Category: math reasoning and applications

Word Problems and Reasoning … Rule 42

In my introductory algebra class, I gave a quiz today; this quiz included this question:

Some milk having 1% fat is being mixed with milk having 4% fat; the mixture will be 100 gallons, and have 2% fat. How much fat is in the mixture?

Now, I always review the quiz right after we complete … so sometimes I get to see some interesting reactions.  Please understand that all of the items on this quiz dealt with ‘puzzle’ word problems (not much real context), so students were not feeling mellow … many were feeling quite a bit of stress.  Besides the stated reactions about the question being ‘tricky’, I got to see students respond when they realized what they were supposed to do.

A typical wrong response to that question was to start working on solving the ‘problem’ they expected to see … how much of each type is supposed to be mixed?  Quite a few of these students wrote the correct equation including the “0.02(100)” for the milk fat in the mixture.  However, not many of these students realized that they had the answer to the question.  Our entire approach to these mixture problems in the class prior centered on “value = rate(quantity)” as a basic concept.

I was pleased that some students just wrote down the answer (2 gallons), perhaps with a note “0.02(100)”.  These students got that basic idea that value = rate(quantity).  I’d say that this 20% compares with the 40% who tried to ‘solve’ the problem but never realized that they had the answer … and the 40% who had no idea what to do.

One of the culprits for the difficulties is the inadequate way percents are done in math classes.  We focus so much on correct answers that we do not make it clear that percent is not ‘how much’ … and that every percent is a rate which is multiplied by a base.  For my question on the quiz, just knowing “percent times base” is sufficient to get the right answer (and show some understanding).

The other culprit is based on the high-anxiety suffered by students when faced with “word problems”.  I’d like to think that my class presents word problems as a reasonable use of language and algebra, even if the problems are either trivial or uninteresting.  Further, I’d like to think this positive approach helps students be more comfortable dealing with these problems.

Some readers might wonder “why do those puzzle problems at all” … perhaps we should “make the content relevant to the students”.  With all of the focus currently on ‘alignment’ and ‘context’, those are reasonable questions.  Based on my understanding of the learning process (along with some sociology), the question is not easily answered.  I am pretty sure that covering ONLY relevant applications is not a good idea for a mathematics course serving a general purpose; it might work in occupational math, or specialized math, but not so much when there is so much diversity among the students.  One student’s relevant problem is another student’s puzzle problem, and another student’s life survival issue; in addition, high context in problems can localize the learning and interfere with general reasoning and understanding.

So, I will continue to work with quite a few puzzle problems in our introductory algebra course — and keep a focus on the basic ideas that allow us to understand and solve them.  My goal is to help students develop a deeper understanding and develop connections.

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Walking the STEM Path IV: Content versus Preparation

Developmental mathematics has a mission to prepare students for college math courses, including those on the calculus trajectory.  Both the data I see and an analysis of the courses suggests that our current courses are not doing very well … so I want to look at this problem from a different perspective.  #STEMpath #Completion #AlgebraicLit

If your institution is like mine, the conversation about intermediate algebra preparing students for college algebra (or pre-calculus) goes something like this:

College algebra covers ‘complex fractions involving binomials and trinomials’, so intermediate algebra should cover ‘complex fractions involving monomials and simple binomials’.

We tend to obsess ‘content’, and presume that a reasonable progression of content creates a good preparation.  This approach uses procedural complexity as a proxy for reasoning at the level needed for calculus success.

Instead of looking at content at such a fine level of detail, how about starting from the target.  In calculus, students need:

  • Procedural knowledge with understanding
  • Flexibility
  • Reasoning, especially related to multiple quantities changing in the same problem

The emphasis in intermediate algebra (and much of college algebra) is on the first half of the first item (procedural knowledge … ‘understanding optional’).  If this is true, then the results we see in the research are not surprising at all.  The question becomes: what is a more effective approach to designing the curriculum?

The ‘calculus list’ above is a list of student abilities.  We should design a sequence of courses deliberately organized to develop those abilities, building a STEM bridge from the basic algebra level to calculus I.  There is no reason to assume that one particular approach to this designing will be superior to others … should intermediate algebra develop all 3 abilities in all content areas included in the course, or should intermediate algebra focus on the first two abilities, or perhaps a mixture of levels where some content areas are done ‘deep’ (all 3 abilities) while others are done ‘shallow’ (first ability only).

We need some field testing of those ideas, but work has already begun.  In the New Life project, our outline of the Algebraic Literacy course takes the approach that we build all 3 abilities in each content area.  Curricular materials for this work are, sadly, not available at this time … I will be sharing 3 sections of material for this model at my AMATYC conference session in New Orleans.  The Dana Center “Reasoning with Functions” (RwF) materials are being developed currently; that model takes a similar approach to the abilities, from what I can see.  One difference is that the two RwF courses form a sequence, replacing both intermediate algebra and pre-calculus; the Algebraic Literacy course would replace an intermediate algebra course only … institutions would still have a pre-calculus course to follow it.

A related design question deals with pre-calculus: one semester, or two semesters (college algebra, then ‘trig’ in some form).  Our default trajectory should be one semester.  The only reasons to need two are (1) our failure to provide a good intermediate algebra course, and (2) the minority of students who MIGHT need a sequence of courses.  We often justify two semesters based on having “too much material”; I suggest that this is a fallacious argument (it’s not about the content … it’s about abilities).

Instead of our current sequence of courses copied from bygone years, we need an efficient system designed to help students move from one place (developmental) to another (calculus).  This is the most exciting work, and the most powerful opportunity, to ever face our profession.

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How to Recognize an Algebraic Literacy Course

The next AMATYC journal will have an update on the New Life Project (over 100 colleges, over 800 sections, and something like 16000 students this fall semester).  In order to prepare that update, I spent a lot of time searching various web sites and following up leads.  That detective work led me to a number of colleges using “algebraic literacy” as a course title … when the course was just ‘intermediate algebra’; I also found some courses titled ‘intermediate algebra’ that were closer to algebraic literacy. #AlgebraicLiteracy #MathLit #NewLifeMath

This post is a brief “field guide” to help us recognize an algebraic literacy course.  Algebraic Literacy (the course) is one of the New Life math courses (AMATYC Developmental Mathematics Committee) developed in 2008 to 2010, based on the professional work from the last two decades.  The material below comes from our wiki ( ).


The Algebraic Literacy (AL) course prepares students for mathematics pathways which include college algebra, pre-calculus, and other courses requiring a background beyond the Mathematical Literacy (MLCS) course.

This is similar to an intermediate algebra course … on the surface.  There is a fundamental difference, however:  intermediate algebra is a derivative of an earlier “Algebra II” course from K-12, while Algebraic Literacy is engineered to meet the mathematical needs of college mathematics (backwards designed).

The focus of the AL course is on building understanding of mathematical systems with a dual emphasis on symbolism and application. The Algebraic Literacy course includes quantitative topics from areas besides algebra, which supports the needs of both STEM (Science, Technology, Engineering, and Math) bound students and other students.

A typical intermediate algebra course is heavily symbolic, with applications playing a minor role (and often using trivial applications with little payoff for preparing students).  The Algebraic Literacy seeks a balance between procedural fluency and higher level skills.  For some Algebraic Literacy courses, the applications form the context within which the mathematics is developed; for others, the mathematics begins first with applications integrated.  In considering applications, the Algebraic Literacy course includes problems with numeric solutions which would be solved symbolically in calculus.


Basic proportional reasoning and algebraic reasoning skills, and some function skills, are required prior to the Algebraic Literacy course.

We do list 5 specific areas of prerequisite skills following this general statement.  However, the Algebraic Literacy course is designed to allow ‘co-requisite remediation’ at the appropriate level: Building on basic algebraic reasoning skills, for example, we aim for deeper understanding and solid symbolic skills.  By contrast, the typical intermediate algebra course presents a conflicted approach: students must show higher levels of symbolic mastery before enrolling but then intermediate algebra reviews many of those skills (without directly dealing with the development of reasoning directly).

More students are able to begin an Algebraic Literacy course than a typical intermediate algebra course.


1. Numbers and Polynomials
2. Functions
3. Geometry and Trigonometry
4. Modeling and Statistics
The content is intended to be integrated and connected.

In the Algebraic Literacy course, we would not see a chapter on “radicals and rational exponents”; we might see a section dealing with fractional exponents in an early sequence dealing with functions, including an application in half-life models … and a later section working on radical notation focusing on domain and range, followed by a section on translating between radical and exponential forms.  Either of these sequences of topics might also include geometry and/or trigonometry, and modeling concepts such as parameters.  Almost all topics will be presented as connected to one or more other topics, both conceptually and in terms of applications.

For most intermediate algebra courses, the content is usually 9 to 12 ‘chapters’ of material arbitrarily divided up … and separated.  A minimum of connections are made to other ‘chapters’.  Overall, the intermediate algebra course does not tell any story; the intermediate algebra course is a long series of vignettes only loosely connected by ‘category’.

By contrast, the Algebraic Literacy course tells a story of mathematical reasoning with both symbolic and application dialogues.  The design of the Algebraic Literacy course is based on being the first step along a path which includes calculus and/or other significant mathematics.  We seek to build covariational reasoning, a step up from Mathematical Literacy, on the path towards a good pre-calculus experience.


This field guide would not be accurate without  emphasizing a fundamental difference: Algebraic Literacy supports other STEM fields in addition to those needing the traditional Calculus Path.  This is primarily a distinction for the two-year college situation, where our programs often include mid-skill to high-skill fields (manufacturing technology, engineering technology, health careers, electronics, computer science, etc).  This inclusive approach is why Algebraic Literacy is not just algebra … geometry, basic trig, and statistics are included.  Most intermediate algebra includes some non-trivial geometry (right triangles, for example); however, you can recognize an Algebraic Literacy course by the presence of non-trivial geometric reasoning and symbolic representations, trig functions at a basic level, and enough statistics to interpret models developed from data.

Recognizing an Algebraic Literacy course involves multiple factors — goals, prerequisites, content, and the nature of the ‘story’.    A instinctive evaluation is based on this:

As a mathematician, can I get excited about teaching this course … is the focus on good mathematics, with the goal of developing abilities as opposed to “Algebra II all over again”?

We will see colleges move in this direction; I hope that you will consider joining the work!

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Meaningful Mathematics … Learning Mathematics

Among the pushes from policy makers is ‘meaningful math’ … make it applicable to student interests.  Of course, this is not a new idea for us in mathematics; we’ve been using ‘real world applications’ to guide some reform efforts for decades. #realmath #collegemath #learningmath

Some current work in reforming mathematics education in college is based on a heavy use of context, where every new idea is introduced with a situation that students can understand.  We know that appropriate contexts with meaning to students helps their motivation; does it help their learning?

Before I share what I know of the theory and research on these approaches, I would like you to envision two types of textbooks or instructional materials (print, online, or whatever).

  1. Opening up the start of a lesson or section, repeatedly and randomly, generates a short verbal introduction followed by formal mathematical symbolism related to the new idea(s) in almost all cases.
  2. Opening up the start of a lesson or section, repeatedly and randomly, generates a variety of contextual situations and mention of a mathematical idea or tool that might be used.

Many traditional textbooks are type (1), while a lot of current reform textbooks are type (2).  Much of the change to (2) is based on instructor preferences, and I am guessing that much of the resistance to change is based on instructor preferences for (1).

Let’s take as a given that contexts that are accessible to students improve their motivation, and that we have a goal to improve student motivation; further, let us assume that we share a goal of having students learn mathematics (though that phrase means different things to different people).  There seem to be a number of questions to answer dealing with how a context-intensive course impacts student learning of mathematics.

  • How uniform is the impact of context on different learners?  (Is there an “ADA-type” issue?)
  • Do students learn both the context and the mathematics?
  • Is learning from a context more or less likely to be used and transferred to new situations?

I know the answer to the first question, based on research and experience: The impact is not uniform.  You probably understand that there are language issues for quite a few students, perhaps based on a class taught in English when the primary language is something else.  However, most of the contexts have a strong cultural factor.  For example, a common context for mathematics work is “the car”; there are local cultures where cars are not a personal possession, as well as cultures outside the USA where cars are either generally absent or relatively new (and, therefore, people know little).  The cultural problems can be overcome with sufficient scaffolding; is that how we want to spend time … does it limit the mathematical learning?  There are also ADA concerns with context: a sizable group of students have difficulties processing elements of ‘a story’ … leading to problems unpacking the context into the quantitative components we think are ‘obvious’.

The second question deals with how the human brain processes different types of information.  A context is a type of narrative, a story; stories activate isolated memories and create isolated memories.  To understand that, think about this context:

You are standing on a corner, and notice a car approaching the intersection.  When the light turns red, the car applies the brakes so that it stops in about 2 seconds.  You estimate that the distance during the stopping process is about 100 feet, and the speed limit is 30 miles per hour.  Let’s look at the rate of change in speed, assuming that this rate is constant through the 2 second interval.

The technical name for a story in memory is “episodic memory”.  This particular story might not activate any episodic memories for a given student; that depends on the episodes they have stored and the sensory activators that trigger recall.  Some students will respond strongly and negatively to a particular story, and this does not have to depend upon a prior trauma.  More of a concern are students who have some level of survival struggle (food, shelter, etc); many contexts will activate a survival mode, thereby severely limiting the learning.  Take a look at a report I wrote on ‘stories’,%20Ignore%20the%20Story.pdf

What happens to the mathematics accompanying the story?  If we never go past the episodic memory stage, the mathematics learned is not connected to other mathematics; it’s still a story.  In the ‘car stopping at an intersection’ story, the human brain might store the rate of change concepts with the rest of this specific story, instead of disassociating the knowledge so it can be used either in general or in new ‘stories’ (context).  Disassociating knowledge is another learning step; many context-based materials ignore this process, and that results in my biggest concern about ‘problem based learning’.

Using knowledge (Transfer … question 3) depends upon the brain receiving sensory input that activates the knowledge.  This is the fundamental problem with learning mathematics … our students do not see the same signals we see, ones that activate the appropriate information.  For this purpose, traditional symbolic forms and contextual forms have the same magnitude of difficulty: the building up of appropriate triggers to use information, as well as creating chunks of information that work together.  We need to be willing to “teach less mathematics” so that we can focus more on “becoming more like an expert with what we know”.  For more information on learning mathematics based on theoretical (and research-based) points of view, see and 

I can’t leave this post without mentioning a companion issue:  Contextual learning is often done by ‘discovery’.  Some reform materials have an extreme aversion to ‘telling’, while traditional materials have an extreme aversion to ‘playing around’.  From what I know of learning (theory and research), I think it is safer to take the traditional approach … telling does not provide the best learning, but relying on discovery often results in even more incomplete and/or erroneous learning.  Just for fun, take  a look at,%20Explaining,%20and%20Learning.pdf

Looking for a brief summary of all of this stuff?

Contextualized learning comes with significant risks; use it with caution and a plan to overcome those risks.
Mathematics is, by its nature, a practical field; all math courses need to have significant context used in the process of learning mathematics.

These ideas about context are related to the efforts of foundations and policy influencing agents (like CCA).  We need to keep the responsibility for appropriate instruction … including context.

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