Math Mania: Justifying your Math Thinking

Math Mania: Justifying your Math Thinking

Are your students struggling in math? Teach your students the skill of justifying your math thinking with this step-by-step guide. elementary | teachers | student | justify

Justifying Your Math Thinking

A few years ago I taught a grade 3/4, which meant that I also had students that needed to complete EQAO (Ontario’s Standardized Tests for gr 3 and gr 6) . Unlike many others, I enjoy teaching EQAO years primarily because I love the support and the resources.  I felt that my students were well prepared for taking this test and sent them off to write.  The following year we got the results.  I was surprised that the data for my students did not reflect their abilities in math.  The open response word problems were not as strong as these students had been capable of. These were strong students, level 3 level 4 consistently, yet their answers were falling below that in the level 2 area.  I just didn’t make sense.  In conversation with the math consultant and my colleagues we dug into the data, and looked at the rubrics to determine that our students were failing in their ability to justify their answers. This was the determining factor between a level 2 and a level 3.  So, we decided to make sure students were explicitly taught how to justify their thinking, explain their process, and communicate their knowledge and understanding.
Prior to this I had explained to students that justifying their thinking meant that if they solved word problems with

“Pictures Numbers and Words”

that this would be enough for students to properly justify their answers in math.  After seeing the data, that my strong students were not able to really communicate their thinking properly. I decided to really focus on this and try to improve these outcomes.
Here’s How I did this
  1. What does it mean to Justify Your Thinking – I gave students a word problem.  Not an easy one but a problem where they would have to do some thinking.  “It was a foggy day on Mr. Brown’s Farm.  He wanted to know how many animals were in the field but could only see the animals legs.  He counted 48 legs.  How many cows, and chickens were in the field justify your thinking” This problem has an open ended question with a variety of answers. I gave students the question and had them work in pairs to solve this problem.
  2. When they were done and had developed their answers we quickly talked about how they solved this question.  I tried to choose students to share that used a variety of strategies and had a variety of answers.
  3. First I asked students to identify for me what steps they took to understand and solve the word problem.  We made this chart. (well this is the pretty version, the initial one was very messy)


    Are your students struggling in math? Teach your students the skill of justifying your math thinking with this step-by-step guide.

    We practised solving word problems following this guide for a few days and reflected on this strategy to solve problems.

  4. Next we needed to begin to explain to others what they were doing to get their answers.  We talked about what it meant to justify your answers. We brainstormed what this phrase meant and what teachers were looking for when we asked students to do this and why.  We made the first chart to describe what it was they needed to do to justify their answers.  We worked again on this for a few days.  We looked for examples of students doing this well we talked about their phrases and compared them to the chart.  Once the majority of students were able to do this we finalised our chart and made the good version for the wall.


Are your students struggling in math? Teach your students the skill of justifying your math thinking with this step-by-step guide. Are your students struggling in math? Teach your students the skill of justifying your math thinking with this step-by-step guide.

Eventually we needed to add to the chart to include that their description of the steps that they use should use words from the question instead of numbers.  Too many students were describing their steps as “First I added 4+4+4 and got 12 then added 2+2+2+2 to get 8, then added them together to get 20 then doubled it” We talked about that this sentence was out of context and did not relate to how you solved the question.  A better response would have been “First I added together the legs of three cows to get 12 then the legs of 4 chickens to get 8.  I added chicken and cows together to get 20 legs.  Then I doubled the number of cows and chickens to get 40 legs.  If I added 4 more chickens I would reach 48 legs.  Therefore Farmer Brown could have 6 cows, and 12 chickens.

I noticed a great difference in the way my students were justifying their responses.  Because they had to explain their steps in a way that related to the question I found that their understanding of the question and the process of how to solve it improved.  Soon simple questions really were simple and the knowledge that they learned from one question was easily transferred to other questions.  Even harder questions that involved multiple steps appeared less intimidating to these students.

If you are looking for ways for students to practise their problem solving and basic knowledge why not use my Mastering Math Sheets.  I use these everyday at the start of my math class. They serve as a the getting started portion of my three part lesson plan.  As well, the word problem component of these sheets can be used as the working on it section of a three part lesson plan.

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30 Days of Place Value

30 Days of Place Value

Everyday for Bell Work in my 4/5 class, I use a mastering math sheet to review basic concepts and previously taught skills.  The page is the same every day but the daily number changes.  These also follow the three part lesson plan format that is popular today in classrooms.  The first page of the sheet is great for the Getting started section of a three part lesson plan.  It reviews key concepts and the fundamentals of the unit.  Additionally because it is repeated every day students are able to gain mastery with these familiar questions throughout the unit.  The second page of the math sheets include a word problem and a reflection section.  These are great for the Working on It and Reflection of the morning math sheets.  Students can work on the problem as a class, as a group, or individually.  Many of the questions are open ended and encourage thought and discussion for students. Word problem questions are also repeated so that students can apply skills learned from previous days with larger numbers.

My first unit of the year is Place Value and This first package has 30 different pages including a blank page that you can print and add your own number or questions.

Mastering Math Sheets - 30 Days of Place Value

30 Days of Place Value

This package gives you 30 different activities that review concepts of 2 and 3 digit numbers and introduce and build up to 5 digit numbers.  These are not a complete unit however they would provide you with 30 days of practice and  problem based math that is easily differentiated and great for multiple ability classrooms.

If you are looking for other place value activities that you could add to also use to supplement this package including games, task cards, and other activities please check out some of my fellow bloggers items here. These are great to supplement your place value unit and would be a great companion to my Mastering Math Sheets.

Order Numbers to 10000
Math Chat Speech BubblesThemed Place Value Practice
Mastering Math Sheet: Mental Math Addition
Happy Counting 🙂

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Why I love Long Range Plans (and you can too!!)

Why I love Long Range Plans (and you can too!!)

Feeling overwhelmed? Check out this 4/5 split long range plan! (Plus you can grab one for your split class, too!) | long range plan ontario | long range plans |

Its done!!!  My 4/5 Long range plans are finally completed.  Having such detailed plans this past fall was a life saver.  Being very pregnant with a toddler at home I didn’t have lots of time to plan these long range plans were a great tool that sat on my desk as a reference.  It helped to stay focused and on task and make sure that I had taught what I needed to before I left for my leave.  Even my LTO appreciated knowing what I had already covered and what she needed to still teach.  As I watched my colleagues without plans struggle to stay on task I was relieved that I spent the time last summer to have these complete and ready to go.

As you can see from my Term #1 plans I include details about literacy planning, Read Alouds, big ideas, Writing Forms, and Comprehension Strategies.  As well as the math units for both grades that can easily be taught together.  New I have included more detail for the social studies and science curriculum including Inquiry Questions, Big Ideas, and Culminating Tasks for both of the grades.  However I have removed Physical Education as I will no longer be teaching this subject.  OPHEA is a great resource and made up the bulk of my physical education program

Complete Long Range PlansIf you are interested in the units in my long range plans you can see them here
Balanced Literacy 
Social Studies
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