Singapore Math Picture

Singapore Math in the Lehigh Valley

This week I would like to talk about math, because I like it.  In fact, it is one of my favorite subjects.  I am very proud of the efforts our school has made in math.  I look forward to analyzing the data we will be collecting over the next few years.  Great commendations should go to the teachers, recognizing their ability to deliver such a promising program.  Specifically, I would like to share about our Elementary math program.

Covers fewer topics in greater depth

Compared to a traditional and Common Core math curriculum, our program, Singapore math, Math in Focus,  focuses on fewer topics but covers them in greater detail. Each math textbook builds upon prior knowledge and skills, moving the students towards mastery and automaticity. We work hard so that the students are ready for their next level. By the end of sixth grade, Singapore math students have mastered multiplication and division of fractions and can solve difficult multi-step word problems, including basic algebraic understandings.

Our Math was found to emphasize the essential math skills recommended by the National Council of Teachers of Mathematics (NCTM).  They do this by focusing in on the essentials that build strong number sense and mathematical reasoning skills.

Three-step learning process

The younger grades spend a significant time building number sense through their work in part-part-whole.  This is the mathematical equivalent to phonics in reading.   The students learn that numbers, quantities, and algorithms are representative of something.  They learn how to take numbers apart and put them back together in a way that gives meaning.

They learn practical steps to solving word problems.  I loved math but loathed story problems.  No one taught me how to approach them or gave me the tools and confidence necessary to attack them.  Our students are taught the bar model beginning in grade two.  Our second grade teacher helps her young pupils see that 100 is comprised of other things.  We can find whole parts and missing parts, just by using the bar model.  Later in the intermediate grades, the teachers and students work through more challenging and complex uses of the bar model.  Just a bit of secret, I used the bar model to quickly solve ratios on the GRE (Graduate Records Exam) that I took for entrance into graduate school.

A bar model used to solve an addition problem. This pictorial approach is typically used as a problem-solving tool in Singapore math.

Three Step Process: Like We Learn

SCS math program teaches students mathematical concepts in a three-step learning process: concrete, pictorial, and abstract. This learning process was based on the work of an American psychologist, Jerome Bruner (1960). In the 1960s, Bruner found that people learn in three stages by first handling real objects before transitioning to pictures and then to symbols. That is why we teach the students in this matter.

The first of the three steps is concrete, wherein students learn while handling objects such as chips, dice, or paper clips. They use something that teachers call manipulatives, anything that the student can move around, touch, or manipulate in order to learn the concept or skill.  Students would learn to count these objects (e.g., blocks) by physically lining them up in a row counting them or grouping them. They would then learn basic math operations like addition or subtraction by physically adding or removing the objects from each row.

Students then transition to the pictorial step by drawing diagrams called “bar-models” to represent specific quantities of an object.  This involves drawing a rectangular bar to represent a specific quantity. For instance, if a short bar represents six books, a bar that is twice as long would represent a collection of twelve. By conceptualizing the difference between the two bars, students could learn to solve problems of addition by adding one bar to the other, which would, in this instance, produce an answer of eighteen books. Multiplication, division and subtraction can be solved by this model.  Additionally, they can use this model to begin to conceptualize algebra with missing parts to a math algorithm (problem).

Bar Modeling allows the student to model the thought process used to solve the math problem.  And, the teacher can provide strong feedback in order to help prevent errant thinking.   The student can move onto more complex mathematical problems with exclusively abstract tools such as symbols and numbers.  
The third part, abstract is simply writing the algorithm with symbols.  3 +4 = ?   or   3 + x = 7.

We don’t spend much time on math facts like the addition facts and multiplication facts.  But, that doesn’t mean that they are not important.  The teachers review them in the morning meeting.  However, this is something parents can do at home with their children to ensure they are prepared by playing a game, flashcard etc…

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Faith of Dissent

I may be somewhat out of step with my fellow Evangelical Christians in acknowledging a certain affinity for St. Benedict and the way of life he represents. Mind you, I am not advocating his theological beliefs.  There are vast differences between our theological mountains.  But, I admire some things about the dedication.

In my time in Europe during my high school years, I made the chance discovery, via the evangelical Lutheran church my host family attended regularly, of the ancient Daily Office.  This Daily Office was associated with the early monasteries and prescribed in St. Benedict’s Rule. These folks immersed themselves in the Psalms and the rest of Holy Scripture as they prayed in addition to hymns and chants.  Daily prayer structures the entire day and the whole of their life in the Benedictine community. I cannot help but wonder what would happen if our school community of Christians were to take up this practice. If our parents, faculty, administration, staff, student and I did so, it might just change the course of history.

We often must stand against the assumptions of the prevailing culture. Given my strong Christian roots and world travels, I recognize the perils of Christians ascribing to their ethnic nationalisms near canonical status.  We see God in view of our culture rather than seeing our culture in view of God.  Advancing our national pride can deflect so many from their primary allegiance to God’s kingdom. Indeed, we need to bring our faith into a commitment that grows deeper each day. God’s Word and common sense dictate that this depth can only be reached by reaching to God first.

H. Richard Niebuhr’s classic Christ and Culture (1951) may offer a key to grasping this ideal. Niebuhr’s posits “Christ against culture” is generally associated with the Evangelical Christians who are generally based on the fidelity to the Gospel, not on ethnic distinctives. In my own experience as a follower of Christ and a Christian educator, I find that many of my fellow Christians, especially students, too easily speak of “engaging the culture”without having a strong sense of why they are doing so or of its associated perils. But I get this. Any effort to transform culture may do nothing of the sort if the Christian community does not first recognize the ways in which it is distinct from the culture it aspires to change.

Christianity, by design, is a faith of dissent. We are to be outliers, persons who stand out from the norm.  We are to be counter-culture.We are to provide the victory.  Often,many churches have permitted the various cultural revolutions to redirect the faith away from the hard path of obedience to one that simply affirms everyone without seeking to transform their affections and their lives by God’s word. It seems easier for Christian schools to declare the love of God without emphasizing the need for the disciplined life as a vessel of this love.Nevertheless, proclaiming a boundlessly permissive love is not the gospel. “Jesus did not die to liberate us from norms for living but to save us from the power of sin so that we might live for his glory in his good creation empowered by the Holy Spirit (Dreher, 1957).”

Our students may or may not recognize that there is a great falsehood we are fed by our post-Christian culture.  We are falsely led to believe that we belong to ourselves and that “one’s individual desires [are] the locus of authority and self-definition.” We are indeed not our own but belong to the God who has saved us in Jesus Christ. Just like those Benedictine monks, we as followers of Christ need to embrace order, obedience, and, above all, daily prayer. Where else could one find a community, than in a Christian school, whose very foundational principles and directives contradict our culture of hedonism and egoism and stand in the gap to save the next generation from the perils of a secular life?

Where indeed. Of course,the vast majority of Christians don’t embrace the Christian school.  They are lured by the lights and offerings of a secular setting.  People visiting this Christian community should see how we have in some fashion exemplified the Benedictine way. We need to be intentional.  We need to educate our children, and embody something of a parallel polis—that is, an alternative to the modern political community.  We need to demonstrate the fulfillment of living within and alongside the prevailing culture but not investing too much hope in it. We do not want to have our students eschew politics entirely, but have them recognize that no matter who is in administration in Washington, no matter how ostensibly pro-Christian, is capable of stopping cultural trends toward desacralization and fragmentation that have been building for centuries.

I enthusiastically agree with Dreher’s  (distinguished professor and author) counsel here: “For serious Christian parents, education cannot be simply a matter of building their child’s transcript to boost her chance of making it into the Ivy League.” And: “The separation of learning from virtue creates a society that esteems people for their success in manipulating science, law, money, images, words, and so forth.” It is right to understand education as formation of the whole person and not simply as the key to a better job. This is precisely why our school wants to develop students who love God with all their minds, bodies, and hearts.

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Culture Wars

The longer I am a follower of Christ, the more I am convinced that we must passionately engage God’s purpose and conform our perspective into God’s view of the world, not our own.   Left to ourselves without the miraculous discernment only possible with the help of the Holy Spirit, we would never be able to understand the world clearly.  This is why those in a Christian school sacrifice so much.  They do this in order to ensure that each subject is enlightened with the Gospel so that our students can see the whole truth and shine it on the lies with which our culture seduces them.cultureware

This is especially important since we are in the midst or an epic cultural war in America and the Western world. Unmistakably, the Christian is being pushed out of our society and relegated to the margins at best.   I just read a quote by George Wegel in First Things, magazine “It’s a Culture War, Stupid…” I was struck with this simple assertion.

Wegel continues to posit that the Church is in the midst of the darkest times that we have seen in millennia and that this dark time is easily seen looming in the short distance.  He calls for the Church to don its spiritual armor, provided to the saints,  and battle for the souls that are damned to eternity without God.   He considers those who persist in denying that the Church is engaged in a culture war are either combatants or victims, whether they choose to accept it or not. He provides several examples.

One example that I have recently read in numerous posts and articles is the lauding of how Iceland has nearly eradicated Down Syndrome from the country.  Wow, impressive.  The Health Services of Iceland and the American media have praised this accomplishment.  Wait! What’s the catch?  What is missing from this fascinating “advance” is that this was accomplished by selectively aborting those babies so that society could be cleansed.  Seem familiar to the goals of some in the 1930’s and 1940’s?  Margaret Sanger, Planned Parenthood founder, would be proud of their eugenic cleansing.   This evil cannot go unchecked by the Church.  I cannot help but think of Isaiah’s words, “Woe to those who call evil good and good evil, who put darkness for light and light for darkness, who put bitter for sweet and sweet for bitter! (ESV)”  We are now praising evil as if it is good in our society; furthermore, society condemns and ridicules the Christian for the ridiculous notion that EVERY life is valuable no matter what.

Wegel (2017) says:

Canada’s vulnerability to the culture of death is exacerbated by Canada’s single-payer, i.e. state-funded and state-run, health care system. And the brutal fact is that it’s more “cost-effective” to euthanize patients than to treat secondary conditions that could turn lethal or to provide palliative end-of-life care. Last year, when I asked a leading Canadian Catholic opponent of euthanasia why a rich country like the “True North strong and free” couldn’t provide palliative end-of-life care for all those with terminal illnesses, relieving the fear of agonized and protracted dying that’s one incentive for euthanasia, he told me that only 30 percent of Canadians had access to such care. When I asked why the heck that was the case, he replied that, despite assurances from governments both conservative and liberal that they’d address this shameful situation, the financial calculus had always won out—from a utilitarian point of view, euthanizing [my friend] and others like him was the sounder public policy.

Like in Canada, America is a mature democracy, where that utilitarian calculus among government bureaucrats and bean-counters wouldn’t survive for long if a similar, cold calculus were not at work in the souls of too many citizens. This one reason why the Church must engage the culture war, not only in Canada but in the United States and throughout the West: to warm chilled souls and rebuild a civil society committed to human dignity.

This is why the Christian school, like Salem Christian School in Macungie, PA, must develop students’ minds so that they love God with all their minds, bodies, and souls.  We passionately desire to partner with Christian parents so that their children see the world as God would see it.

For example, in our World Studies courses we provide instruction that sees it as a civic responsibility to influence the world with the Gospel. We want to stand with parents, churches, and society in order to win the war for our youths’ souls.  The Christian school will stand in the the gaps so that America does not reduce a human being to an object whose value is measured by “utility”.  Even a secularist can see that it is to destroy one of the building blocks of the democratic order—the moral truth that the American Declaration of Independence calls the “inalienable” right to “life.” That right is “inalienable”—which means built-in, which means not a gift of the state—because it reflects something even more fundamental: the dignity of the human person. (Wegel, 2017)

When we lose sight of that, we are lost as a human community, and democracy is lost. So the culture war must be fought for the sake of our country and the sake of these souls lost if they do not know Jesus. And a Church that takes social justice seriously must fight it.  Please keep your Christian school in prayer.  We are on the front line.  The Christian educator is there to battle for the hearts and minds of their students.  They are there to not only shelter them from the onslaught; but, provide them tools to fight the Good fight wherever God takes them.

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Self Advocacy and Data Use

Last week I discussed the emerging culture of data use at Salem Christian School. I would like to continue this conversation. However, I would like to expound on student use of data as a pursuit of excellence. We are excited to see the programmatic growth and the teaching and learning developments at Salem Christian School. We are seeing the benefits of adaptive lessons based upon using the classroom assessment and national tests. The key to our vision of data use is to benefit the student not a bureaucratic sense of accountability. A school, Christian school or not, is about the students. One of our visions is to help students become life-long learners. And, we consider a healthy self-advocacy is important to being a life-long learner. Of course, as a Christian school we hope that this would provide the tools necessary to accomplish this vision. This is why we will be continuing to help the students use their information and feedback to build better understandings.
As we move forward we will be trying to incorporate the best of research. The researchers, Hamilton et al (2009), suggests the following steps to carry out this recommendation:
  1. Explain expectations and assessment criteria.
  2. Provide feedback to students that is timely, specific, and helpful.
  3. Provide the means and tools for students to respond to the feedback.
  4. Use students’ data to inform instructional and programmatic changes (p. 9).
There are a few things that need to be considered as we move forward.
  • Students need to understand the goals. If teachers are trying to show students how to examine their own data, they must understand the goals and the criteria that will be used to assess the goals (Hamilton et al, 2009). Teachers must explicitly tell students the goals to be attained through each unit or lesson and how these goals tie into state standards. This is why the essential questions are posted in the teachers’ plan within the student and parent portals in addition to the homework.
  • Teacher feedback is crucial. If students are to use their own data to improve achievement, teachers need to provide the feedback and information for them to use. The classroom instruction should be “designed to help students understand their own strengths and weaknesses, explaining why they received the grade they did and identifying the specific content areas the student should focus on to improve their scores” (Hamilton et al, 2009, pp. 20-22).
  • An opportunity to ask questions and reflect is important. It is important that students have time to process and learn from the feedback in class (Hamilton et al, 2009).
We must be aware that there could be some challenges. According to Hamilton et al (2009), there can be a couple of roadblocks to using this recommendation. Parent, students, and teachers should be aware so that we avoid the hazards.
1. “Students view the feedback they receive as a reflection on their ability rather than an opportunity for focused improvement” (p. 24).
  • Hamilton et al (2009) suggests that teachers give students feedback that is explanatory and provides students with a chance to improve. Teachers should also emphasize the students’ level of performance on a task in relation to the learning goals and avoid making global statements about the student’s ability. In addition, teachers should encourage goal setting because students may be more willing to view feedback as a source of useful information if there is a larger goal that they are working to achieve. (p. 24)
2. “Teachers within a school have different approaches to providing feedback to their students” (p. 24).
  • Here, Hamilton et al (2009) suggests that teachers should engage with students in ways he or she finds effective; however, teachers may benefit from professional development on how to provide concrete and constructive feedback that informs student learning through students’ own data. (p. 24)
3. “Teachers are concerned that they do not have enough instructional time to explain rubrics or help students analyze feedback” (p. 24).
  • Hamilton et al (2009) says that time spent explaining assessment tools and strategies for analyzing feedback is essential to helping students understand their own achievement. Hamilton et al (2009) also suggests that incorporating time for students’ analysis of their own data into routine classroom activities may help students develop a habit of learning from feedback, making them more independent as the year progresses. (p. 26)
Hamilton, L., Halverson, R., Jackson. S., Mandinach, E., Supovitz, J., & Wayman, J. (2009).  Using student achievement data to support instructional decision making (NCEE 2009-4067). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education Retrieved from .

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