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Week 7: Prerequisites and accessibility

Part 1

Pick a particular, small topic in math ed - something you could reasonably introduce in just a few lessons. For example, "Equations" is too big, but "Quadratic equations" is more like it.

List prerequisites for your topic.

Part 2

Write a definition of what "prerequisites" are, using your topic as a source of examples. You may need to search some sources about the concept of prerequisites.

Part 3

Describe how technology can help you circumvent or remove prerequisites you named, making your topic more accessible. What are some pros and cons of removing prerequisites from one topic? What if you designed a whole curriculum with a goal of having little or no prerequisites for each topic?

Task Discussion

  • Green Machine   April 29, 2013, 3:30 a.m.


    Part 1

    Topic: Transvesrsalsn

    Prerequisites: supplementary & complementary angles, parallel lines, basic geometry concepts

    Part 2

    The scholars should possess a firm foundation in the background of geometry; specifically in the concepts of angles and lines. They should also be able to perform basic math computations mentally.

    Part 3

    I’m having a hard time wrapping my mind around the idea of removing prerequisites through the use of technology. Within the aspect of a collegiate institutions, I can agree with removing them from the curriculum. By that age, you should have the mental capacity to independently study. They also slow the academic process in certain aspects. A curriculum with no prerequisites, in my eyes, could be possible under the guise that each individual is able to analyze information and critically think.


    I believe that prerequisites are important in secondary education and below institutions. They may be utilized in the form of diagnostic tests, do now, or intervention strategies. Once all the scholars have met your expectations then you might be able to ease up on the prerequisites.

  • Lisa Ritt   March 3, 2013, 8:39 p.m.

    PART 3 continued:

    Although this is one example of something I don't think needs a prereq necessarily, I do agree with much of what most of you guys have written in that more complex areas of math need a foundation of knowledge is simpler forms of the same thing. Showing the relations between math concepts I find is wonderful for learning and I'd hate to lose it. But, we also shouldn't OVER do it!

  • Lisa Ritt   March 3, 2013, 8:36 p.m.


    PART 3:

    Here are some practice stem and leaf and other plotting practices that you can easily find online. I honestly don’t understand why the curriculum I’ve seen requires the concepts of mean, median and mode being prerequisites to Stem and Leaf plotting. I actually think it’s becoming something that confuses students when they get to this concept. Students are working on find the mean and median and mode…then they move into Stem and Leaf…and they have a really hard time looking for these 3 items on a Stem and Leaf plot…its like the regular numbers disappear…so the concept of the 3 things they just worked so hard to understand gets lost in Stem and Leaf (this has been what I witnessed and experienced thus far). It has almost felt like you have to ask students to forget all the mean, median, mode stuff when you are learning the Stem and Leaf.

    Here are some good Stem and Leaf practice spots:

    this one has everything wrapped together:

    There is also controversy over how to stem/leaf a 3 digit #. Again, confusing for students. (see problem 4 with the parenthesis, etc)…uggg! Now mix this in with mean, median, mode in an 8th grade urban district…. It feels like a lost cause. (sorry ..I’m moving off subject here!!!)

    I guess my argument thru far has been to remove the prerequisite. Some pros for keeping the Stem and Leaf concept with the mean, median, mode would I guess be that these are a simple way of putting numbers in order vs a stem and leaf plot. However, I’m FOR getting rid of this particular prereq! I believe this concept can and should be taught alone as a way of data collecting. Its something that some people will like and some won’t and they can use it or not use it. 

  • Maria Droujkova   March 9, 2013, 5:50 p.m.
    In Reply To:   Lisa Ritt   March 3, 2013, 8:36 p.m.

    Lisa, as I was reading through your stories from Part 1 to Part 3, I was holding my breath. What will the investigation uncover? Will Lisa like this infographic better? Drama!

    And wow, what a story. If you feel your curriculum sequencing is a "lost cause," you are probably not alone. There may have been historic reasons for that sequence at some time. What you see now does not seem logical, though, does it?

    When you analyze prerequisites for yourself, and I really would like to encourage every math teacher to do so, you will often want to re-design the sequence of topics. In some cases, you'll be able to do it. For example, what if you taught stem-and-leaf earlier? 

    What are stem-and-leaf plots for? They give us the overall shape of the data, and help to catch the outliers. As you say, they are not good for looking for the whole trio of mean-median-mode. However, you do see the modes quickly on that type of plot. So, maybe students can make stem-and-leaf plots of some data sets that have prominent modes. Students can notice the modes, you can give them the word for what they notice, and then they can do more exercises. For example, "Create a data set such that its stem-and-leaf plot shows three modes." Then they can look at other types of plots and data representations, and ask themselves if they still see the modes - or if they see something else special! For example, a median is really easy to see in a simple ordered list, or a box plot. 

    So you can make the plots into prerequisites for mean, median and mode, not the other way around. Explore Learning has a pretty accessible gizmo taking more-or-less this approach:

  • Lisa Ritt   March 10, 2013, 7:52 a.m.
    In Reply To:   Maria Droujkova   March 9, 2013, 5:50 p.m.
    Great idea! I LOVE THIS IDEA! thank you!!! :) -Lisa Lisa Rittler email: cel ph# (215) 740-6036 Our OceanCity, NJ condos are available all year!
  • Lisa Ritt   March 3, 2013, 8:11 p.m.

    part 2 continued:
    here are some explanations of stem and leaf plots:



  • Lisa Ritt   March 3, 2013, 8:07 p.m.

    PART 2:
    "Prerequisites"...after researching several definitions, I'd define for math ed purposes as a foundation of math knowledge or math concepts you need to have a solid understanding of prior to learning a new concept with relates to the prerequisite. 

    Every text book I've seem or online examples of Stem and Leaf plotting show the lesson of mean, median and mode coming right before it and wrapped around the lesson of Stem and Leaf. Stem and Leaf seems to be thought of as a more complicated concept mostly because in examples of when to use stem and leaf, you are using a larger amount of information and numbers being tallied together. 

  • Lisa Ritt   March 3, 2013, 8:02 p.m.

    PART 1:
    STEM AND LEAF PLOTTING: I'm doing field work in a classroom where the kids are learning "stem and leaf" number plotting and really struggling with it. I'd say a typical prerequisite would be to learn the mean,median and mode prior to learning about stem and leaf. I personally am NOT a big fan of the stem and leaf plot so I figured, why not do this project on it..I'll see if I like it anymore when I'm done : )


  • Katherine Hanisco   March 3, 2013, 2:49 p.m.


    Part 1

    The topic I chose is calculating the area under a curve in calculus. The prerequisites for this topic are algebra and some calculus. 

    Part 2

    Prerequisites are the topics you need to understand before tackling a new topic because it uses those as a foundation from which to build. In order to learn how to calculate area under a curve, students need to have a deep understanding of functions, know how to manipulate functions algebraically, and understand the concepts of limits. It can be really difficult for students to learn calculus without a strong background in algebra, since calculus frequently requires skills such as combining like terms, simplifying, factoring, etc. If students get tripped up at those steps, it takes away from focusing on the calculus, rather than the algebra. Additionally, a deep understanding of limits is essential to understanding calculus.

    Part 3

    Technology can remove these prerequisites by giving visual demonstrations of how to fund the area of a curve. Students don’t need to know all the specifics of algebra to be able to understand the visual of a series of rectangles under a curve as an approximation of area. Technology can also remove the prerequisite of limits since a graph could be animated in such a way that it demonstrates the concept by starting with a few rectangles, then adding more and more. This shows how the more rectangles you use, the closer your approximation is, and with this visual, students could probably make the mental leap to the idea of shrinking the rectangles’ widths down to nothing.

    In that same vein, technology could also be used to demonstrate the concept of limits using Archimedes’ method of exhaustion to approximate pi. Students could use a circle with inscribed and circumscribed polygons to understand that one overestimates the circumference and one underestimates. Technology would allow students to watch as the number of sides increased to get a very good visual demonstration about how the true value is between the perimeters, and the more sides the polygons have, the closer it is to the true value.

    I like the idea of younger students getting some exposure to calculus before completing the required sequence of courses because there are a lot of results and applications that would make material more interesting and relevant for them. I think that a whole curriculum without prerequisites might work for students who don't intend to pursue math education. A lot of the standard math curriculum is designed to prep students for calculus and a lot of students will never take calculus. I think there could be some interesting and meaningful math classes where students learn about a variety of topics without necessarily getting the traditional prerequisites for each. However, I wouldn’t want to remove all prerequisites from calculus since those skills are essential for students who intend to pursue advanced mathematics education. 

  • Maria Droujkova   March 9, 2013, 5:26 p.m.
    In Reply To:   Katherine Hanisco   March 3, 2013, 2:49 p.m.

    Katherine, your first answer made me think about standartization of courses. You can name "algebra" as a prerequisite, and other course members will probably understand what you mean (I do for my part), because the school course of algebra has been designed to be pretty much the same across the US. There is a push for even more standards, like the Common Core: At the same time, the internet allows us to use materials from different countries, or design our own non-standard materials (such as my "algebra for toddlers").  The more diversity we have, the harder the job of labeling prerequisites will become!

    Love your examples in Part 3, and the connection with the method of exhaustion, which is links to the topic of circumference Sue explored in Week 5: Whether different track students should have the same math requirements is a hard question: how do they switch tracks later? Can they catch up if they take "math for poets"? Raising these questions in terms of prerequisites is such an interesting angle - thank you!

    Here is the low-tech version of "rectangles under the curve" approach you designed, with Legos:

    Lego Integration


  • Gina Mulranen   March 2, 2013, 11:54 p.m.

    Part 1

    The topic I chose is simplifying radicals. To find prerequisites for this topic, I researched a lot of different online lesson plans and tutorials to see where the lesson includes a review of a concept before teaching how to simplify radicals. The prerequisites for this topic that I noticed were most commonly referenced were factoring integers (factor trees), perfect square numbers, what a radical represents, and exponents and radicals as inverse operations.

    Part 2

    My own definition of a prerequisite is a skill or concept that the student needs to have already learned and mastered in order to be able to successfully learn a new concept. For example, the concept of radicals is essential in order for students to understand why they are looking for perfect square factors in the radical they are simplifying. If students cannot find the perfect square factors, they can use factor trees to find the prime factorization and they take out any pairs of like terms (square numbers). Students would also need to know about how radicals and exponents relate as inverse operations in order to check their work after they simplify the radical.

    This website had a good introduction of what a radical represents before teaching how to simplify the radicals:

    Part 3

    I think technology can help remove certain prerequisite skills like factoring a number. There are websites that can do that computation for you, like this one: I also think that technology can be modified to INCLUDE review of prerequisite skills so students have the opportunity to show what they already know before learning the new concept. This type of review could be a quick quiz with worked out solutions when the quiz is submitted so the students can review the steps and check their answers before they start. That way the skill is fresh in their memory and they can relate the steps they did to solve that problem to the new concept.

    I think that a pro of removing the prerequisite of factoring a number would be helpful when students are investigating a new concept or factoring large numbers that can consume a lot of time. However, I think it is a bigger con because of the students do not know how to factor a number or what a radical represents, pulling out the perfect square factors is not going to make much sense. I think this encourages more step-by-step based math, not mathematical thinking.

    I think that stating the prerequisites before or during a lesson is very important when introducing new material because of the connections the students should make with the old and new information. By building off of what the student already knows, they will have a better understanding of the topic and be able to pick up the skill by updating the existing information, not creating a whole new memory. I remember talking about the importance of addressing prerequisite knowledge in a course for my Psychology minor in my undergrad work. These connections will enhance memory retention of that topic because of the multiple connections made in the brain. Therefore, I think that a whole curriculum without prerequisites would not be successful.

  • Maria Droujkova   March 7, 2013, 6:40 p.m.
    In Reply To:   Gina Mulranen   March 2, 2013, 11:54 p.m.

    Review lessons - oh, good way to research prerequisites! With online resources available, this is a smart step for teachers preparing for a new topic. And yes, as you say in Part 3, clearly listing prerequisites is a service for students that can provide clarity for their review, but also for deeper math learning (good point). 

    Separating skills and concepts is a strong side of your definition. As you mention in your Part 3, tech can remove a lot of skill prerequisites, in particular. But as you illustrate, it can (and does) create a disbalance between conceptual understanding and the ability to get answers out of computers.

    How can technology help to address the other side - conceptual prerequisites? In other words, can technology help us understand what we weren't able to understand without it? Big questions...

  • Gina Mulranen   March 9, 2013, 8:28 a.m.
    In Reply To:   Maria Droujkova   March 7, 2013, 6:40 p.m.

    I think technology can help students understand a concept, depending on the way it is delivered. Some websites are designed to just teach the student the concept through videos and assessments that provide instant feedback on whether the student mastered the skill. I think this is the structure for most math-based tutorial websites that I have seen. The challenge is creating an activitiy where students can achieve a better conceptual understanding of the topic before practicing and mastering the skill. I think the manipulatives in this site help students discover and learn concepts on a deeper level. I think that these are great activities because they provide directions on how to use the manipulative and questions that relate to what the student should be noticing. For example, in this manipulative, the students are dragging different functions and seeing how the constants change in the function. Notice the question, "How does adding a constant to a function affect its graph?" This helps prompt students on what they should be discovering. This would be a great conceptual pre-requisite activities before starting to graph functions based on what the students already know about the shapes of certain functions.

  • Maria Droujkova   March 9, 2013, 4:57 p.m.
    In Reply To:   Gina Mulranen   March 9, 2013, 8:28 a.m.

    Gina, you write:

    "better conceptual understanding of the topic before practicing"

    - THAT is a big design challenge for tech people. NLVM is one of better examples of meeting that challenge. I have a lot of respect for Joel Duffin, the lead designer of that project and spin-off projects like eNLVM.

  • SueSullivan   March 1, 2013, 8:55 p.m.

    Part 1:


    My topic is the Cartesian Coordinate System and I found some prerequisites listed at, which is an informal lesson plan.  The student prerequisites are:
    Arithmetic: Students must be able to:
       perform integer and fractional arithmetic
    Algebraic: Students must be able to:
      work with very simple linear algebraic expressions
    Technological: Students must be able to:
      perform basic mouse manipulations such as point, click and drag
      use a browser for experimenting with the activities
    Part 2:
    I feel that, theoretically, prerequisites are the skills educators feel students need in order to be able to make sense of concepts that will be presented to them in the upcoming lesson.  Requiring students to have mastery of certain skills/concepts is an attempt to ensure that the upcoming lesson will the in their zone of proximal development.
    It is sometimes unclear why prerequisites are required, as so many people (local, state, federal) are involved with curriculum-regulating legislation.
    Part 3:
    In this case, technology (such as a calculator or software) could greatly reduce the need for the arithmetic and algebraic prerequisites (I'm assuming the prereq's want students to be able to perform these tasks without tech calculation assistance, otherwise why would they mention it? Or does such a statement assume 'with tech help'?)  Graphing software (such as Geometer's Sketchpad) can perform the arithmetic and algebra that introductory-level students need for introducing the Cartesian Coordinate System.  Advantages of using tech to reduce/remove these prerequisites are time savings (less time spent doing arithmetic/algebra computations) and motivation for students who haven't mastered the prereq skills yet - they can see how the prereq skills are applied (an answer to "why do we have to learn this stuff"?).  The most important advantage to using tech in this instance is more accessibility for people with disabilities.  Students whose disabilities interfere with their ability to do arithmetic or algebra can use tech to perform the computations and still be included in the lesson in a meaningful way.  The disadvantage to all of this is that students might be able to perform the computations using tech, but have no idea of how/why the tech is performing the computations (i.e. how to do it without tech (note that this may not apply to special-needs students for various reasons)).  Specifically, being able to "work with very simple linear algebraic expressions" is essential to understanding the concept of using the Cartesian Coordinate System to visualize the behavior of expressions/functions.  Students can use tech to generate function values, but will gain nothing from the lesson unless they understand what is actually happening mathematically.  Teachers need to ensure that this does not happen.
    Teachers can use tech for graphing instead of graph paper.  I think that students wouldn't have too hard of a time switching from tech to graph paper if tech was unavailable (they should know how to this anyway).  Personally, I find tech (if available) advantageous because it saves paper not only from error waste (it only erases so many times before tearing) but from students (such as my 5th grade self) swiping it for their own personal projects (such as paper airplanes and origami).
    As far as the technological prerequisite, I don't find any significant disadvantages to using a mouse to 'click & drag', as long as students are taught the mathematics regulating the behavior of the object being manipulated.  I believe that this prerequisite is fair; each and every student should be taught these tech basics.  Yes, the Digital Divide is an issue (for this reason, I believe that all homework should be able to be completed without tech (unless the teacher knows for sure that students have safe and reliable tech access)).  However, learning to perform the basic mouse manipulations and use a browser is something that most students would hopefully pick up rather quickly; even if computers were limited, these skills could probably be acquired in very little time.  As with all tech, teachers must always be ready to provide support; "How To" posters containing instructions/screen shots could be posted in the room as well (or used to try and compensate for a small amount of 'tech time').  I believe that the skills addressed by this prerequisite should be part of core standards (in an age-appropriate way).
    Personally, I feel that while some prerequisites are appropriate for all populations (such as the previous tech prereq), many prereq's put certain populations (those with disabilities, those of lower SES) at a disadvantage (cognitive/physical limitations; Digital Divide).  I feel that designing a whole curriculum with little or no prereq's would motivate learners, because they would never be told "you can't learn about 'this' because you can't do 'that' "(despite the fact that many maybe never had the opportunity to learn about 'that' or have a physical/cognitive disability that interferes with doing 'that').  Yet, assessment as we know it (i.e. as dictated by local, state, federal standards) would become difficult, if not impossible, as these systems address learning/assessment in a linear fashion.  Eliminating prerequisites gives learners freedom to pursue their passions and choose their own educational path, which would definitely result in non-linear movement of students within the educational system.  Students might be ready for this, but I don't think the rest of the world is - there are so many questions without definite answers and 'the system' is often very, very afraid of the unknown.
  • Maria Droujkova   March 7, 2013, 6:27 p.m.
    In Reply To:   SueSullivan   March 1, 2013, 8:55 p.m.

    Sue, small world - I designed the games that go with that lesson, when I worked at Shodor. One of my goals with the Maze game in particular was to make coordinates accessible to younger kids, with fewer prerequisites!

    ZPD is very relevant here, and as you explain in your Part 3, quite changeable by using of tech.

    There are several math ed researchers who are on a secret quest to figure out how to teach topics with minimal prerequisites... Why secret? Because, as you say, it's a question of power and freedom. 

    (I am only half-joking here).

  • MgnLeas   Feb. 26, 2013, 7:54 p.m.

    Part 2 (I know it’s out of order but it is how I thought about this assignment.)

    A prerequisite “something required beforehand or necessary as preparation for something else <the course is a prerequisite for advanced study>”. (I got this definition from So a prerequisite is knowledge one should have prior to learning something else. I should know how to read a recipe before attempting to cook something.

    Part 1

    The topic I thought of was writing proofs on congruency of triangles. Students need to know what triangles are and what congruent means. There are also many theorems the students must know before writing proofs; including definition of midpoint, definition of angle bisector, reflexive property, vertical angles are congruent, and several more. So now, my specific definition of prerequisite with respect to writing proofs of triangles would be something like, ‘The students must know what triangles are and what is means for objects to be congruent. Students should also learn theorems involving triangles before writing proofs.”

    Part 3

    Technology could help students in not having to remember all the theorems and reasons that make triangles congruent. (ASA, AAS, SSS, CPCTC, ect.) These theorems are specific and to be honest, I never used them outside of the geometry classroom. I have had to prove other things congruent or equal so the idea of writing the proof was something I used again but never these specific theorems. Some pros of removing these prerequisites would be that we could get right into writing the proofs. The students could use different software to help show that triangles are congruent. For example, in Geometer’s Sketchpad they could construction two triangles and show they are congruent by being able to move one and the other moves the same way, therefore remaining congruent. (After thinking about it more this might not be the best topic to remove the prereqs.)  I think some cons are the students will not be able to write the proofs in a specific enough way. Also they could use wrong rules if they do not actually have to learn them first. Although, while playing with the software they would most likely learn the rules as they are doing it. But I digress. I am unsure of writing a curriculum with little to no prerequisites. We need to know things so we can build on that knowledge. ‘You must crawl before you walk’ is a saying that comes to mind. I can show my two year old my calculus 3 work (Which he has seen) but without the prior knowledge of algebra, it is not teachable to him. Yes he can put y= a function into the calculator and see the graph but he does not really know why it is how it is. I feel like prepreqs help students to be able to know what they are ready to learn next. Kids write made up stories before they write specific narratives and other forms of papers. Some knowledge is needed before moving on; I do not feel that technology will enable us to eliminate them.

  • Maria Droujkova   March 7, 2013, 6:21 p.m.
    In Reply To:   MgnLeas   Feb. 26, 2013, 7:54 p.m.

    Meagan, I see you are going back and forth on the desirability of prerequisites for formal proofs. This may have to do with the bigger issue: what are these proofs for? 

    Imagine a circle, and its diameter. Imagine a point somewhere on the circle. Connect that points to the ends of the diameter. I am telling you the resulting triangle is always a right triangle. Am I lying? Is it a true fact? You'll probably want to be convinced that is so (or not) instead of taking my word for it. But maybe not by those acronyms! Lockhart used this example in his "Lament...' to make a point we should do away with formal school-geometry proofs.

    Thanks for the great example of how a proof may look in Geometer's Sketchpad (moving things around). Geometer's Sketchpad has its own language of proofs; a very different language from "two-column" formal proofs on paper, but it accomplishes some of the same goals.

    Here's a tongue-in-cheek book "Calculus for infants" - but it's not your Calc 3 by far!


  • MgnLeas   March 7, 2013, 8:52 p.m.
    In Reply To:   Maria Droujkova   March 7, 2013, 6:21 p.m.

    So as I read Lockharts paper, I became a little sad that I do not know more of the history behind math. As I look back to my math career the classes I learned the most in were my high school calculus class, and my college graph theory and combinatorics class. The first because my school did not offer calculus before this year and me and 8 other students went to the school board to have it added. My teacher sat with her college notebook and we learned it together. We the students were able to do what Lockhart talks about, we investigated, got frustrated and the had our ah ha moment. Our teacher and us were in it together! The second, our professor would give us a statement and say prove it. We would work on it and get so mad. I worked on one for like four days and had thought I finally had it, the professor found fault in the second sentence! It was maddening and I hated him so much for not just giving me the answers. However, after I finally got it I thanked him for not giving me the answer. Finding the result for myself was such a feeling of accomplishment. I discovered the solution for myself. These are the teachers that have inspired me and I hope to be as good as them. Lockhart makes great points about taking the art and fun out of math, we as teachers need to fight to put it back. In his way of thinking prerequisite are irrelevant I feel. You simply take one problem and let it lead you where it leads you. If that means that working with triangles leads you to a calculus idea you go with it. I will definitely be looking more to the historyof math and trying to incorporate it as much as possible when I teach. Thank you for this article.

  • Maria Droujkova   March 9, 2013, 7:47 a.m.
    In Reply To:   MgnLeas   March 7, 2013, 8:52 p.m.

    "Our teacher and us were in it together!" - this is a great feeling, isn't it? I think Paul Lockhart tries to keep the feeling alive when he teaches.

    I was sad when I read the "Lament" too. He made a book version, co-authored with Keith Devlin, which is much more optimistic. It has a whole another part on how good things can be! And Lockhart has another book out, which I have not read yet, but it sounds great: