From Hands-On Problem Solving: A Minds-On Approach
By Jennifer Lawson

For grades 1–4
Coming Soon for grades 5–6

Problem solving is central to learning mathematics. By learning to solve problems and by learning through problem solving, students are given numerous opportunities to connect mathematical ideas and to develop conceptual understanding. Problem solving forms the basis of effective mathematics programs and should be the mainstay of mathematical instruction.
—The Ontario Curriculum, Grades 1–8: Mathematics

Although students may have a good understanding of mathematical concepts, they can still have difficulty applying their knowledge in problem-solving activities. They may not view problem solving positively and may not have internalized a model that can guide them through the process. In order to change this perception and help students solve problems, Hands-On Problem Solving includes an effective four-step problem-solving model—think, talk, solve, and share.

FOCUSING ON STEP 2: TALK

The process of communication is an essential element of mathematics. Once students have been presented with a problem and provided time to think (step 1) about it, the next step is to have them share their ideas, as teachers probe their thinking with critical questions.

Encouraging students to ask questions is a critical component of the “Talk” step. Engage students in such dialogue by regularly asking them:

• What questions do you have about the problem?
• Is there a question you would like to ask me (the teacher) or your classmates?

The talking stage of the problem-solving process is also a time to have students do the following:

Identify important information in the problem.
Encourage students to point out any essential facts in a word problem. Have students use highlighters to mark this information or use coloured pencils to circle important words when working on a problem. For example, in the following problem, important information is shown in bold text. You may wish to discuss why it is not necessary to consider the 5 days in this problem and for whom the soup cans are being collected.

Jinan and Tegan are collecting cans of soup for a local soup kitchen. They collect soup cans for 5 days. Jinan collects 13 cans of soup, and Tegan collects 9 cans. How many cans of soup do they donate to the soup kitchen?

Identify unimportant or extraneous information in the problem.
Many problems present information that is not required to solve a problem. This kind of information can distract students from focusing on essential information and can therefore influence their success in solving the problem. It is important for students to identify the information that can be considered non-essential to the problem-solving task at hand. Have students use a pencil to underline the unimportant information when working on a problem. For example, using the same problem as above, the non-essential information is underlined. The non-essential text sets the context, but is not vital to solving the problem. Neither the 5 days nor the soup kitchen are important since the total number of cans collected has been identified.

Jinan and Tegan are collecting cans of soup for a local soup kitchen. They collect soup cans for 5 days. Jinan collects 13 cans of soup, and Tegan collects 9 cans. How many cans of soup do they donate to the soup kitchen?

Name the answer.
It is important for students to be able to identify and articulate what it is they are attempting to find out. For example, if asked to name the answer for the preceding problem, the response should be “cans of soup.”

Share possible strategies.
Encourage students to talk about various ways to approach a problem, including the use of manipulatives, pictures, symbols, and calculators, as well as any personal strategies they have. This is a time and place to encourage risk-taking and the sharing of new ideas!

During the “Talk” stage of the problem-solving process, it is important that students discuss only the process and strategies—not share possible solutions to the problem.

It can also be beneficial for teachers to engage students in talk during the problem-solving process. For example, students may be asked or told:

• What do you notice?
• Are there any challenges that we need to discuss?
• Take a walk about, and discuss your work with other students. Ask questions of each other.

This process helps to build a community of learners.

We learn to write by writing . . . Provide frequent opportunities for writing throughout the day. Students need time to write in reflection of their learning, to record new ideas, generate stories and poems, and to communicate information. (Brownlie, Feniak, McCarthy, p.13)

Learners often find writing to be a challenge. Writing in another language may involve even greater challenges both for beginners learning to write in English, as well as for students honing and advancing their writing skills as they learn to communicate about content in English.

If the focus is on the instructional climate in the classroom and it is one where caring relationships and learning with and from each other take precedence, then ESL students and all students will learn best. Mistakes need to be treated as okay, expected and part of learning.

Why learn to write without constant correction and editing?
Beginners who write with constant editing and correcting may shut down, impeding rather than supporting learning. When beginners are learning to write in English they need to be able to take risks and make mistakes without fear of constant correction which may seriously inhibit taking these risks. We want beginners to experiment with writing as they are learning and to focus on the process not product.

How can we encourage ESL learners to write and increase their writing skills?
To learn to write, ESL students need to be taught to write. They need exposure to many thoughtful writing activities throughout the school day so they have the time it takes to learn to write well. Classroom teachers and resource teachers help ESL students write best by creating a classroom community of writers where:

• Ample time is given for writing and everyone writes throughout the day

Writing takes time. Note taking, organizing sentences and paragraphs, thinking about what to write and how to express ideas takes time. Give students the time they need to learn to write across projects/content.

• Peer to peer support in pairs and small groups is highlighted and encouraged

Students learn a great deal from each other. They are also more engaged in the classroom when they participate actively in learning with and from each other. Take every opportunity to involve students in writing activities.

• Students use the writing process for learning

Descriptions of the writing process involve any/all the following tackled in authentic situations – draft, write, edit through conferencing with a peer or adult, and publish. When we teach students how to do this well, they learn to edit their own work and can organize their own writing.

• Writing activities are intentionally modelled and scaffolded

Model how to structure writing so the process is clear in an open ended way that provides a framework that over time can change with a growth in skills. Providing a formula which never changes may inhibit growth. For example, talk about a big idea within a content area, have students work in mixed ability groups and take a few minutes to write questions about the idea. Share questions orally, record the questions. Or, the teacher writes the big question and discusses it aloud with the students writing down key ideas. Take notes aloud with the class, organize them together and write for a purpose. Model the entire process all the way to the completed writing example – don’t skip or rush steps.

• Vocabulary is intentionally highlighted through asking questions, and note taking

Focus on discussing the specific vocabulary students need to know to answer questions so they can learn how to use it in writing activities. Think aloud about vocabulary for example: Which word(s) are most important in this text? How would this word be used in writing?

• Thinking and writing are connected

Learning is meaningful to students when they are most engaged in reflection, analysis, and synthesis. As examples: when they compare and contrast people, places, events and ideas, or they reflect on their own learning identifying what was personally most important, or they write for a variety of audiences and purposes in different ways to persuade, challenge, summarize main ideas or innovate.

Teachers and other resource staff can work together to support writing, sharing expertise and providing more than one set of hands to help students grapple with the writing process. When students get stuck, one staff person can give short, specific mini-lessons to help a small group.

In Instruction and Assessment of ESL Learners: Promoting Success in Your Classroom, we demonstrate the writing process and give examples of writing activities that work for ESL students.

Guest Contributor
Vicki McCarthy, PhD., Author

P&M Press is pleased to announce that 7 Generations has been nominated for Best Illustrated Book of the Year and McNally Robinson Book for Young People Award – Older Category. Congratulations to author David Alexander Robertson and illustrator Scott B. Henderson.

The awards will be presented at the Manitoba Book Awards gala, on Sunday April 28^th at the West End Cultural Centre and hosted by Ismaila Alfa.  Doors open at 7:15 p.m., with the ceremony beginning at 8:00 p.m. The event is free and open to the public.

The shortlists and recipients are selected by a variety of juries, comprised of writers, publishers, and other book industry personnel from across Canada.

For more information about this award, please visit www.mbwriter.mb.ca.

A great way to introduce The Learning Pathway to students at any grade level is to have them count collections of items in the classroom. By observing students as they count, you have the opportunity to “see” how they are applying their mathematics skills. For example, students may know the rote-counting sequences of 2s, 5s, and 10s, but count their collection one at a time. This is an opportunity for you to step in and discuss with the learners how to apply their knowledge of skip-counting to help them efficiently count a collection. As well, teachers can look for students who use their knowledge of dice patterns and ten frames, rather than random piles of items, to visually organize their collection. By knowing how students apply their math skills, you can determine where on the learning pathway they are.

Teachers will find that using The Learning Pathway:

• Helps them group students into signposts (according to their mathematical knowledge)
• Provides a frame of reference for what to look for when students are counting
• Helps them document what strategies students are using when they are counting
• Guides them to ask purposeful questions by looking within the signposts for the other math that students need

The real-life examples below provided teachers with opportunities to make connections between what students do when counting collections and how The Learning Pathway helps inform their assessment and instructional decisions. By using The Learning Pathway as an instructional guide—what to “look for”—when students are counting, teachers made instructional decisions that moved the teaching forward within one lesson.

Example 1

In this activity, the students pointed to each interlocking cube as they counted cubes. According to The Learning Pathway, they are in the 1:1 correspondence signpost. However, the students kept losing track of the count. The Learning Pathway provided next steps to help the students find a more efficient way to count the cubes (e.g., breaking the interlocking cubes into 5s or 10s).

Example 2

In this example, students showed knowledge of the 100 array, as well as of “doubling.” (They made two layers.) When the teacher saw her students were capable of using arrays and doubles, she placed the students in the correct signpost for next instruction.

Example 3

In this example, students counted by 5s, 10s, and identified the benchmark of 100.  Noticing how students chose to group their collections as well as count them gave the teacher a great deal of useful information for both instruction and assessment. By determining what signpost students were in, the teacher used The Learning Pathway to determine the math they could receive next. The pathway can also be used to guide the type of questions to ask students, for example: ”How would you determine half of this collection?”

Example 4

In this example, students began counting their collection by putting the tiles into piles of 10. The teacher realized that these students knew their ten-frame patterns but were not applying their knowledge. Asking the students “How do you know there are 10 in each pile?” led the students to re-arrange their collection into the quick image representation of a ten frame.

A few minutes later, when the teacher came back, they discussed other benchmark numbers they knew, which led to the group deciding to use sets of 20.
The teacher used The Learning Pathway to guide her instructional decisions of the next steps for questioning. She had many options, but the pathway helped her to determine the appropriate instructional step for this group of learners. As a result, she had the students do the following:

• Connect to place value (2, 4, 6 to 20, 40, 60)
• Discuss arrays: How did they see the 20? As 2 groups of 10, 4 groups of 5, or 5 groups of 4?
• Discuss the commutative property of multiplication
• Partition an uneven numbered set into equal groups
• Discuss the relationship between repeated addition and multiplication

Assessment and the ESL Learner

What teachers really want to know is how well their students can think, read and respond, problem solve, and express their ideas orally and in writing. (Brownlie, Feniak, McCarthy, p. 25).

Before beginning any assessment with ESL students, school teaching and learning teams can better support students by asking themselves a fundamental question: What is the purpose of the assessment? In other words, what do we need to know and why?

In Instruction and Assessment of ESL Learners, classroom teachers, resource teachers, ESL teachers, and administrators collaborate to assess ESL students in three areas:

1) learning about students new to a school or a classroom
2) meeting data collection demands related to issues of funding
3) supporting teaching by providing valuable information about how successfully students are learning

1) Learning about students new to a school or a classroom

Once new students have been oriented to the school/classroom and are no longer crippled by culture shock as new immigrants or refugees, it is time to learn more about them to help plan for learning and instruction.

A collaborative school team can administer quick, informal assessments of proficiency that will give information about the new student’s oral language (listening and speaking), written language, and reading comprehension. The purpose here is to provide a starting point for planning lessons and activities – an initial informal assessment. Over time, the team can better monitor the progress of ESL students through curriculum-based assessments within the context of authentic classroom activities, meaningfully adjusting instruction as appropriate.

Completed results are often kept in student files for further reference. They are updated as needed to monitor progress and used to revise the instructional focus to support learning once more is known about what the students can do.

2) Meeting data collection demands related to issues of funding

Where data collection is deemed necessary to meet funding obligations, school teams will benefit from working together to make this data collection as efficient as possible. Here the purpose is to follow the criteria established by the funding source, while reducing the labour intensity of the task as much as possible for staff. Generally, this assessment involves monitoring the ESL students’ reading, writing, listening, and speaking. Often, data collected over time must show both student improvement, as well as a demonstration of the need for continuing student support. Sometimes connections are made to reporting.

Ideally, data collection for funding purposes could be better tied to learning and instruction in more authentic situations, designed to take away as little teaching time as possible and to focus more on process than on product.

3) Supporting teaching by providing valuable information about how successfully students are learning

Assessment tied more closely to learning focuses on the process, directly linking assessment to what is going on in the classroom. The purpose here is to promote and enhance learning, not to measure a product. Teachers and ESL students work collaboratively to inform and support both teaching and learning in authentic situations. This assessment is generally less formal and is curriculum and/or project based. It is less time consuming and gives assessment a positive use and focus in the classroom. It involves a few, very specific, planned instructional goals chosen as most significant by the teacher or the team, for student mastery. Often students are involved in learning what they can do to learn more effectively, and this engagement helps move them toward mastery.

Developed and used thoughtfully, this assessment creates a sensitive and powerful process for both teachers and students; teachers give specific, ongoing descriptive feedback to students, and students develop self-awareness as they come to understand over time what it is that is required of them to master content and be effective learners.

In Instruction and Assessment of ESL Learners, practical examples of each of these three areas of focus are provided to demonstrate ways of assessing ESL learners. As stated in the book, “Our purpose is to fuel learning, not to arrest it.”

Guest Contributor
Vicki McCarthy, PhD., Author

For more information about this book or to purchase a copy, please visit www.pandmpress.com

On The Same Page, a project of The Winnipeg Foundation and Winnipeg Public Library, encourages all Manitobans to read, and talk about, the same book at the same time. The program includes book giveaways, author appearances, and special events inspired by the feature book.

When: Tuesday, January 22 at 7pm
Location: Millennium Library, Carol Shields Auditorium
Address: 251 Donald St. in Winnipeg, MB

For dates and locations of future events, click here.

We are pleased to announce that the graphic novel, Sugar Falls: A Residential School Story, is listed by the Saskatchewan Ministry of Education as an approved resource for English Language Arts. Congratulations to author David Alexander Robertson and illustrator Scott B. Henderson.

Click on the book cover and buy your copy today!

Our populare Hands-On program is based on a straightforward philosophy: Student-centred activities that are based on authentic, hands-on experiences encourage learning.

Hands-On is a flexible and affordable instructional program that has been custom written for your curriculum. Program creator, Jennifer Lawson, has expanded the Hands-On series to include some exciting new titles:

• Hands-On Problem Solving for teaching strategic problem solving in mathematics
• Digital Blackline Masters for science, social studies, and mathematics
• Hands-On Interactive includes an engaging interactive learning environment for interactive whiteboards or personal computers (mac or pc)

For more information or to place an order, click on the titles listed above.