Collaborative Math Inquiry: Car Caravan Findings K-8

We gathered with teachers from our school board from Kindergarten to Grade 8 to discuss the results of the Car Caravan Math Task. Here is an account of our findings.

What struck you?

• Students seemed very frustrated they couldn't just apply one formula
• students not used to open ended questions and seemed overwhelmed
• Need to narrow down the wondering in order to be able to tackle it and move forward
• Teachers also at times frustrated that students were not thinking deeply and not producing questions or taking the problem very far.
• Some teachers were surprised that some ESL students were very engaged and using math language to talk about the colours and the proportional language used : ex. more than
• The task was so open it pushed us to think about what we were documenting and pushed us to think about how we question students and focus on presenting more open-ended questions.
• Teachers found students were debating a lot and had trouble working in groups and collaborating.

What are the big ideas we discovered by presenting this problem?

• Big idea: Getting students to justify their thinking. Often the questions used were:
• How do you know?
• Can you prove it to me?
• Are you sure?
• Students ended up first trying to use strategies and concepts that were taught in the previous couple weeks.
• Learning context has a huge impact on the students. Some important questions to ponder prior to planning to help focus on educators’ intentions:
• Are we letting them struggle with a problem or giving them the strategies too quickly?
• What kind of student groupings will be best?
• Are we providing materials that are consistent?
• Ex. on the computer it may look different than when it is printed.
• How does the environment affect their wonderings?
• Ex. The lighting

Kindergarten Small Group Discussion:

• Difficulty getting out the math until educator seemed to find the right questions
• Ex. It`s race track! If it`s a race track , which one is winning?
• Some KG students talked about cars, colours, it`s a rug.
• Lots of talk about what the picture represented.
• Some noticed the background first.
• They were trying to figure out what it was versus wondering
• Focus on colours: ex. 2 shades of blue
• Some discussion about shape
• Teacher asked : Are there a lot of car? - provoked their thinking about what is a big number. One student remarked that they would need to “count fast” - and he started to count by 10s.
• Asking questions was challenging for the teacher because the students went on many tangents in their discussion and it was hard to bring them back and focused.

Kindergarten discussions after viewing Students' work from K-8:

-

• You could see all the strands covered across all the grades
• The problem could be presented at the beginning of the year and brought back again to ask other questions and reach other strategies
• EQAO years – affect their answers and wonderings
• We heard other teachers commenting on kindergarten inquiries saying that in Kindergarten it seems to be more of an exploration
• Noticed how kindergarten students were drawn to colours but that math can be integrated later

What would you do differently?

• Some of us wanted to rethink the questions we were using to figure out how to draw out more math
• Need to MODEL more mathematical wonderings

Number Talks, Take 5!

"Do not give up when things don't go perfectly well; make adjustments and keep moving forward" (p. 31)

And that is just what we plan on doing. We did, however take another break and dropped our Number Talks very readily this past month when our new strategies didn't produce the results we had expected. With report cards due, we all felt pretty justified in focusing our attention elsewhere. But now that they are all done, I finally had the energy to read Chapter 2 of Number Talks to find some fresh ideas. This last tip was at the very end; and yet it is the most important one. Staying positive and persistent when faced with challenges, are necessary qualities if we want to accomplish our goals. And it just so happens that promoting a positive mindset and/or GRIT is a board-wide endeavor this year. Keeping a positive mind set is the only way the students will succeed as well as us teachers.

So last week we restarted our Number Talks with some adjustments. Firstly, we decided it was best to continue to do Number Talks in small groups. This time, we did them first thing in the morning during Tabletop Activity Time when students are most alert. To avoid distractions of other activities at the tables, I brought them into a quiet corner of the room on the carpet and I sat facing outward so I had a view of the rest of the room, while they sat facing me. As Sherry Parrish suggests: "I have found it that it is easier to build a cohesive community and a focused discussion when students shift away from their regular routines and are removed from typical desk distractions" (p. 17). This shift in location, made a huge difference in terms of their attention and engagement. And within one week, students began to await their turn to join in a Number Talk. This past Monday morning I saw one student walk in and wander, so I asked if she wanted to join me for a special activity in the corner. She replied: "I've seen you doing the Number Talks in the corner." I asked if she was excited that it was her turn and she just smiled from ear to ear. Parrish notes: "One way to elevate the status of number talk time and signal its importance is to have a specific location in the classroom where the students gather for this purpose" (p.17). 

Click Here for a blog post on Small Group Number Talks in the Junior Grades

First, I reminded each group of the routine for our Number Talks. First, I would show a card of dots very briefly and then ask them "How many dots do you see?" and "How do you see them?". Then they would place a thumbs up on their chests if they were ready with an answer. I reminded them that it is not important to figure out an answer quickly, but rather that we give everyone a chance to take their time to think and come up with an answer. Since I began teaching, the importance of " wait time" has always been stressed; however this strategy of including a gesture is neat because it makes the wait time obvious to the students themselves. By making "wait time" known to students, they may better understand that speed is not the most important part of success in math. As Sherry Parrish explains: "Many students have the misconception that they are not good at math because they are not fast. We can easily reinforce this misconception if we begin gathering answers as soon as the first students are ready. " (p. 18). Using the procedure of giving a thumbs up is important to "send the message that all students are expected to think and contribute during this time" (Parrish p. 18). 

Set students up for success by grouping them
 by ability and providing tasks at their level.

Next, I explained I would invite those interested in sharing their answer to raise their hands. Keeping the groups small, to a maximum of three students, meant that students were not only more engaged but more comfortable sharing their thoughts. I also made sure to group the students based on similar abilities. Because I kept the groups small (max. 3 children) and carefully selected students with similar needs, all students were eager to share and even the shy ones participated, though they still spoke softly. Parrish also notes that by grouping the students based on their needs, it allows for those that are ready to be challenged (p. 25). I am not yet sure what kind of challenge I will present to those that are stronger but for now I will record their answers and then take the time to analyze them and plan for our second set of small group Number Talks. 

One of Parrish's 5 steps towards teaching for understanding (p. 28) suggests limiting number talk time to a maximum of 5-15 minutes in order to maximize engagement. Back in January, we had decided to review all the cards for number 4, so the Number Talk definitely went on for longer than 5 minutes. Even though it was in a small group,I found even I, myself, was quite bored since it was always the same answer. Although I was recording the answers and analyzing them throughout the Number Talk, this documentation took up way too much time and thus, the students were distracted. So finally I told them, this time our talk would be short so no one would get bored and they could go back to a table activity of their choice after only 5 minutes. Before we began, I set a timer so that we would not get carried away. We were able to get through just 3 cards (one set) with everyone sharing each time. The timer has been ringing just as we are discussing the final card. Limiting the talk to 5 minutes in Kindergarten is a rule I would definitely recommend. 

A few other tips I liked from Chapter 2 (p. 30):

• Offer a strategy from a previous student 
• If no one offers a strategy, give an example and ask if it will always work (checking for reliability).
• It is all right to put a student's strategy on the back burner.
• If you don't understand a student's strategy and they've already restated it, you can have students turn and talk about what they understand and brainstorm questions to pose for clarification.
• During this time, listen to students and get a sense if everyone is confused or you can build upon this idea.
• If not, tell the student you need time to think about it before continuing.
• This helps keep the conversation pace versus slowing it down and losing students' focus.

I especially like this last phrase because it reassures the student his/her ideas are still valuable and as Parrish says, you can meet later with the student to work through their strategy.

And finally, Parrish's last tip (p.31): 

Possible Next Steps:

• Use one half PLC day to meet with my colleague from my second class to explain Number Talks and plan to implement them.
• Read Ch. 3 of Number Talks for more specific strategies in the K-2 classroom to be shared with colleagues.
• Use my final half PLC day to meet with my first Kindergarten team and discuss possible next steps below.
• Devise new plans for recording students' thinking so that it can be easily shared with classmates. (Right now we are simply making our own handwritten anecdotal notes to share among the teaching team.)
• Discuss possible activities/questions Parrish suggested to add to develop more accountability in our Number Talks. (p. 25)
• Ask students to use finger signals to indicate the most efficient strategy.
• Require students to solve an exit problem using the discussed strategies. 

Collaborative Math Inquiry Lesson Plans: Car Caravan Math Task

As part of the Collaborative Inquiry in Learning Mathematics (Funded by EOSDN), we have been given some PLC days in order to plan for our math tasks. Today, our primary team met to discuss and plan for the Car Caravan task. This task has been given to all teachers in our board from Kindergarten to Grade 8. We will be gathering documentation on target students and coming together to analyze our findings with the goal of promoting a growth mindset, more positive attitude towards Math in general and encouraging the development of proportional reasoning.

Here are our plans for the Car Caravan Task:

Day 1 

1) Introduce the Car Caravan problem by showing a small group (including the target students) the following photo and asking:

• What do you see?
• What are you wondering?

Document their wonderings with video and/or anecdotal notes. Videos can be shown to the whole group for discussion.

(Gr. 2 students willl all be put into small groups to work on their own.)

Our intention: to allow target students to develop a wondering. By presenting in a small group, the target students may have more confidence to speak and  share. This will allow them to have a voice and develop a more positive mindset versus potentially withdrawing  if the problem was presented in a whole group and feeling intimidated.

2) Present the problem to the whole class and present the video of the small group’s wonderings and discussion. Ask the whole group if they have any other wonderings.

3) Analyze documentation and list of wonderings. Choose a wondering to focus on for the following day, based on target students’ interests. Gather potential materials to assist in solving the problem tomorrow.

Day 2

4) Challenge of the Day: Present the chosen wondering of the day at the end of community time as an optional centre time activity, guided by the educator.

**If the target students do not willingly join the challenge, (perhaps they are hungry and want to go to the snack table right away), the educators will invite them to join later. If they are resistant, then give them time to do another activity and invite them back later. Document their response and engagement.

5) Provide the small group with the photograph and the specific wondering (written on a paper). Ask them what materials they need to solve the problem. And guide them to discuss and attempt strategies and work on a solution.

6) Consolidation: During sharing time, we will have small group of students present their strategies either orally or through a video/pictures.

Pedagogical Documentation Look-Fors:

• Describe what you see and hear.
• Do they have a curiosity about the task?
• Look at their understanding and ability to explain their thinking.
• Look at their persistence.
• What other factors contributed to the students’ engagement : Did anything else happen that morning? or the night before?

• What materials did the students suggest.
• What language are you (the educator) using to pull out the information?

Next week we will meet for another PLC day as a primary team and use the following success criteria to analyze and evaluate both educators and students learning. We are very excited to put our plans into action and discover all the learning that will take place!

Kindergarten Success Criteria

Success Criteria Teachers	Success Criteria for Students
Using descriptive feedback more than just saying wow and passing it along	Having them explain their thoughts and they can have a voice – they can be the teachers too
Planning open ended activities for all entry levels	Demonstrates interests and curiosity by taking risks and attempting to solve problems in play areas
Praise the students based on their abilities rather than their output	Shows willingness to collaborate with others when solving problems.
Allow the students to use their own strengths to communicate their thinking	Displays persistence when facing challenging problems and when experiencing failure
Using intentionality in placing the manipulatives and allowing the students to be creative in their explanations.	Seeks opportunities to extend their mathematical learning and make connections to real life situations
Using your language to help pull out the proportionally reasoning language (You can go and wash your hands if you are wearing more red on your shirt than any other colour)	Allow the students to use their own strengths

Learning French Sight Words

I gave the group of students a list of French sight words for the first time and asked them to take a few minutes to see if they could figure out how to read some of them. We had a very interesting conversation so I decided to share it below!

Mme: So, have you been able to figure out any of the words?

Student 1: pointed to le and said: This is “Il”!

Mme: How did you know?

Student 1: I recognized the letter.

Mme: So you sounded it out?

Student 1: Yes

Mme: Does everyone agree this is Il? (pointing to le).

Student 2: I think it’s La and this (elle) is Il.

Student 3: No, this one is La! (pointing to la)

Mme: How do you know?

Student 3: Because llll...aaa.

Mme: Does everyone agree?

Everyone: Lll...aaa, la! Yes!

Mme: So are these two words the same? (Pointing to le and la).

Student 1: No, the ending is different.

Mme: What’s similar?

Student 4: They both start with the same letter but end with different ones.

Everyone agreed.

Mme: So, if the first sound is L, can this word (le) be Il?

Student 2, Student 3, Student 4: Noooo!

Student 1: Yes!

Mme: Let’s look at the first letter. What is the first sound?

Everyone: LLLL

Mme: And the next one?

Everyone: EEE (letter name), Leee!

Mme: That is the letter name. If you recall, the French sound is different from the English sound so we were focusing on the English one. What is it?

Everyone: e (short vowel sound)

Mme: I told some of you the French sound last week. Does anyone remember?

Student 2: e (French vowel sound)

Mme: So let’s blend the sounds together.

Everyone: L-e... Le!

Student 4: This one (elle) is Il!

Tia: I agree with Alyss.

Student 1: I do too!

Student 3: Me too!

Mme: That’s great. We will work on that next time for sure. For now, we know 2 of our French sight words - le and la! Le’ts see if we can find them in our French books!

Professional Literature Review: Capacity Building Series: Maximizing Student Mathematical Learning in the Early Years

Source:

http://www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/cbs_maximize_math_learning.pdf

Summary:
This monograph explores how educators can take advantage of students’ mathematical knowledge and experience which they bring to the classroom (p. 1). From the research, in order to establish an environment with rich mathematical opportunities, educators must understand the early mathematics learner and have a good understanding of the mathematics of teaching (p. 2).
Some key characteristics of early mathematics learners:
- students enter school with and abundance of mathematical understandings
- play is “the gateway to engaging in mathematical inquiry... and need not complete for time in the classroom [with other subjects as it should be embedded into the play itself].
-- play that involves mathematics & play with mathematics itself (p. 2)
- manipulatives are great tools to help students build and explain their understanding
- teachers need to allow time for students to reflect and discuss their play in order to make connections between their mathematical ideas and how they are represented in play.

Teachers’ mathematical knowledge is actually linked to student achievement.

“Teachers need to be able to reason through and justify why certain procedures and properties hold true, to talk about how mathematical language is used, to see the connections between mathematical ideas and to understand how they build upon one another” (p. 3).  Even early math concepts of number sense and numeration are actually very complex and it is very important to understand these complexities in order to be able to teach the mathematical skills.

This monograph then identifies some general suggestions and practical tips which I have included at the end of this review. The key ideas are that math is embedded into the day and across the curriculum subjects, that students are encouraged to talk and learn the language to communicate their mathematical thinking, and teachers should be promoting positive attitudes towards math.

Ontario Mathematics Kindergarten Curriculum Connections:
The Ontario Full-Day Kindergarten Curriculum section on Mathematics links directly to the idea that early learners bring prior math skills which need to be further developed in every-day life activities at their level: “Young children use mathematics intuitively and develop their understanding of mathematics through their individual approaches to learning, as well as through their prior experience of their linguistic, family, cultural, and community backgrounds. It is therefore important that children’s existing conceptual understanding of mathematics be valued and that children be introduced to mathematical concepts in an appropriate manner and at an appropriate time in their development... Rich mathematical problems involve important mathematical ideas and arise out of real-life situations, and can be approached in a variety of ways so that all children can be involved in exploring solutions. Solving such mathematical problems requires persistence, since they do not have one easy-to-find correct answer. Through active participation in mathematics investigations, including problem solving and discussions, children develop their ability to use mathematics as a way of making sense out of their daily experiences.” (p. 92 ELKP)

Another aspect in the monograph is that math needs to be integrated across subjects and each concept should not be compartmentalized (p. 7). “When developing their Full-Day Early Learning–Kindergarten mathematics program from this document, Early Learning–Kindergarten (EL–K) teams* are expected to weave together the mathematical processes and related expectations from the five mathematics categories, as well as relevant expectations from other areas of learning (e.g., science and technology, language, the arts). It is important that the study of various aspects of everyday life should permeate young children’s mathematical experiences... Children demonstrate their understanding of these counting concepts in all five areas of mathematics – for example, a child might demonstrate his or her understanding of one-to-one correspondence while analysing data on a graph made by the class.” (p. 93 ELKP).

Personal Reflection:

• What resonated with you?
  Teachers’ mathematical knowledge is actually linked to student achievement. I did not find this surprising at all. I find, in fact, this can be said for any subject matter. If a teacher is knowledgeable and confident, students will ultimately be more engaged and thus learn more. And of course it is difficult to teach something if one is lacking the knowledge/skill/confidence.

“As educators observe student problem solving, they can document what children say, do and represent in order to make both planned and “in-the-moment” decisions about how to respond, challenge and extend student thinking” (p. 4). This notion of observing students and guiding their learning by posing appropriate questions and bringing in the mathematical learning and language at the same time, this is exactly what we are doing in full-day kindergarten play-based programming. And by using pedagogical documentation, I am becoming very aware of my own thought processing and the language and questions that I am using to help guide students in the moment and make both their and my learning visible.

The suggestion to model and encourage “Math Talk” also resonated with me since my colleagues and I are conducting a Number Talks for the first time and inquiring into their application and influence on students. Language and communication is extremely important and therefore I fully agree that fostering an environment and encouraging students to discuss their mathematical thinking is imperative if they are going to be able to successfully solve problems in the real-world. Consolidation time must be integrated into the day to regularly give students a chance to articulate their mathematical thinking and assist them in making connections within their strategies and solutions and make more generalizations (p.3). “Suzanne Chapin proposes five effective talk moves which help create meaningful mathematics discussions: revoicing, repeating, reasoning, adding on, and waiting” (p. 4 & 5). These are also excellent practical tips to use in “Math Talk”.

• What challenged you?
  I found it very interesting that “early mathematics skills were more powerful predictors of later academic achievement in both mathematics and reading than attentional, socioemotional or reading skills” (p. 1).  I always thought attentional skills had a high impact on student success. I am not surprised that early math skills are indicative of future achievement but I would not have thought they are more powerful predictors than reading level. However, upon reflection, it makes sense that early math skills are powerful because math is about problem-solving and this permeates all subjects both in the world of academia and in life. Thus, this fact should really influence our teaching and place a high priority on the importance of guiding students to further develop their mathematical knowledge, understanding and communication skills related to solving everyday problems tailored to their interests and developmental level.

I wasn’t surprised to hear that “while technology was being used in many classrooms, its potential to promote student thinking was being greatly underutilized” (p.2). According to Yelland and Kilderry, many computer tasks in primary classrooms had the equivalent complexity of “pencil and paper mathematics tasks, were teacher-directed and anticipated correct answers with narrow solution strategies” (p. 3). There are so many computer games that focus on one particular question with only one possible answer and seem to be the drill type of questions we are supposed to be moving away from. This type of activity is mindless and rote and without any real-life context.
It is recommended that educators should offer activities that are “multidimensional mathematical tasks ensuring both student input into the direction of their learning and supporting more varied learning outcomes (p.3). These types of activities are certainly ideal but most definitely challenging to invent and unfortunately, this monograph does not offer any examples. I must admit, even many of the technology resources that myself and my classmates discovered were mostly drill type of games.
I read an article on the SAMR model in the last year, which I roughly recall but it commented on the stages of using technology to innovate our education. Providing students with the opportunity to use technology to complete a task in an alternative way was on level of using technology but to actually invent a task that could not otherwise exist without technology is truly the most innovative way and I would love to know how to do this in the kindergarten classroom and link it to ‘playing with math’. I find we tend to resort to these simple, one-solution type of games for children since it is fast and easy for students to use. It is much more time-consuming to actually teach early primary learners how to use the technology in order to show their thinking. It think this would be the necessary step in order to provide students with opportunities to use technology to represent their mathematical thinking. Perhaps this might be a good inquiry for myself: to teach students at least one means of using technology to represent their mathematical thinking. I could present it to a small group and see how it goes.

• How might this text be helpful/useful in our teaching practice?
  This text gives many important suggestions and tips to get started such as immersing yourself in the curriculum and reading up on the grades below and above your current grade. It is also highly recommended to study these mathematical ideas with colleagues and “together inquire about how your understanding impacts your related teaching” (p. 4). Hence, why there are so many collaborative inquiries being conducted and funded by ministry programs.

Four main suggestions in order to improve student learning and achievement:

1. Identify and use everyday mathematics knowledge to plan instruction
2. Encourage and foster “math talk”
3. Facilitate experiences that allow for mathematization of everyday knowledge
4. Model and nurture positive attitudes, self-efficacy and engagement.

Five practical tips for creating a mathematics-rich environment:

1. Use problems that have meaning for children (both practical and mathematical).
2. Expect that children will invent, explain and offer critiques of their own solution strategies within a social context.
3. Provide opportunities for both creative invention and practice.
4. Encourage and support children by providing carefully scaffolded opportunities which allow them to deepen their understanding and use meaningful and elegant solution strategies and confidently engage in the mathematics.
5. Help children see connections between various types of knowledge and topics, with the goal of having each child build a well-structured coherent knowledge of mathematics.