Engaging with a Mathematical Thinking Grid

This is a follow up blog post to: Day 1 of Reasoning about Multiplication and Division with Place Value.


On Day 2 of the lesson, I realized that though impactful, the interviews were taking a long time and I needed to move the lesson along while still maintaining a focus on analyzing the reasoning of others and developing one’s own thoughts.  


I asked students to consider each of the questions posed in the task.  When considering I wanted them to:

  1. Develop a solution,
  2. Explain how it was connected to the original problem of 13 x 17 = 221 and
  3. What was the math used in making that connection.(Note:  We are using the task reasoning about multiplication and division and place value from Illustrative Mathematics).




Thinking Grid  (Link to grid students wrote inside of).

I gave students about 25 minutes to work on developing their ideas around these.

Writing down their solution and explanation.

I then asked students to cut out the boxes and insert them into the corresponding page protector with the matching expression (no name was on their individual rectangles as to protect a safe learning environment). If students failed to get to a few of them, I wasn’t concerned.  I did not wait for every student to finish all tasks, as it would have left lag time and I was to move into analyzing.


I then gave a folder to each table group (I have seven groups).  Groups now began isolating their conversation on one question.  Before looking at any of the student work samples, there were to come to consensus (verbally) on the above three directions for the given problem.  

Group coming to consensus

Once the group came to consensus, they pulled out the student work samples for that problem.  They were instructed to:

  1. Find one that you agree with (or mostly agree with) and explain what was the mathematics or reasoning that you agreed with.
  2. Find one you disagreed with or thought could have been improved.  What was the mathematics of reasoning that you disagreed with?
Finding a work sample to analyze.

They were then to post their analysis of the two student work samples on a paper as their measurement for the day.


As I began to read their explanations, I noticed that many of the explanations were void of language that hid mathematical thinking (such as instead of saying multiplying by 10, students often say add a zero).


What I enjoyed about this lesson:

  1. Students had repeated exposure to thinking in various formats, verbal and written.  
  2. Students got to interpret at a level they were comfortable with.  For same this meant analyzing right or wrong solution.  For others, it entailed developing analysis of another student’s reasoning.
  3. No matter the level, all student were engaged in high level analysis.  
  4. They were only made accountable for a product after repeated exposure so inherently, students felt pretty successful.
  5. Most of all, the quality of the conversations at the groups were dynamic.  Students could not wait to clarify their thinking for their table mates.  The precision of their language needed in analysis, was very obvious.


I use this strategy often when I want students to engage with misconceptions without making them so public that we hurt a safe learning environment.  I find that when introducing an idea, going to quickly to a public conversation may shut down some students thinking that are either struggling or not at a point where they are ready to engage in a whole class discussion.  After two days worth of lessons, the group discussions were of rich quality.
Through these two lessons, I was able to adequately determine where a student was at and if they had enough understanding of how/why they can manipulate a math fact when decimals are involved.


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