Bridging Procedural and Conceptual Understanding

Yesterday I was putting together a few math projects when a Tweet caught my eye. The Tweet below started a short conversation that I thought was interesting.

David’s Tweet had many responses.  Most responses revealed that educators tend to side with solving one problem ten different ways rather than having students solve ten similar problems.  I started to reflect on how teachers give assignments that ask students to complete repetitive problems that often reinforce procedural mathematical thinking.  I also started to think how in an effort to provide practice, teachers may focus on procedural aspects first and then move towards practical application.  I find this happens frequently with math concepts at the elementary level.  What I don’t find often is the viewpoint that practicing procedural aspects can be embedded in solving specific problems multiple ways.  This type of thinking reminds me of number collection boxes.

Regardless of the assignment I want to be able to give specific feedback.  A larger problem that involves multiple steps can provide opportunities for teachers to pinpoint where misconceptions are and give direct feedback.  This isn’t always possible with ten similar shorter problems.  Below is an example of a few problems that you may find in a fifth grade classroom.  I don’t condone using these types of problems as they are definitely utlized, but I think we need to ask what’s being assessed when students complete this type of problem?  Students are simply asked to find the volume and show a number model.  I appreciate how the problems ask students to show their number model, but these types of problems seem to measure procedural understanding.  Do students know the formula?  Yes, well then they can answer many of these problems, even 10 in a row.



I think the above problems have a place in the classroom, but shouldn’t necessarily be the norm.  Usually these types of problems are found on homework sheets.  The problem below which was adapted from a recent fifth grade test is more challenging, but gives students opportunities to showcase their own mathematical understanding and persevere.  Some would say that these two problems are completely different.  I would agree, but similar concepts are being assessed.  They do look different and the second requires more skills to complete.  Students need to be able to use their procedural understanding and apply it to the situation.  Also, one key element that’s missing from the first problem is the student explanation.  Students are required to show their mathematical thinking in the second problem.  This is big shift and can reveal student misconceptions more clearly than the first problem.  I struggled with the decision, but eventually had students work in groups to complete the problem below.  Students were allowed to use any of the tools in the classroom to find a solution.



At first, all groups struggled with this problem.  Near the end of class all the groups presented their findings.  What’s interesting is that all the groups had different answers and ways in which they came to their conclusions.  I was able to offer opportunities for students to see and ask questions about different math strategies.  During the next class I was able to pull each group and give feedback.  This activity took a good amount of time to complete, but I feel like it was worth the commitment.

Through this experience and others I’m continuing to find that it takes a “bridge” to connect the procedural and application pieces.  At times I feel like there’s an assumption that if students are able to answer 10 similar procedural problems that they will be able to simply apply that knowledge in a multi-step problem.  This isn’t always the case and sometimes the bridge doesn’t fully form immediately.  Performance tasks, similar to the problem above can be one way in which teachers can help the transition from procedural understanding to practical application.  Being able to apply that knowledge to a math performance task can be a challenge for some students.  When teachers focus so much on the procedural, that’s the only context that students see and practice.  A blend between procedural and application needs to be established within the classroom.  I feel like activities like this help bridge this gap.

How do you bridge mechanical and conceptual understanding?

Math and M.C. Escher


Math and M.C. Escher

Math and M.C. Escher

During the last week of school my students started to explore topography concepts. Topography usually isn’t the first thing that is thought of when someone mentions the word math. That’s why I find it so interesting.  I truly enjoy teaching this topic because it often brings out the best from my students.  I find that most upper elementary students tend to thrive when given geometric shapes and asked to explore, rotate, translate or even turn them inside out.

I generally introduce the unit with M.C. Escher.  The class learns a bit about the life of Escher and his contributions to the world of art.  Moreover, we discuss how art and math are related. This is often a deeper conversations as students start to expand on the notion that mathematics can be found throughout our world.  Topics like the golden ratio and Pi often get brought up during this time.

After learning about Escher’s life and his influencers, the class looked at his different artistic creations. Usually my students recognize at least a few different creations.  Students seem to gravitate towards his optical illusion pieces or the famous Waterfall work.  As each work of art was discussed the more students found mathematics as an integral part of Escher’s work. After reviewing the different pieces of lithograph art, the class watched a short video on how Escher’s design and math are connected.

After the video the students were asked to have a conversation about how math can be found in most art.  The words symmetry, rotations, slides, translations, reversals, surfaces, and perspective were all brought up during the discussion.  What’s nice is that the vocabulary was brought up naturally as students spoke to one another.   I was able to highlight the words and facilitate the discussion as needed.

Eventually the discussion ended and the class moved to the next activity.  I planned to have the students create their own Escher-like artwork.  The students reviewed how to have “Escher-like eyes” when creating their own pieces.  I was proud of the student responses and the imagination that came forth during this discussion.  The class then reviewed the directions to create their own Escher-like creations.

The students went through the directions and asked questions.  Once the expectations were clear I passed out a 8 inch by 8 inch square to each student.  Students created their own tessellation template.  In the future I’m probably going to cut the square dimensions in half so the patterns become more evident.

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Eventually the students used the template to create an Escher-like creation.  Students showcased their work to the class using the vocabulary mentioned above.  The students were able to bring their work home on the last day of school.  All in all, this is a lesson I’m intending on using next year and a definite #eduwin in my book.


How do you incorporate art and math?





Exploring Rates in the Classroom


Exploring Rates in the Classroom

Exploring Rates in the Classroom

The topic of mathematical rates was introduced earlier this week.  Personally, I tend to find this unit enjoyable as there are many opportunities to connect the topic outside of the classroom.  To introduce the topic my classes go home and find examples of rates in their kitchen’s pantry.  The next day the class shares out what they found. This usually leads to an in-depth conversation about rates and patterns.  After our conversation I felt as though more examples and experiences were needed.

That evening I found some masking tape in my desk.  I decided to create a race path around the classroom.  The path varied in width and it purposely had a few sharp turns.  A roll of masking tape was used as well as a few proactive comments to the janitorial staff to not pick up the tape overnight.  When students came into the room the next day they saw this:

The raceway

The raceway

When the students walked into the room they were surprised.  A few started to jog around the track and ask questions about the room.  Already I was fairly excited as the students were pumped to see what I was up to.   I explained to the students that we were going to use the track to discuss rates, patterns and measurement.  The class then measured out the track and found that it was 66 feet long.  We had a conversation about how this track could be used to emphasize rates. I then introduced the students to the sheet below.

Screen Shot 2014-05-31 at 7.43.43 AM

Rates Sheet

Students were starting to see the big picture of this activity.  Students then took turns and quickly walked the course.  While they walked I had a few students become referees to make sure that no one stepped outside of the path.  I used an online counter and displayed the results as students quickly walked.  Once all the students completed their route and wrote down their results the class reviewed how patterns can be developed with rates.  Students were able to find the amount of feet traveled per second and then used that information to find how fast they walk one foot.  I was finding that students were trying out different mathematical strategies to find a solution. I gave them opportunities to work with each other to find solutions.  I asked clarifying questions when needed, but for the most part the students were on track. When the class finished this part of the sheet I gave them the second part.

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Converting feet to miles

This portion of the activity was more challenging.  Students were able to find the total amount of seconds, but converting the seconds to minutes was a struggle.  Many students asked how they could convert 12.9 minutes to minutes and seconds.  I was proud to see that students understood that 0.9 doesn’t mean 9 or 90 seconds.   This was a great opportunity to explore the concept of converting decimals to actual minutes.  The class used different calculations and found that 12.5 would actually be 12 minutes and 30 seconds.  As progress was made students started to find a conversion strategy to correctly convert the decimal to seconds.

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Click for Word template


As a class we shared our results and found patterns and the mean.  This activity worked so well that I used it with a few other classes this week.  I’m finding that students are developing a better conceptual understanding of rates while participating in a learning experience that I hope they don’t forget too quickly.

How do you introduce rates in the classroom?



Understanding Volume

Constructing Conceptual Understanding

Constructing Conceptual Understanding

This past week second and third grade students at my school are learning about measurement. The students are making progress. The classes have become more fluent with understanding perimeter and area, and are now starting to explore the concept of volume. Throughout the process students have used various manipulatives, such as prisms and nets to deepen mathematical understanding.   Even with all the activities  some students that are still struggling with the concept of volume. In about three weeks or so students will be assessed on this particular topic. Providing extra sessions for students to develop a conceptual understanding of volume is important. I wanted to find or create a math task that gave students intentional time to review geometry and measurement terms, while at the same time allow opportunities for students to create different products. After reviewing different options I decided on having students use the project detailed below.

Students were given a full sheet of colored centimeter graph paper.  They were then asked to read through the directions.

Directions: Create a net for a rectangular prism using the graph paper provided. The rectangular prism you build should have a volume of 20 cm3. Cut out your net and build a rectangular prism using glue or tape. Write the dimensions of the prism you built in the charts below.

Looking back, it seems like there were more than enough questions about what was expected.   Students always seem to have questions when there are multiple solutions/products.  After I answered their questions I took about 10 minutes to model the activity with the students.  This was important as it cleared up expectations for the activity.  I then passed out the assignment.

Click for sheet

Click for sheet

Students then used the centimeter grid paper to create a rectangular prism net.  They then filled out the top portion of the sheet.

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Some students had to use multiple attempts to create a net with a volume of 20 cubic centimeters. This was great opportunity for students to show perseverance and find a solution that worked.  I went around the classroom and asked students questions to help them think of a solution.  The students then cut out the nets and constructed their prisms.  The students then presented their rectangular prisms to the class.



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How do you construct meaning in geometry?

Representing Fractions with Thinking Blocks

Many classrooms in my school are in the midst of reviewing fraction concepts. Throughout the school students are finding fractional pieces, converting fractions to decimals, and identifying fractions on number lines.  For the past week students in second grade have been identifying fractional parts.  Earlier in the week students completed the page below during a math station.  Students did well on the first two pages, but struggled a bit when identifying fractions on a number line.

Representing 3/4 in many different ways

Representing 3/4 in different ways

This was a challenge for some students as many are more familiar with identifying fractions within objects (in a circle/rectangle).  Moving from identifying fraction to placing them on a number line can be a stretch.  Many students have already started to decompose numbers and have completed “fraction-of” problems.  These types of activities have helped reinforce the number line and fraction connection.  Next week students will be assessed on the fraction unit and many classrooms move into geometry concepts.  Before focusing in on geometry, I wanted to give student an opportunity to visualize fractions and use them with more complex word problems.

As I was looking for supplemental material I came across a Tweet by Paula (@plnaugle). She referenced Thinking Blocks  as a resource that she uses with an interactive whiteboard. I looked into the site and thought that it might be useful for my grades 2-3 classes since the app allows students the opportunity to solve fraction problems visually.  Specifically, I downloaded the fraction app on the school iPads.   Yesterday a second and third grade class used this app in their classroom as a guided activity.  The app was introduced to the class and I modeled the different steps involved in solving the problems.



The students were then asked to find a comfy place in the room and complete a minimum of three exercises.  What’s nice is that the problems are picked at random, so students aren’t on the same problem at the same time.  There’s also a feedback box that assists in guiding students towards labeling the correct parts of the fractions.

Click to enlarge

Click to enlarge

I helped the students as needed, but many were able to use the virtual manipulatives and generated feedback to stay on track.  Some students completed three problems, while some went beyond and tried out five.  After about 12 minutes the class gathered and we reflected on the perseverance that was needed and celebrated successes. This activity gave students an opportunity to make mistakes and persevere.  I’ll be keeping this app in my repertoire for the future.

Using Comics in Math Class


Comics/cartoons have been used to communicate important issues for many years.  Education has even been part of the comic movement.  Susan O’hanian has demonstrated with her website how cartoons can communicate and start important conversations.  I’ve tinkered with comics in my math classroom this year.

I believe that humor has a place in the classroom.   Comics can bring in a humor aspect, as well as practicality and motivation that can engage students.  My students are starting their algebra unit this week and I’ve been looking for new ways to introduce combining like terms and solving for unknown variables.  In the past I’ve used Hands-on-Equations and different types of narratives that explain how like terms can be combined.  After a lot of searching,  I ended up using pages 5 – 8 in this pdf to help introduce the concept this year.  Students responded well to the comic and I believe it helped them complete the activity in the document below.  I used one of the practice sheets as a model and the second sheet was completed and shared in student groups.

Combining Like Terms Activity

When I look back at some of my favorite K-12 teachers, many of them were able to connect and build rapport with students quickly and use humor appropriately.  I’m going to explore how to use  comics a bit more in my classroom over the next few weeks.  I may even have students create their own through comic creator apps on the iPad.

How do you use comics in the classroom?

Coffee and Mathematical Understanding

Coffee and Math Connection

Coffee and Math Connection

I spent the majority of this past week visiting with friends and family.  My destination ended up being in snowy and icy northern Michigan.   The house that I was staying in lost power for five straight days.  Thankfully the house had an efficient fireplace and a small gas-powered generator that ignited a few space heaters to keep one of the rooms fairly warm as outside temperatures hovered around 20 degrees.

In normal circumstances, the first person that wakes up in the house starts to brew the coffee for the family.  Since we had no power the coffee machine wasn’t an option.  You see, my family definitely enjoys their coffee. Being official coffee addicts my family has a decent understanding of the integral parts of the coffee-making process: hot water, filter, coffee grounds and cup. The one missing ingredient in this process was the hot water.  One of the family members found a pan and began to boil water on top of the fireplace.


The water was then used to complete the coffee making process.  Success!  All of the family was able to sit around the fire and drink our coffee.

coffee maker

As you can imagine or already know, they’re many ways to make coffee.  My family knows this and that understanding led us to a solution that was adequate.  We substituted a different process in the coffee making flowchart and arrived at decent tasting coffee. Regardless of the process used, the user ended up with the same solution.  In the end some type of hot coffee was served.  Understanding the key components of any process allows opportunities to substitute yet arrive at the same solution. Because my family knew the process in-depth we we’re able to substitute the missing item and still have the same result.  I feel like this type of thinking applies to the classroom.

Having a limited understanding of place value and number sense can limit opportunities for students.  Students need to be exposed to an array of methods to complete problems, not just shortcuts.  Only understanding the formula/shortcut doesn’t necessarily show mastery of a particular concept.

At the upper elementary level students are expected to find the product of  3+ digit numbers.  If students have been exposed to using only the traditional method to solve these types of problems they know how to multiply large numbers using one method.  Although that process might be effective for them, it doesn’t cement a deep understanding of multiplication.  Students often have problems when decimals are introduced when finding the product of these types of problems.  Having a more in-depth understanding of place value and multiplication can give students the tools to solve more complex math problems.

On the other hand, if a student has been given opportunities to use repeated addition, partial-products, lattice and traditional methods, students might have a better understanding of the role that place value has in the multiplication process.  Having that understanding of place value will help students when they approach decimal computation and throughout their academic career.  Having multiple tools/strategies also encourages students to be independent and choose the correct method to find a solution.  Even more important, students that are then able to apply their mathematical understanding to practical situations (beyond the test) can often immediately see the benefits.

In the end my family used the tools that we had (fireplace, metal pot, water) along with other materials (coffee filter, ground coffee, cup) to create coffee that we all could enjoy.  If we were fully reliant on just the coffee maker and electrical power, we wouldn’t have had coffee to drink.  Understanding the details of the process and that there are multiple ways to find a solution is an important skill to have as adults and as students in the classroom.

The power came on and I believe we all found a newly acquired appreciation for the electrical grid in Michigan.  Our coffee story is unique and yet I feel as though it’s mathematically relevant as teachers will be back in the classroom to start the second half of the school year.  Enjoy the rest of 2013 and I look forward to a successful 2014.

Math: In Response to Your Question

I’ve been exploring the use of multiple solution problems in my math classes.  These types of problems often ask students to think critically and explain their mathematical processes thoroughly.  To be honest, these questions can be challenging for elementary students.  Most younger students expect or have been accustomed to finding one right answer throughout their academic career. Unfortunately, state and local standardized assessments often encourage this type of behavior through multiple choice questions.  This type of answer hunting can lead to limited explanations and more of a focus on only one mathematical strategy, therefore emphasizing test-taking strategies.  Encouraging students to hunt for only the answer often becomes a detriment to the learning process over time.  Moving beyond getting the one right answer should be encouraged and modeled.  Bruce Ferrington’s post on quality over quantity displays how the Japanese encourage multiple solutions and strategies to solve problems. This type of instruction seems to delve more into the problem solving properties of mathematics. Using this model, I decided to do something similar with my students.

I gave the following problem to the students:

How do you find the area of the octagon below?  Explain the steps and formulas that you used to solve the problem.

Octagon Problem

At first many students had questions.  The questions started out as procedural direction clarification and then started down the path of a) how much writing is required? b) how many points is this worth? c) how many steps are involved? d) Is there one right answer?  I eventually stopped the class and asked them to explain their method to find the area of the octagon, basically restating the question.  I also mentioned that they could use any of the formulas that we’ve discussed in class.  Still, more questions ensued.  Instead of answering their questions, I decided to propose a question back to them inorder to encourage independent mathematical thinking.  Here are a few of the Q and A’s that  took place:

SQ = Student Question         TA = Teacher Answer

SQ:  Where do I start?

TA:  What formulas have you learned that will help you in this problem?

SQ:  Do I need to solve for x?

TA:  Does the question ask for you to solve for x?

SQ:  Should I split up the octagon into different parts?

TA:  Do you think splitting up the octagon will help you?

SQ:  How do I know if the triangle is a right angle?

TA:  What have we learned about angle properties to help you answer that question?

Eventually, students began to think more about the mathematical process and less about finding an exact answer.  This evolution in problem solving was inspiring.  Students began to ask less questions and explain more of their thinking on paper.  At the end of the math session students were asked to present their answers.  It became apparent that there were multiple methods to solve the problem.  Even more important, students started to understand that their perseverance was contributing to their success.  The answer in itself was not the main goal, but the mathematical thinking was emphasized throughout the process.

Afterwards, students were asked to complete a math journal entry on how they felt about the activity.

Image Credit: Kreeti

Taking Math Outdoors

Math Outdoors

Recently I had an opportunity to attend an outdoor education trip with our elementary students. The trip took place over three days and was located in a very remote part of the state, away from high rises, city lights, cell phone signals, and televisions.  The trip focused on learning about birding, forest ecology, Native Americans, orienteering, and pioneering.  For many students this trip is a different learning experience.  It’s outside of the classroom and therefore a different learning environment for them. Acclimating to this environment took a bit of time for staff and students.

The adults were responsible to teach many of the concepts during hikes on campus.  Being outside is a great opportunity to introduce or highlight academic concepts that are generally taught through abstract means.  While talking about math outdoors, students expressed interest and asked questions that often led to additional mathematical questions.  Students that might not usually be fully engaged in a math lesson at school were shining on the hike. This experience led me to reflect on our current mathematical practices.  At times there’s a disconnect between what’s happening in the classroom and what’s occurring right outside of the doors to the school.  Teachers often attempt to bridge the gap, but self-directed student questions often come from real world experiences and curiosity.  Curiosity is often followed by questions.  Finding answers to those questions can lead students to find their passions (eg. #geniushour).  This motivation can be encouraged but not genuinely bought or sold.  Students decide how engaged they want to be and internal/intrinsic motivation often leads to learning experiences.

Below are some (of what I can remember) of the questions/topics that were discussed while on the trip:



Math Debates in Elementary Classrooms

Learning through Conversations

Over the past few months I’ve dedicated a good amount of time to to having math conversations. These math conversations occur when the class is unsure of how to solve a problem or when disagreement ensues over what particular strategy should be used to tackle a problem.  The math conversations (or debates) allow students the freedom to openly discuss logical reasoning when solving particular problems.   These conversations can be sparked by the daily math objective or follow another student’s response to a question.  It’s not necessarily planned in my teacher planner as “math conversation” in yellow highlighter, but I do make time for these talks as I feel that they bring value and encourage student ownership.  The conversations also give insight to whether students grasp concepts and are able to articulate their responses accordingly.  Mathematical misconceptions can also be identified during this time.

During these conversations I have manipulatives, chart paper, whiteboards, iPads and computers nearby to assist in the discovery process.  I emphasize that there’s a certain protocol that’s used when we have these discussions.  Students are expected to be respectful and listen to the comments of their classmates.  To make sure the class is on task I decide to have a specific time limit dedicated to these math conversations.  Some days the conversation lasts 5 minutes, other days they may take upwards to 15-20 minutes.  When applicable, I might use an anchor chart to display the progress that we’ve made in answering the questions.  I should also mention that sometimes we don’t find an answer to the question.  Here are a few questions (from students) that have started math conversations this year:

  • Why is regrouping necessary? (2nd grade)
  • What can’t we divide by zero? (3rd grade)
  • Why are parentheses used in math? (3rd grade)
  • Why do we need a decimal point? (1st grade)
  • When do we need to round numbers? (2nd grade)
  • Why is a number to the negative exponent have 1 as the numerator? (5th grade)
  • Why do you have to balance an equation? (5th grade)
  • How does the partial products multiplication strategy work? (3rd grade)
  • Why do you inverse the second fraction when dividing fractions? (5th grade)
  • Why is area squared and volume cubed? (4th grade)

Above is just a sampling of a few of the math conversations that we’ve had.  Afterwards, students write in their journals about their experience finding the solution to the problem.

Of course this takes additional time in class, but I believe it’s time well spent.  The Common Core Standards  focus on depth of mathematical understanding, rather than breadth.  This allows opportunities to have these conversations that I feel are beneficial.  They also emphasize the standards of practice below.

  • CCSS.Math.Practice.MP1 – Making sense of problems and persevere in solving them.
  • CCSS.Math.Practice.MP3 – Construct viable arguments and critique the reasoning of others

Photo Credit:  Basketman

Do you have math conversations in your class?