking requires analysis of patterns and relations in multiplication and division.

Computational fluency and flexibility with numbers extend to operations with larger (multi-digit) numbers.

To focus the students’ thinking, connection-making and our discussions, the question we posed for the students to investigate was: What is the relationship between multiplication and division?

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For each class we began with a game, to activate students’ thinking, get them talking about mathematics and to practice computing multiplication facts. In one class we played Product Gameboard and in the other, the card game Salute.

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IMG_9770After discussing the strategies the students used in each game, a problem was introduced to each class. Both of the problems were taken from the book: Good Uestions for Math Teaching. Using different strategies, I facilities meaning-making of the problems with the students and then the students began to engage in problem-solving. They had an opportunity to “turn and talk” and share their strategies and were encouraged to approach the problem in different ways.

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The students all began with the same problem but could adjust the number of students in the school (in the problem context) they were working with. They used whiteboard to show their different approaches to solving the problem.

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As students shared their solutions and strategies, we asked the students to listen to each other and build on or connecting to each others’ thinking as part of the discourse.

In the second class, a related problem was presented.

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What was interesting to notice as students engaged with this problem is that none of the students paid attention to the “four grades”- it was not required information to work through the problem but would have added an extra layer of complexity and context. We did pause near the end of our time together and this was pointed out, and if I had been with the class the next day, I might have had them re-visit this problem, being mindful of the “four grades” context.

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What students did pay attention to in terms of sense-making for this problem was the types of sports students might be playing and the number of students that would make sense for each team. The students found a context (tennis) that made sense of having one person per team and two per team (doubles). The students shared their different solutions on the large whiteboard which we used as a starting point to compare and contrast their different solutions and strategies and have the students make connections to how both multiplication and division are related and could be used to engage with both of the problems posed to the classes.

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Teaching through the big ideas was also a topic of conversation during an afternoon of Hamilton’s professional development day in January. We will be continuing our conversation at Hamilton’s pro-d day in May and continue to think about ways to nurture ways for students to make connections between mathematical concepts and strategies.

~Janice

making pentominoes

Posted on: February 27th, 2017 by jnovakowski No Comments

Many of our schools have the brightly coloured flat plastic pentominoes tucked away in storage cupboards. I have always like pentominoes due to their affordances for puzzles, problem-solving and spatial reasoning.

One thing I’ve noticed is that they have not been particularly appealing to our younger learners and I wondered if it was because of their two-dimensionality. I thought something that was more tactile for them and that they could build with might be more appealing.

When I saw these cubes at the dollar store, my mind went to building pentominoes based on a task I had read about in the Taking Shape book by Joan Moss et al.

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How many different ways can 3 blocks be put together, with all edges and faces flush? How will you know if you have found all the ways? Are two objects congruent if their orientation is different?

With 4 blocks?

With 5 blocks? (pentominoes)

I have done this task with teachers, both as part of our BCAMT Reggio-Inspired Mathematics Collaborative Inquiry Project and as part of a session on the mathematics curricular competencies at our district conference.

Teachers have found this task touches upon so many areas in our curriculum – spatial reasoning, geometry, problem-solving, visualizing, reasoning and analyzing, communicating, etc.

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As an extension to the building task, using the little wood cubes, I glued sets of pentominoes together, using an image of the 12 pentominoes I found online to help me as a guide. I also left lots of blocks not glued to be used for building the different arrangements.

Pro-tip – don’t use liquid white glue on the coloured blocks….the dye runs and you’ll have a mess on your hands and elsewhere!

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One of the unique aspects of pentominoes is that they are able to fit together to form various rectangles. How many different rectangles can you make using some or all of the pentominoes?

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~Janice

elementary math focus afternoon 2017

Posted on: January 17th, 2017 by jnovakowski No Comments

We hosted this year’s Elementary Math Focus Afternoon on January 16 at Byng Elementary. Over 250 educators attended, from 14 schools.

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There were a range of sessions to choose from and a huge thank you goes out to all the teacher facilitators who shared with their colleagues. A special thank you to our colleagues from Surrey and Delta who shared with us.

Elementary Math Focus Afternoon Jan 16 2017 program FINAL updated Jan 13

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Rebeca Rubio shared some of the many math resources and kits from the District Resource Centre.

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Tracy, from the Canadian Federation for Economic Education, shared resources to support the financial literacy component of the math curriculum, particularly around the Talk With Our Kids About Money initiative.

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The schools attending each contributed a display of materials, documentation or resources sharing an area of professional inquiry amongst their staffs.

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QR code Math Tags were available with links to IGNITE videos, websites and blogs.

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Math Tags 2017

 

General Handouts:

BC K-5 Mathematics Big Ideas (one pager per grade)

BC 6&7 Mathematics Big Ideas

K-5 Math Connections between Core and Curricular Competencies

6-9 Math Connections between Core and Curricular Competencies

The Sum What Dice Game Jan2013

Product GameBoard

BCFinancialLiteracyResourcesShared

 

Session Handouts:

Fred Harwood Grid Algebra 1

Fred Harwood Grid Algebra 2

Barker & Schwartz Picture Books Math & Literacy

Bebluk & Blaschuk Formative Assessment

High-Yield Routines September 2015

Linear Measurement final  from Marie Thom’s K-2 Measurement session

Primary Math Routines (Carrusca, Wozney, Ververgaert)

DST Formative Assessment for All

Jacob Martens Numeracy Competencies Presentation

Sentence Frames for Math ELL

ELL Tier 2 words poster

Carrie Bourne Mental Math Poster – Faire 10

Carrie Bourne mental math poster – valeur de position

(contact Carrie for more Mental Math Strategy posters en francais)

MIchelle Hikida Grades 1-4 Mathematical Inquiry

Michelle Hikida Symmetry

Sandra Ball’s Power of Ten Frames presentation and handout

 

A big thank you to the Byng staff for hosting and to all the facilitators for sharing their experiences and inspiring their colleagues in their sessions.

~Janice

geometry tiles

Posted on: January 5th, 2017 by jnovakowski No Comments

Inspired by a post on Christopher Danielson’s (yes, the author of the book and teacher resource Which One Doesn’t Belong?) blog called Talking Math With Your Kids, I created a set of geometry tiles. Always up to a crafty challenge, I thought…hmmm, I could make those! Christopher has created a one-pager of instructions as part of his Math on a Stick project for the Minnesota State Fair.

I found some balsa slats at Michael’s (teachers get a discount with a teacher card, just ask) and cut them to a 2 to 1 ratio which fortunately, gave me a set of smaller tiles of the same proportions.

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I painted front and back sides and edges with diluted acrylic paint although they could also be left plain. I marked the midpoint of one long side with a sharpie and used regular adhesive tape to tape off from the midpoint to each corner.

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I used black acrylic paint to paint in the triangles creating by the taping.

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And when the tiles were dry, I played around with what I could create with them. So much composing and decomposing of shapes!

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I used these geometry tiles as part of our mathematical tablescape at our Provincial Numeracy Project meetings to oohs and aahs.

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I’m looking forward to seeing how some students investigate these tiles!

~Janice

 

 

 

looking for math outdoors

Posted on: January 4th, 2017 by jnovakowski No Comments

During my last visit of the year to the Kindergarten classes at General Currie Elementary, it was a snowy and icy day so we decided to venture outdoors with some iPads to capture images of things that inspired our mathematical thinking. We had a quick talk with the students about how to look for math outdoors – looking up, looking down, looking all around. We talked about what math might look like outdoors – the counting of items, the shape of things, patterns in the environment, as well as sources of inspiration for thinking about math.

One of the first mathematical ideas we played with was shadows – how does your position affect your shadow? what determines the height of your shadow? what do we need to think about if we wanted to put our shadows in height order?

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As with the case of all our school sites…there is a story that lives there. General Currie was one of the first one room school houses on what was originally called Lulu Island. We stopped briefly at the historic building that is still on the new school’s site and talked about the time elapsed – what school might have been like, what the neighbourhood might have looked like, etc.

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We ventured on to the field and took photos as we walking along noticing nests in trees, tracks in the snow, all sorts of ice and frozen leaves.

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The ice was a source of fascination and many questions for the students. They were also very interested in some footprints they found and wondered about the size of different footprints or tracks.

We came back into the classroom and the students used the app Skitch with one of the photographs they took. They labelled, circled or used arrows to show where they noticed math or what inspired a mathematical problem or question.

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Classroom teacher Kelly Shuto then showed some of the students “skitches” to the class to inspire further questions.

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The following week Kelly tweeted out about the class photo book they had created, based on the idea “What math lives here?”

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In this crisp wintery weather, what will your students notice outdoors? What math lives in the frozen puddles and tracks through the snow? How far do animals need to travel to find food? What might your students wonder about?

~Janice

extending counting collections

Posted on: January 3rd, 2017 by jnovakowski No Comments

Counting collections has become a regular mathematics routines in many of our classrooms in Richmond. One of the important aspects of a routine is that students have opportunities to revisit and extend their mathematical thinking experienced through the routine over time. Although counting collections are kept “fresh” for the students over the year by introducing new materials to count, teachers have been asking me for ideas for other ways to use the counting collections they have accumulated. Inspired by a tweet about a blog post by Tracy Johnston Zager and a personal passion around the importance of problem-posing, I want to encourage teachers to create opportunities for students to pose mathematical problems, inspired by counting collections.

img_9163 I visited the grades 1&2 class at Garden City Elementary again at the end of November. Since my last visit, the class had continued to engage with counting collections and I talked to Cheryl Burian, the classroom teacher, about extending counting collections with problem posing. I read the book Cookie Fiasco with the class (from a  great new series of Elephant and Piggie books) in which some animal friends find different ways to share some cookies. During the story, we paused and considered what new problem emerged and the students discussed different ways to solve the problem. I explained that this sharing context was one type of math problem that could be considered with a collection of some sort.

The students counted some collections and recorded their counts on a math graffiti board or chart. During our debrief, we spent some time analyzing the counts.

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The students then chose one of their collections to inspire a math problem to be solved by their classmates. The gold pirate coins were a common source of inspiration and many students also drew upon the sharing context from the story to inspire their problems.

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Another way to play with the idea of counting collections is to see the “units” counted in different ways. The grades 3&4 class at Grauer Elementary counted some new collections the day before the holidays (which was also pyjama day at the school – just to explain some of the photographs!). The class has been learning about multiplication and thinking about different ways to represent the concept of multiplication such as in grouping and arrays. Although the students used the term “skip counting”, I introduced the term multiples to them. The question I asked them to focus on as they engaged in counting collections was: “What is the connection between counting and multiplication?”

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One pair of students decided to practice their 7x tables (their words…) and so grouped their glitter balls in groups of 7 on paper plates. As they began to count, they  noticed since they had organized their plates in two rows that they could visually see a ten-frame and decomposed their total number of plates into a group of ten and then a four. The video below has them explaining their thinking.

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Counting by 7s

Some of the collections I brought to the class were specifically curated to inspire students to think about multiples. I had bought several strands of holiday beaded garland and cut them into groups of 2, 3, 4, 5 etc beads. I anticipated that the students would either count them by 1s (each strand) or by multiples (the number of beads in each strand).

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Here are two videos of students’ counting of the collections:

Counting by 5s

 Counting by 10s

Other pre-grouped collections that could be used are items like packages of crayons (I have seen them in 8s and 12s) or geometric shapes where the number of sides could be counted as multiples. In both cases, the item (or shape) can be counted as a singular unit or a multiple unit, creating different entry points for students as they engage in counting collections.

Richmond teachers (and others!) – if you give one of these ways to extend counting collections a try with your students, let me know and send some photos and insights along!

~Janice

introducing clothesline to the kindergarten students at General Currie

Posted on: November 29th, 2016 by jnovakowski 1 Comment

Last Tuesday, I made another visit to the kindergarten classrooms at General Currie Elementary. During each visit I introduce a new mathematical “routine” to the students and teachers and then extend the routine with some related learning experiences.

I introduced the “clothesline” introduced to me via Twitter by Andrew Stadel last year. There is a website dedicated to sharing information about clothesline math HERE. Most of the work I have seen done with the clothesline is at the middle school level and I can see great uses for it in exploring equivalent fractions, decimal fractions and percentages with our intermediate students. In looking at the kindergarten mathematics curriculum  for BC, sequencing and representing numbers from 0-10 is an important learning standard and connects to the use of the clothesline, a form of interactive numberline.

We began with just the numeral cards and the students came up on a a time (in random order) to place their cards on the clothesline. They were asked to state their reasoning for why they put their cards where they did.

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After the 0-10 cards were in place, we took them off and then I shuffled them with the ten frame and tally cards and handed one card out to each student. Again, the students came up one or two or three at a time and placed their cards, explaining their reasoning. When there was an equivalent representation already in place, they just placed the card on top of the other.

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The tent cards I created can be downloaded here:

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When I asked the first class of kindergarten students one way of showing “seven”, one little guy held up seven fingers. I hope to take some photos of the students finger combinations next week when I visit to include these on a set of cards.

I can also see great potential for the clothesline to look at multiple representations of numbers in grades 2-5 to help students think about place value.

After each class worked with the clothesline, the students could choose from several related learning experiences, all that focused on sequencing numbers or representing quantities to 10.

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The students were highly engaged with the materials and were able to share their thinking about why it was important to know how to order numbers –  ”to count, to be organized”. In one of the kindergarten classes we looked around the classroom for ways that numbers in order or sequence were used. The students found the 100-chart, the calendar and the clock.

Next week, we are going to do some number talks with dot cards and ten frame  cards and investigate the idea of parts-whole relationships in numbers by decomposing and composing quantities.

~Janice

what does it mean to be a “low” math student?

Posted on: November 23rd, 2016 by jnovakowski 2 Comments

So typically on this blog I share stories of what is happening in Richmond classrooms and about professional learning experiences for Richmond educators. This post takes a different tone…one that I hope will provoke thinking and discussions about the intersection of language and students and math.

Here goes…

I am often engaged in conversations about mathematics teaching and learning where I hear from teachers, “I have so many low students,” and it makes me wonder what is meant by “low”. I am sure I have used the term myself in the past but I have been increasingly more aware of the impact of labels and language on not just the professional conversations we have but also on how this impacts our relationships with our students. I have begun to challenge teachers on their use of this term and stop them as they say it…”What exactly do you mean when you say ‘low’?” I don’t mean to put teachers on the spot or to to make them feel uncomfortable in our conversations but I think the language we use in conversations about students is really important and we need to be mindful about this.

My prickliness about how we talk about children was amplified when I had my own children, both of whom have their own personal strengths and stretches. I can’t imagine how I would feel, or how my sons would feel, if they were ever described as “low”. What impact does this language of  ”low” have on our students as learners and on ourselves in our role of teacher? How does this thinking affect our mindset about learning?

So what does it mean to be a “low” math student…

Does it mean that the student does not have an understanding of foundational concepts in mathematics? Did the student not have access to teaching at his or her just right level? Was the student absent from school or ill for extended periods of time? Was the student not assessed thoroughly to inform instruction? How can the student be supported to gain foundational concepts and confidence in mathematics? What structures are in place in your class and in your school to support core foundational understanding in mathematics?

Does it mean that the student has difficulty learning math because of memory, health, attention, behaviour or learning difficulties? When in class, does the student have difficulty paying attention, focusing, sitting? Does the student seem unable to retain information the way it is being provided? Does the student have behaviours that are affecting his or her learning and engagement? What practices, materials and structures are in place in your classroom or school that provide choices and adaptations in time/pacing, materials, place/learning environment, quantity of work output expected and depth of content knowledge?

Does it mean that the student has a different story than his or her classmates? Has the student had breakfast? slept? Is the student living in a safe home environment? Does the student have to care for siblings or parents? Does the student need to work to add to the family income? Does the student have regular absences? Why is that?  What might be affecting his or her image of self as a learner and as community member in your classroom? As teachers, are we acknowledging and checking our place of privilege and power and how this might be affecting our students? What is the student’s story and how might this be affecting his or her learning of mathematics? What supports does this student in your classroom and school need to be successful?

Does it mean that the student does not have access to resources to support learning and success at school? Does the student have the tools and resources (human and physical) he or she needs at home to support learning? Are assignments and studying accessible and equitable for all students regardless of their home or financial situations? What supports can the teacher and school provide so all students have equitable access to the resources needed to support their learning? Afterschool homework clubs or peer tutoring? Choices in assignment and homework formats?

Does it mean that the student’s written work, homework and quiz and test scores do not indicate achievement of learning standards? Is written work or practice not completed during class time? Are homework assignments not turned in or completed, or attempted? Does the student seem to understand the mathematics during performance tasks and class discussions but is not successful on quizzes and tests? What different opportunities are students provided to communicate their thinking and learning? (It does not have to be written down to “count”!)

In all of the above scenarios, it may seem that I suggest that it is the teachers’ and schools’ responsibility to ensure student success in mathematics. Well, it mostly is – that is our job. Of course we need to have students and parents as part of this story, but when they may not seem to be, we, as a system, need to think about how to bring them alongside instead of using fixed terms such as “low” as an excuse, and explanation or a dismissal of responsibility.

How can we re-frame how we talk about our students and how we talk about learning mathematics?  There is a strong movement in mathematics education coming from various voices including Dr. Jo Boaler of Stanford University. This movement is based on the belief and conviction that ALL children can learn mathematics. Dr. Boaler’s work around mathematical mindsets is shifting how educators, parents and students think about the learning of mathematics. More information can be found here.

I attended a Learning Forward dinner event at the end of April and one of the question prompts the secondary teachers from Surrey gave us to provoke discussion was:

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This issue of deficit language resonates with me and I think by re-framing the language we use will re-frame how we see ourselves as educators and how we see the students in our classrooms.

Inspired by Linda Kaser and Judy Halbert and the four fundamental questions of the NOII, I wonder how many of our students feel that their math teachers believe that they can learn? We know its important that teachers convey that they care for their students and that they believe they can be successful. How does our language need to be re-framed in our classrooms so our students believe this to be true?

Instead of describing our students as “low”, what different language could we use? Learning. Developing. Growing. Competent. Full of promise and potential. How does using strength-based language shift our conversations and interactions with our students and with each other as professionals?

My hope is that we can describe our students as curious and engaged mathematical thinkers and learners – what is the story that needs to unfold in our classrooms if this is our goal?

Math matters. Language matters.

~Janice

With thanks to Faye Brownlie, Shelley Moore, Jane MacMillan, Lisa Schwartz and Sarah Loat for their feedback and contributions to my thinking for this post.