Every time I sit down with a student to introduce a new concept, it goes one of two ways. Either the student begins to make conceptual understanding and progresses through the objective or we hit a wall and realize the student has more work to do.
Both situations can cause problems. In the first scenario, I'm left wondering how far do I push the student. What if he/she masters the objective before everyone else? Do I enrich? Do I provide more problems? Do I continue to the next level in the objective?
In the second scenario, I'm left wondering what to do with a student who just wasn't ready. This student may not have the foundational knowledge or be developmentally there yet. How do I support the student? Do I push through the concept and come back to it again? Do I stop teaching the concept and go back?
We know that within a classroom, we can have several of these different students and one decision may impact everyone a little bit differently. Making these decisions is a challenging task that doesn't have a one-size-fits-all solution.
One thing that I believe can help is to look at our grade level objectives and determine where this objective falls within a student's math experience. If you study a scope and sequence of most math objectives, you tend to see a pattern with a topic being introduced, practiced and then maintained across several grade levels. Using this pattern, we can determine the best course of action to take when we have several different learning levels for one objective.
Conceptual Understanding: Intro
What you typically see when studying the way concepts are taught in mathematics is that there is a first introduction to a concept. This is usually where students will be introduced to a topic for the first time using concrete items. They are often encouraged to look for patterns and make sense of a concept using real-world connections.
When you identify this objective within your grade level, you know that it's OK to spend time here exploring. You know you are only laying the foundation and play and experimentation should be included in the lesson. A few suggestions for conceptual understanding topics include:
providing students with several different manipulatives to model the same concept
asking students to show the process or concept in a variety of ways
not moving on even if the student seems to grasp it within a few minutes. Instead focus on how you can encourage depth into a concept to make sure students understand all the facets of the objective.
An example of this is when students are introduced to multiplication and division for the first time in Grade 1. They are not expected to begin multiplying or dividing. It's about seeing the idea of equal groups and what that means. It's seeing that these equal groups can be counted to find how many in all or that objects can be shared into equal groups.
Mastery Learning: Proficiency
You will notice when looking at standards again, there is usually a grade level that has a huge chunk of time devoted to one specific process or objective. In this grade level, students are expected to spend time mastering the concept through practice. This may include learning the standard algorithm or learning multiple ways to solve a problem. Either way, this objective is a huge learning point for that grade level and time must be spent mastering the concept.
Before we go any further, I want to talk about the meaning of mastery and proficiency. A student can master a concept but only be proficient. This might mean that a student understands the concept and is able to do it according to the grade level expectations. To truly master a concept, this student should exceed the objective. This student should be able to apply the concepts to an unknown situation and make sense of how to solve the problem. There has been a lot of conflict over assessments that include a novel question, because teachers feel that it is unfair to students. However, this novel question isn't for everyone. It's for determining the students who fit that top tier of mastery- not proficiency. (Something to consider for your assessments- Does a 100% represent proficiency or mastery? But that's a whole other topic for a whole other time!)
When you identify this objective in your grade level, here are a few things to consider:
You will need to go through the entire concrete-pictorial-abstract sequence. You may also have students who are at different phases of that sequence. When assessing students, it's OK to allow them to use the manipulatives but it should be considered as needing support.
These are the objectives that are going to take up a lot of your time. These objectives will be represented with longer chapters.
Check the rest of your curriculum. A lot of times these objectives come around again in a different way. This means that students will have other opportunities to work on this concept. So if you have students who are right on the cusp of mastery, continue practicing but MOVE ON. Know that you will have more time to work on the concept again in later chapters.
An example of this is developing addition and subtraction algorithms in 2nd Grade. Students have already learned the conceptual understanding of addition and subtraction in Grade 1 but now have to apply it with 3-digit numbers with and without renaming. Students walk through each problem type and do a lot of work with base ten materials and writing the algorithm. The concept comes back later in the year with mental math computation, calculating money and even some measurement concepts. The entire year is spent adding and subtracting.
Maintenance Learning
For each grade level, you will always find a few concepts that seem like review. They may be just a quick few lessons that take something students spent a lot of time learning in previous grade levels and apply it to a different place value or a different concept. In Grade 4, there is always addition and subtraction of greater numbers.
Identifying the maintenance topics in each grade level help you determine how to teach those concepts. These are great opportunities for differentiation. This is where small groups shine, because 9 times out of 10, students will be at different points along this continuum. When teaching a maintenance objective:
Use small group instruction to review previous grade level content for those students who did not master it in the previous grade level. This may require you to study how the concept was taught in previous grade levels or to use previous grade level materials. This is one reason why members of my video library have access to grades K-6 videos. This allows teachers to review previous grade level content in just 5 minutes.
Begin at the concrete to aid in the foundational understanding and move to pictorial whenever possible. These students need to learn to use the pictorial to help them to move to the abstract with some support.
Use small group instruction to provide opportunities to push students who were merely proficient to mastery through application-type activities. This might include project-based learning (I have several available in my SHOP that are perfect for this.), games or even puzzles. This is the perfect opportunity to encourage students to play with the concept and develop new understanding.
So how do you determine which concepts fit under each heading: conceptual, proficiency, maintenance? You study your scope and sequence. A lot of textbooks include a chart within the front matter of the book that outline these concepts. Look for patterns and see where concepts repeat.
If you are like me and want it in a snapshot, I have created a document that outlines those concepts. I have recorded the concepts that are mainly taught at each grade level and tried to show how they come across several grade levels. Again, this is not exact to a specific textbook series but allows you to look for patterns. You can print out your own FREE version by accessing the link below.
Download here:
Learning about the big topics for each grade level takes some time but is totally worth it. I'm hoping that by using this document, it will help you to make educational decisions about how you should spend your time on each concept. Where do you keep moving and where do you stop and trudge through? It's important to know the difference! Leave me a comment about how you support your students in each level of their understanding!