Transdisciplinary Math – An epiphany and a plan!

For the past few weeks I have been helping my teams review their math scope and sequence and decide which math is transdisciplinary and fits within a Unit of Inquiry and what math is better taught in stand-alone units. This process always seems to lead to the same conclusion….

Teaching math in a transdisciplinary way is hard. 

Teachers seem to believe in the purpose and power of teaching math in a relevant and significant context and want to do it… but most seem not too sure about how to do it.

As I get ready to transition back into the classroom in the fall, this is something that has started to occupy my mind as well. How DO you do it? The last time I was a PYP teacher I can self-admit that teaching math within the context of my UOIs was not a strength of mine – in fact, I’m not sure if I did it at all. So naturally, this is an area I want to get much better at. But how? 

And then I had an idea! It hit me this weekend while I was watching BBC’s Africa series.

Since teaching math in a transdisciplinary way was on my mind, I couldn’t help but notice that every vignette was OVERFLOWING with opportunities for math inquiries!

“The average size of a giraffe’s tongue is half a meter.”

“Only one out of 1000 turtles make it to adult hood.”

“One million birds migrate over the Sahara each year.”

“Each chick weighs only 20 grams.”

“The adult grows to be 5 times the size of the baby.”

“Silver ants can only survive in the sun for 1 hour.”

Every few minutes there was some piece of information about an animal or a landscape or a natural phenomenon where you needed to understand the math concept being referenced in order to fully understand what was being said. And that is when it hit me! All of the movies, books, articles, graphics etc. we use in our Units of Inquiry probably already contain opportunities for math – we just need to be looking for them and know what to do with them!

So here is my plan for next year!

Step 1- Introduce a text related to the central idea or the central concepts.

As usual, choose (or invite your students to help choose) a resources to explore the big idea in your current Unit of Inquiry. Introduce the text in an open-ended way. Allow the students to engage with the text in a natural and organic way. Read the book. Watch the movie. Listen to the song. Look at the info graphic. Allow the students to enjoy it and ask questions, make connections and offer thoughts. I’m thinking of using a back channel like Today’s Meet to allow students to communicate their thoughts, reactions and questions with their learning community while watching, listening or looking without interrupting one another. You could also provide post-its so students could record their thinking if a device is not available.

Step 2 – Revisit the text with a math focus

The next day, revisit the same text, but this time let students know that they will be looking at the text as mathematicians. Re-read the book. Re-watch the movie. Re-listen to the song. Re-look at the infographic. But this time, stop and pay specific attention to the “math moments”. If the video says “Giraffes’ tongues are half a meter long” pause the video and ask students, “What does that mean?” “How long is that?” “How can we find out?” “How can we show it?”. Any time a number, a measurement, a statistic, a pattern, or a concept is mentioned stop, point it out and explore it.

Step 3 – Follow where it takes you

When you stop to explore the math within a UOI text, be prepared to follow the inquiry. If it takes 10 minutes great. If it reveals other math concepts, skills and vocabulary that need to be explored first, back up and inquire into those. If your students need to bust out some manipulatives, look online, consult other mathematicians – do it! Allow what ever time is needed to explore and truly understand what the math means in that context.

Step 4 – Don’t stop at math! 

After the initial open-ended viewing and the math-specific viewing… keep going! You could apply the same strategy for many different purposes. Explore the same text a third time with a writer’s lens and hone in on the techniques the writer used. Explore the same text with a musician’s perspective and focus on how different segments of music contribute to the message. Explore the same text from an artist’s point of view to analyze colour, line and shape that was used. This would be a great opportunity to connect with single-subject teachers and share some of the texts with them to be looked at and deconstructed multiple times, in multiple ways, through multiple disciplinary-perspectives. Your whole week could be deconstructing one text in different ways for different purposes!

Eventually, I believe you will be able to get to the stage where instead of telling students “here is the math” when exploring a UOI text, you will be able to ask them “where is the math?”.  I also have the sneaking suspicion that if you allow students to document their thinking during the initial, unstructured exploration of the text there will be some math-related questions that are recorded about the quantities, measurements and statistics that are referenced. So you wouldn’t even need to point out the math, you could allow students’ own questions to be the driving force of the math inquiry.

So I challenge you… go back and look at some of your UOI books, videos, graphics etc and notice the opportunities for “math moments” and more!

How do you explore your UOIs through the discipline of math?

What are your best approaches to inquiring into math within the context of a UOI?

Inquiry Based Math Strategies

During our half day of Personalized Professional Learning, I hosted a workshop on inquiry-based math strategies, but not everyone who wanted to attend could attend… so I thought I’d recap the workshop here for those of you who could not make it – and for those of you at different schools who might be interested in this topic as well.

The structure of the workshop was very hands on, so in the absence of you being able to actually engage with the materials and manipulatives, I will provide a combination of notes, photos, questions and reflections that will hopefully allow you to engage in some of the same ideas, just in a different way.

Tuning in – What do already know?

Think about or jot down your current understanding of each of the inquiry-based math strategies listed below:

• math time capsule
• open ended centers
• magic question
• open-ended questions
• number talks
• math congress
• visible thinking routines
• inquiry cycle

If you have a thorough understanding of each of these strategies, you probably do not need to read on. If you think your current understanding has room to grow, read on!

Open-Ended Centers

I’ve already written a post about open-ended math centers and how they work in our early years classrooms. During the workshop today, each group had a bin with the three essential ingredients of an open-ended math center: manipulatives, writing utensils, and a placemat/whiteboard.

Here are some pictures of how teachers tested out a few open-ended math centers:

The Magic Question

I’ve also written about my favourite inquiry question – What do you notice? In the workshop we looked at how this question can be used for math specifically.

Take a look at this multiplication chart. What do YOU notice?

Open-Ended Questions

Answer this question: Compare the following fractions using < > or =

1/4   ____  1/2

Now answer this question:

What is the same as a half?

Reflect on the difference between answering the first and second question. What are the benefits of asking open-questions in math?

Number Talk

Take a look at the following image. How many dots are there?

How did YOU figure it out? Here is a picture of all the different ways the participants of the workshop figured it out.

Math Congress

Step 1 – Present the problem: A sports store has a number of bicycles and tricycles. There are 60 wheels in total. How many of each kind of bike could there be?

Step 2 – Work towards solving the problem. Markers and chart paper work best!

Step 3 – Share discoveries and strategies with fellow mathematicians. Make sure fellow mathematicians are invited to ask questions, make connections, comments and conjectures!

Visible Thinking Routines:

Use the Visible Thinking Routine “Claim, Support, Question” to share some of your thinking about decimals.

There are also many other Visible Thinking Routines that are helpful in approaching math in an inquiry based way!

Inquiry Cycle:

Kath Murdoch’s inquiry cycle is a great way to make any math more inquiry-based.

A CCSS math standard: Know and apply the properties of integer exponents to generate equivalent numerical expressions.

What do YOU already know about this?

What do YOU need to find out about this?

How could YOU find out about this?

Math Time Capsule:

Now, think about or jot down your understanding of each of the inquiry-based math strategies listed below. A math time capsule is a great way to show growth and progress in math – whether it’s over the course of a unit, a year… or even of the course of a workshop!

• math time capsule
• open ended centers
• magic question
• open-ended questions
• number talks
• math congress
• visible thinking routines
• inquiry cycle

How did your understanding of these strategies grow and change?

In the actual workshop, after each strategy, we took some time to discuss how the strategy could be applied/adapted to different content and different age levels. Too often when we are looking at strategies we are focused on the actual strategy within to confines of the example that is used. This leads to the conclusion that “That doesn’t work for the grade/content that I teach”. Instead, I challenged the participants in the workshop – and I challenge you in the same way – to focus on the essence of each strategy and how that same approach can be used in different ways, for different ages and for different strands of math.

Here are a few examples of how the same strategy can be adapted for different content and different ages:

Math time capsules – In Grade 5 you might give students the summative task on the first day and then again on the last day to show all of the growth and progress they experienced. But in KG, you may conference with a student and voice/video record everything they know about shapes, and then record them again at the end of a unit to capture growth in their understanding.

Magic question –  In KG you might show a ten frame and ask “What do you notice?”. In Grade 2 you might show a hundreds chart and ask “What do you notice?”. In Grade 4 you might show a multiplication chart and ask “What do you notice?”.

Inquiry cycle – In Grade 1 you may use the inquiry cycle to structure a whole class inquiry into measurement. What do we know about measuring objects? What do we want to know about measuring objects? How can we find out more about measuring objects? In Grade 6 you might use the inquiry cycle to structure self-directed, personal inquiries towards calculating volume of 3-d shapes. What do I already know about finding volume of 3-D shapes? What do I still need to find out? How can go about that?

The possibilities are endless. If you focus on the “why” a strategy is effective and “how” a strategy helps foster thinking and exploration… then the “whats” become infinite! I also shared this google doc with some of my favourite inquiry-based math resources (books, blogs and Tweeters!) Feel free to have a look!

What are your favourite inquiry-based math strategies?

How Children Learn Math: Bringing it Altogether

Last year I wrote a post about what the PYP believes about how children learn math:

1. Students construct their own meaning about math
2. Students transfer their meaning into conventional symbols (vocabulary, notations, algorithms)
3. Students apply their understanding to problems and real world contexts

I also shared some examples of how teachers at our school have been helping students to construct their own meaning about mathematical ideas and concepts. A lot of the teachers I work with are feeling confident about that first stage in the math cycle! However, they are still wondering how to bring all 3 of the stages together.

In an attempt to step back and see the big picture of how the stages fit together, our teaching teams generated a list of all the math strategies they use in their math programme.

Once the list was compiled, they asked these three questions:

What strategies give students the opportunity to construct their own meaning?

What strategies help students to transfer meaning into symbols?

What strategies provide students the chance to apply their understanding?

Then, they sorted each strategy into each phase of the math cycle. (Along with some amazing debates, disagreements, discoveries and many references to the PYP Math Scope and Sequence document!) We discovered that many strategies fit multiple stages in the math cycle depending on the question you ask or how you present it. We also spent a good chunk of time discussing how many of the strategies that allow students to construct their own meaning at the beginning of a new unit or new concept, would also be good at the end of the unit to allow students to apply their understanding using conventional symbolic representations.

It is interesting to note that no two teaching team’s chart looked the same. Another point for acknowledging that all learners construct their own meaning in their own way!

Here is the chart our Grade 3 team developed:

Now when we are planning a stand-alone math unit, we have an anchor chart that will help us purposefully select math strategies to support students as they to move through all three stages of the math cycle.

How do you bring the three stages of how children learn math together?

Open-Ended Math Centers

At our school, we strongly believe in the benefits of inquiry, exploration and play based learning – for all of our students, but especially our youngest students in KG (kindergarten). One of the best strategies our teachers implement are open-ended math centers.

Here are a few reasons why we love open-ended math centers:

• open-ended math centers have no start or finish, which means there are never students who are ‘done early’ and never students who need to ‘finish their work’
• open-ended math centers allow students of different abilities to self-differentiate and explore the math concepts and skills they are developmentally ready for
• open-ended math centers allow students to construct their own meaning, collaborate with their peers and engage in authentic conversations about math
• open-ended math centers allow teachers to observe and collect assessment data in a non-threatening, non-stressful environment.

Take a look at some of our open-ended math centers in action. What do you notice?

How many of the following Common Core State Standards for Kindergarten Math are being explored?

How many of these Common Core State Standard Mathematical Practices are being developed?

Here is a sneak peak into how we plan for open-ended math centers:

Manipulatives	Writing Tools	Boards/Placemats	Teacher Questions/Prompts	CCSS
Dot cards (p.34 guide for effective in kindergarten)	white boards markers, pencils	white boards	How many dots are there? Which has more? Has less?	6
Any ( peoples, farm animals, cubes)	white board markets, pencils	white boards, papers, dot cards, stampers, two circle placemats	How many are in this circle? How many are in that circle? Which group has more? Less? How could we make it equal?	6
number line, counters	white boards markers, papers	white boards, papers	How can you show me this number? Can you show me a number bigger than this number? Less than this number?	6
counters	white boards /white board markers	ten frames	What number did you build? How many more do you need to make ___? How do we make ex: 11 ?	3
shape blocks	pencil, markers	paper	What do you notice? What are you drawing? Tell me about that shape? How are these shapes the same?	3
building blocks number cards	playdo		How do the numbers look different? How do they look different? Choose two numbers. One of them is a lot more than the other. What are they and how could you write them?	3

We are always growing in our own understanding of math centers and play-based learning, so we would love your feedback about our open-ended math centres. We would also love to hear about and see what early math learning looks like in your classroom! 

Learning Time Capsules – shifting the focus from achievement to progress

Here is an example of how one of our Grade 4 teachers is shifting his students’ focus from achievement to progress through the use of a math “time capsule”.

Diagnostic: This teacher looked at all the big concepts the Common Core outlined for fractions in Grade 4 and created open-ended questions to allow students to show what they already knew or thought they knew about each big idea. Students were encouraged to be risk-takers and try every question!

The teacher then tracked students’ prior knowledge on an excel sheet. This allowed him to plan full group, small group, guided and individual math inquiries based on needs.

Formative: After a few weeks of inquiring into these fraction concepts, the teacher gave back the same task and highlighted questions that students were required to try (based on the concepts that had been learned over the past few weeks in class). Green meant they showed competent understanding the first time they tried the question (during the diagnostic), but could still show extended understanding if they added to it. Pink meant they had not previously attempted it or showed a developing understanding and would need to add or change their answer. Students were encouraged again to be risk-takers and try the questions that were not highlighted, as their new knowledge and understanding might help them figure out concepts that had not yet been explored as a class.

The teacher then added this formative data to the excel sheet to show the progress each student had made in each area, who was ready for a challenge and who needed more support.

Students were also given the chance to reflect their own understanding of the concepts learned in class so far and indicate which areas they were feeling confident in and which they wish to work on more. The teacher also filled in the same feedback sheet which highlighted his perspective on what the student did well and what they could still practice. This feedback was shared with parents along with recommendations for support at home.

Summative: At the end of the Unit, the teacher gave the same task back and the students were instructed try every question in order to show their final knowledge and understanding. Again the questions were colour coded so students knew which of their answers showed a competent understanding and which answers needed to be added to, changed or attempted. Prior to handing out the time capsule, the class came up with a student generated rubric for each question, indicating what would show a competent or extended answer. Students had access to both an electronic and paper copy of the rubric to help them understand how to be successful at each question

After completing the time capsule, students completed the Visible Thinking Routine “I used to think… Now I think” to reflect on how their thinking about fractions changed from the beginning of the unit to the end of the unit. The time capsule, self-assessment rubric and meta-cognitive reflection were all sent home so students could share their progress with their parents.

When the focus is on achievement, students have no choice but to compare their achievement to the achievement of others. But when you place the importance on progress, students focus on how their knowledge and understanding grows and changes over time. Each time the students added to or changed their time capsule it was a visual representation of how their knowledge and understanding had grown and changed. Each student felt successful in his or her own way because they could see the progress they made over the course of the unit.

How do you help your students focus on progress and growth?

We tuned in!

Before our Units of Inquiry started, grade-level teams inquired into “tuning in” (with the help of  this post from Kath Murdoch). Many teachers walked away with a new, or deeper, understanding of the purpose behind the “tuning in” phase of inquiry. Teachers were excited to put their new learning into practice… here is how it turned out in our Grade 1 to 5 classes:

Grade 1: Peaceful relationships are created through mutual understanding and respect.

Students tuned in to problems and solutions:

Students tuned in to the concept of numbers:

 

Grade 2: Citizens build communities.

Students tuned in to the concepts of “community” and “citizenship”:

Grade 3: Decisions impact conseqeunces.

Students tuned in to “decisions” and “consequences”:

Students shared important decisions they made in their life:

Students tuned in to decisions made by readers:

Students tuned in to the decisions they make as mathematicians:

Students tuned in to the number of decisions they make:

Teachers tuned in to the type of decisions they make:

Grade 4: Relationships are affected by learning about people’s perspectives and communicating our own. 

Students tuned in to the concepts of perspective and relationships:

Students tuned in to different representations of numbers:

Grade 5: Relationships among human body systems contribute to health and survival. 

Students (and teachers) tuned in to the concept of systems:

Students tuned in to what they think they know about body systems:

I love that the students’ thinking is front and centre!

I love that the students’ thinking is visible!

I love that students were able to demonstrate their thinking in a variety of ways!

I love that teachers tuned into conceptual understandings, not just topic knowledge! 

I love that transdisciplinarity is evident!

I love that teachers were acting as inquiries themselves… doing reconnaissance to find out about what their students bring to a Unit of Inquiry!

The feedback from teachers about “tuning in” has been great! Teachers are excited because they have learned about their students’ prior knowledge, their misconceptions, their interests and their questions. It has not only provided them with diagnostic assessment data, but also a road map that illuminates “where to next?” based on students’ needs and interests! I can’t wait to see where these inquires lead!

How do you “tune in” to your students’ thinking?

Classroom Set-Up: How much should we be doing without students?

Every year many teachers spend hours upon hours setting up their classroom to ensure it is picture perfect before the students arrive.

But I wonder, by doing so, are we taking away some great learning opportunities for students? In PYP classrooms, we start the year with blank walls to ensure there is lots of space to display students’ questions and students’ thinking, but what other classroom set-up jobs should we be sharing with students? Involving students in classroom set-up is not only a great way to build a sense of community and send the message that it is our classroom, not my classroom,  but their are also some great opportunities for math, literacy and transdisciplinary skills… if you’re looking for them!

Here is a list of some typical classroom set-up jobs that involve literacy, math and transdisciplinary skills that could be shared with students:

Covering bulletin boards: measurement, surface area, cooperation, problem solving, group decision making, planning skills

How much paper will we need to cover this bulletin board? How can we figure it out? What tools could we use? Is there another way to figure that out? 

 

Bulletin board borders: measurement, perimeter, repeating patterns, adding, multiplying, creativity, planning, organization, fine motor skills

How much border will we need to go around the outside of the bulletin board? How can we figure that out? How wide should the border be? How do we know if we have enough? What designs can we put on the border so it is appealing to the eye? 

 

Sectioning Bulletin Boards: Shape and space, measurement, division, fractions, arrays, problem solving, cooperation, analysis, spatial awareness

How many equal sections do we need? How big will they be? How many rows and columns can there be? How can we be precise? How can we section them off?

 

Arranging desks/tables: equal groups, shape and space, multiplication, division, problem solving, listening, speaking, planning, gross motor skills, safety

How many different ways can we arrange our desks into groups? How many different ways can we arrange our desks into equal groups? How can we set up our tables to maximize the number of chairs that fit around? Which arrangement gives us the most space? 

 

Name tags: Literacy, printing, letter formation, capitalization, non-verbal communication, respecting others, planning, organization

How we can show which cubby belongs to who? Why do we need to label cubbies? What do we need to remember when we write our names? How can we make sure our letter are the proper size and shape?

 

Classroom Library: sorting, genre, organization skills, labelling, counting, adding, estimation, planning, group decision making

How can we organize our books? Is there a different way to organize them? Where should we put them? How should we label them? What are the fancy literacy names for these kinds of books? Where can we find out? How many do we have in total? How will people know where to put them back?

 

Toy Shelves: sorting, labelling, organization, systems, cooperation, making group decisions, planning, speaking, listening,

How can we sort our toys? How can we keep them organized? What should we label each bin? How will students know where to put them back?

 

Student-Made Class Alphabet Strip: Literacy, letter formation, letter sequencing, letter sounds, upper case and lower case, writing, synthesis, fine motor skills, team work

How do we make this letter? What word starts with this letter sound? Which letter comes next?

 

Student-Made Class Number Line: Counting, sequencing, quantity, number formation,writing, synthesis, fine motor skills, team work

How do we make this number? How much is that number worth? What number comes next? What is the name of this number? How do we spell it? How do we make/spell this number in our other language?

 

 

Class Schedule: Measurement, writing time, lapsed time, adding/subtracting/dividing time, fraction, percent, analysis, evaluation, planning, time management

How can we split our classroom time? How can we make a schedule that has x minutes total for literacy/math each week? How long is in between first recess and second recess? How can we show that this class is 45 minutes long? How can we display our schedule?

 

 

As with most things in PYP/inquiry-based teaching it can seem that the teacher’s role is minimal. Quite contrary! In order for a teacher to share the classroom set-up duties with students, there is much thinking, planning, organizing and orchestrating needed on the teacher’s part in order for this to be successful and meaningful to students. Here are a few guiding questions to help with this:

1. Which tasks are appropriate to share with the age of students I teach?

2. Is there purposeful literacy, math or transdisciplinary skills in this task for my students?

3. How can I organize this process? (What materials should I have ready?  How long will it take? How should I split the students in to groups?)

4. What questions can I ask to guide the process and maximize student thinking?

5. Is the juice worth the squeeze? (Do the benefits of having students involved in this task justify the time it will take?) 

To be perfectly honest, I have never tried this myself… but I wish I could have before I left the classroom! I think there are so many authentic literacy and math skills needed to set up a classroom that require social, communication, thinking and research and management skills – both by the students doing them and the teachers planning them!

Have you ever tried this before with your class?

Do you have any advice for teachers trying this for the first time?

What other classroom set-up jobs would you add to the list?

 

Transdisciplinarity. (It’s a word…I think)

Transdisciplinarity.

The Golden Goose of the PYP.

Every PYP teacher’s dream…

“Wouldn’t it be amazing to spend the whole day on our Unit of Inquiry”

“How great would it be to have no stand-alones?”

“I wish all my math and literacy was authentically integrated!”

This past year, our school adopted the Common Core State Standards for Literacy and Math and we have spent the year trying our best to integrate the standards into our PYP.

We survived… and we did a pretty darn good job!

Now, as we begin to wrap up this year and think about next year, our teachers were chomping at the bit to “transdisciplinary-ify” (that ones definitely not a word) our language and math standards further. So we used one of our half day PD sessions to inquire into transdisciplinary learning. 

First our staff did a growing definition to tune into what they thought transdisciplinary learning was all about.

3 minutes to write your own definition on a post-it:

5 minutes to combine your definition with a partner onto a recipe card:

10 minutes to construct a collective definition with your whole teaching team on a sheet of blank paper:

After giving each team a chance to share, we looked at what they PYP says about transdisciplinary learning:

We used this video as a provocation to get teachers thinking about the endless possibilities of transdisciplinary learning:

 

Then we unpacked 3 levels of transdisciplinary connections:

Level 1:
What concepts, knowledge or skills are essential in order to:
– understand the central idea
– inquire into the lines of inquiry
– complete the summative

Level 2:
What concepts, knowledge or skills can enhance…

Level 3:
What concepts, knowledge or skills can be taught
within the context of your Unit of Inquiry?

Finally we gave teams the rest of the afternoon to look at the Common Core State Standards with fresh eyes, through the lens of transdisciplinarity.

The result was amazing! All of our teams ended the day with a more transdisciplinary math and literacy scope and sequence. Some teams even ended up with no stand-alones!

Hopefully these documents will help to guide our thinking more next year about how math and literacy can serve the units of inquiry. It will be interesting to come back and look at these documents at the end of next year and reflect on how we can refine them again to further enhance the opportunities for transdisciplinary learning.

As we often say to our students…once you’re done, you’ve just begun!

Shifting the Culture of Math

What brought 1,015 people back to school last Monday night?

Math!

Math is traditionally a subject that is disliked and oftentimes, feared… by students… by parents… and sometimes even by teachers. This cannot be. If we want to help our students become confident and competent mathematicians, we need school cultures that love and celebrate math.

We want students to love math. We want teachers to love teaching math. We want families to be involved in and value the way we teach math. Our first-ever, Family Math Night was an important step in beginning to achieve that.

Here is a little glimpse of what made our Family Math Night such a success:

We had Grade 4 volunteers keeping “Live Stats” of how many people came from each division.

 

We had an open-ended question for each PYP strand of math that could be answered by anyone… from our youngest students to our teachers and parents!

“What can you do with this data?”

“What can you measure about a watermelon?”

“What is special about the number 10?”

“The answer is 10. What could the question be?”

“What do you know about this shape?”

 

We had every classroom display the amazing math happening in their class.

We had take-home resources for our parents.

We had interactive activities in the common spaces.

We had interactive, showcase classrooms for each grade where families could do math together.

We had a bingo cards where students would get a stamp for every interactive, showcase class they visited. (Any student who got three in a row could enter their card into a draw for a pizza party with 3 friends on the roof!)

We had displays and activities to show how math is integrated into Visual Art, Music and PE.

 

We posted 21 awesome careers that involve math.

The response was amazing! The following day I had students coming up saying…

“Math Night was amazing!”

“Thank you for organizing Math Night!”

“I had SO much fun at Math Night!” 

“Can we have Math Night every month?”

Having students, teachers and parents actually like math is one of the best ways we can help students become better at math. So let’s keep giving them reasons to like math!

 

How Children Learn Math

At our grade level math meetings, we have been discussing a very important question.

How do children learn math?

A powerful, and oftentimes perplexing question! Luckily, we are an IB school and the Primary Years Programme clearly lays out what they believe about how children learn math.

Simply put:

First, children need an opportunity to construct their own meaning about mathematical ideas and concepts. Then, they need the guidance and support to transfer their own meaning into conventional symbols, vocabulary and algorithms. Finally, they need a chance to apply their understanding. These three phases are fluid and children move back and forth through them in an on-going cycle.

 

The teams I work with have really taken interest in the phase of the cycle where students construct their own meaning. I think this may be because when we were students learning math, most of our teachers often jumped right into the second phase – conventional symbols, notations and algorithms.

“Today we are learning about fractions. This is what a fraction is…. This is how you write a fraction… This is what the different parts of the fraction are called…. Now you go try.”

Our teachers are discovering that there are many benefits to allowing students to construct their own meaning first, before jumping into symbols. Not only are their students understanding math concepts more deeply and communicating their understanding more clearly, but when it comes time to “teach” to conventional skills, notations and vocabulary it end up taking less time and sticking much better!

So now for the million dollar question in teaching…

What does it actually look like in the classroom?

Here are a few examples across the grades of how teachers are helping students to construct their own meaning about math:

KG – Number Sense

This is a picture of an open-ended math center where students choose to play and no instructions are given in advance. Specific math manipulatives and writing tools are set out to try to lead students down specific paths and teachers and teaching assistants prompt with lots of questions.

 

Students have not been told that this is a ten-frame and have not been showed how to use a ten-frame. Yet, as the students play their curiosity drives them to experiment with how many squares fit inside and how many numerals fit inside. Allowing students this chance to interact with a ten-frame and construct their own ideas about it will make it much easier when it comes time to discuss “What mathematicians call this” and “How mathematicians use this”.

Grade 1 – Data Handling

In this grade 1 class the teacher gave each student a single piece of white paper and told them to pick a bunch of things and go find out which thing is the most liked in the class.

Students have not yet been taught how to collect data. In fact, the class had never discussed data in anyway before this activity. Students had to figure out their own method to record the choices and how many people liked each choice.

 

Some students wrote the names of each classmate beside each choice.

Some student drew a picture of each person next to their choice.

 

Some students took a “short cut” and made a tick to represent each student to save time.

The teacher now had 20 different student-generated strategies for collecting data and as a class they could discuss the pros and cons of each strategy. From here, a lesson about taking tallies is much more purposeful because students have a context of when to use tallies and why they are helpful.

Grade 2 – Fractions

How many ways can friends fairly share this pizza?

 

Before Grade 2 students ever heard the word “fraction” they were splitting things into equal parts. Students were cutting up squares, circles, and rectangles in multiple ways to “fairly” share them with different amounts of people.

 

This provided a great context to introduce fraction vocabulary. Using the cut up pizzas, the teacher could ask many purposeful questions to help her students transfer their understanding of “fair shares” into the conventional understanding of fractions.

What is the fancy math word for ‘fair’?

What would a mathematician call something that has two equal parts?

What would a mathematician call something that has three equal parts?

If mathematicians call something with four equal parts fourths, and five equal parts fifths… does anyone want to guess what mathematicians call something with six equal parts? Ten equal parts? One hundred equal parts? ONE MILLION EQUAL PARTS!?!?!?!

Grade 3 – Area and Perimeter

Find something in the class and find the distance around the outside of it.

This grade 3 class was trying to figure out the distance around the outside of classroom objects without ever hearing the word perimeter or being taught the formula P = 2L + 2W.

 

These students were using logic to figure out that they could add up all the sides to find a distance around something. Some even discovered they could take a short cut when measuring a rectangle because both sets of sides were the same. One student even realized that when figuring out the distance around a square you could simply multiply the side by 4 to take a really quick short cut. Students were using rulers, cms, inches, blocks, paper clips… you name it. One student even cut a piece of string that could fit around the outside of her circular object and then held it against a ruler.

From here, the teacher is able to take their understanding and trade it for the conventional vocabulary and formulas for finding perimeter.

Grade 4 – Pattern and Function

There are 60 wheels. Some are on bicycles. Some are on tricycles. What combinations of bicycles and tricycles could there be?

 

For this question, students were able to find the answers in any way that made sense to them. Some students drew wheels and put them in bundles of two and three. Some students split the wheels in half and figured out home many bikes and trikes would be in each half. Some students did repeated addition. Others did repeated subtraction. After one hour the most number of solutions found by one group was 4.

 

So when the teacher said she had a “math short-cut” to show them how to find 12 answers in 15 minutes, she had 100% buy in from her students. How different their level of interest would have been if she started with “Today I am going to show you how to use a table to find patterns.”

Grade 5 – Number Operations

Why does 6 x 1/2 = 3?

 

At the Grade 5 level, students are expected to have a solid understanding of multiplication. Students are also expected to have a solid grasp on fractions. However, multiplying fractions would be something brand new. So before this teacher taught her students how to multiply fractions, she gave them the chance to construct their own meaning first.

 

Some students went to the manipulatives shelf and took 6 “halves” and concretely put them together to discover that they could make 3 wholes. Others knew that multiplication was repeated addition, so they added 1/2 six times to discover the answer was 6/2 which is the same as 3. Other students approached the problem a different way. They drew a set of 6 objects and crossed out half, to show that half of 6 is 3. Now the teacher would be able to introduce the standard algorithm for multiplying a fraction by a whole number, and if that algorithm doesn’t make sense to someone, they have other self-generated and peer-generated strategies for finding the answer instead.

 

I’ve been hearing a lot about Math Wars lately, where people are either advocating for the importance of understanding in math (reform) versus the importance of skill computation in math (traditional).

I think we can have both.

If we allow students to construct their own meaning first, then help them transfer their understanding into symbols, then have them apply their understanding I believe we can have well-rounded mathematicians who can not only “do” math, but also deeply “understand” math.

What do you think?