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What did we learn? By making our own inquiries about the topics brought up in class, and discussing the topics in relation to the activities and readings from each topic, we will develop a firm understanding of each topic. Read along to hear my understanding of each topic we discuss throughout the learning logs.
October 10th 2018: Learning Log #1 "Number Sense and Numeration"
Are the students counting by memory, or are they associating meaning? There is a large difference between the two ideas, and for a student to progress they must be counting with meaning.
What does Strong Number Sense mean to you?
What does Strong Number Sense mean to you?
For a student to have a strong number sense they require a strong foundation, starting with counting. As simple as counting may appear it is a fundamental math concept that is the basis of all other mathematical models and understandings. As Payne and Huinker note, counting begins in children as young as 18 months old (46). This skill must be developed and practiced from that moment on. Children will learn to recite their numbers using "rote counting," which means they are saying the numbers without assigning the numbers to a specific object (44). Rote counting is comparable to young children reciting the alphabet. They may have memorized the order (or at least the majority of the order) and recite the numbers on command for adults, or other children.
How will I know if my students have it?
How will I know if my students have it?
To measure a student's understanding of number sense, the teacher must find ways to evaluate the basic principles of counting. By working through each principle with the student, the teacher will be able to pin-point any struggles throughout the process to developing number sense through counting. There are five principles to develop number sense:
- Using Numbers
- Students will demonstrate their ability for different uses of numbers e.g. on a birthday invitation. Numbers are used for the date, the address, and your age (44)!
- Counting
- "Counting begins in three portions" (45). 1. "accurate number-word sequence... one, two, three" 2. "stable incorrect portion... six, eight five" 3. "nonstable incorrect portion... the sequence order changes each time" (45).
- Connecting Counting and Cardinality
- The understanding that the "last counted word might be the answer to the how-many question" (47). Not just assigning a number to an object, but realizing that counting is a total of the objects.
- Thinking in Groups
- Students can understand they have "two shoes when they have one on each foot," which is the process of subitizing (48). Students can identify groups of numbers without counting each individual number/object.
- Writing Numerals
- Before being able to write, students are able to read one- and two-digit numbers (52). When students are learning how to write numbers, they are able to match the number to the quantity it represents (52).
How will I ensure they develop this?
How will I ensure they develop this?
This video is just one example of using manipulatives to demonstrate number sense. Students learn best through doing, and manipulatives gives them an opportunity to build the relationships between the numbers.
More Examples of Manipulatives That Help Students Develop Number Sense and Numeration
More Examples of Manipulatives That Help Students Develop Number Sense and Numeration
When students have a chance to create a physical representation of the numbers, the concepts become less abstract and more concrete. By practicing Number Sense and Numeration throughout math lessons by associating number relationships with physical representations, students will continue to strengthen their number sense and numeration skills as they progress from Kindergarten and onwards.
Magnetic Triangles
Geometric Shape Blocks
Ones Blocks
Connecting Cubes
Cardinality
Cardinality
Here is an example of a child that is struggling to develop cardinality. This child is almost 3, and is not expected to have a full grasp of cardinality just yet. The child does not yet know how to count objects to find a total, but he does understand the sequence of counting, as he counts "1, 2, 3, 4, 5..." Therefore we know he does understand the concept of counting, and is using the "accurate number word sequence" (45). However with practice, and by repeatedly asking the child "how many?" the child may come to the realization that the final number in the sequence when counting objects, is in fact the total of the objects.
Subitizing
Subitizing
This Halloween themed activity is an excellent addition to a math lesson. However the way this activity is displayed requires students to understand the base ten squares. Without some knowledge of how many squares are in each row (5), the students would have more of an issue to subitize the pumpkins as they would be distracted by the squares. Once students are comfortable with the base ten squares, it becomes a consistent tool to use while subitizing various objects. By filing only a few squares at a time with pumpkins students won't need to count the pumpkins since the students will be aware that there are 10 squares in total. Using the base ten squares is a building block that students can continue to use to problem solve addition and subtraction questions.
Writing Numerals
Writing Numerals
Once students can count, subitize, and recognize numbers, what comes next? Writing the numbers for themselves! Just like practicing the alphabet, it is important for students to practice writing their numbers. There are tons of adorable songs that help students remember the way to write the numbers, like this one here. Although being able to remember how to write the numbers is not as important as associating a value to the numbers. When a student writes the number 1, they must also know what that number represents. Otherwise the student is simply memorizing how to write the number, without understanding what it means.
But can students memorize all of this?
But can students memorize all of this?
No! Because I don't want my students to memorize, I want them to understand. With a strong understanding of the fundamental concepts, students will never need to memorize their math concepts. Memorization has its place, and maybe memorizing the multiplication tables, or how to spell "photosynthesis," is a valuable skill to memorize. However to memorize fast facts, and to understand fundamental concepts, is two different things. Memorization will only get a student so far. When things start to get too hard, or the problems change slightly, the student who relied on memorization will begin to struggle and will likely give up on trying. However the student that built their mathematical skills by understanding fundamental concepts are more likely to succeed when the problems progress and become more challenging.
Comparing memorization vs. understanding to fixed vs. growth mindset:
- Memorization is equivalent to having a fixed mindset; you cannot progress much further as your skills will only go as far as you have memorized. Just as a fixed mindset will keep a student in one place, whether or not that place is positive for learning. Without movement/progression in math, students will never succeed.
- Understanding is equivalent to having a growth mindset; you can progress much further as your skills will continue to grow with your experiences. By learning through understanding students will learn through successes and failures, and this level of learning is only supported in a growth mindset.
References
Payne, J. N. and Huinker, D. M. "Chapter 3: Early Number and Numeration" pp 43-71.November 7th 2018: Learning Log #2
So your students have Number Sense, but they can't multiply. What now?
So your students have Number Sense, but they can't multiply. What now?
In the "traditional" classroom, the classroom I experienced, multiplication was not taught in connection to addition. Rather it was introduced as a completely new concept. As we have discussed in class, context is just as important in a mathematical equation as the answer. If there is not a clear reason for the student to develop an answer, then why would the student be motivated to find the solution?
Multiplication is best understood with context, much like all other math concepts, multiplications should not be taught as a stand-alone concept. What does it matter to know that [2 x 7=14] if a student cannot work out:
"if your family is going on vacation for 2 weeks, and you need to bring a pair of socks for everyday you will be away, how many pairs of socks should you bring?"
This is a problem that students may relate to, if they have gone on vacations and possibly packed for themselves before leaving for the trip. However the best context for a word problem, is a relatable context for the students. Do not ask 2nd grade students to determine how much gas they need in their car, but you could ask 2nd grade students how many minutes in a week they are allowed to watch T.V. if they are allowed screen time for 30 minutes a day.
"Learning to solve problems and learning to memorize information require profoundly different ways of thinking and behaving" (Bushman, 308).
"Learning to solve problems and learning to memorize information require profoundly different ways of thinking and behaving" (Bushman, 308).
Context in Math
Context in Math
Why does it matter if a student can complete the entire multiplication table in a "mad minute," but they cannot solve a simple word problem? I personally experienced this struggle as the focus was memorizing the multiplication tables, rather than working on problem solving skills.
Word problems were used as "challenge questions" to stump the smart kids in class. They may have been bonus questions on a test, or they were additional challenges if you have completed all of the required questions during the work period.
When these problems are proposed as challenges for students, students become discouraged to try when they are unable to come to a solution on the first try. Unlike the standard "mad minute" worksheets, word problems do not line up the numbers of the problem into the correct order. Therefore the true challenge of a word problem is not in finding the solution, but in determining what the question is truly asking, and how the equation would be written to illustrate the question.
"Engaging in sense making while solving problems is one way that children deepen their understanding of mathematics" (Buschman, 304).
"Engaging in sense making while solving problems is one way that children deepen their understanding of mathematics" (Buschman, 304).
But what if the teacher is struggling to problem solve too?
But what if the teacher is struggling to problem solve too?
Believe it or not, teachers are not worldly geniuses that know everything about everything. Sometimes we have personal struggles too, and those struggles may include math. Unfortunately as teachers we are very prone to bringing in our own biases into the classroom, and one of the largest biases we carry may be negativity towards math, and problem solving. This is one of the largest issues with traditionally taught teachers trying to instruct students in the "new methods" of today's math class.
"Common Core Math is GARBAGE!"
"Common Core Math is GARBAGE!"
This man watched a video about "common core" math, the open array, and considers the traditional multiplication style to be more valuable than the common core style. This comparison is the negativity that exists between students and adults, and this comparison is still very present in the classroom, as many parents helping their children with math problems at home will be more familiar with the traditional style than the open array.
But what does it all matter as long as we get to the correct answer? Who is it hurting if an individual finds the traditional style to work for them, versus the open array, and vice versa. Where the problem truly remains is in this battle between the "right way" and the "wrong way" of teaching concepts to students.
There is more than one way to solve a problem!
There is more than one way to solve a problem!
To set our students up for success, we need to consider that not all students think the same. I always struggled with mental math skills during elementary school, which made the mad minutes and traditional method very difficult. I found it challenging to work with large numbers that were not even, or multiples of 5s or 10s. Once I was shown how to problem solve using arrays, I was opened up to a world of success. Finally I was taught how to break down the numbers into numbers I could work with more easily. After learning that it is okay to work with the numbers, math became a whole new experience for me. No longer did I dread seeing 362 x 47 anymore! For now I had the tools to work through the problem with minimal stress, and less opportunities for errors.
If it was not for my teacher to introduce me to the concept of arrays, I never would have known they existed. Thankfully my teacher didn't just tell me they existed, she also taught me how to properly use them... imagine that! There is no greater opportunity in a student's career than to be given the adequate tools, with the proper instructions for use. Once I knew how to use arrays I was unstoppable. But This was only possible because my teacher worked with me through the rational of arrays, and how that model is able to work. If it weren't for the proper instruction I had received, this model would not have been helpful to me. Therefore the understanding that we learn differently is important, just as that different methods of problem solving require unique explanation to understand how the model is able to function successfully.
"When teachers carefully choose tasks that require students to engage in mathematical thinking and problem solving, then draw out students' thinking through their questioning, and finally encourage reflection and sense making, their students make significant gains in mathematical understanding and attain higher levels of achievement" (Rigelman, 314).
"When teachers carefully choose tasks that require students to engage in mathematical thinking and problem solving, then draw out students' thinking through their questioning, and finally encourage reflection and sense making, their students make significant gains in mathematical understanding and attain higher levels of achievement" (Rigelman, 314).
In my experience during elementary school, and during our classes, open arrays have been my go-to model to problem solve multiplication questions. The user is able to basically draw a map to differentiate the place values and numbers being worked with in the equation. This interactive image, a Thinglink, illustrates the benefits to using the open array, as it breaks down a "complex" problem into "simpler" problems that will be more familiar to students.
This is not the only model we have looked at for working with multiplication or division problems. In the course readings, and during out time together in class, we have found a variety of models and methods to get to the correct answer. Although I find the open array to be my favourite model to work with, I would not force my students to work with the open array if it did not match their natural problem solving reasoning. I am looking forward to celebrating the different methods my students will take to come to the answer. For the purpose to problem solving is not just to find the correct answer, rather it is about the skills used to find the solution. If students possess a logical understanding of the concepts, there should be no issue in weighing one model above another. It is not the model students decide to use that is important, rather it is there understanding of the concepts, and their rational for problem solving that is important.
References
Buschman, L. "Teaching Problem Solving in Mathematics" pp. 302-309. Rigelman, N. ""Fostering Mathematical Thinking and Problem Solving: The Teacher's Role" pp. 308-314.February 13th 2019: Learning Log #3
Fractions: how to make the impossible seem possible
Fractions: how to make the impossible seem possible
A concept that many students struggle with, and I personally struggled with, can be conceptualized to become possible. I class we have explored many alternatives to the traditional methods of teaching, to help students understand the concept.
Some examples of fraction kits that could be made with students in class, or made for the students to use in class as a manipulative. A faction kit like this helps students see the fractions as concrete ideas rather than an abstract concept.
Some fraction kits go beyond the simple boxes or squares, and create a meaningful kit using slices of pizza. The pizza fraction kits become something the students are familiar with, which makes the idea of a fraction more familiar.
Other fraction kits leave the student to work with their imagination about the shapes/colours/etc. Whichever kit is used, as long as it is introduced with meaningful connections the students will be able to use the tool to problem solve fractions.
How Meaningful Connections Help Students in Math.
How Meaningful Connections Help Students in Math.
Just like the fraction kits, when students are able to connect to something the concepts seem easier. If you were to pose a question to students, in a context they are unfamiliar with, using an operation that is new to them, they will have a very low chance of solving the problem
For example, giving students a question such as this would be unfair due to the context needed to understand the question.
Sam is driving from Orillia to Barrie and back, and only has 3/4 of a tank of gas. If she burns 1/4 tank of gas every 20kms, and the distance between Orillia and Barrie is 40km, will she be able to drive from Orillia, to Barrie, and then back to Orillia without stopping formore gas?
Although this question involves simple addition of fractions that a student would be able to solve, it is in a context that is unfamiliar to students since they do not drive. A question that would be more meaningful would involve scenarios that are more likely to occur in their own lives.
An example of a fair question, with meaninful connections would be this...
After Sam's birthday party there was a lot of leftover pizza. There was 3/4's of the hawaiian pizza left, 1/6th of the pepperoni pizza left, 2/3's of the deluxe pizza left, and 1/2 of the cheese pizza left. Sam wants to know how much pizza she has left all together.
This question involves more critical thinking than the first example, but using the context of pizza is more familiar to students so they can visual the problem easier. As well if they have a pizza fraction kit, or a circular fraction kit, they would have an even easier time to solve the problem as they could use the pieces to make whole pizzas from the leftovers.
"Children engage in meaningful thinking about fractions when the contexts relate to their personal situations" (Sharp, 27).
"Children engage in meaningful thinking about fractions when the contexts relate to their personal situations" (Sharp, 27).
There are also great videos available to demonstrate the visual representations of fractions. Students now have access to so much information online, and when that technology and information is brought into the classroom, it will help students build that understanding.
There are more videos to introduce more concepts around fraction, such as addition, subtraction, multiplication, etc. that could help give students another way to think about fractions.
This video explains the make-up of a fraction, in relation to objects that students would be familiar with like pizza, pie, and donuts.
For the students it can be helpful to have multiple methods of explanation to understand a new concept. As teachers it can be difficult to create a million and one ways to teach about fractions, which is way it is beneficial to bring in outside sources that may spark an understanding for our students.
There is no perfect way to teach fractions.
There is no perfect way to teach fractions.
We use tools, manipulatives, and games to teach but it is not the only route. We use what is available to us, and what is appropriate for the students. When we use our knowledge of our students to guide our teaching style, it will result in students that can and will excel. By trying out techniques we learn as educators what works, and what does not work.
"That no single most powerful form of presenting an idea exists is clear from the variety of strategies that students use. Teachers must develop a repertoire of alternative approaches and forms of presenting ideas. Teachers can derive these from several sources, including their students' strategies, research, exemplary materials, other teachers, and reflection on their own experiences of what works and why" (Teaching Children Mathematics, 457).
"That no single most powerful form of presenting an idea exists is clear from the variety of strategies that students use. Teachers must develop a repertoire of alternative approaches and forms of presenting ideas. Teachers can derive these from several sources, including their students' strategies, research, exemplary materials, other teachers, and reflection on their own experiences of what works and why" (Teaching Children Mathematics, 457).
Get your hands dirty, and make it fun!
Get your hands dirty, and make it fun!
When teaching fractions the majority of the lessons should involve opportunities for some hands on learning. Activities such as using blocks to make a shape that is "one third blue and one fourth green" is open for students to get creative, and allows students to excel if paper and pencil learning is not theit strongest suit.
Teachers could use activities such as these to have students work with partners, or individually, to demonstrate their learning outside of number lines, word problems, and mad minutes.
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Assessing Those "Hands-On" Tasks
Assessing Those "Hands-On" Tasks
This can be a challenge, how can you prove to the principal, their parents, on their report card that they understand fractions after playing some card games and making shapes? There are a lot of opportunities to see how the students are able to work with fractions, and by having fun with it the students are more likely to be engaged in their own learning.
By leading some games and having smaller focus groups that you can sit in with and play along with, you could be assessing the students without them knowing. Playing a game of "fraction war" with a few students:
- Is the student able to make a proper fraction?
- Is the student able to identify the numerator and denominator?
- Is the student able to identify which fraction is bigger?
Each of these success criteria can be assessed through observation. Keep a single-point rubric for the class, and check in with the groups to see where the students are in their learning.
How about having students building shapes with "one third blue and one fourth green"? Simple. Have the students use manipulatives to make the shape, then they could draw their shape on to chart paper and present their shape to the class to justify whether or not they met the requirements.
- Is the student able to make a shape with one third blue, and one fourth green?
- Is the student able to explain how they decided to make their shape?
Each of the success criteria could be assess through observation as well using a single-point rubric too.
There is so much than can be done with fractions to make them relevant, and understandable. Don't get caught up in having big complicated assessments to prove the students understand. Keep it fun and engaging and the students will be more likely to learn through play!
There is so much than can be done with fractions to make them relevant, and understandable. Don't get caught up in having big complicated assessments to prove the students understand. Keep it fun and engaging and the students will be more likely to learn through play!
References
Sharp, Garofalo, & Adams. "Children's Development of Meaninful Fraction Algorithms" pp. 18-27. Teaching Children Mathematics. "Connections in Fractions" pp. 452-457.March 18th 2019: Learning Log #4
Problem Based Learning
Problem Based Learning
The introduction of problem based learning in the classroom is an excellent tool for students to further develop their mathematical sense. When compared to the "traditional" method of memorizing facts and equations, the problem based learning will encouraging students to understand the purpose and use of math in real-life contexts.
"They most likely experienced algebra as a collection of techniques for factoring and simplifying expressions, solving equations, and so on. They probably did not explore what the expressions or equations might mean, where they came from, and why anyone would manipulate them"
"They most likely experienced algebra as a collection of techniques for factoring and simplifying expressions, solving equations, and so on. They probably did not explore what the expressions or equations might mean, where they came from, and why anyone would manipulate them"
(Blanton & Kaput, 2003).
Our teachers learned in the traditional method, but need to relearn and teach today's students in the problem based method. This quote from Blanton and Kaput speaks about algebra, and the use of memorized equations without understanding how those equations came to be.
Yes. You can do math through memorization, but when faced with a challenge, will you know how to solve the problem?
Like this cartoon shows, through memorization we do not gain an understanding of mathematical concepts.
By learning that "x" represents an unknown, and that the unknown is ever-changing depending on the sitauation, is something that can be memorized.
But understanding how, or why, we manipulate the equation to find the unknown is a concept that needs to be developed and explained to students to ensure they have a reason for doing math!
Without a reason to isolate "x" why are we doing it? Why does it matter?
Problem Based Learning Gives Learning a Purpose
Problem Based Learning Gives Learning a Purpose
In my experience with teaching math to students, and my own experiences learning math, everything makes more sense when there is a reason to do it. When you feed into the wild and eccentric student mind you will find out what interests them, and how to communicate with them in a meaningful manner. Finding the common ground of interests to begin mathematical conversations will lead to richer and more meaningful discussions.
"Children appear to be natural problem solvers, and problem solving is one of the primary functions of the human brain"
"Children appear to be natural problem solvers, and problem solving is one of the primary functions of the human brain"
(Buschman, 2004 p. 305).
And this is why it is important to give the students context that applies to them. Your elementary students are not driving, so why would you ask them a questions about when they would meet someone in traffic?
Why not give them a more realistic questions, even something they could act out, or have possibly experienced in their own lives.
Problem Based Can Be Experimental
Problem Based Can Be Experimental
Math does not always need to be a paper and pencil. Why does the measurement unit always have to force the students to measure the height of their friend, or the length of their desk? Why can't they measure the correct amount of ingredients for a cake, and experiment with what ingredients are needed, and how much of each ingredient leads to the best tasting cake?
There are numerous resources online that encourage teachers to stop with the "Mad Minutes" and Tests, and bring mathematics into the real world.
Whether your students have baked before or not, they will learn the importance of precise measurements when they learn how to follow the measurements laid out in recipes.
Problem Based Learning In Action
Problem Based Learning In Action
The students investigate and take ownership of their own learning. The teacher is simply a facilitator to help guide the students.
The students worked out a strategy they would use, then solved the problem independently. Then they work with a partner to explain their strategy and choose one strategy to use together to solve the problem. Then the teacher chooses a pair of students who will demonstrate their strategy to the class.
The rest of the class will follow the strategies and methods used to further develop the full class' abilities to solve the original problem.
Amber Carney, the teacher from the above video, finds success in problem based learning in her grade 2 class, as it allows her students to see there is more than one way to solve a problem. This builds confidence in the students to strike out on their own to develop their understanding without the fear of "doing it wrong" or "failing"
Her class appears comfortable learning in a problem based method, but that likely took a lot of practice. The skill of problem based learning is one that needs to be developed among the students, and by the teacher. Although problem solving might be a natural skill, you cannot expect students to know how to do it without any guidance.
Without guidance, students might not know how to "show their thinking" but with practice, and help, students will soon thrive with the problem based learning.
With the progression of problem based learning to replace the traditional methods of memorization, students are likely to get more out of their time in class than students before them. Reflecting on my own time in elementary school, to my experience with instructional methods of today, I wish I was in elementary school now! The purpose of learning is much more clear.
Now, students learn for a chance to grow their understanding of the world around them, instead of memorizing information to pass the test next week.
Learn for the sake of expanding your knowledge. When you understand what you're doing, the lesson stays with you longer.
Don't memorize the solutions; understand the problem.
References
Buschman, L. 2004. "Teaching Problem Solving in Mathematics" pp. 302-309. Blanton and Kaput. 2003. "Developing Elementary Teachers: 'Algebra Eyes and Ears'"Report abuse