Joe Rosenstein


Problem Solving and Reasoning with Discrete Mathematics

Why This Book?

Briefly …

  • Although the content of this book is discrete mathematics, its goal is to use the content as a vehicle for improving students’ reasoning, modeling, and problem- solving skills
  • this goal is very much in line with the “Mathematical Practices” perspective of the CCSS – and, with respect to focusing on this goal, this text goes beyond the standards themselves
  • the core of any worthwhile set of standards should be the “Mathematical Practices” … and that is the focus of this text
  • the current version of the CCSS is overly focused on preparing students for calculus — and thus largely omits the content of this text — and neglects the needs and development of a substantial percentage of high school students
  • this book will provide enrichment for the abler students who want to go beyond the narrow approach of the CCSS, and an appropriate alternative for the students who do not need and are not ready for Algebra 2
  • high school teachers are being challenged to accomplish the impossible – to teach Algebra 2 to all students – and to focus on teaching to narrow and procedure- based assessments
  • many high school students are being challenged to accomplish the unreasonable – to prepare themselves for future courses they will never take
  • all students should have the opportunity to work on real problems, on meaningful mathematical tasks – like those in this book – not only on routine exercises
  • all students should have an opportunity to appreciate the value and usefulness of modern mathematics and to be engaged in, and enjoy, doing mathematics

On the one hand …

Everyone seems to agree that it is important that students go beyond the basics, that as a result of taking courses in mathematics they should acquire skills in reasoning and problem solving, that they should come to understand the important role that mathematics plays in modeling and shaping their world.

The question is, “How should they achieve these skills and understandings?”

While it is possible that they can become expert problem solvers and reasoners as a result of taking traditional math topics, it is unfortunately not likely that this will happen because many teachers have become accustomed to teach traditional topics in the traditional way and will continue to do so, no matter what the national standards say.

And, unfortunately, though the national standards want students to achieve these goals, and state so explicitly in the “Mathematical Practice” portion of the standards, they fall short in indicating how teachers are to achieve these goals.

Problem Solving and Reasoning with Discrete Mathematics enables teachers to focus on teaching students problem solving and reasoning while teaching the content of discrete mathematics, a collection of topics that lend themselves easily to a problem solving, modeling, and reasoning approach.

From that perspective, Problem Solving and Reasoning with Discrete Mathematics is designed to help schools achieve the CCSS goals of improving students’ problem solving, reasoning, and modeling skills.

That is one side of the balance beam in the website’s icon. Problem Solving and Reasoning with Discrete Mathematics can help teachers achieve these important goals of the standards.

On the other hand …

The other side of the balance beam is based on the conclusion, shared by many math educators, that the current math standards are inappropriate for all students. The idea that all students should learn the content of Algebra 2 is simply absurd. As a result, despite the claims made by supporters of the standards, the CCSS is not providing students with the math that they will need for college, career, or citizenship. It is simply trying to prepare all students for calculus, a subject that most students will not need.

When the New Jersey Commissioner of Education tried to institute a statewide Algebra 2 exam, I testified before the legislature’s joint education committee. I asked the legislators, “What is 64 to the two-thirds pow er?” They quickly apprehended that this basic fact of Algebra 2 was not something that was needed by all students and, perhaps as a result, the proposal was quietly tabled.

Why is the content of discrete mathematics appropriate for all students? It is a wide-ranging collection of mathematical topics that are related to everyday experiences and that are accessible to all students, including those who have not mastered fractions and algebra. Here is a short list of simply-stated problems that fall under the rubric of discrete mathematics:

  • Which way of connecting a number of sites into a network involves the least cable?
  • What’s the best route for a courier who must collect deposits at 25 ATM machines?
  • What is the smallest number of colors need ed to color the 48 states in the continental United States if states that share a border must be colored with different colors (so that all borders can be clearly distinguished)?
  • How many different pizzas can you have if each pizza must have at most three of the eight available toppings?
  • How many tickets do you have to buy to make sure that you have a winning ticket in the contest that involves correctly selecting six numbers from 1 to 36?
  • How should the 465 seats in the United States Congress be apportioned fairly among the states after each census? What methods have been used to try to accomplish this?

It is unfortunate that the authors of the CCSS did not realize the value of discrete mathematics in achieving their goals and essentially eliminated two important domains of discrete mathematics – vertex-edge graphs and their applications and systematic listing and counting – from the standards. They did this despite the recommendation of the National Council of Teachers of Mathematics in its 1989 standards document that “discrete mathematics should be an integral part of the school mathematics curriculum.”

Why did they do that? Because of their slavish adherence to the false assessment of American mathematics education as “a mile wide and an inch deep” and the false conclusion that American students fall far behind their counterparts in other countries.

Since the international assessments only assess topics common to the curricula of all countries, of course our students will do worse on those topics than students who spend all of their time on those topics. Rather than accepting the claim that our curriculum is too broad, we should be arguing that the curricula of other countries are too narrow.

Has it been shown that engaging in discrete mathematics facilitates the improvement of problem solving and reasoning for all students? No. Indeed, no studies have been done that support the conclusion that any particular approach works.

However, New Jersey’s state mathematics standards included topics in discrete mathematics from 1996 until 2008, (when its state Board of Education surrendered to the CCSS tidal wave) and New Jersey consistently scored among the top states on the math portion of the National Assessment of Educational Progress. (Indeed, if diversity of states’ population had been taken into consideration, New Jersey would have been considered first in the nation.) Perhaps the presence of discrete mathematics in the standards played a role in New Jersey’s success.

Problem Solving and Reasoning with Discrete Mathematics

Copies of this book are no longer available, but I hope to print additional copies soon.  Email me and I will notify you when copies are again available.