Mathematics Learning: A Journey, Not a Sprint

By Matthew Larson posted 12-20-2017 09:25


One of the questions I am frequently asked goes something like this: “What do you think about acceleration?” The question is often asked in the context of teachers working with parents who want their child to skip grade level work or entire courses so they can get into the next grade or course more quickly. The parental goal more often than not is to ensure their child can complete calculus in high school. As a school district math curriculum administrator, I faced “acceleration pressure” from parents nearly weekly.

Should we support acceleration? This question, like many questions in mathematics education, does not have a binary answer. The answer is “it depends.” Sometimes acceleration is appropriate and sometimes it isn’t. What does the answer depend on? Here the answer is clearer: it depends on the student’s demonstrated significant depth of understanding of all the content that would be skipped. If a student demonstrates significant depth of understanding of some but not all the content that would be skipped, then this is more appropriately an opportunity for enrichment rather than acceleration.

Last fall NCTM released a new position statement titled Providing Opportunities for Students with Exceptional Mathematical Promise (recently endorsed by the National Association for Gifted Children) In this position statement NCTM argues that “when considering opportunities for acceleration in mathematics, care must be taken to ensure that opportunities are available to each and every prepared student and that no critical concepts are rushed or skipped, that students have multiple opportunities to investigate topics of interest in depth, and that students continue to take mathematics courses while still in high school and beyond.”  

At the elementary level, and even into secondary school, speed completing computational tasks or carrying out routine symbolic manipulations cannot be the basis for acceleration. Too many parents, and others for that matter, still have a narrow definition of mathematics as computation and symbolic manipulation.

We must emphasize to parents, teachers, counselors, administrators, and students that the goals of learning mathematics are multidimensional and balanced: students must develop a deep conceptual understanding (why), coupled with procedural fluency (how), but in addition they also need the ability to reason and apply mathematics (when), and all while developing a positive mathematics identity and high sense of agency. All four goals are critical components of what it means to be mathematically literate in the 21st century.

There is evidence that students who speed through content without developing depth of understanding are the very ones who tend to drop out of mathematics when they have the chance (Boaler, 2016).  Acceleration potentially decreases student access to STEM careers if it results in students dropping mathematics as quickly as possible, rather than cultivating and developing the joy of doing and understanding mathematics. This is important to point out to parents, as dropping out of mathematics is clearly not an outcome parents want to encourage.

Acceleration can also potentially reveal inequitable practices, bias, or structural obstacles within a school or district. It is critical to remember, as argued in the NCTM position statement, that “exceptional mathematical promise is evenly distributed across geographic, demographic, and economic boundaries.” If the demographics of students accelerated in mathematics in your school or district are not evenly distributed across racial and economic boundaries, then reflection and analysis are necessary to determine why, and actions should be taken to remove whatever bias or structural barriers led to this inequitable outcome.

Here are some questions you can ask: What are the demographics of students in eighth grade algebra? Do they match your district’s overall demographics? What are the demographics of students in calculus or AP Statistics? How do the demographics change from eighth grade algebra to AP Statistics or calculus enrollment? Was the instructional climate not supportive of each and every student? Was the instructional focus not on developing depth of understanding? Were students accelerated into eighth grade algebra on the basis of computational proficiency, but without the conceptual foundation necessary to be successful in the long run? These are some of the critical questions that need to be asked and addressed.

The recent Mathematical Association of America [MAA] Study of College Calculus and related article in the Mathematics Teacher addresses the issue of acceleration to calculus in K-12 schools. In his article addressing the MAA study, Bressoud (October, 2015) questions the wisdom of the rush to calculus in U.S. schools. Bressoud specifically advocates for “an alternative to calculus in high school that focuses on strengthening students’ understanding of algebra, geometry, trigonometry, and functional relations while building problem solving skills …” Again, the recommendation is that our first and foremost goal should be to  ensure students’ depth of understanding of grade level or course-based mathematics before recommending a student be accelerated.

A joint position statement of MAA and NCTM states that the “ultimate goal of the K-12 mathematics curriculum should not be to get students into and through a course in calculus by twelfth grade but to have established the mathematical foundation (and disposition toward mathematical work) that will enable students to pursue whatever course of study interests them when they get to college.”  In addition, it is worth noting that not every student’s mathematical journey is best served by calculus. Statistics is arguably the subject within mathematics utilized by the widest variety of careers and is clearly a critical tool of analysis necessary for full participation in our democratic society.

As Jo Boaler (2016, p. 192) has written, “mathematics learning is not a race, and it is mathematical depth that inspires students and keeps them engaged and learning mathematics well, setting them up for high-level learning in the future.” But for some parents the metaphor of mathematics as a race is deeply embedded in their expectations for mathematics teaching and learning. We have to remember that many of the parents of our students attended school themselves at a time when they experienced a mathematics curriculum that was repetitive and less challenging. We need to help parents see that standards today are more rigorous, focused, and coherent.

As we work to change cultural expectations for mathematics teaching and learning—which  admittedly will take time—one  strategy we can use with parents when addressing acceleration and the “race” metaphor is to clarify what type of “race” might be more appropriate to mathematics learning. If parents want to see mathematics as a race, then we need to convince them that it is a marathon and not a sprint. Having a great 100-meter time is rarely helpful if we are on a lifelong journey to deepen our understanding of mathematics and use it productively in our lives.

Mathematics should be taught deeply and in a balanced way, with equal attention paid to procedural fluency, conceptual understanding, reasoning and problem solving and the development of a positive mathematics identity.  When these goals are achieved, students will benefit from mathematics learning that will serve them for their entire life.

Acceleration should not happen at the expense of creating gaps in student understanding by skipping foundational learning standards. Skipping or rushing through instruction in ways that fail to develop depth of understanding may lead students eventually to drop out of mathematics, cut off their future mathematical opportunities, thus denying them the potential to be fully actualized as members of our democratic society.

We need to engage students in ways so that they can embrace their own mathematical journeys and be empowered by mathematics in their own lives. Yes, acceleration may be appropriate if a student has demonstrated significant, deep, and complete understanding of grade level or course-based mathematics. We certainly want each and every student to be appropriately challenged.

I encourage you to post a reply and share with others how you have addressed the “acceleration issue” in your school or district.





04-28-2018 22:56

Something that really stood out to me when reading this blog was when you say "students must develop a deep conceptual understanding (why), coupled with procedural fluency (how), but in addition they also need the ability to reason and apply mathematics (when)" because as a future math teacher, these are all really important things that I will need to be able to address when teaching my class.

01-01-2018 04:00

This is a very significant discussion. I work at an international school and we teach the Cambridge Curriculum till Grade 10 and IB in Grade 11 & 12. We have to often have the discussion along "creating lifelong learners" of mathematics with parents. Parents at times don't see that taking an extra year to learn certain topics does not imply that their child is not capable mathematically.

12-28-2017 12:24

Great discussion.  I would add that if doing what was best for learning math or any subject was the primary function of public schooling  then typical schools would look and work a lot differently than they do.  

I question if accelerated mathematics and other higher level courses serve other functions along with advancing students toward calculus in HS.  If schools in the area were I live don't offer accelerated mathematics, or other elite academic programs, they won't market themselves well to families who are looking for a nice area with "great schools" for their children.  School leaders are under significant pressure to offer these programs so their schools can compete with other districts.   Add keeping up with other districts and seeking status for schools and communities, to the list of reasons students take advanced math.   I admit this view is cynical, but I think it is worth considering.

12-25-2017 14:09

Matthew,  thank you for your insights. I think it is really important to focus on the depth of understanding before considering acceleration. I doubled-up on my Algebra I and Geometry my Freshman year so that I could take Calculus my senior year. I really needed the foundation to be successful in mathematics. For me, this was the best choice, and I still found myself needing to study long hours when I was working on my B.S. in mathematics. I think acceleration would have been detrimental in my case.

12-22-2017 05:29

This is such important conversation to have and to socialize... I have used many of the resources shared in the post and the discussion thread, and I find I am also in a constant loop of responding to parents (often supporting admin to do so) specific to this topic. Some of the work that may need to happen is to identify what an "outlier" is in terms of mathematics learning, and if we have outliers (true, exceptions) how do we meet their learning needs? I say this with a huge macro view behind it: schools that dive deeply into naming and framing what their math program is (and is not...) can rely heavily on that alone. I think about reaching all learners, habits of mind / Mathematical Practices, progressions across the standards, rigor/coherence for all, equity in T&L math, etc. Two of my favorite quotes from @Matthew Larson at the P2A conference a few years ago are: Who is doing the math? and Who is teaching, what/which math, to which students? Powerful to pause and ponder. I hope this discussion continues...

12-21-2017 19:36

In April 2016 the University of California (BOARS) released this statement on acceleration in math and its impact on their admissions process.

Two highlights:

Does completing calculus in high school have special weight in UC admissions decisions?
Generally not. No single course, including calculus, determines an admissions decision... While rigorous coursework is a positive factor, poor performance in an advanced class such as calculus usually does not strengthen a student's application.

Should calculus be offered in high schools?
BOARS strongly recommends that high schools maintain calculus as an option for enthusiastic, well-prepared students. However, BOARS also strongly urges students not to race to calculus at the cost of full mastery of the earlier math curriculum... Choosing an individually appropriate course of study is far more important than rushing into advanced classes without first solidifying conceptual knowledge. Indeed, students whose math classes are at a mismatched level – either too advanced or too basic – often become frustrated and lose interest in the topic.

12-21-2017 18:53

Matt, this is a very good piece.  If you want a testimonial, I skipped a grade and strongly encouraged my parents not to let my sister do the same thing.  There are many reasons, but I firmly believe this as a rule.  There are special cases and I have never forgotten a counselor and school refusing to allow a young man in mathematics to skip to calculus from algebra ii because "he cannot be advanced in mathematics if he is not advanced in everything."  In that case, the young man was denied what he needed.  Thanks for posting this.

12-21-2017 13:12

Thank you for this clear and thoughtful position statement. As a teacher, I am in almost total agreement. Skipping over entire grades, in particular, can be dangerously short-sighted. As for tracking, as a sixth grade teacher making those recommendations within our district's system, I have monitored the demographics carefully for years and would strongly encourage others to do the same.

I've also tried really hard to communicate to families that our district's course options for tracking 12-year-olds are about their apparent needs at that age, not their overall potential. In case it's helpful to other middle school teachers, here's some typical language I've used: .

However, as a mother of a kid who used to have sky-high math potential and loved talking about math, but who's been bored in math class for years and now just goes through the motions, I think we still have work to do as a profession to ensure all kids are getting to experience joyful problem-solving and thinking in math class. My kid's scored really high on standardized math tests the whole time, so he doesn't show up as a problem in this age of obsession with "accountability," but I can clearly see his science classes have been sparking his curiosity and higher-level thinking way more than his math classes have been.

12-20-2017 17:43

Thanks for sharing this Matt.  I wrote a blog post about this called The Case Against Acceleration, where I shared that people often are motivated to accelerate students because:

  1. Students are bored or not sufficiently challenged
  2. People think the content of some grade levels are so similar that they can be combined.
  3. That middle school math is easier than high school math.
  4. Students won't be able to take Calculus without acceleration.

I have concerns with each of those reasons and I share an option I prefer.  Hope it's helpful.

Thanks again for your leadership.

12-20-2017 17:42

I agree with everything written here. I would add that the social maturation of a student is also relevant. Students may have a sufficiently deep understanding of the material to be skipped; however, their mathematical learning, going forward, will probably not be with their peer group. In this case, students who do not have the maturity (or maybe it's confidence) to interact with older students may be limited in the help they can acquire when struggling with new material.


12-20-2017 15:50

So glad to see an article on this.  For our HS math department we wrote a letter to parents giving some guidance to accelerating or not from 8th grade to HS courses, especially geometry and algebra II. We gave examples of what could happen, what has happened, and pros/cons. We dramatically cut down on requests that were premature and have hopefully prevented some pain and conflict. There are those who qualify, but many who do not.  As Mr. Larson says, "it depends."

12-20-2017 14:26

I wrote about this:
The article includes suggestions for what may constitute reasonable acceleration.

See also this guest post on my blog, from a Penn prof: 
More Calculus, Less Understanding?


Henri Picciotto
"There is no one way."

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