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Brookmede Public School and École catholique St. Antoine
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A sub shop sells 13 different subs and six drinks. If you want to buy one drink and one sub, how many possible combinations could you choose from?
Do you think pairs of Grade 3 students could solve this problem?
Teachers at Brookmede PS have embraced a powerful and effective approach to teaching math. Students are engaged, on task and having fun solving complex problems without learning traditional algorithms.
Gone are the days of teacher-centred math instruction when students memorized algorithms (rules) and were drilled on what they learned through a seemingly endless series of increasingly difficult worksheets. Gone is the uncertainty about whether students understand underlying concepts and can apply them to complex problems. And (dare we say?) gone is the plain old boredom of students who have been turned off math.
In recent years literacy and numeracy have been major focal points in schools throughout the province. In May 2002 the government announced it would spend $25 million to expand the year-old Early Reading Strategy and to create an Early Math Strategy to help primary students improve mathematics understanding and develop the math skills necessary for the 21st century. In 2003 the Ministry of Education asked expert panels to examine numeracy and literacy issues for students at risk.
This spring the Students-at-Risk Expert Panel on Numeracy completed its examination. Janine Griffore chaired the French-language side of the panel. "Our work was designed to support the acquisition of numeracy skills for Grade 7 to 12 students at risk - those 25 or so per cent who may not earn their diplomas," say Griffore. "But we think our report and its recommendations are beneficial to every teacher and student in the province." The report aims to provide school boards and teachers with both a framework and specific strategies to ensure that all students, especially those at risk, acquire the numeracy skills they will need to succeed on their chosen paths beyond secondary school.
It seems that one of the biggest challenges is a negative attitude toward math. Many people - teachers included - perceive math as difficult, avoidable and the domain of a smart and select few. There's a sense that most of us can survive without strong math skills. Have you ever heard (or said), "I didn't do well in math and look at me, I'm successful."
It's no surprise then, that the report's key recommendations emphasize the importance of numeracy in students' everyday lives. Good numeracy teaching must be concrete, hands-on and show students the relevance of what they are learning. And foundations for numeracy must be laid early.
This year's students-at-risk panel raises similar points to those noted by the expert panel on early math in its report last year, which stated that "Many elementary teachers have mathematics anxiety and this anxiety needs to be addressed in professional development activities so that teachers may aquire a positive attitude towards mathematics. An effective teacher needs mathematics knowledge, a comfort with and confidence in mathematics, an understanding of how children learn mathematics and an understanding of effective instructional and assessment strategies."
The report recognizes that teachers need regular school-based professional development (PD) support in collegial learning communities, rather than one-time PD sessions.
Irene McEvoy, Peel District School Board's (PDSB) Instructional Co-ordinator for Mathematical Literacy, says, "We've taken to heart the ministry's move towards improving instruction in mathematical literacy. We know that teacher knowledge of math and effective teaching skills are critical to student success and we are working to address this."
The PDSB has used ministry funding to train one teacher at each of its elementary schools in early math strategies. The teachers then share what they've learned with colleagues. The board also offers various supports for teachers to improve their math knowledge and teaching skills, including book talks on best practices and on teaching resources, summer institutes, regular PD sessions and AQ courses.
"Our goal is to have good math programming in place - starting in Kindergarten - delivered by teachers who have solid knowledge of mathema-tical content, pedagogy and assessment, as well as the ability to identify needs for and deliver differentiated instruction in the regular classroom," says McEvoy.
Easy as one, two, three
One of the vital elements in ensuring a school provides good programming is strong administrative support. A principal with a passion for math and a desire to create a strong learning community is a bonus. Bonnie Jaakkimainen, the principal of Brookmede PS, believes that math is key to students' education. "Math is not simply about numbers, it's also about language. Math and literacy cannot be separated and the best teaching and learning require students to both communicate and problem-solve," she says.
Teachers Jessica Reiter, Lorri Scott and Sherri Barrow, along with their board literacy-resource-support teacher Karin Milne, took a math AQ with Dr. Alex Lawson, an expert presenter to the early math panel. They are now passing on what they've learned to colleagues at Brookmede.
Jaakkimainen explains the effect that this one course has had in their school. "In many of our classes we have completely changed the way we teach math by adopting a three-step problem-solving or constructivist approach." The three steps involve:
Students in Kindergarten, Grade 3 and Grade 5 classes are given one engaging, complex and familiar problem, like the one about the subs and drinks. Pairs of students of similar mathematical ability work together to find the answer. But the answer is not the sole focus. Students must be able to explain how they solved the problem and must do so using at least two different methods.
Grade 3 teacher Jessica Reiter explains that it took several weeks at the beginning of the year for students to become competent working in pairs and explaining their strategies. "We used simple problems and manipulatives like the 100-pocket chart to develop basic numeracy skills such as addition and subtraction. And we used reliable problems, such as those in Marilyn Burns's books."
Once students have a conceptual understanding of an operation, teachers introduce symbols (the multiplication sign) or tools (a T-chart). When students display a conceptual understanding of an operation, games are used to reinforce basic facts. "We don't teach algorithms, we simply provide problem-solving opportunities for developing concepts and skills," says Reiter. "Most importantly, we leave it up to the students to decide how to find the answer."
Pairs of students were given the problem, some chart paper and markers. Weaker pairs were given the same task with smaller numbers. All were instructed to solve the problem using at least two different strategies. Pairs had to explore and test each strategy and be able to articulate as well as show visually how their strategies worked.
Students eagerly accept the task. They are familiar with the routine and what is expected of them. They keenly pick up their materials and scatter throughout the classroom, working on the floor or at desks, as suits the various partnerships and learning styles. The talk is focused on solving the problem. At first, the bright ones simply want to record the answer, but when they realize they must show how they arrived at the answer, they refocus. They may know the answer sooner than other students, but having to explain how they arrived at it slows them down - forcing them to examine and articulate their process.
Partners in learning
Jaakkimainen reminds us of the importance of pairing students. "Students need to work with partners at a similar ability level. For the first time, many weaker students are responsible for their own learning; they can't rely on a stronger partner to do their work for them. Students with similar abilities fuel each other. They have the same questions and often struggle at the same place, but together they can usually solve the problem using a strategy that works for them."
Most students start by drawing pictures to represent the food, mapping out their thinking either in words or numbers and pondering just how many combinations are possible. After a few minutes of focused thinking, some stronger students quickly begin listing and totalling a growing number of possible combinations - using numbers and foregoing visuals altogether. But most students remain visual, drawing 13 subs and 6 drinks, then drawing and counting 78 connecting lines. When students realize that the task of drawing and counting lines is inefficient, they may stop and discuss other strategies.
In Reiter's class, several began breaking the numbers into manageable chunks that could be multiplied (10 x 6 plus 3 x 6). Others continued with direct modelling - drawing pictures, connecting the lines and counting. Most pairs were able to come up with two or more solutions in the allotted time. But the lesson doesn't end when the chart paper is full and everyone has two strategies and an answer.
Next comes the third and most critical stage - the debrief, when students explain their work to the class and consolidate their learning. Board co-ordinator Irene McEvoy explains that many teachers and students initially have difficulty with this step.
"It's critical if students are truly going to learn deeply and see connections to their prior knowledge and experiences," says McEvoy. Students listen to their peers' explanations and might return to their problem, applying what they learned to complete their work - pursuing or modifying their original strategy as necessary.
Teachers facilitate the presentations, asking questions to clarify and pointing out errors as necessary, but at no time are students told the correct answer and approach. And since every student eventually produces a correct answer, all answers and abilities are valued.
Also, because they must show two strategies, students cannot simply opt for the easiest solution. If they use a similar approach several days in a row, the teacher can suggest they try a strategy presented by a peer. The introduction of problems with increasingly larger numbers challenges students to become more efficient in their solutions.
This problem-solving approach is also being used in Senior Kindergarten (SK) and Grade 5 classes at Brookmede. In SK, pairs of children were motivated to solve a problem involving suckers. (See sidebar, page 43.) Kindergarten teacher Sherri Barrow has pairs of children use paper and pencil to find the answers and show how they got them.
As is often the case with group work, not all students parti-cipate equally. But with regular use of this problem-solving method, teachers can begin to identify and monitor weaker children, correcting inappropriate approaches and helping where necessary.
Barrow notes, "In the end, students are accountable for their own learning because they must explain how they solved the problem. If I feel students don't fully understand I can question their visuals, asking 'Why did you put that there?' or 'What does that represent?'"
Meanwhile, stronger pairs quickly begin to work independently - asking and answering their own questions and, as was the case with the Grade 3 students, even devising additional strategies.
Power of five
Margaret Allen used the same problem-solving approach in her Grade 5 class for the first time this year. "The differences between this year and last are stark. My students are more engaged. They use mathematical language with each other. They can explain how they solve problems and they are better able to apply what they are learning. They've been empowered and that's tremendously satisfying for me. Best of all, we all look forward to math time, unlike last year when it was a much less popular time of day, for both me and my students."
"This new approach is not really difficult," says Jaakkimainen. "But it's not something one can do overnight either. Our teachers are now teaching each other - sharing ideas, strategies and resources."
Jaakkimainen is excited about the future. "I can't wait to see how our students perform on this year's Grade 3 EQAO math test. I'm sure they'll do well. But what's more important is that they are really enjoying math and fewer students are struggling."
The exemplary program at Brookmede PS is clearly laying a foundation for math that students will be able to build on in Grades 7 through 12 and beyond.
Meanwhile, schools such as École élémentaire catholique St-Antoine in Noëlville - a rural community southeast of Sudbury - of the Conseil scolaire catholique du Nouvel-Ontario (CSCNO) are finding ways to engage their Grade 7 and 8 students in the world of math.
Paul Henry, the principal at the school and a member of the province's expert panels on numeracy and literacy, says that St-Antoine has integrated technology, literacy and numeracy throughout much of its curriculum. He points out that technology has a lot of appeal for students and is therefore a very useful resource for teachers who want to improve students' numeracy and literacy skills.
Robots and real life
St-Antoine's Grades 7 and 8 students have a program that integrates math, science and technology. Building robots that operate on computer-generated commands, students use math skills to calculate distances and inclines and to determine proba-bilities for lifting and dropping objects.
"They use math skills every day in this program, and they quickly see that math is necessary in all areas of their studies, not just when it's time to take out their math books," says Henry.
This school board places special emphasis on preparing elementary students for the Grade 9 EQAO test and provides teachers with extra math training and resources - including some from the Ottawa-based Centre franco-ontarien de ressources pédagogiques. One of the programs, Avos-Marques, teaches students to answer math questions using high-school-level approaches and formulas.
Henry notes, "The program challenges students to enjoy numeracy and to see it as an everyday must-know skill, and it encourages them to explore alternate solutions to problems."
One particularly interesting program launched at École St-Antoine places students with local businesses. All Grade 7 and 8 students spend an average of one hour per week with a local employer during the school year. Henry explains, "When the project was launched, the business people showed an uncommon degree of enthusiasm. They saw it as a way to participate in educating our young people."
This year, businesses included the credit union, local grocery and hardware stores, a pharmacy, St-David Catholic Church, the library and the local arena. Students used numeracy skills in most placements and, according to Henry, "they were surprised that math was so common and necessary in the workplace."
Students working at the arena had to calculate the building's square footage to determine how much new flooring material the operators would order. Students working at the credit union counted deposits and helped prepare budgets and forecasts.
In the classroom, teachers' repertoires of textbooks and worksheets have also expanded to offer students practical and engaging learning activities. Students do industry-related math calculations, solve real-life problems and are encouraged to work in unique ways, including via team projects, math debates and contests. They also regularly coach each other and tutor students in lower grades.
As a member of the numeracy and literacy panels, Henry is well aware that many students have trouble when they encounter the predicting components of Grade 9 math and the EQAO test. To address the problem, his Grade 7 and 8 teachers have created special integrated activities to help students sharpen their predicting skills.
During science experiments students use graphing calculators (traditionally not used until high school). Once they know how to make graphs by hand, they learn to use sophisticated calculators to predict outcomes. "When we reinforce the same skills across various subjects, students learn them better."
"These teaching and learning activities show students the myriad skills they need to be successful in life," Henry concludes. "The experience shows them the importance of numeracy - both in their studies and their everyday lives."
Janine Griffore, chair of the French side of the Students-at-Risk Expert Panel on Numeracy, is optimistic. "In three years I hope schools will have implemented interdisciplinary teaching and learning activities that help students acquire skills and demonstrate what they have learned - as is already the case with literacy." If this happens, she is hopeful that we'll have fewer students at risk.
"I want to see a strong culture of sharing exemplary work within schools and among boards," Griffore adds. "Teachers are already doing excellent work and we need to share it."
Kindergarten sample problems
Students were asked to solve a problem. "I have three bags of suckers. In each bag there are four suckers. How many suckers do I have altogether? Do we have enough for everyone to have one?"
The students figured out that there would be enough for just the senior students - but to be fair they thought everyone should get one. They counted how many children would not receive one and decided the teacher should go shopping and buy more.
Two of everything
This problem uses the story Two of Everything by Lily Hong. In the story, whatever goes into the magic pot doubles: one hairclip turns into two, etc. The teacher read the story and then posed the problem. The students were asked to figure out how many items would come out of the pot if two things were put in and if three things were put in.
- Sherri Barrow, Brookmede PS
Grade 3 suggested activities
Use Marilyn Burns's Lessons for Introducing Multiplication, Grade 3, page 66. Children pretended they worked for a chocolate factory. They had to figure out how many different square or rectangular boxes could be made to contain 6 chocolates. Each chocolate measures 2 by 2 cm. Then they did 12 and 24 chocolates.
Amanda Bean's amazing dream
Use Marilyn Burns's Lessons for Introducing Multiplication, Grade 3, page 23. In the book Amanda Bean's Amazing Dream by Cindy Neuschwander, there is a picture of a group of lambs on bicycles, wearing caps. We gave the children the following problem: How many caps, legs and bicycle wheels are in the picture?
Use Marilyn Burns's Lessons for Introducing Multiplication, Grade 3, page 48. Problem: Which has more wheels - five bicycles or seven tricycles? Children need to find out which has more, and justify their answer.
Two of everything
Use Lessons for Algebraic Thinking, Grade 5, by M. Wickett, K. Kharas and M. Burns, page 3. We read the book Two of Everything by Lily Toy Hong (see Kindergarten problems), which tells the story of a magical brass pot. Children were asked to use a T-chart to show the pattern of doubling. The lesson also introduced the concept of algebraic symbols, and the symbol representing multiplication. Later, we used the idea of the magical pot as a tripling, then a quadrupling pot.
An ice cream shop sells vanilla, chocolate and strawberry ice cream. It has sugar cones, large cones and small cones. How many different combinations of single-scoop ice cream flavours and cones can be sold at this shop?
Our class orders 26 slices of pizza each week. How many slices of pizza will we order in 7 weeks?
- Jessica Reiter and Lorri Scott, Brookmede PS
Recipe for a 60-minute constructivist math lesson
Introduce the problem (5 to 10 minutes)
Present the students with an engaging problem. The focus of the lesson will be the strategies used to solve the problem and students' justifications and explanations.
Do not model strategies for how to solve the problem. Allow children to generate their own strategies and to make meaning while finding solutions.
Working partners (30 minutes)
Partner children of similar math ability and ask them to work together to find the solution to the problem. As students become more experienced they should use at least two different strategies to come up with the solution. Insisting that students use more than one strategy helps deepen their understanding and helps them move toward more sophisticated ways of solving problems, eventually arriving at efficient algorithms of their own.
When students of similar abilities work as partners they become more confident. They are more willing to take risks, they can challenge and extend each other. Students become more responsible for their own thinking when they can't defer to someone they see as smarter.
Pairing two higher-needs students means they will not be able to rely on someone else to do the work for them. Left to do their own thinking, they are able to solve problems and explain at their own level.
Over the year, these pairs form professional relationships with one another. Provide students with chart paper and pens on which to show strategies and solutions that explain or illustrate their thinking.
Debrief - share strategies, explain thinking (15 minutes)
This is the most important part of the lesson. Students have an opportunity to share their strategies with the class. Moderate with prompts, summarizing when necessary. Students have a chance to reflect on their strategies and those presented by their peers. Generally, students come up with a working strategy of some sort. All responses are valued, so a range of strategies is shared at debriefing. With exposure to a range of strategies over a period of time, students develop more efficient methods of solving problems.
When a teacher teaches an algorithm (how to solve the problem) students just follow a set of rules, often not making any meaning while doing it. They are not thinking on their own.
When children develop strategies based on their own developmental level of understanding, they acquire a firm grasp of math concepts and an ability to communicate and explain that understanding to others.
- Bonnie Jaakkimainen and the staff of Brookmede PS
Lessons for Introducing Multiplication, Grade 3, by Marilyn Burns (Math Solutions Publications, 2001, www.mathsolutions.com)
Lessons for Algebraic Thinking, Grade 5, by M. Wickett, K. Kharas, M. Burns (Math Solutions Publications, 2001, www.mathsolutions.com)
Elementary and Middle School Mathematics, Teaching Developmentally, by John A. Van de Walle (Virginia Commonwealth University, 2004)
Amanda Bean's Amazing Dream: A Mathematical Story, by Cindy Neuschwander (Scholastic, 1998)
Two of Everything, by Lily Toy Hong (Albert Whitman and Company, 1993)