Math & Proportional Reasoning in FDK

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As a student, I was always afraid of math. To me, it was hard and that was that. I can’t recall too many teachers who tried to make it fun or meaningful to me and instead, I just remember a lot of textbook work and rote learning that never quite ‘stuck’ with me. Math just seemed so abstract and since I relished in creative tasks, math was just not interesting to me and even a bit scary because of how far away from reality it seemed. I can honestly say that I learned more about math after my school years than when I was in school.

The real mathematical learning came when I had to use numbers in everyday situations and when ‘playing with numbers’ took place within authentic experiences. I learned a lot, for example, by simply becoming a cashier at my first part-time job. I had to add, subtract, multiply and divide quickly and on the spot. I learned how to round numbers, estimate but also how to be a careful counter. Being off, even by a few cents, meant I had to do some major backtracking in order to catch my mistake and that definitely wasn’t how I wanted to spend the last half hour of my shift. As I’ve gotten older and began earning money, paying bills and making larger purchases, I’ve continued my learning, now really ‘feeling’ the direct consequences of any miscalculations or poor spending choices. As a result, most of my mathematical knowledge and understanding of numbers has come from my own day to day living – not from school.

But becoming a teacher has made me realize that my experience of math as a student was extremely unfortunate and it’s definitely not how I want my own students to experience math. Had I been given the chance to learn math through play and hands-on investigations, more meaning would have been given to the math facts I was learning about. Perhaps math wouldn’t have been so scary. Better yet, I might have grown to love the challenges math provided.

As a teacher, I want to create learning experiences that can be accessed by all of my students, regardless of their abilities or learning styles. I know that the explorations need to be as authentic as possible for children to ‘connect’ to them and find practicality in them. Rather than frame math as a separate and abstract subject or set of skills, math needs to be woven into all parts of the day to show students that math really is everywhere. For instance, just as I enjoyed creative pursuits as a student, I want to integrate this component into math-based activities to attract and inspire other children who have similar interests. Most of all, when I invite children to participate in math-based activities or when I intend to ‘draw out the math’ in a spontaneous teachable moment, the main thing I want to accomplish is to make it fun. It sounds so simple and so obvious but I think it’s so important.
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In this post, I will highlight some of the math games and activities that my students took part in this past spring. Hopefully you will be able to see how I made special efforts to keep learning about math hands-on, authentic and fun.

Proportional Reasoning in FDK

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Currently, there is a huge focus on teaching proportional reasoning within our education system. It was a focus of my school’s improvement plan over this past year and has been the basis for which teachers, including myself, have been designing mathematics programs around.

What is proportional reasoning?

So, what is proportional reasoning and why is it so important? According to Ontario’s Ministry of Education (2012) document, K-12 Paying Attention to Proportional Reasoning, proportional reasoning, though difficult to define, is basically the ability to think about numbers in relative terms as opposed to absolute terms. When we consider the relationships between numbers and compare quantities or values, we are involved in proportional reasoning and in effect, proportional reasoning is infused into all strands of mathematics. It is the crucial foundation necessary for a thorough understanding of numeracy.

What is involved in proportional reasoning?

Proportional reasoning is often thought of as a web of interrelated concepts. There are so many different ways that one can demonstrate the ability to think proportionally, which is why our goal as teachers of early learners – and older students – needs to be to supply a range of proportional reasoning experiences. The concepts involved in proportional reasoning include:

  • Understanding rational numbers (Mainly dealing with fractions)
  • Multiplicative reasoning
  • Relative thinking
  • Understanding quantities and change (Understanding how a change in one quantity can coincide with a change in another)
  • Comparing quantities and change
  • Measuring, linear models, area, volume
  • Unitizing (Seeing quantity sets as units. E.g., a quarter can be seen as 1 quarter but also as 25 cents simultaneously. This concept is important because it’s the basis for multiplication and the place value system)
  • Spatial reasoning (Understanding a unit as equal intervals of distance. E.g., there are two rulers, each having 30 centimetres on them. This is a necessary concept for understanding unitizing.)
  • Scaling up and down
  • Partitioning (Equal splitting of a whole)

In the examples shown in the photos below, you will see many of these concepts targeted. You will also note that because my students are early learners, I have provided them with many experiences that are qualitative (and not just quantitative) in nature. Qualitative problems allow students to participate in proportional reasoning without exclusively working with numbers since this can conflict with some children’s developmental levels in this age group.

Math Explorations in our Class!

Investigating 10 Frames, Cube Trains & Part-Part-Whole
This activity was set up to allow the children to explore the number and quantity of 10 in multiple ways. They were given laminated mats that they used dry-erase markers with. They were also given two different coloured cube trains of 10. They practiced filling up the 10 Frames using different combinations of the coloured cubes. They also wrote the corresponding numbers of the two “parts” to make the “whole” (10).
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The child shown in the middle photo filled out the mat in the way we had anticipated (the numbers correspond with the coloured in 10 frame). The child shown in the two end photos, however, enjoyed demonstrating all the different ways she knew how to make 10. Her numbers didn’t correspond with the way she filled in the 10 frame but what I love is how she didn’t feel obligated to stay within certain boundaries (much like would be the case with a worksheet). Instead, we encouraged her to continue showing us the different combinations she could think of. She would wipe her board clean and continue exploring, showing us that this activity is even more effective when left open-ended.

Interacting with Magnetic 10 Frames
This invitation was set up at a magnetic white board. We encouraged the children to work together to think of the different ways to fill in the 10 frame and then come up with different addition equations. This activity was modeled many times during whole-group lessons and we supported the children when they began to do this on their own by scaffolding and assisting as needed. As you can see, the children became quite independent at completing this activity and loved the challenge. Although this activity is less ‘exploratory’ in nature, we didn’t see a problem with including it in our program because it was still very play-based, the children loved doing it (something special about working at the tall magnetic white board and using dry-erase markers I suspect) and it was only one of many different ways the children could work with numbers.
 photo IMG_3012w_zpsefnufxzn.jpgComparing the Weight of Objects
In this learning experience, children were presented with a bucket scale, various ‘measurement units’ (the fruits) and a basket of different school supplies. We wanted to see how the children would use the materials to explore the concepts of weight. Most of the children came here and put a school object in one bucket and then used the fruits to try and balance the scale. Once it was balanced, they discovered how many fruits that object was equivalent to in weight. Other children came here and tried to see which school objects weighed the same, less or more than one another. Either way it was played, the children were involved in a ton of proportional reasoning-related ‘talk’.
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 photo IMG_3046w_zps2hae5p1o.jpgCreating Different Lengths Using Playdough
For this activity, the children were guided in rolling the playdough to create different sized worms. Since we were on the verge of a full-blown Worm Inquiry, this seemed appropriate and the children were quite interested in worms at this point. This activity involved an element of creativity and kinesthetic appeal, drawing in almost all of the children, regardless of ability or level. Madison and I ‘drew out’ the math language from them as they played here. We would ask them questions about their creations and how they knew that one was longer, shorter or the same than the other. For more advanced level children, we were able to encourage them to figure out how many of one small worm was the length of a much longer worm.
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 photo IMG_3035w_zpsg1dgi76r.jpgExamining Each Other’s Heights & Relating to Length
This activity got the children moving and using their entire bodies to explore height and length. They used themselves as measurement units by working in pairs and measuring the length of each other on butcher paper. They made ticks where the top of their partner’s head was and where the bottom of their partner’s feet were. They then worked together to draw on a worm that covered the length from one tick down to the other. We called them, “kid-sized worms”. When we had a few worms finished, the children compared the lengths of the worms. They put them in order of longest to shortest and also just looked at how each worm turned out relative to the others.

What I found interesting was that not all of the children made the instant connection that one worm was smaller in length than another worm was because the child who made that worm was shorter than the other child. It took some rich discussions and many thoughtful questions on Madison and I’s part to help the children figure that out for themselves. What I thought would be something so obvious and instinctive, turned out not to be so for these early learners. I think it was a great example of why proportional reasoning is crucial for mathematical development and also taught me not to make such assumptions on learning.
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 photo IMG_3047w_zpsmvuqlwta.jpgUnderstanding Differences in Quantities Using Names
At this station, math was integrated into language and the children also practiced writing their own names and their peers’ names. They were presented with student names on cards and recording sheets that included a table with sections for greater than, less than and equal to. The students first figured out how many letters were in their own names to use as a base of comparison. They filled in the table with peers’ names who had a greater number of letters, less and to consider in relation to measuring the other names.
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 photo IMG_3037w_zpspwsnahr2.jpgExploring Capacity & Volume 
At the water table, we supplied various materials to encourage exploring the capacity of different sized containers and comparing the volume of water in each. To give it more of a focus and make it a game for those who were interested, we included letter sponges. The children found the letters from their names and squeezed water from those sponges into containers to see how much their names filled the containers. We talked with the students about how the height of the water from one name compared to the height of the water from another name. Some children even pointed out that the size or shape of the container had an impact and that sometimes it appeared a name produced less water because of the apparent height of the water in the container when really it produced more water because the container was wider. Great conversations occurred at this table!
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The game “Tall Towers” was created by Madison and I after some thought was given as to how we could use our cube ‘bases’. This game features many mathematical elements and encourages children to engage in: predicting (e.g., what number they will draw or who will get a taller tower), estimating (e.g., how many more they need to reach the same height as their partner), comparing (i.e., looking at both towers and envisioning differences in height), counting (e.g., number of cubes they need to find and add to their tower, number of cubes on their tower in total, quantity difference between towers, etc.), reading/identifying numbers (i.e., on the number cards) and spatial reasoning (e.g., when perceiving the difference in towers).
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Remember – Keep it Fun  & Meaningful!

Hopefully these examples have demonstrated various ways that mathematical concepts and proportional reasoning can be targeted in FDK. It is important to note that while these examples feature math-based activities (many of them still integrated with other subjects like language, art and science), math concepts in FDK are also largely taught spontaneously while the children are involved in open-ended play. The concepts listed earlier as being a part of proportional reasoning are concepts easily integrated into all parts of our day. Here are some other examples of ways we have tapped into these concepts and took advantage of an opportunity while the children engaged in play:

  • During snack, talking about how much of something someone has ate or drank (e.g., “You have eaten half of that cookie.” “Finish most of your sandwich before you have your treat”)
  • Discussing parts of the room (“You’re right. There is more space to dance on the carpet area than in the cubbies.”)
  • Playing outdoors (“Yes, you’re right the Primary Yard is larger than the Kindergarten Yard. Where do you think would be a better place to play soccer?”)
  • Talking about changes in growth (“Yes, every week the plant grows a little bit taller because we are feeding it a little bit of water of every day”)
  • Noticing differences in height, length and width of spaces and objects (“You’re right, Ms. Madison is 5 cubes taller than me” “This long table has more space on it to colour than the small round table.”)
  • Building a new dramatic play area (“Which box would make a better front entrance? Yes, this one is taller and we can cut a better door out of it”)
  • Playing with playdough (“I have one ball of play dough. How many kids should play here so that everyone gets a good amount to play with? Does everyone have an equal amount? What if one more person comes – how can we make sure she has an equal amount?”)
  • Using loose parts for building or creating (“Does everyone at this table have a fair amount of materials? How can we make it more fair?” “Whose creation has the most red gems?” “Who covered more space?”)
  • Drawing and painting and reviewing art pieces (“This drawing includes a lot of details but this one is less detailed. Which drawing helps us understand better? Why?”)

The possibilities really are endless! This approach makes the learning authentic, meaningful and memorable to the students because they become active participants in their learning. Instead of me standing in front of them, telling them all what they will be learning and showing them an exact way to do so, I am allowing each student to navigate their own learning at times that makes the most sense for them and in experiences that fit their developmental levels and interests. They will remember the concepts learned because the learning benefited them in those moments and occurred within relevant experiences. The key is to ‘tune in’ as the educator and recognize the opportunities to support and expand the learning as it arises naturally.

Perhaps this post has given you some ideas for your own class or young learner(s). Please feel free to comment below and share some other great play-based math activities or ways that you have made learning about math authentic and meaningful to an early learner. I look forward to expanding my own repertoire!

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