Word Problems
Co-constructing Understanding in Mathematics
Overview of the SWST Initiative
Socio-demographic Context
The research setting of this report is a middle school situated in an urban part of Peel. There are approximately 625 students who come from a wide variety of multicultural backgrounds with over 35 different home languages spoken, from over 75 countries around the world in attendance.
The Students
The context of the Inquiry was 4, grade 6 math classes. In order to provide focus for the inquiry, 2 to 3 marker students were selected from each class to guide the interventions we planned. These "students of mystery", did not have any known learning disabilities, and according to their teachers, were achieving below their expectations.
The Teachers
3 Host Teachers volunteered to collaborate with the SWST in the Inquiry. They were responsible for delivering the math curriculum to these 4 homeroom groupings.
Research Referenced
Ontario Ministry of Education
Monographs
Academic
Web Casts
Leaders in Educational Thought: Special Edition on Mathematics
Marian Small, Lucy West, Cathy Bruce, Alex Lawson, Daniel Ansari, Doug Clements, Dan Meyer
Open Mindset: Starting with Student Strengths
What were students already doing, or positively responding to?
- open-ended, rich tasks
- independent use of problem-solving framework, PS4
- use of teacher prompts/questioning to guide problem-solving process when students feel “stuck”
- students seemed engaged when working in groups to co-construct mathematical understanding (error analysis exercises, placemat: developing best solution)
- students were referring to one another more as they checked for understanding
- debriefing/Consolidation was when some students used math language, built and solidified understandings, cleared up misconceptions
- the explicit teaching of math vocabulary (cards, student generated glossaries)
- students are able brainstorm comprehensive list of study and test-taking strategies
What were their areas of need?
- to develop grit/perseverance and become more comfortable with ambiguity and not knowing when solving word problems
- some needed further scaffolding of the PS4 framework in the "Try" area
- the more fluid use of the PS 4 framework to other mathematical situations
- the reading (comprehension) of the math problem (understand its structure: scenario, what numbers matter, what actions needed to be taken?
- prioritization of information (what is important? What is the deep structure?)
- visible application of test-taking strategies (manipulative use, highlighting key words)
- engaging in “self-talk” in order to understand the deeper underlying structure to a word problem
- asking the right questions to clarify misconceptions
- metacognition (thinking about the mathematical thinking)
Theory of Action
Our initial observations and conversations led us to the following theory of action which served as a working theory that organically evolved based on new data gathered from student observations.
Curricular Focus
Mathematical Processes
- apply developing problem-solving strategies as they pose and solve problems and conduct investigations to deepen their mathematical thinking
- demonstrate that they are reflecting and monitoring their thinking to clarify their understanding
- communicate thinking orally, visually and in writing using everyday language, developing mathematical vocabulary and variety of representations
Rich Word Problem Challenge
This rich task was performed over 3 days. On the first day students were individually challenged to solve Charlie's Gumball problem. A video showing one possible way of solving the problem was shown to the whole class. During the focus class' math period, all three Host Teachers were present to observe student problem-solving behaviours, and if necessary, to offer "hints". On the second day students worked in groups to compare their solutions. They were challenged to use their collective thinking to generate the best solution to the same problem. On the third day students were invited to take a "Gallery Walk" and to compare their own group's solution to others. They then annotated each others' work with their personal responses.
Charlie’s Gumball’s
http://www.youtube.com/watch?v=7SYZWDYmLf8
Charlie has a giant bag of gumballs and wants to share them with his friends.
He gives half of what he has to his buddy, Jaysen. He gives half of what is left after that to Marinda. Then he gives half of what’s left now to Zack. His mom makes him give 5 gumballs to his sister. Now he has 10 gumballs left.
How many gumballs did Charlie have to begin with?
The PS 4 (Problem-solving 4 ) Framework
During the first month of school all grade 6 students were introduced to the PS4 problem-solving framework. This was used by students throughout the year to help them navigate through a word problem.
Step 2: WHAT? What information is given?
Step 3: TRY! Try to solve the question.
Step 4: WHY? Why is the answer correct?
Collecting Data: Student Reflections on Word Problems
Students thoughts were collected early in the Inquiry to help us understand their attitudes about solving word problems. The following comments came from one of the classes that had done this word problem. This feedback directed the interventions the teaching team planned during our co-planning sessions.
Teacher Prompt:
Exit Ticket: Reflect and write about your experience.
· What still puzzles you about the word problem?
· What strategies worked well for you?
· How did you feel while working through the word problem?
· What questions do you still have?
Student Reflections:
Student: MYA
It was kind of hard. Like out of 10, a 5 or 6. I don’t get nervous but sometimes I just don’t know what to do. It is easy to explain if I know what I’m doing and understand it. I don’t really have any questions but to know how to do a word problem and practice.
Student AZ
Working on that question with the gumballs has hard for me because I’m not perfect at word problems. The words get jumbled up in my head and then get harder and harder. Working with a group didn’t make it easier because sometimes I wouldn’t know what they are talking about. I felt nervous working on a word problem. Watching the video didn’t make a difference because I didn’t understand all of it.
Student: H
The word problem had WAAAY too many words. Reading the question over and over and over. Also, the help of friends. I didn’t feel so good when I was doing the problem because I had a headache trying to think of ways to try and solve it. Why did the teachers have to watch every move we made? I felt REALLY uncomfortable when the teachers were watching our every move, and it was sort of creepy.
Student: AS
I did not like these word problems because I did not understand fully and working in a group did not help me with these. But, the video did help me a little with these word problems. I felt flustered when I did the word problem.
Student: SD
At the beginning I got frustrated about solving the question. When I was working I started to understand, and finally I got the answer. The teacher helped me (get unstuck) by giving me hints. The hint that was most helpful was the “doubling” hint.
Student: BL
Well, word problems were never a problem for me, not yet. My mom started giving me extra-hard training on problem solving when I was small, but sometimes the problems are so hard, they make mince meat out of my brain. To me, the problems are identifying what type of “math” I should use like division, multiplication, addition or subtraction, and trying to make the question out of all the description. Like this one, there so many words and the description was so long, it muddled my head. And the question- what did he begin with WHAT!!!! These type of things balls me up. I just want to find a way out of this.
Student: N
When I read the paper I was confused. So I read it over and over until I got it. I was using manipulatives which helped a lot. Then, I got the answer so I started to add. I was almost stuck till the answer popped, and then boom, it popped in my head and it was right, I hope. For me to explain was the hardest part. I still want to know if the answer was right. The answer I got was 120.
Student: HR
-I kind of like word problems
-I think for a bit before
-I used blocks for help
Student: RA
I really didn’t understand the problem in the beginning after we played the video it really helped. Then after I saw it maybe if I guess and check then after I show base ten blocks that really helped be because I took so many base ten blocks. I took 120 of them then I did one for me and one for Jayden. Then I got 60 for him. Then I knew that was the half of 120. Then I thought that Zack, Marinda, sister would have it between 60. So, I subtracted is from that cuz 5-sister 10-to me. I had 55 the I did the same one to me and one to her I got 30 for her the I did Zack. I got 15 for him and 15 left. 5 to sis and 10 left to me that’s what I did.
Student: NA
When I first say the word problem I felt like I had no clue how to do it. After Ms. M helped me then I understood what to do. But before that I didn’t understand how to get the half of 4 numbers. The strategy of dividing to solve the problem helped me the most. It felt like a lot of work at first, but then after Ms. M explained it, then I was like oh that was not too much work. The question that I still have is how to solve it in the way of graphing the info.
Student X
When Miss G, Miss A and Miss M came into the class and started teaching us math, I was excited. Me and my group worked really hard to get the answer. When my group worked together it was fun and we figured the answer together really fast. The hard thing was when we didn’t work as a group and when I didn’t know what to do. I felt confused, but when I saw the video, it was easier for me to do the answer because it told me a way to figure out the answer.
Student AP
· Something that still puzzles me is that I knew what to do but I just didn’t know what numbers to use.
· One strategy is working backwards.
· When I was working through the word problem I felt the word problem was hard at first, but when Ms. M explained it to me it made sense.
· Something that helped me was when we watched the video.
Student D.F.
Guessing and checking because when I started to understand it better the guessing and checking was really easy for me. So, I got the answer faster than I expected. I didn’t really like word problems before but since we did the problem and “worked” in groups it makes a little more sense.
Student: A.G
1. How to know what to do first?
2. Working with a group it felt good. So my peers can help me.
3. I felt crappy because it was hard, you did not know what to do first.
4. I did understand it a lot when I worked in a group!
Student E
As soon as Ms. M and Ms. G walk in I thought it was some sort of presentation. At the first glance of the piece of paper, the question seemed simple. Working backwards was probably the quickest strategy, and that is the first strategy I thought of. The question was easy to solve but gets confusing to show the classmates. I used pictures working backwards and PS4. Almost all individuals can solve the question with working backwards and if they fully understand the question. There was no stress incorporated while working. Overall, the question can be solved in first glance to me. The video is pretty much saying what the paper said so I prefer the paper. Word problems for me are not much. It is pretty much solving an equation except with words.
Student S
One thing I did not understand was when I saw some of the other’s strategies. I started to wonder how they got the answer but afterward I understood. I changed my mind on disliking word problems because now I have more ways to solve problems. Before I did not like word problems but now I can use fractions and pie graphs. When I first heard the problem and watched it, it was easy for me. The word that kept on helping me was “HALF”. It got very very easy to do the word problem and I also had fun; and no, I like word problems.
Student: N.P.
· A thing that puzzles me is how do you check your work when you get a question like this?
· Work backwards and guess and check
· I felt I wasn’t sure about solving the puzzle because I didn’t know what to add.
· I don’t have any questions but I can understand the question when I watched the video.
Burning Questions
- How can we help students engage in meaningful “error analysis” so that they can not only identify the error, but fluidly transfer their understanding to new questions?
- How can we help students understand the deeper structure within word problems?
- What are the most efficient ways to help students apply their understanding of mathematical vocabulary to the problem-solving process?
- How can we help students connect the use of manipulatives to deepening their mathematical understanding?
Learning Goals
During our co-planning and co-debriefing sessions the team determined the overall learning goals for our Inquiry which provided us with a focus for our intervention strategies.
- We can identify the key mathematical concept learned today.
- We will use accurate mathematical language to analyze a problem.
- We will choose an appropriate model in order to best show our understanding of the math ideas in the problem.
- We will break the word problem down into steps, and connect each step to a mathematical operation.
- We will work collaboratively to build our understanding,
Developing Success Criteria
- I have used correct math language, notations.
- I chose manipulatives/models to clearly show my understanding, or that validate my mathematical reasoning.
- I am able to compare and analyze different models/strategies.
- I can justify why I chose a solution/strategy/model to use.
- I can work with others to build my understanding (listening, sharing, questioning).
- I can recognize the similarities and differences in word problems and use that information to solve a new problem.
What Actions/Interventions Did We Take?
Accountable Talk: Placemat
The groups then walked around "the Gallery" and compared their own solution to that of other groups. Each group then offered feedback in the form of "sticky notes" annotating their response and thoughts on the work displayed. The feedback was valuable because the clarity and efficacy of the solution became the focus of the peer feedback. This gave students another authentic opportunity to use math vocabulary within a rich context.
Error Analysis
Error Analysis:
After a couple of opportunities of analysing provided errors. We followed up by asking students to select a problem they had had difficulty with from a test. They needed to choose a question that did not house a "mistake" (i.e. miscalculation) but that showed an error in thinking, and then solve it. We were interested in determining whether or not they were able to recognize their own gaps, and then identify the necessary steps to close them.
Steps to Error Analysis:
Students worked independently on their own sheet of paper.Next, they paired-up to share their work, and through discussion, collectively built or checked for understanding.
Error Analysis Question
Strategies for Developing Mathematical Vocabulary
Assessment for Learning
The Class Learning Goal Tracker
In order to address students' need for receiving more immediate support, we
decided to ask students at the end of a key math lesson what they believed the learning goal(s) for the session had been. This would help us to check if the planned learning targets had been met.To implement this we developed the Class Learning Goal Tracker. Before exiting the math class each student placed his/her individual sticky note indicating what they had learned, or what was still unclear on to his/her designated spot on the class chart. This would provide us with the immediate assessment for learning data, we needed to revisit, scaffold or build upon key concepts. This data also allowed for some on the spot target teaching.
On Rotating Shapes Student D I learned how to rotate around a point Student M I learned how to rotate shapes 180 and 90 degrees. | On Transformational Geometry Student N I learned how to rotate a shape and I need help with rotating and sliding at the same time. | On Patterning Student M I need more help on patterning. Student R I did good at Algebra. I had some trouble with patterning but not much. I understand most of it. I learned how to make different pattern rules. Student MA I learned how to make rules for different patterns. I learned how to make the next patterns following the rule. |
On Rotating Shapes
I learned how to rotate around a point
Student M
I learned how to rotate shapes 180 and 90 degrees.
On Transformational Geometry
I learned how to rotate a shape and I need help with rotating and sliding at the same time.
On Patterning
I need more help on patterning.
Student R
I did good at Algebra. I had some trouble with patterning but not much. I understand most of it. I learned how to make different pattern rules.
Student MA
I learned how to make rules for different patterns. I learned how to make the next patterns following the rule.
Surveying Our Students: What did they have to say?
Towards the end of the inquiry, we decided specific information from our students was needed to help us understand their personal attitudes towards learning mathematics; and their thoughts on the efficacy of the strategies we had been exploring. Ninety grade 6's completed a survey that the teaching team had created.
Our Key Learnings
On increased ownership of learning:
Increased Confidence:
All 4 teachers observed increased levels of self-confidence amongst the students. We interpreted the increased quantity and quality of student questions asked as important evidence. We also heard them using key vocabulary to make their thinking more visible.
Increased use of manipulatives:
By the end of the inquiry cycle students were more readily electing to use manipulatives/models (e.g., white boards, fraction circles, fraction towers, base ten blocks) to work through word problems. And they were less dependent on teacher prompting to do so. Indeed almost all of the surveyed students ranked working with manipulatives as a word problem strategy that was either helpful, or very helpful.
The teaching team felt that over the course of the the 4 months, using manipulatives in the math classroom had been reframed in the students' minds, and it was no longer associated as a strategy that marked the struggling math student. Students were demonstrating that by using manipulatives they were building and verifying mathematical understanding.
On Use of the Problem-Solving Frameworks (PS 4):
Over the course of the school year, students were able to flexibly refer to the problem-solving framework introduced to them in the fall. When they did use it, they had a default structure to help them outline their mathematical thinking process. This allowed students to think about the math in the word problem, as opposed to expending working memory on how to express their thinking.
We noticed that students who were initially struggling with a word problem were able to get further into finding a solution when they used PS4. This was particularly the case in situations where students described not knowing what to do. In fact one of our marker students in a unit test situation was able to achieve a level 3 in her response to a patterning and algebra word problem when using the PS 4 framework. However she scored a level 1 in the section of the test where she filled in missing numbers (fill in the blanks) to a provided pattern. Even though the word problem might appear more challenging, she was able to use the framework to understand the question, and put it into a context that made sense to her. Whereas, the level 1 indicated that she seemed to have difficulty connecting the "fill in the blank" questions, to the concepts that the class had been working through.
We were surprised that students did not consistently apply PS4 to diverse types of mathematical questions. Some seemed to associate using it only with word problems. We found it interesting that not all students were able to see that this framework could also be applied to number sentences. Some students also may have opted not to use it because they felt "it took too long". They were unable to see, or value, its connection to successfully navigating a problem.
However, it is important to note that their response in the survey showed that students did find the framework effective as over 75% listed the PS 4 as being helpful to very helpful when solving word problems.
On the Efficacy of Co-constructing Knowledge:
Working with others led to an increase in student engagement and participation levels. We noticed that the end of the math period seemed to regularly take us by surprise, and students often requested more time than the allocated math period to work through problems. Students were also able to articulate to the SWST the positive impact discussing mathematical ideas with peers had on their learning.
Student T: I like it (working in groups) because we share strategies.
Student H: It (the placemat) helps to understand how others think.
Student W: It helps because we share strategies and we find where we go wrong.
Student R: I find I understand better when I work in a group, than when I work alone.
We observed that "Accountable Talk" became the "default" approach to problem-solving which led to a deepened mathematical understanding, and increased use of math language. At the same time, we were not sure if this directly translated into an increase in student test scores for our marker students. We only started to notice the positive impact to grades of this work by mid-May. We believed that with more time to internalize the thinking processes, the marker students whose marks appeared static, or shifted back and forth, might also show improvement in test performance.
On Developing and Applying Mathematical Vocabulary:
Students became more metacognitively aware of the importance of expanding their math vocabularies, and took measures to do this (vocab. cards, glossaries). However, we observed that they were not always certain about how to apply this knowledge to solving word problems. For example, when encountering difficulties in creating equivalent fractions, students needed to be prompted to return to their definitions and examples of the relevant terms. We wondered whether students saw creating the glossaries, and vocabulary cards as being isolated from working with the mathematical concepts.
On "Fixed" vs. "Growth" Mindsets
We were interested in seeing what attitudes students had about learning mathematics. The teaching team felt that if students could start to see that making "mistakes" was a means of building knowledge, this attitudinal shift would empower them to persevere, and thereby, more deeply investigate word problems.
In response to the survey question:
Is it important to get the right answer? Why or why not?
24% did not answer the question.
43% demonstrated a growth mindset, and articulated that "mistakes" were useful because they helped students deepen their understanding.
Student Mn:
No, it is not important to get the right answer because if you get the wrong answer, you can learn the real answer. You can also learn the strategies used to get the right answer. Not everything is about getting the right answer, you learn more about getting the right answer.
Student K:
It is important because it shows that you know about math. But also if you get the wrong answer if is helpful to figure out where you went wrong and to fix it. So, you can get the wrong answer, and still know what you're doing.
Student H:
No it isn't because if you use the right strategy it is better than getting the right answer with the wrong strategy. Using the right strategy shows that you understand the question and the main math ideas.
Student M:
Sometimes if you get it right the first time you don't have any idea what could be the process. If you get it wrong you can get it wrong and learn more from your mistakes.
Student Kf:
I think the question is not about the right or wrong answer. It's more about Thinking.
Student Hb: (recognizes that anxiety impacts her performance)
Not always. It makes you feel stressed out and sometimes I can't do the work when I'm stressed out, or I can't think of getting the answers at times.
30% demonstrated a fixed mindset. They expressed a belief that getting the right answer and high marks was the purpose of learning math, and that it would directly impact their future success.
Student A: Yes, because I want to prove to people that I am smart and that I don't give up on everything.
Student M: Yes it is because you get a better mark.
Student K: Yes, because if you get the wrong answer you're not learning to do it right and you'll be wrong all the time.
New Questions
- How can we help students apply their knowledge of math vocabulary to solving word problems?
- How can we help students close the gap between a more deepened mathematical understanding and test performance?
- How can we help students transfer the skills that help them succeed in other areas to mathematical problem-solving?
- How can we help students apply what they know about test-taking strategies to test-taking situations?
- Can we use Bansho/Board Writing as a means of mobilizing student knowledge?
Soula Katsogianopoulos. Hon BA, BEd, MA
Peel District School Board
Ontario, Canada