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 I hope that life is treating you and yours well.

I love when teachers are honest about their challenges with teaching mathematics. If one teacher is thinking it, many more teachers are probably thinking the same thing. Hearing the thoughts and concerns of teachers helps me identify how to offer the best support, overcome a challenge, or develop a new perspective.  Please keep being honest and asking questions.

You may already know this about me but, I do not demonstrate for students what I know about mathematics. I pose problems to students and ask them to demonstrate what they know about mathematics. Then, I take what they know and refine, refine, refine their work until they can approach problems solving with confidence, flexibility, and depth of understanding. Using an open-ended approach with students will lead to students using a variety of strategies. Analyzing and correcting student work that involves wide-ranging strategies will require far more effort than just checking off of a box.  Last week a teacher admitted to me that looking at student work at the end of a long day could be overwhelming, confusing, and exhausting.

Thank you! Thank you for speaking your truth. Now I know I need to help teachers create systems that will help them analyze, organize and correct student work.

The first thing I do is separate the work into 2 piles: 1) correct answers and
2) incorrect answers

Then, I use four lenses to make sense of student work: comprehension, organization, number sense strategy, and connect and extend.


Does the student comprehend the problem?

A student can be effective at adding, subtracting, multiplying, and dividing, but if a child does not know when and why to apply which operation their calculation skills won’t save them.

Look at the example below.

You can see the student is efficient with their calculations. However, the student did not read through the problem thoroughly and determine the correct approach. Asking students who make these types of mistakes to read and retell the problem or explain their strategy based on more than keywords in the text is crucial for their future success. I would check in with these students for (3 – 5 minutes) every day for at least a week before sending them off to solve.


Is the student’s work organized and easy to understand?

Messy work leads to messy results. By providing students with systems for organizing their work, they will have the skills to avoid unnecessary mistakes.

Labeling your work as you go, crossing off items as you count them, and organizing models into arrays can make it simpler to identify answers and avoid mistakes.

Number Sense Strategy

Does the student use an inefficient or ineffective number sense strategy?

Two main challenges affect students’ number sense strategies: using procedures that don’t make sense to students and counting individual units one by one versus making groups of the units. You can see examples of both issues below.

On the left, the 5th-grade student used the standard algorithm to multiply decimals (that is a 6th grade standard by the way). It is clear the student is unaware of where to place the decimal when using the standard algorithm. Misplacement of the decimal is a common mistake for students when multiplying decimals with the standard algorithm. This common error is why the Common Core standards recommend students in 5th grade use alternative strategies to multiply decimals to avoid this type of error.

On the right, the student added up 0.02 over and over again to get to 0.32. The final answer is correct. The decimal point is in the correct place. However, this student could make a mistake if they had more decimals to add up. Making groups of items or units can help students who use these types of strategies understand how to use more efficient and effective number sense strategies. I would ask the student if they circled groups of 5, how much would be in each group.  My hope is that the student will shift from counting one by one and move to (5 x 0.02) + (5 x 0.02) + (5 x 0.02) = 0.1 + 0.1 + 0.1  + 0.02 = 0.32

Connect and Extend

Is the student ready to extend their strategy to something new or more complex?The thing about mathematics is that just when you think you know the answer, there is always another layer to uncover. One of the biggest challenges I have when working with students who primarily use textbooks to learn mathematics is that skills are taught in isolation.  Students are unable to see the connections between mathematical ideas. Mathematicians are peeling back layers and making new connections every day.

If we look at the student work samples below, it is clear that these 5th-grade students have a strong understanding of multiplying unit fractions and equations. I would love for these students to compare and contrast their work with one another. I would love for them to have a conversation about the relationship between multiplying fractions and dividing fractions. What else is there for these students to learn?

When I look at the students’ work, I think about making a connection between the ratio table and the coordinate plane/grid.

Understanding the meaning of slope (mx+ b = y) and making rotations on coordinate grids are major challenges for middle and high school students. The sooner students can explore how and when to use coordinate plans, the easier their mathematical future will be and the more mathematics will make sense to them.

Is there more work involved? Maybe, but where else will students have the opportunity to explore mathematics and develop a repertoire of skills that they can take forward and use for the rest of their lives?


How do you see using these 4 lenses for student work in your own classroom?

I hope this blog gives you a system you can apply in your own classroom. As you can see from the above examples students have different needs that must be met in order for them to grow as mathematicians. Repeating the same lessons or procedures for every student will only help the student progress so far.

One way we can move students forward is by looking at student work after school, creating small groups of students who have the same needs, and providing a short 5-10 minute lesson that will address their needs (comprehension, organization, number sense strategy, connect and extend), while the rest of the class works on their new daily assignment or perhaps reviews a past concept. Sometimes students will have more than one need. Sometimes their needs may fall into multiple categories. Choose one lens to view the students’ work through and refine, refine, refine.