From Models to Numbers – Making Connections in Mathematics

Students solve mathematical problems in a variety of ways.  Some students need to touch and manipulate cubes or tiles to help them visualize the problem.  Some may use software to investigate and explore different models.  Some may use symbols to represent their solution algebraically.

But how do we get those students who are at the concrete stage to represent their models algebraically?  Do students who represent their solutions symbolically really understand what their algebraic expressions mean or have they just learned to manipulate symbols?

Intermediate mathematics should emphasize making connections between the concrete and algebraic models.  If we rush to teach rules or algebraic manipulation without first imparting an understanding of what these rules or expressions mean, then what are students really learning?  The purpose of making connection is to understand the development of the rules, the formulas and the algebraic expressions.

In grade 7, students are expected to learn how to add and subtract integers.  When they are presented with a question like:  “What is the value of  7 – (-3) ?“,  we may get a solution that looks like this…


Although the solution is correct, what does this student really know?  If, instead of paper and pencil work, we use tiles and an integer mat, we can focus on and assess students’ understanding of the zero principle.


By going through many examples and seeing the pattern of subtracting a negative,  students  begin to understand what the rule “two negatives = a positive” means.

We can also use visual or concrete models as tools to aide in the understanding of problems in areas such as fractions.  In the following problem, the grid on the left mirrors the calculations on the right. The grid is used to not only generate the solution (1/8) but to add meaning and intuition to each step of the algebraic derivation on the right.

Simaya’s dad cut her birthday cake into 12 equal parts and Simaya ate one of the pieces. Later on, there were only a few pieces left, so her dad cut each of the original pieces in half. Simaya then ate 1 of those smaller pieces. What fraction of the cake did she eat in total? Explain your reasoning.


In this Patterning & Algebra question, students are asked to complete the chart and find the 20th term in this growing pattern.


Some students may ignore the pattern of dots and focus on developing a rule which generates the sequence of numbers 2,6,12,20,… at the bottom of the chart. While their rule may work, the connection between the numeric sequence and the dots sequence is lost. They would have difficulty justifying their rule. By focusing on both the pattern of numbers and the pattern of dots at the same time, a student connects a dot rectangle with a number in the sequence and can potentially derive and justify a general expression for the nth term in both sequences.  Below are two different models which help students visualize and justify two ‘different’ equations for the nth term in the sequence.


Paper folding is a great way to concretely demonstrate mathematical concepts.  It’s inexpensive and requires minimal preparation.  The following exercise motivates and develops the formulas for the area of a trapezoid and parallelogram. Notice how the exercise implicitly includes concepts such as conservation of area under transformations.


Making connections naturally leads to better mathematical intuition and an understanding of the derivation of rules, formula and algebraic manipulations.  A focus on the final answer to a problem, without a motivating model to connect the solutions steps is like seeing only the last ten minutes of a movie. The end doesn’t make nearly as much sense compared to watching the movie from start to finish.  Besides, watching the whole movie is more fun, usually something interesting happens along the way.


  1.  Area Formulas by Paperfolding –
  2.  Mathstar –

From Start to End – Maximizing Instructional Time in the Intermediate Classroom

We are well into our school year and have now created a mathematical environment in our classroom.  When we walk in, we see walls covered with interesting puzzles, book shelves full of math story books, games available to sign out on weekends, and math tools in bins readily available when needed.  When we walk in, we hear pairs or triads discussing math problems, questions being asked and students articulating their mathematical thinking.

Now that we’ve created this environment, how do we use our math time to ensure we are utilizing this instructional time effectively?

TIMSSA video study from the TIMSS report had given some insight as to how time was spent within a mathematics class in the US, Japan and Germany (Fig 2).  Japanese lessons demonstrated a more active learning environment where students worked on problems, struggled with the problems and then articulated their thinking by sharing different representations of the solution.  Lessons from the US and Germany were more teacher-led by providing instructional steps, examples and opportunities to do seat work that were similar to the questions done in class.  So how do we structure our class to incorporate a more student-centered lesson?

Here in Ontario, many are familiar with the 3-Part lesson format.  TIPS4RM refers to the parts as:

  • Part 1 – Minds On
  • Part 2 – Action!
  • Part 3 – Consolidate Debrief

Part 1:  Minds On  “Activating Prior Knowledge”  ~ 5–10 min

The Minds On is the hook to activate prior knowledge that is needed for the task they will be doing in Part 2.  A math string is an example of a Minds On task.  For example, a question might be:


Given the following math string, how would you visualize these mathematical sentences?

CookieCrazeA Minds On can also be in a form of a problem.  In this problem, students were given the Cookie Craze problem.  Clickers were used to respond to the problem.  Discussions followed about how the mathematics was generally represented but further discussed later in Part 3: Consolidate Debrief.

Part 2:  Action!  “The Investigation” ~ 15-20 minutes

MangoesThe Action! is when students explore and investigate a new concept.  Students can work in pairs or triads to solve a problem.  Same-ability groupings allow every student to have a voice and be actively engaged in the process.  Often when students are partnered with another student of a different level, the stronger more assertive student tends to do the work.

ManipsA variety of manipulatives such as fraction rings and circles, are provided so students can choose tools that are appropriate to their learning style.  Chart paper and markers are used to record their thinking when solving the problem.

Part 3:  Consolidate Debrief  “Summarizing the Learning” ~ 20-30 minutes

soln1The purpose of the Consolidate Debrief is to connect the mathematical concepts to the actions they did in Part 2.  Students summarize their learning by sharing their strategies, comparing and contrasting solutions, identifying common misconceptions and raising other math questions that came out of the lesson.  This consolidation can be done through a math congress where students present and justify their work to their peers.


Questioning is a large and important part of the Consolidate Debrief.  In Classroom Instruction that Works: Research-Based Strategies for Increasing Student Achievement (ASCD, Alexandria, VA), Robert Marzano describes instructional strategies that are significant factors that increase student learning.    The following chart connects some of these strategies to actions that are done during the math congress.

Instructional Strategies During the congress…
Identifying similarities/differences Ask “What is similar about these solutions?  What is different?”
Summarizing Ask, “Can someone describe what you think this group did to solve the problem?” and “What did you do when you got stuck?”
Nonlinguistic representations Ask “How does the concrete representation connect to the algebraic expression?”
Cooperative Learning During the task, students work in same-ability pairs/triads to actively involve all learners.  These groups present together to justify their solutions.
Providing Feedback Teachers facilitate discussions providing feedback and encouraging students to also provide feedback to solutions they see.

Now that we have a structure for our lesson, how can we support each other in our schools to have consistency among all our classes? 

Administrator support is key to making this happen. Scheduled time is needed during the day for teachers to plan together and teach together.  Here are some suggestions that can help make this happen:

  • Pairs of teachers co-teach together.  This is inquiry based, reflective and collaborative. Teachers take on the role of lead co-teachers and teach one of their classes together or bring their classes together to teach the lesson. Co-teachers have “live-time” professional discussions about what they are observing, doing and make collaborative decisions to best meet the needs of the students. The teachers are learning from each other as their students are learning from them.
  • Collaborative Planning.  Teachers of a grade, division or subject meet and plan lessons that include trying out the task/activity/problem and anticipate what student responses might be. The group could consider a focus for their professional learning such as “questioning” or time on task for each portion of the 3 part lesson. The focus is on planning and instruction and moves the community of teachers in a school along their continuum of learning.
  • Professional Learning Communities. This professional learning community can take place right in the classroom.  Teachers could have colleagues (teachers, administrators, consultants, coaches) come to co-teach with them.  Other teachers can observe this co-teaching session  and still be involved in the debrief and planning for the next lesson.  The co-teaching model should be ongoing whether it be once or twice a week or once a month.  It’s a great model for students to learn from.

Happy planning!


  1. Fosnot, Catherine. Context for Learning: Investigating Fractions, Decimals and Percents. Portsmouth, NH:  Heinemann, 2007.
  2. Krulik, Stephen, Jesse A Runick.  Roads to Reasoning 8, Chicago, IL:  Wright Group/McGraw-Hill, 2002.
  3. Martinez, J., “Exploring, Inventing, and Discovering Mathematics:  A Pedagogical Response to the TIMSS”, Mathematics Teaching in the Middle School,  Vol 7, No. 2, (October 2001), pp 114 – 119.
  4. Marzano, R., Debra Pickering and Jane Pollock.  Classroom Instruction that Works:  Research-based Strategies for Increasing Student Achievement . Alexandria, VA: ASCD, 2001
  5. National Council of Teachers of  Mathematics  – Illuminations

Building a Culture of Mathematics – Creating a Mathematical Environment in the Intermediate Classroom

As the end of summer approaches, teachers begin to feel that tinge of panic and excitement as they begin to think about their new class in September.  As teachers, we have many questions that come to mind when developing our math program.  Questions like,

How will I cover all the expectations?  How will I engage my students in math?  How can I encourage them to take risks?  How do I get them to write/explain in math?

But most of all…  How do I get my students to be interested and like math?

As teachers examine their programs with their school teams, much of the focus is on how to divide strands for each reporting term.   But does the planning include ways to help students become better risk takers, ways to teach how to articulate their thinking through oral and written explanations, and ways to enjoy and appreciate mathematics?  At the end of the school year, what do we want our students to learn from our class?

If we don’t think about developing risk takers then how will we really know what they know or don’t know?  When students say, “I know it but can’t explain it.”  then what part did they understand?  Do we want mathematics to be a subject they just had to take at school or do we want them to say, “Wow, I remember in math when we did…”.  This is the type of impact we want to have on our students.

Mathematics has been viewed as a subject that is unrelated to students’ interests.  We need to do a better job at having students (and parents) view mathematics as a real, tangible subject.  Something they can be passionate about, can argue about, relate to and see as a beautiful subject.  We need to create a culture that lets our students understand what math is really about.

It is quite common to hear people say, “I’m not good at math.” and responses like “Yeah, neither am I.”.  But how often do we hear people say, “I’m not good at reading?”  Why does our society so readily accept this attitude about math?  I remember sitting in conference and hearing Dr. Ed Barbeau, mathematics professor at the University of Toronto, say that intermediate math should be about recreational math.  I was fascinated by this statement and understood it to mean that students should have more opportunity to problem solve, collaborate with one another, debate and justify their solutions and learn to develop strategies to problem solve. The idea of incorporating more time to just explore problems really had an impact on how I taught math.  Perhaps the reason many feel they are not good at math is that they never really had an opportunity to explore mathematics and develop  higher level skills to make math a habit of mind rather than a list of skills to be learned.

So how do we program and create an environment so students are more involved in the math they do?  According to child development experts [2, 6], adolescent students thrive on arguments and discussions.  They are introspective and often critical of their own thoughts. They need to feel relevant and are concerned with justice and fairness. They are very self-aware and worry about what others think of them.    We should take advantage of adolescent motivations and concerns, and use them to help us teach students mathematics.

Anthony and Walshaw [1] have developed the following set of principles of effective pedagogy of mathematics.


We can use this guide to create a diverse program that makes mathematics a habit of mind. The principles address content as well as ways to think about and approach math.  We need to think about what a classroom environment supporting these principles looks like.

Below are some suggested ways to help you create a culture of mathematics in your classroom for September.

Representing Mathematics in the Classroom

  • Post a variety of puzzles on the walls in class or in the hallways.  Students might as well be intrigued by puzzles on the walls while waiting to come into class.
  • ThinkFunHave games that require persistence, thinking and problem solving available during breaks and for weekend loans. Games like Rush Hour or other Thinkfun games are perfect to develop these types of skills.  Be sure to incorporate this into your program.
  • Provide print materials like puzzle magazines or math story books available for students to read.  Why not read math related materials during independent reading time?
  • Ask students to describe their culture’s number system.  Post these on the walls for future reference.
  • Include a word wall for mathematics to be used during discussions and debates.
  • Post question prompts like “This solution is similar/different from this solution because…” or “I agree/disagree with your solution because…” or “I like how you represented your solution by…”.  Intermediate students like to debate and discuss, so let’s take advantage of this trait!
  • Post newspaper clippings, advertisements, photographs, art or anything related to a mathematics inquiry.  Encourage students to examine fairness and question the use of statistics in the media. Draw connections between math, patterns, art and architecture.
  • Have a variety of tools available in the classroom for students to represent their thinking.  Manipulatives and technology provides hands-on experiences to help bridge concrete representations to abstract models.

Developing Risk Takers and Questioning Skills


I noticed that my grade 8 students left many questions blank on the first test I gave them.  At first I thought they just didn’t understand the math. I later realized that understanding was only part of the problem. They were afraid to take the risks needed to investigate and solve problems. Students gave up when a solution was not immediately apparent.  I decided to use the game Mind Trap at the beginning of each class to encourage students to take risks and ask questions.  This game consists of questions that are set in a crime context. Students become more comfortable struggling with a problem, learned to ask better questions and identified specific properties of the problem to help them solve the crime.  It’s a great resource for intermediate students.

Life Size Mathematics

North Option OISE/UT students dramatizing Frogs

North Option OISE/UT students dramatizing Frogs

Take mathematics off the textbook page and bring it into the real world.  Students are engaged when they are part of the mathematics.  Frogs is an excellent way to dramatize mathematics and demonstrate problem solving.  In this task, students wear Hawaiian leis to represent green and yellow frogs.  Teachers act as facilitators to help identify patterns for when students get stuck.  Towers of Hanoi is another great puzzle to do using cardboard boxes of varied sizes.  These are great ways to build collaboration and introduce them to reasoning at the beginning of the year.

Problems for Pairs or Triads

Students need more opportunities to regularly explore math problems.  Working in pairs or triads creates a safe community where they can struggle with a problem, contribute their own thoughts, build on one another’s ideas and debate in a more controlled setting.   When students are given opportunities daily to discuss and work together on problems, they eventually feel more confident and are able to take more risks.    Puzzles like Shikaku [5} or Paint by Numbers (also known as Pic-A-Pics) are great logic puzzles for students to develop their reasoning.

MoreGoodQuestionsGoodQuestionsQuestions from More Good Questions [4] and Good Questions for Math Teaching [3] are great questions to further develop their problem solving and connect to the curriculum they are learning.

Talk Time

Adolescents love to talk so let’s give them this opportunity!  Not only do we want them to talk in their math pairs or triads, but as a whole group as well.  Provide many opportunities to debate, justify and ask questions to one another.  If they don’t understand another group’s solution, have them challenge them and ask questions to clarify their thinking.  Use question prompts that are posted on the walls as reminders to help them focus their questions. It’s such an amazing thing when you hear a heated debate about mathematics in your intermediate classroom!

In conclusion, if we want our students to value reasoning, be active participants, and to enjoy mathematics, we need to build a culture of mathematics by providing an environment to help them see and be part of the math.  As teachers, we need to model problem solving and work out problems together with the class.  If teachers are more excited and engaged about math then students will be as well.   Doing this at the beginning of the year will set a positive attitude and immerse students in a place that develops those habits of mind for mathematics.  When you hear your students say, “I remember in math when we did…” then you know they’ve got you excited about math too.


  1. Anthony, G. and M. Walshaw, “Characteristics of Effective Teaching of Mathematics:  A View from the West”, Journal of Mathematics Education,      Vol 2, No. 2, (Dec 2009), pp 147-164.
  2. Health Canada, “Growing Healthy Canadians – Transition to Adolescents”,
  3. Schuster, L. and Anderson, N., “Good Questions for math Teaching:  Why Ask Them and What to Ask, Grades 5-8”, Toronto:  Pearson Education, 2005.
  4. Small, M. & A. Lin, “More Good Questions:  Great Ways to Differentiate Secondary Mathematics Instruction”, Toronto:  Nelson Education Ltd, 2010.
  5. Wanko, J., “Deductive Puzzling”, Mathematics Teaching in the Middle School,  Vol 15, No. 9, (May 2010), pp 524 – 531.
  6. Washington State Online Foster Parent Class. “Child Development Guide”,