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## Direct Instruction of Mathematics Vs. Computer Based Mathematics Instruction

If I were to ask you which teaching strategy works better for mathematics in the classroom, direct instruction (teaching), or computer base instruction what would you say? That is such an important question today because school districts are dumping millions of dollars into computer based instruction in mathematics and other areas of the curriculum. So I decided to do a little research of my own on ERIC. To my surprise the first article I read was about a study done in a middle school in Minnesota. They were using ALEKS (Assessment and Learning in Knowledge Spaces), a technology-based mathematics curriculum and was piloted in the 2012-2013 school year. Two groups were compared: regular classrooms where direct instruction was the mode of delivery of the math lessons, and classrooms where the delivery of math instruction was through the computer based ALEKS program. The comparison of test scores on the Minnesota Comprehensive Assessment (MCA-III) was made at the end of the year and an additional district developed test based on mathematical concepts. Surprisingly, the findings showed grades six and eight student achievement was significantly higher from the direct instruction group when comparing test results on the district developed concept tests. On the other hand, seventh grade students in the direct instruction group achievement was significantly higher on the official MCA-III. So it turned out to be a mixed bag.

So I went to another study “Unpacking TPACK in Mathematics Education Research: A Systematic Review of Meta-Analyses” published in 2016 by Jamaal Young. He examined 65 initial studies that compared computer based technology in mathematics education. He narrowed it down to 13 that met the criteria he needed to do the study and found that of the mean effect size (growth) in mathematical achievement was from small to medium.

So in summary, computer based technology is definitely having a positive effect in mathematics instruction, but the significance is not huge at this point. We need to consider that there are many programs out there at this point and some are better than others. If we want to see these programs improve, we would be wise to give feedback to the companies that make them and suggest improvements as we see them. We also learned that direct instruction is still very powerful in teaching mathematic concepts and we should not altogether abandon that approach. Personally, I see the need for both. Some concepts will be better taught and reviewed with computers, and other topics with direct instruction. You be the judge!

If you would like to see math lessons built on the 13 principles of effective math research click here: Effective Math Lessons Store

Sources:

Mertes, Emily Sue, “A Mathematics Education Comparative Analysis of ALEKS Technology and Direct Classroom Instruction”, ProQuest LLC, Ed.D. Dissertation, Bethel University

Young, Jamaal Rashad, “Unpacking TPACK in Mathematics Education Research: A Systematic Review of Meta-Analyses”, International Journal of Educational Methodology, v2 n1 p19-29 2016

## Do You Ever Wonder How Your Math Series Compares?

Do you ever wonder how the math text you are using measures up to other publications on the market? Well, there is research available for some of these math series. Granted publications change as new adoptions come up, yet it is interesting to see who of the publishers are in the lead.

A study was conducted over a 3 year period involving 1,309 1st graders in 39 elementary schools, the four-state study is considered the largest experiment to test some of the nation’s most widely used commercial math programs. It was commissioned by the Institute of Education Sciences, the primary research arm for the U.S. Department of Education.

To determine how much math the students learned, the researchers used a nationally normed exam that was developed for the federal Early Childhood Longitudinal Study.

The results of the study were published in 2009. Two programs for teaching mathematics in the early grades—Math Expressions and Saxon Math—emerge as winners. At the end of 1st grade, investigators found, children in classes using the Saxon and Math Expressions curricula scored 9 percentile points to 12 percentile points higher on those tests than their counterparts in other classrooms. That is quite a significant gap above the other contenders.

So how were the top two different than those that were lagging behind?

The Saxon curriculum, published by Harcourt, is a more traditional, scripted program in which teachers offer explicit instruction on effective mathematics procedures. The Houghton Mifflin Co.’s Math Expressions curriculum, in comparison, integrates a more emphasis on student reasoning with direct teaching that is aimed at moving students to more-advanced mathematical strategies. The Investigations program was considered the most student-centered of the four curricula, while Scott Foresman-Addison Wesley Mathematics is a basic-skills curriculum that combines teacher-led instruction with a variety of different materials and teaching strategies.

So what can we learn from this? Direct Instruction (Teaching) is still a very important tool in teaching mathematics. Hands-on can certainly enhance students’ understanding of the concepts, but it is time to end the debate whether hands-on is better than direct instruction or visa versa. We should use both. Second, when it is time to adopt a new series, how about suggesting that the committee look at the research before making the final decision.

If you would like to see in-depth direct instruction math powerpoint presentations that highly involve the students, click on the Products menu tab at the top of this article.

To see a more in-depth article about this study click on the picture above.

Source: Viadero, Debra, “Study Gives Edge to 2 Math Programs” __Educational Weekly__, Vol. 28, Issue 23, Pages 1, 13, __Published in Print: March 4, 2009__

## Common Core Math – Working or Broken?

In 2007 the Federal government appointed a group of educational professors, researchers and stakeholders (school administrators, parents, and teachers) to study and advise on ways to foster greater knowledge of an improved performance in mathematics among American students. This group reviewed 16,000 research publications, listened to 110 public testimonies, reviewed 160 organizations’ written commentaries, and 743 teacher surveys. There were 3 main findings that came out of the this study:

- The K-8 math curriculum should be streamlined to emphasize the most critical topics in early grades.
- Rigorous research on how children learn should drive math instruction, emphasize conceptual understanding, procedural fluency, and automatic recall of facts.
- High quality instruction uses both student centered and teacher centered strategies.

This was basically the birth of our Common Core math curriculum today. My question to you as teachers is are all three of these guiding principles being followed in your math Common Core curriculum? As an outsider looking in #1 has definitely happened as I see many topics being skipped or moved up to the next couple of grade levels. On #2 I see great emphasis on conceptual understanding, but almost to the point of sacrificing procedural fluency and automatic recall of facts. Just remember the latter two are just as important as conceptual understanding. On #3 I see much more student centered activities today, but some teachers have gone overboard in eliminating direct instruction (teacher teaching in front of the classroom accompanied with good “checking for understanding”) altogether.

One last point that really bothered me about this study is what the guidelines left out. There isn’t a word in there about problem solving. What is the purpose of even doing math if we are not using it to solve problems. Maybe they just figured that was a “given”.

My point here is to be careful not to “throw the baby out with the bathwater.” Make sure as you teach mathematics you keep in mind the huge body of research that has already been proven that works well in helping students to learn mathematics, and that all three of these guiding principles are included in your instruction. And oh yes, don’t forget to include some problem solving.

Source: Brown, Carolyn, “A Road Map for Mathematics Achievement for All Students Findings from the National Mathematics Panel” __Center for Comprehensive School Reform and Improvement__ Published 2009

## Meta-Findings from the Best Evidence Encyclopedia

**Robert E. Slavin**

**One of the Authors of “Effective Educational Program: Meta-Findings from the Best Evidence Encyclopedia”**

The Best Evidence Encyclopedia is a free website created by the Johns Hopkins University School of Education’s Center for Data-Driven Reform in Education (CDDRE) under funding from the Institute of Education Sciences, U.S. Department of Education. It is intended to give educators and researchers fair and useful information about the strength of the evidence supporting a variety of programs available for students in grades K-12.

The Best Evidence Encyclopedia provides summaries of scientific reviews produced by many authors and organizations, as well as links to the full texts of each review. The summaries are written by CDDRE staff members and sent to review authors for confirmation.

The Best Evidence Encyclopedia covers reading and math reviews and allows the opportunity to describe both substantive and methodological patterns across a broad set of studies involving elementary and secondary grades. In an article entitled, “Effective Educational Program: Meta-Findings from the Best Evidence Encyclopedia” by Robert E. Slavin and Cynthia Lake the following conclusions were drawn as to the most often identified best practices in the classroom that affect student achievement. Here is what they listed:

- Strategies likely to improve student learning are those that improve the quality of instruction.
- Increased student active participation.
- Helping students to learn metacognitive skills.
- Improved management and motivation approaches.
- Comprehensive programs such as Classwide Peer Tutoring and Missouri Mathematics Program.
- Extensive professional development.

To visit the Best Evidence Encyclopedia website click on the picture below:

## What Research Says About Teaching Math Vocabulary

**The Research**

Teaching the math vocabulary is a must according to numerous research studies if you want your students to excel in mathematics. Unfortunately, few school teachers use effective vocabulary instruction in their math lessons. “Students may excel in computation but their ability to apply their skills will suffer if they do not understand the math vocabulary used in instructions and story problems” (Brun, Faye; Diaz, Joan M.; Dykes, Valerie J.2015).

Another study recommends that students learn best when the definitions of words are appropriate to the age level. “Teachers should provide student friendly explanations of the word rather than dictionary definitions.” (Beck, McKeown, & Kucan 2002)

Research also supports that the vocabulary that they will need as the concept is being taught needs to be understood before the teacher gets to that point. “Pre-teaching vocabulary in the mathematics classroom removes cognitive barriers that prevent children from grasping new content. When taught only at point-of-use, vocabulary words are often lost or misunderstood (Chad, David)

**Recommended Practices**

So here are some recommendations of how this should be done to reach the most effective way of learning the math vocabulary:

1. Pre-teach mathematics vocabulary.

2. Model vocabulary when teaching new concepts.

3. Use appropriate labels clearly and consistently.

4. Integrate vocabulary knowledge in assessments.

## Research on Developing Math Fact Fluency

There is a huge amount of research out there (Ando & Ikeda, 1971; Ashlock, 1971; Bezuk & Cegelka, 1995; Carnine & Stein, 1981; Garnett, 1992, Garnett & Fleischner, 1983) that in order to master math facts to the automaticity level students must proceed through three stages: 1) procedural knowledge of figuring out facts; 2) strategies for remembering facts based on relationships; 3) automaticity in math facts—declarative knowledge.

**Procedural knowledge**

Common Core does a good job with the first level where students need to be able to figure out correctly the answer: counting, count by, arrays, drawing objects or pictures, decomposing and composing numbers, using ten frames etc.

**Strategies for Remembering facts**

This is often the step that teachers, or publishers either do poorly on or skip altogether. Samples of strategies in addition could be doubles, doubles + 1, in-between doubles etc. Samples of strategies in multiplication might be the nine pattern (subtract 1 from the number being multiplied for the first number, subtract that number from 9 for the second number), the five pattern (cut the number being multiplied by 5 in half, if it is an even number put a zero behind it, if an odd number change the ½ to a 5 e.g. 3 ½ = 35). There are strategies and or mnemonics available for every fact that needs to be taught. All of the math fact lessons in Effective Math Lessons teach those strategies and mnemonics. Click on the words to see these lessons: Addition, Subtraction, Multiplication, Division

**Automaticity in math facts**

Students who are automatic with math facts answer in less than one second, or write between 40 to 60 answers per minute. Ashlock (1971) showed that children must have “immediate recall” of the basic facts so they can use them “with facility” in computation. Not knowing the facts in this way will greatly hinder their ability in later years when students are taking math classes in junior high, high school or college. When automaticity is developed, one of its most important traits is speed of processing. Think how important that would be for many of today’s careers. Automaticity comes from frequent practice through timed tests, flash cards, or today, iphone, ipad, or computer math fact games.

In retrospect, ask yourself as a teacher if you are making sure your students go through each of these steps in order to master the basic facts. Effective Math Lessons can help you greatly in this area, especially with step 2) Strategies for Remembering Facts.

## Starting the Year Off Right

August is here and many schools have already started with many more to follow in the next few weeks. Research shows the first day of class is critical for setting the pattern for the year (Tikunoff, Ward, and Dasho). They go on to say that effective experienced teachers take more time in setting up the behavioral and procedural standards for the year compared to those new to teaching. When this is done, the students know what to expect and a great deal of behavior problems can be avoided in the future. In addition with good classroom procedures in place, less time will be lost and more time will be spent on learning. Teach your students these rules and procedures just as you would a math lesson. By that I mean make sure they understand what is expected and then practice them. So here are a few things to discuss with the students:

- Introduction of yourself
- Introduction of the courses for the year
- Schedule
- Classroom rules
- Rewards and Consequences related to the rules.
- Procedures for pencil sharpening
- Labeling and turning in papers
- Policies on water bottles, cell phones,
- When and how to use computers
- Homework policy and procedures
- What to do with free time
- Accelerated Reader (if you have it)
- Grades
- Home Reports
- Skills for success

There may be more you can think of that you need to cover. Take the time thoroughly to go over these things and avoid problems later.

To help you get off to a good start I have a free powerpoint template that covers the things mentioned above. All you have to do is fill in the specifics. Highlight the text you want to type over and type your rules, and your procedures. You can do this very quickly and will be ready to go the first day of school.

**Click here to download the Back to School student presentation.**

## What Research Says About Direct Instruction

**Direct Instruction**, (not the commercial program) refers to (1) instructional approaches that are structured, sequenced, and led by teachers, and/or (2) the presentation of academic content to students by teachers, such as in a presentation or demonstration. In other words, teachers are “directing” the instructional process or instruction is being “directed” at students.

The largest educational research ever made was called** Project Follow Through**, completed in the 1970s. It cost over $600 million, and covering 79,000 children in 180 communities. This project examined a variety of programs and educational philosophies to learn how to improve education of disadvantaged children in grades K-3. Desired positive outcomes included basic skills, cognitive skills (“higher order thinking”) and affective gains (self-esteem). Multiple programs were implemented over a 5-year period and the results were analyzed by the Stanford Research Institute (SRI) and Abt Associates (Cambridge, MA). The various programs studied could be grouped into the three classes described above (Basic Skills, Cognitive-Conceptual, Affective-Cognitive).

The program that gave the best results in general was true **Direct Instruction**. The other program types, which have come and gone as educational strategies (having labels like “holistic,” “student-centered learning,” “learning-to-learn,” “active learning,” “cooperative education,” and “whole language”) were inferior. Students receiving **Direct Instruction** did better than those in all other programs when tested in reading, arithmetic, spelling, and language. **Direct Instruction** improved cognitive skills dramatically relative to the control groups and also showed the highest improvement in self-esteem scores compared to control groups.

That being said, good **direct instruction** should include the following:

- Establishing learning objectives for lessons, activities, and projects, and then making sure that students have understood the goals.
- Purposefully organizing and sequencing a series of lessons, projects, and assignments that move students toward stronger understanding and the achievement of specific academic goals.
- Reviewing instructions for an activity or modeling a process—such as a scientific experiment—so that students know what they are expected to do.
- Providing students with clear explanations, descriptions, and illustrations of the knowledge and skills being taught.
- Asking questions to make sure that students have understood what has been taught.
- Practicing the skill with the students called
**guided practice**. - Assigning more practice with the skill called
**independent practice**or homework.

Some may want to criticize the fact that this research is forty plus years old, but let me point out this was the largest study ever made in the history of educational research and it will never happen again. Why? Because the money is not there, nor ever will be there again to repeat it on such a large scale. So according to this, you cannot go wrong with using** direct instruction** in the classroom.

## Recognizing Student Achievement in Mathematics

When students experience the satisfaction of mastering a challenging area of mathematics, they feel better about themselves (Nicholls). The teacher can enhance this with praise, visual reward, or personal recognition. As a result, students learn to like and look forward to math because mastering it gives them a feeling of achievement. How do I know this? This very thing happened in my class. When I asked students what their favorite subject in school was, the majority said mathematics. I always praised the students with little reward slides after guided practice when I saw they were mastering the topic of the lesson. I did not have to do all kinds of clever games and activities to get them to love math, they loved it because they mastered it. This is another reason why lessons have to be very thorough and 100% on target. Everything in the lesson needs to be aimed in getting them to understand the concept. Then at the end of the lesson, have a little celebration to recognize their hard work in learning the concepts.