Most college students struggle with the vocabulary of our disciplines. In their various electronic exchanges, they do not use a lot of multisyllabic, difficult-to-pronounce words. And virtually all college courses are vocabulary rich – unfamiliar words abound. Most students know that the new vocabulary in a course is important. They use flash cards and other methods to help them memorize the words and their meanings for their exams. Two days later, the words and their meanings are gone.

Word sort is a strategy that helps students learn and better remember new vocabulary. Students work in small groups, with each group given an envelope containing key terms on separate slips of paper. Students are instructed to discuss what they think the relationships among the words might be. The strategy was developed for use in science courses, where terms have more precise meanings and fit more readily into categories. Students do this initial sort before reading about the terms or hearing them defined and discussed in lecture. After exposure to the words in the text or lecture, students get back into their groups and re-sort the words, comparing their new arrangements with the ones they first constructed.

Lots of iterations of the basic strategy can be used. For example, individual students can be given the collection of terms and told to define and relate them after having done the reading as a homework assignment. Before turning their work in for some modest number of points, students might share with other students in a small group what they’ve done. Or the instructor might use a particularly good categorization in a final review of the material or position that chunk of content with what’s to be learned next.

As might be expected, some students (in this article it was a small group) object to the approach. These are the students who think that the instructor should just tell them the definitions and their relationships. Having to figure it out for themselves means that the students are doing the work the teacher should be doing. What these students fail to understand is that the process of discussing – saying the words aloud and using them in sentences – makes the words more familiar and therefore easier to remember. Exploring how the words relate to each other means that the students are building a framework that puts the words in context, also making the words easier to remember in both the short and long terms.

If students work with the terms and their relationships before being given their definitions and relationships, they are forced to draw on their prior knowledge and experience. Students discover that they often do know something about the terms and their relationships, and teachers need to include more activities in courses that challenge students to draw on their prior knowledge. Students do not arrive in college courses as blank slates – they have taken (in this case) science courses previously. That tasks like these challenge students is a good thing. Students benefit when they are put in situations where figuring out answers is up to them.

Reference:

Nixon, S. and Fishback, J. (2009). Enhancing comprehension and retention of vocabulary concepts through small-group discussion: Probing for connections among key terms. Journal of College Science Teaching, May/June, 18-21.

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As a math teacher, I have struggled with what to do with homework. Homework is a conduit of communication between the students and the teacher. If a student knows the material and can show competency without homework, what is the use of the homework? Should every student have to complete the same amount of work for mastery? I guess we should break down homework types in mathematics. The most popular homework is rote practice. A student is given a list of problems broken up into multiple levels of difficulty, which allow for practice and mastery. A second type of homework is the story problem, which is a train wreck according to students everywhere. A third type of homework is experimentation and discovery.

Rote practice. Rote practice has become the four-letter-word of mathematics, yet remains one of the most common classroom assignments. It is frowned upon because rote practice doesn’t show how the mathematics is applied and used outside of the math class. Being able to add a list of fractions together is a great test of patience and determination, but will it translate to the kitchen when the student bakes a double batch of cookies, or worse, a half recipe?

Although many teachers frown on working on a list of problems, there is some validity to providing direct instruction practice so that when the student understands the application, the mathematics doesn’t stand in the way of finding an answer. With a list of pure mathematics to which the student has the answers, it is incredibly important to check for more than completion. If a student works all of the problems and gets full credit for completing the homework, but all of the work is wrong, what has she learned? When the student takes a quiz or test and does the same process as in the homework, and has the wrong answer, is it really the fault of the student? Was there a point in doing the homework? If the homework isn’t a vehicle of communication between the teacher and the student, then no, there wasn’t a point.

Story problems. Story problems are a double-edged sword. Some problems that are written are pure gold. They show a valid application of the mathematics in a way a student may need to use in real life. For instance, given a recipe that serves eight with lots of different fractions, translate the amounts for 16 people or four people. A more challenging problem may be 12 people, but that makes the problem much less realistic. A train traveling at any speed is good for a quick warm up in class, but as homework the topic has too much of a stigma. I’ve had many students freeze up for no other reason but the word train appears in the problem. Once you get up to trigonometry, train problems are back on the table.

When creating story problems, less is more. Given a homework assignment with over five story problems, the students who need the practice will freeze up and be overwhelmed. The students who understand the math will either do the problems because they enjoy the math, or skip the homework because they understand the math, and don’t like “English.”

Creating a few story problems that get to the heart of the material is the best way to engage students in applying the math. At the beginning include several steps to help the students investigate the question. What information is pertinent? What is extraneous? What formula should we use? What concrete information are we given? What unknown are we looking to find? If this is started early on in the year and continued throughout the year, by the end the students should feel more comfortable with story problems and know how to solve them. Even if you teach the same students two year in a row, starting off simple and leading students through the problem solving will always benefit the learners.

Experimentation and discovery. New textbooks often start with experimentation and discovery, and, to me, that is a big problem. Students need to have basics to fall back to before they can discover a higher level concept. That said, if students can play with the math in with experimentation, they are much more likely to remember the topic later. They have invested time into the problem and are committed to the solution. A good extension to this is a project that brings life to the math and allows students to show their individuality and creativity.

Bringing it all together. Let’s see how all of this can be used together. I teach a class that has a unit on linear equations at the start of second semester. I want to make sure that once we get there, lines are the only challenge. (For more information about this activity, see next Friday’s post, which will provide teaching tips and a printable handout.) I try to use a form of sustained learning that focuses on the material throughout the year. Starting in the first quarter, I will hand out a packet with 10 to 15 problems that hyper-focus on a mathematical topic. I usually start with one-step algebraic manipulation. I demonstrate exactly how I expect the students to solve the problems from method of solution to checking the work. When the student hands in the packet, she either gets full credit or I hand the packet back to her for corrections. No score is recorded until the packet is perfect.

I move from one-step equations with addition and subtraction to one-step with multiplication and division. Before moving to two-step equations, I might add a packet combining all forms of one-step problems. After I moved through several layers of algebraic manipulation, I’ll move to formulas and their manipulation. Eventually I will have a packet on the formula for slope. I will have a packet on substituting into equations, and have a packet on point-slop form of a line. Then I’ll combine the two topics and have students take the point-slope form and solve for y. Throughout all of these packets I present the information and math basics, without saying anything about linear equations or algebraic manipulation.

When I get into the unit on linear equations, I can have application and story problems to figure out the equation, either directly or with scatter plots. I end the unit with a project. I ask the students to create a picture of a face using straight lines. If the class is upper level, I may include parabolas, hyperbolas and circles. Despite the number of lines they use for the creativity level of the project, I only grade 20 lines. Of that 20, only five can be vertical or horizontal lines. This allows for creativity, but doesn’t demand a huge amount of grading. I also give them a sheet that asks for endpoints, slope, point-slope and slop intercept form of each line. This really helps in grading and following their work.

Homework is for practice and communication. The point of homework is to give students a chance to practice on their own and communicate to the teacher what they do and do not know. If the theme of practice is rote, which has four letters is not a four-letter word; it is of immense importance that the teacher reviews the work and comments. Story problems are great for applying the math. It really highlights that math is just a tool, not an insurmountable obstacle. With story problems it is important to go over the work in some manner, be it by grading or just reviewing with the class. There are often several ways to approach a story problem so a class discussion becomes a learning moment where students can teach each other on how they approached the work. Experimentation and discovery is great, especially in a project. In this type of homework it is possible that a great amount of work has gone into it. Grading is important, but the real nit picky math check should be saved for the rote homework.

Huckleberry Rahr taught high school mathematics for nine years throughout the Midwest and one year in Papua New Guinea. During that time, Rahr earned a Masters of Education from Cardinal Stritch University. She spent two years working in the private sector before returning to education as a part-time instructor for ITT Technical Institute in Madison, Wis. Rahr is a member of Inside the School’s editorial board.

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According to the National Council of Teachers of Math, all students, k-12, need to make decisions about what units and methods are the best for measuring something. A unifying concept for high school science is the idea of change, constancy and measurement.

But when you talk to students about how small an atom really is or the breadth of the universe, they have a hard time visualizing just how small or large the scale really is.

Nikon has developed an interactive Website, the Universcale, which places light years and femtometers (a quadrillionth of a meter) into perspective. The site features a bar at the bottom of the screen that lists the units of measurement from the smallest to the greatest. Above the measurement bar are silhouettes that illustrate an object that is commonly measured with each unit. Objects line up from least to greatest, with invisible quarks at the small end, the great pyramids in the middle, and the vastness of outer space at the far end. Humans are included on the scale, as are rabbits, fleas, and dust mites.

At each measurement stop (picometers, nanometers, centimeters, meters, kilometers) the site offers an explanation of what the measurement term means and what it commonly measures. Measurements are listed in scientific notation and a grid displays the relative grandeur of each object.

Another great site, though much simpler in design is SensibleUnits.com. Type in any unit of measure and the site will offer equivalent measurements in relatively common objects. Two centimeters is the same as the length 2.5 grains of basmati rice, lined up end to end, or the width of two CD cases, stacked. Ninety-seven feet is the same as the span of 22 moose antlers. For numbers over 1,000, the site will translate the number into scientific notation, but it does not accept it for entries.

When used together, the Nikon site and the Sensible Units site reveal some interesting results. According to Nikon, the Great Pyramid of King Khufu is 147 meters tall, or 1.3 American football fields, according to Sensible Units. The 15-meter Tyrannosaurus is the equivalent of 2.5 stretched out human intestines or 7.6 Kobe Bryants.

Both sites could be used, separately or in conjunction, to make math and science measurements more accessible.