Mathematical Rigor vs. Acceleration
- Jessica Kaminski
- 7 hours ago
- 5 min read
“The program is just not rigorous enough…”
“He isn’t doing rigorous math in class…”
I hear from teachers and parents all the time who want to switch curricula that their current program just isn’t rigorous enough. I've learned that we all have a different definition of this word so I ask them to explain what they mean. These are some of the responses I hear:
“It’s just too easy.”
“He completed the assignment really quickly.”
“She should be learning 6th grade concepts earlier.”
Those responses focus on one specific quality of mathematics- the answer. It tells me that these educators are looking at rigor as something that is calculated by speed and accuracy. Their solution is to find a curriculum that skims over more topics so it feels like more is being accomplished.
By definition, that’s not really rigor. That’s actually acceleration.
What is mathematical rigor?
Let’s use it in another context. My family loves to hike. My 3 kids have been hiking even when I was carrying them in the baby carrier. When we travel anywhere, we take a look at a hiking app and try to find hikes around us that our children could do based on their age and ability.
Many of the apps will label the hikes as easy, moderate or rigorous. When you begin reading what a rigorous trail actually looks like, it includes steep trails and rocky terrain. It would take a hiker deliberate attention and more time to complete. It’s usually not a trail for a novice but for someone who understands how to hike that type of terrain.
If I wanted an accelerated trail, I would likely look for something that was flat and easy. It might even be a shortcut between more difficult trails. The goal would be to go easily and quickly.
Now let’s relate it back to math. Over the past few decades, it has been painfully evident that students need rigor in mathematics. Just look at the latest NEAP results to see that only 39% of fourth graders are proficient in mathematics. Many students, including myself, have received a superficial math training where they learned rules to memorize. Students were not taught to understand why these rules were formed or why they were needed. We just followed the procedure, got an answer and were told that was correct.
The problem with this type of teaching is that it is not creating thinkers but just learners who regurgitate information. (Yes, I purposely used that word. Students are literally just repeating what you tell them with no thought to its meaning.) Students are not going to be able to connect mathematical ideas and make breakthroughs with this superficial type of understanding. We have to provide them with a rigorous education.
The definition of rigor must consider all the components of math understanding from being able to do the math to being able to understand the math. This can also look different based on the grade level and the complexity of mathematics. A common definition was created by the Common Core that stated rigor in mathematics includes 3 key components: conceptual understanding, procedural skills and fluency, and application of those skills.
Even if the Common Core isn’t a source you like to follow, this definition isn’t just from their research. Richard Skemp introduced us to Instrumental Understanding, knowing the rules and procedures, and Relational Understanding, knowing the how and why. His research showed that there should be a balance of both in mathematics education to help students truly understand the concepts.
Other resources such as Bloom’s Taxonomy or even the Depths of Knowledge show us that there is a higher level of understanding that should be emphasized to push students beyond surface level understanding in mathematics. All of these resources are pointing us to a deeper definition if we truly embrace it.
How we encourage mathematical rigor in our instruction?
As I mentioned before, many educators tend to look to their math curriculum for the answer. That's only part of the solution. Your Student Book provides the problems students will answer. Run from any Student Book that has millions of problems and way too many chapters that just skim the surface. Avoid Student Books that don’t give opportunities for students to explore the meaning behind algorithms or formulas. This would all point to signs of an accelerated curriculum and not a rigorous one.
Once you have a great Student Book, look at the Teacher’s Guide. This is where a lot of the rigorous material will be housed. Are there great questions that challenge students to go deeper? Are there extension opportunities? Are there journal options that challenge students to create and critique?
What you might notice is that a lot of rigor comes from the instruction itself and not just the curriculum. The curriculum is a tool for that amazing educator who is going to push students when they need the push and differentiate when needed. This is why we need teachers who have a strong content knowledge and know how to interpret mathematics. They create rigorous expectations based on state standards and expectations.
What does mathematical rigor look like?
It's a lot easier than you think. Take your basic math problem and invite students to go deeper. Here are a few examples:
This was a 3rd grade lesson on adding and subtracting 4-digit numbers with 1-step word problems. Students were encouraged to solve the problem using a bar model to show why they chose that operation.
To make the lesson more rigorous, we invited students to show multiple ways to solve the problem. You see the student rounded to check the answer in a few different ways.
We also asked students to consider another word problem they could write using the same mathematics. This showed us whether they understood the context of the operation.
In this lesson, students were learning the algorithm for multiplying 2-digit numbers by a 1-digit number. We asked students to split the paper into three sections and show the other forms of the algorithm that reinforce place value. Then, we asked students to draw connections between them.
As I mentioned above, we can add mathematical rigor by adjusting our expectations. Instead of merely asking for answers, we can encourage explanations. We can invite students to draw diagrams and tell us more about how they got their answers.
Your curriculum should have great problems that progress students through concepts. They should build from concrete to more abstract concepts. You definitely want a program that is going to give you good basics to create your math block.
However, it's your questioning and expectations that's truly going to get the rigor you need in your math lesson. Every time you put a question up for students to solve, I want you to think:
Am I encouraging conceptual understanding? Do they show it with manipulatives or visuals?
Am I helping students work to procedural fluency? Do they understand why the algorithm is a short cut?
Can they apply the mathematics to any situation? Do they know when to use this strategy and when it would not be appropriate?
By thinking of these questions, you are now embedding the very definition of rigor into your lessons. If you want a friendly reminder of some of those prompts, I invite you to snag my free poster download below. Simply print and go!
If you are looking for even more resources for mathematical rigor, consider one of my math notebooking courses. These $39 courses include everything you need to help students implement math notebooks in a way that easily invites rigor.
Math Notebooks for K-1 includes printable centers, games and word walls to help students begin to use inventive writing to share their thinking.
Find out more by clicking on the image.
Math Notebooks for Grades 2-8 dives deeper into what notetaking looks like in the math lesson. We look at different types of notes students can take and how to utilize math notebooks for students to share their thinking.
Find out more by clicking on the image.
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