One of the most rewarding and challenging teaching assignments during my teaching career was the 11 years that I spent teaching fourth through eighth grade mathematics in a small school. Rewarding in that my classes were small, having the same students multiple consecutive years, and both students and parents valued education. Being the only teacher of mathematics in grades four through eight and having the same students multiple consecutive years created, challenge of having no other teacher to bounce ideas off of and share with.

But the bigger challenge was in making sure that each year built the foundation for the following year – that is, in knowing about the progressive nature of mathematics, and how certain concepts and skills must be learned in each grade level to continue the study of mathematics in the next grade level and beyond. I knew that if my students didn’t learn the concepts and skills they needed to know this year in class I would have no one to blame but me. And I also knew we’d be spending valuable time during the next school year learning prerequisite skills the students should have learned the previous year rather than learning new content and progressing with their study of math.

But you know that no matter how hard you try, there may be some concept or skill that a student will learn or master only long enough for the assessment and then forget it. You also know that solving a particular problem may require several math concepts or skills learned within or across grade levels that students may not remember or have never mastered to begin with.

Take for example this problem available at https://www.learnbop.net/S/7.EE.B.4b-2 where students are required to graph the solution set of an inequality derived from a word problem and interpret it in the context of the problem.

Think about the mathematics concepts and skills a student needs to know and be able to do in order to solve this problem. There is a range of prerequisite concepts and skills within and across grades that a student must know in order to solve this problem. They must know how to:

- represent solutions of such inequalities on number line diagrams. (6.EE.B.8-3)
- use variables to represent quantities in a real-world or mathematical problem and construct simple equations and inequalities to solve problems by reasoning about the quantities. (7.EE.B.4)
- convert a rational number to a decimal using long division. (7.NS.A.2d-1)
- solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. (7.EE.B.4b-1)

The full report data shows that only about 30% of students can answer the initial Bop (problem)—which requires solving word problems leading to inequalities (7.EE.B.4b-1) correctly on the first attempt— since they don’t choose the graph that represents a finite set. In looking at Step 5 it’s evident that they are struggling with representing solutions of inequalities (6.EE.B.8-3) in that about 80% of them cannot answer this step on the first attempt. Their misunderstanding of this prerequisite of representing solutions of inequalities clearly impacts their ability to choose the correct graph that represents the solution to the Bop.

Here at LearnBop, we know that students may struggle with prerequisite concepts and skills within and across grades that are needed to solve a problem. So our step-by-step tutorial guides a student through solving the problem offering critical thinking statements along with hints, visuals, and instructional LearnZillion videos that support them in reviewing concepts they may have forgotten or not yet mastered that are necessary for solving the problem.

Interested in a free 30 day trial of LearnBop for your class(es) click here – or sign up your school and get 60 days of free access.

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